**Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption**

Vasiliy Fefelov, Vitaly Gorbunov, Alexander Myshlyavtsev and Marta Myshlyavtseva

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51802

## **1. Introduction**

388 Thermodynamics – Fundamentals and Its Application in Science

International, Metals Park OH, 1987).

Hall, London, 1992).

[31] H. Okamoto, T. B. Massalski, Phase Diagrams of Binary Gold Alloys, (ASM

[32] D. A. Porter, K. E. Easterling, Phase Transformations in Metals and Alloys, (Chapman &

The lattice models naturally arise in different fields of physics, chemistry and other sciences. First, it is physics of the solid state and physicochemistry of the surface. Among the many well-known lattice models the magnetic, alloys, liquid mixture, adsorption models are usually mentioned. The lattice models can be both classical and quantum. In this chapter only the classical lattice models focusing on models arising in physicochemistry of the surface will be considered. For the beginning let's give the most common formal definition of the classical lattice model.

Let there be given some finite or countable set M. Its elements will be called sites or nodes of the lattice and numbered index i. Each site is associated with the vector ( *<sup>i</sup> c* ) having *<sup>i</sup> k* components. Each of the components can take a finite countable or uncountable number of values, i.e. without loss of generality, it can assume that the components of this vector take either integer or real values in some finite or infinite interval. The specific values of the *<sup>i</sup> c* vector components determine the *i*-th site state. The *M* set state uniquely determines the state of all its elements. The system states can be divided into allowable and unallowable ones. Each allowable state of the *Mci* set is match to the real number *E c <sup>M</sup> <sup>i</sup>* called the energy per lattice site. (Formally, unallowable state can be assigned value *EM* ). Accepting the Boltzmann probability distribution of the system states as an additional postulate, one can receive object called a classical lattice model. All the models considered below are particular cases of the introduced generalized lattice model.

Determination of the lattice model dimension may be connected with the number of sites where can be the particle performing a random walk on the lattice in n steps. For *d*dimensional regular lattice the number N is proportional to the volume of a box with an edge n, where ~ *<sup>d</sup> N n* , i.e. the higher the dimension, the closer neighboring sites are located.

© 2012 Fefelov et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Following this approach 1,2,3,…, *d*-dimensional lattice models (for example, *d*-dimensional hypercubic lattices with interaction between the nearest-sites) are naturally obtained. 1,2 and 3-dimensional lattice models are of most importance for specific applications to the natural sciences. However, besides models whose dimension is equal to some natural number models which dimension defined the same way as above is fractional or infinite are of great interest. The simplest example of a model with a fractional dimension is the Ising model on the diamond-like lattice.

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption 391

, (1)

is the chemical

sense of the common definition. The latter statement is a very important because the

One of the simplest examples of the system for which LGM is quite adequate model is the monomolecular adsorption model. It is well known that chemisorption has localized character as in this case there occurs chemical bond between the adsorbed molecule and the substrate. For substrate being face of monocrystal chemisorbed molecules are located in well defined places of the periodic lattice formed by the surface atoms. Therefore, the simplest LGM introduced above is completely adequate model of monomolecular chemisorption on

The thermodynamic Hamiltonian of the simplest LGM with one type of particles on regular

potential; the occupation number *<sup>i</sup> n* is equal to unity if the adsorption site is occupied and zero in the case of a vacant site. The Hamiltonian (1) for *d* = 2 in the first approximation

In some cases this description corresponds to reality but in most cases it does not [1]. It is known that even at adsorption of the most simple gases such as Ar, Kr, Xe, nitrogen, carbon monoxide, oxygen and others the adsorbate molecule size is usually larger than the distance between neighboring active centers of adsorption [2-6]. In this connection at the adsorbate molecule adsorption on one active center it simultaneously occupies one or several

The assumption of one-center adsorption becomes even more inadequate at the description of adsorption of linear and quasilinear molecules such as the simple saturated and unsaturated hydrocarbons [7,8]. In the paper complete review of the experimental results on the adsorption of simple hydrocarbons on the surface of metal monocrystals (Pt, Pd) was given and, particularly, the adsorption of such molecules was shown to occur parallel to the solid surface in connection with which multicenter nature of the adsorption becomes apparent.

In addition in this review some types of ordered structures of adsorption monolayer С4 – С<sup>8</sup> alkanes experimentally observed at low temperatyres on the surface of monocrystalline platinum are discussed. The necessity of describing of the n-alkane adsorption in the framework of the models with multisite adsorption is also supported by the fact that at the interpretation of experimental adsorption isotherms by using the known multisite Langmuir model and other analytical models there exists good correlation between the model parameter k (the number of active sites occupied by adsorbate molecule) and the real number of segments in the molecule of the adsorbate [7,8,10-12]. It should be noted that the above analytical models of multisite adsorption are used to determine the specific surface of porous solids and its topography (in the case of energetically heterogeneous surfaces).

*ij i n n i H nn n*

 

,

is the lateral interaction energies of the nearest neighbors;

describes the monomolecular (one-center) adsorption on a monocrystal face.

property of one model can be obtained that of another one.

lattice (for example, hypercubic) has the form:

monocrystal face.

neighboring active centers.

where

Concluding the general description of lattice models it is worth to introduce the concept of a homogeneous lattice model. Models properties of all their sites are identical are called homogeneous lattice models. It follows the obtained lattice model has a geometric realization possessing the property of translational invariance. As it is mentioned above, the models relating to physicochemistry of the surface are of main interest, respectively, these are primarily two-dimensional ones. It is readily to conclude that all the two-dimensional homogeneous models can be divided to three classes having in the basis the geometric realizations: the square, triangular and hexagonal lattice. Note that the lattice models having different types of sites but whose geometric realizations possess translational invariance can serve the generalization of concept of lattice model homogeneity. From the computational point of view these models are very similar to homogeneous ones. Thus, the most common definition of the classical lattice model, its dimension, homogeneity and translational invariance have been introduced.

Lattice gas model and its various generalizations are one of the most important lattice models of modern statistical physics. Despite its relative simplicity these shows fantastic variety of non-trivial physical phenomena. First, this refers to the phase transitions of various types whose study is of great interest from the standpoint of the general theory. Note that the adlayers on the monocrystal faces represent the physical realization of many interesting and important models for *d* = 2, in particular, these admitting exact solutions. The fact causes an increased interest of theorists to such systems.

Usually the lattice gas means the molecular system which differs from the continual gas following feature: molecules of such a system can occupy only such places in the space in which their centers are located at the sites of one-, two- or three-dimensional geometric lattice. The interaction energies of molecules located in different configurations respect to each other are the parameters of the model. In the general case, these parameters also depend on the relative orientation of the molecules. These interactions called lateral naturally can be divided to pairs and many-particle. Pair interactions are additive while many-particle ones are nonadditive.

It is easy to see from the standpoint of common definition of classical lattice model the lattice gas model (LGM) stated above refers to the same class of models as the Ising model. Indeed, considering the dependences of the energy of lateral interaction of molecules on their relative orientation one can obtain the lattice model in which the vector *<sup>i</sup> c* is a scalar accepting two values (when all the molecules are the same type). From the above it immediately follows that the simplest LGM is isomorphic to the classical Ising model in the sense of the common definition. The latter statement is a very important because the property of one model can be obtained that of another one.

390 Thermodynamics – Fundamentals and Its Application in Science

model on the diamond-like lattice.

invariance have been introduced.

many-particle ones are nonadditive.

Following this approach 1,2,3,…, *d*-dimensional lattice models (for example, *d*-dimensional hypercubic lattices with interaction between the nearest-sites) are naturally obtained. 1,2 and 3-dimensional lattice models are of most importance for specific applications to the natural sciences. However, besides models whose dimension is equal to some natural number models which dimension defined the same way as above is fractional or infinite are of great interest. The simplest example of a model with a fractional dimension is the Ising

Concluding the general description of lattice models it is worth to introduce the concept of a homogeneous lattice model. Models properties of all their sites are identical are called homogeneous lattice models. It follows the obtained lattice model has a geometric realization possessing the property of translational invariance. As it is mentioned above, the models relating to physicochemistry of the surface are of main interest, respectively, these are primarily two-dimensional ones. It is readily to conclude that all the two-dimensional homogeneous models can be divided to three classes having in the basis the geometric realizations: the square, triangular and hexagonal lattice. Note that the lattice models having different types of sites but whose geometric realizations possess translational invariance can serve the generalization of concept of lattice model homogeneity. From the computational point of view these models are very similar to homogeneous ones. Thus, the most common definition of the classical lattice model, its dimension, homogeneity and translational

Lattice gas model and its various generalizations are one of the most important lattice models of modern statistical physics. Despite its relative simplicity these shows fantastic variety of non-trivial physical phenomena. First, this refers to the phase transitions of various types whose study is of great interest from the standpoint of the general theory. Note that the adlayers on the monocrystal faces represent the physical realization of many interesting and important models for *d* = 2, in particular, these admitting exact solutions.

Usually the lattice gas means the molecular system which differs from the continual gas following feature: molecules of such a system can occupy only such places in the space in which their centers are located at the sites of one-, two- or three-dimensional geometric lattice. The interaction energies of molecules located in different configurations respect to each other are the parameters of the model. In the general case, these parameters also depend on the relative orientation of the molecules. These interactions called lateral naturally can be divided to pairs and many-particle. Pair interactions are additive while

It is easy to see from the standpoint of common definition of classical lattice model the lattice gas model (LGM) stated above refers to the same class of models as the Ising model. Indeed, considering the dependences of the energy of lateral interaction of molecules on

accepting two values (when all the molecules are the same type). From the above it immediately follows that the simplest LGM is isomorphic to the classical Ising model in the

is a scalar

their relative orientation one can obtain the lattice model in which the vector *<sup>i</sup> c*

The fact causes an increased interest of theorists to such systems.

One of the simplest examples of the system for which LGM is quite adequate model is the monomolecular adsorption model. It is well known that chemisorption has localized character as in this case there occurs chemical bond between the adsorbed molecule and the substrate. For substrate being face of monocrystal chemisorbed molecules are located in well defined places of the periodic lattice formed by the surface atoms. Therefore, the simplest LGM introduced above is completely adequate model of monomolecular chemisorption on monocrystal face.

The thermodynamic Hamiltonian of the simplest LGM with one type of particles on regular lattice (for example, hypercubic) has the form:

$$H = \varepsilon \sum\_{\{n,n\}} n\_i n\_j - \mu \sum\_i n\_i \,\,\,\,\tag{1}$$

where is the lateral interaction energies of the nearest neighbors; is the chemical potential; the occupation number *<sup>i</sup> n* is equal to unity if the adsorption site is occupied and zero in the case of a vacant site. The Hamiltonian (1) for *d* = 2 in the first approximation describes the monomolecular (one-center) adsorption on a monocrystal face.

In some cases this description corresponds to reality but in most cases it does not [1]. It is known that even at adsorption of the most simple gases such as Ar, Kr, Xe, nitrogen, carbon monoxide, oxygen and others the adsorbate molecule size is usually larger than the distance between neighboring active centers of adsorption [2-6]. In this connection at the adsorbate molecule adsorption on one active center it simultaneously occupies one or several neighboring active centers.

The assumption of one-center adsorption becomes even more inadequate at the description of adsorption of linear and quasilinear molecules such as the simple saturated and unsaturated hydrocarbons [7,8]. In the paper complete review of the experimental results on the adsorption of simple hydrocarbons on the surface of metal monocrystals (Pt, Pd) was given and, particularly, the adsorption of such molecules was shown to occur parallel to the solid surface in connection with which multicenter nature of the adsorption becomes apparent.

In addition in this review some types of ordered structures of adsorption monolayer С4 – С<sup>8</sup> alkanes experimentally observed at low temperatyres on the surface of monocrystalline platinum are discussed. The necessity of describing of the n-alkane adsorption in the framework of the models with multisite adsorption is also supported by the fact that at the interpretation of experimental adsorption isotherms by using the known multisite Langmuir model and other analytical models there exists good correlation between the model parameter k (the number of active sites occupied by adsorbate molecule) and the real number of segments in the molecule of the adsorbate [7,8,10-12]. It should be noted that the above analytical models of multisite adsorption are used to determine the specific surface of porous solids and its topography (in the case of energetically heterogeneous surfaces).

Obviously that the adsorption of more complex (in regard geometry, the chemical structure – the presence of double/triple bonds or several functional groups) of molecules is even more nontrivial [14-25]. First of all this is manifested in that complex organic molecules (cyclic hydrocarbons, aromatic systems etc.) depending on their geometry and chemical structure can form set of different ordered structures on the solid surface.

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption 393

It is difficult to overestimate the applied significance of considered systems in the field of chemistry and biochemistry where they are used as active elements of chemical and biological sensors [36], in heterogeneous catalysis [37] and as coatings for biomedical implants [38]. Polymorphism of organic thin films and the ability of molecules to constitute different crystalline forms leads to is it is very difficult to control growth and properties of such systems. Moreover, it was recently shown that the structure of the organic film significantly affects epitaxial growth of crystals on the film [39]. This fact can be used to

In all these cases location of the molecules on the surfaces is a decisive factor that determines process of growth of the film and its physical properties. Therefore, a detailed understanding of elementary phisical and chemical processes occurring in such systems is the primary motivator at the investigation of molecular self-organization on the solid surface. The development of realistic models of such systems can allow completely to control the process of self-assembly of organic and other molecules on the solid surface and to come nearer to dream of nanotechnology – to gather material possessing the necessary properties with atomic precision. Based on the above it is clear why the interest of researchers specified both practical and theoretical considerations is now shifting towards

Let's consider the class of lattice models describing the so-called multisite adsorption. The simplest LGM considered above in the two-dimensional case is a model of monosite adsorption (active centers of adsorption, generally speaking, do not coincide with the surface atoms). However, as noted above at the description of many adsorption systems one is forced to abandon from idea about the monosite adsorption. In the framework of the LGM multisite adsorption is described as a system of prohibitions on certain configurations. At the same time a relatively simple lattice geometrically equivalent to the crystal lattice of the surface is persisted but the number of possible states of the site determining which part of the complex molecule is located above this site and how adsorbed molecule is oriented is increased. Further, the main results obtained in the framework of the models with multisite

The simplest model of multisite adsorption is a dimer adsorption model. In the first approximation the dimer model described adsorption of molecules consisted of two the same atoms, for example H2, N2, O2 etc. when temperature is relatively low and molecules

Statistical thermodynamics of the dimers lattice models has a long history. This is one of the earliest lattice models which take into account the own size of molecules in the frame of the lattice gas model. Apparently, the first model of the dimer has been studied in the context of the entropy of the adlayer in 1937 [40]. As it turned out, the dimer model has deep connections with the Ising model and many other important models in statistical physics. In the early sixties of the twentieth century an exact solution of the dimer model on a square

more complex lattice models some of which will be discussed in this review.

control the morphology of nanocrystalline systems.

adsorption will be presented.

cannot dissociate.

**2. The models of dimer and** *k***-mer adsorption** 

Along with the possibility of multisite adsorption that of different orientations of molecules with respect to the interface is one of the most interesting features of these systems. Indeed, over the past ten years series of experimental works devoted to the study of organic selfassembled monolayers on metal surfaces has been published. The general conclusion of these studies is that the molecule orientation in the adlayer is a function of external parameters such as concentration, pressure, temperature, electrode potential and others[14- 25]. Moreover, very interesting ordered structures have been experimentally found in some of similar systems. Those are structures which simultaneously contain the molecules with different orientations in the adlayer. For example, in [18] the authors investigated behavior of the adsorption monolayer of molecules p-Sexiphenil on the Au(111) surface in ultrahigh vacuum using the method of scanning tunneling microscopy. It was shown that five various ordered structures two of which contain molecules with different orientations in the adlayer – molecules oriented parallel to the surface and tilted to the surface at an angle can be formed. In [24] phase transitions in the adlayer of acid trimezin molecules on the Au(111) surface have been investigated using the method of scanning tunneling microscopy, and one of them proved to lead to formation of the ordered structure containing the acid trimezin molecules oriented both parallel and perpendicular to the surface. Another striking example of the adsorption system in which the adsorbate molecules can have different orientations with respect to the surface is the adsorption of cyclic unsaturated hydrocarbons on the reconstructed semiconductor surface especially on the reconstructed face of the Si(001) [25- 27]. This is connected to the fact that cycloaddition reaction [2+2] with formation of di-σ bond Si-C with the silicon surface atoms results from chemisorption of unsaturated organic molecules on the silicon surface. In chemisorptions of more complex hydrocarbons containing several unsaturated bonds the particle adsorbed on the surface can have several configurations depending on the number of di-σ bonds Si-C [28]. Moreover, the stable πcomplex being resulted from interaction between unsaturated hydrocarbon molecule and silicon atom has been experimentally discovered recently [29,30]. In light of the above it is clear that all specified features of the behavior arising at detail studying of adsorption of simple or complex molecules can be investigated only in the framework of models taking into account the multisite character of adsorption and the possibility of different orientation of molecules both with respect to each other and with respect to the solid surface.

Practically, theoretical study of organic self-assembled adsorption monolayers or thin organic films is of great interest generally in connection to the set of possible applications in which thin organic films are used anyway [31]. The potential field of application of such systems is an organic optoelectronics, in particular, electroluminescent devices [32], photovoltaics [33], organic field-effect transistors [34]. Similar systems are used as coatings on computer hard drives to provide protection against corrosion and low friction [35].

It is difficult to overestimate the applied significance of considered systems in the field of chemistry and biochemistry where they are used as active elements of chemical and biological sensors [36], in heterogeneous catalysis [37] and as coatings for biomedical implants [38]. Polymorphism of organic thin films and the ability of molecules to constitute different crystalline forms leads to is it is very difficult to control growth and properties of such systems. Moreover, it was recently shown that the structure of the organic film significantly affects epitaxial growth of crystals on the film [39]. This fact can be used to control the morphology of nanocrystalline systems.

In all these cases location of the molecules on the surfaces is a decisive factor that determines process of growth of the film and its physical properties. Therefore, a detailed understanding of elementary phisical and chemical processes occurring in such systems is the primary motivator at the investigation of molecular self-organization on the solid surface. The development of realistic models of such systems can allow completely to control the process of self-assembly of organic and other molecules on the solid surface and to come nearer to dream of nanotechnology – to gather material possessing the necessary properties with atomic precision. Based on the above it is clear why the interest of researchers specified both practical and theoretical considerations is now shifting towards more complex lattice models some of which will be discussed in this review.

Let's consider the class of lattice models describing the so-called multisite adsorption. The simplest LGM considered above in the two-dimensional case is a model of monosite adsorption (active centers of adsorption, generally speaking, do not coincide with the surface atoms). However, as noted above at the description of many adsorption systems one is forced to abandon from idea about the monosite adsorption. In the framework of the LGM multisite adsorption is described as a system of prohibitions on certain configurations. At the same time a relatively simple lattice geometrically equivalent to the crystal lattice of the surface is persisted but the number of possible states of the site determining which part of the complex molecule is located above this site and how adsorbed molecule is oriented is increased. Further, the main results obtained in the framework of the models with multisite adsorption will be presented.

## **2. The models of dimer and** *k***-mer adsorption**

392 Thermodynamics – Fundamentals and Its Application in Science

Obviously that the adsorption of more complex (in regard geometry, the chemical structure – the presence of double/triple bonds or several functional groups) of molecules is even more nontrivial [14-25]. First of all this is manifested in that complex organic molecules (cyclic hydrocarbons, aromatic systems etc.) depending on their geometry and chemical

Along with the possibility of multisite adsorption that of different orientations of molecules with respect to the interface is one of the most interesting features of these systems. Indeed, over the past ten years series of experimental works devoted to the study of organic selfassembled monolayers on metal surfaces has been published. The general conclusion of these studies is that the molecule orientation in the adlayer is a function of external parameters such as concentration, pressure, temperature, electrode potential and others[14- 25]. Moreover, very interesting ordered structures have been experimentally found in some of similar systems. Those are structures which simultaneously contain the molecules with different orientations in the adlayer. For example, in [18] the authors investigated behavior of the adsorption monolayer of molecules p-Sexiphenil on the Au(111) surface in ultrahigh vacuum using the method of scanning tunneling microscopy. It was shown that five various ordered structures two of which contain molecules with different orientations in the adlayer – molecules oriented parallel to the surface and tilted to the surface at an angle can be formed. In [24] phase transitions in the adlayer of acid trimezin molecules on the Au(111) surface have been investigated using the method of scanning tunneling microscopy, and one of them proved to lead to formation of the ordered structure containing the acid trimezin molecules oriented both parallel and perpendicular to the surface. Another striking example of the adsorption system in which the adsorbate molecules can have different orientations with respect to the surface is the adsorption of cyclic unsaturated hydrocarbons on the reconstructed semiconductor surface especially on the reconstructed face of the Si(001) [25- 27]. This is connected to the fact that cycloaddition reaction [2+2] with formation of di-σ bond Si-C with the silicon surface atoms results from chemisorption of unsaturated organic molecules on the silicon surface. In chemisorptions of more complex hydrocarbons containing several unsaturated bonds the particle adsorbed on the surface can have several configurations depending on the number of di-σ bonds Si-C [28]. Moreover, the stable πcomplex being resulted from interaction between unsaturated hydrocarbon molecule and silicon atom has been experimentally discovered recently [29,30]. In light of the above it is clear that all specified features of the behavior arising at detail studying of adsorption of simple or complex molecules can be investigated only in the framework of models taking into account the multisite character of adsorption and the possibility of different orientation

structure can form set of different ordered structures on the solid surface.

of molecules both with respect to each other and with respect to the solid surface.

Practically, theoretical study of organic self-assembled adsorption monolayers or thin organic films is of great interest generally in connection to the set of possible applications in which thin organic films are used anyway [31]. The potential field of application of such systems is an organic optoelectronics, in particular, electroluminescent devices [32], photovoltaics [33], organic field-effect transistors [34]. Similar systems are used as coatings on computer hard drives to provide protection against corrosion and low friction [35].

The simplest model of multisite adsorption is a dimer adsorption model. In the first approximation the dimer model described adsorption of molecules consisted of two the same atoms, for example H2, N2, O2 etc. when temperature is relatively low and molecules cannot dissociate.

Statistical thermodynamics of the dimers lattice models has a long history. This is one of the earliest lattice models which take into account the own size of molecules in the frame of the lattice gas model. Apparently, the first model of the dimer has been studied in the context of the entropy of the adlayer in 1937 [40]. As it turned out, the dimer model has deep connections with the Ising model and many other important models in statistical physics. In the early sixties of the twentieth century an exact solution of the dimer model on a square lattice was obtained in the case of the so-called "close-packed limit", i.e. all lattice sites belong to one and only one of the dimers [41-43]. In particular, the entropy per one lattice site was calculated. Interest in the dimer model persists to this day. Exact solutions for the dimer model were recently obtained with close-packed limit in the case of the twodimensional non-orientable surfaces, such as the Möbius strip and Klein bottle [44,45]. The problem of the packing of dimers in the presence of vacancies is much more complicated and largely solved numerically [46,47].This is mainly due to three factors: 1) there are no statistical equivalence between the particle and the vacancy, and 2) any occupied site indicates that at least one of the neighboring sites occupied too and 3) it is impossible to determine exactly whether there will be adsorption on the isolated vacancy. Exact solutions for the dimer model on lattices of dimension greater than two is currently unknown. For three- and more than three dimensional lattice models it is the overall situation characteristic not only for the dimer model, but also for the simpler one-centered model such as the classical Ising model and its many generalizations.

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption 395

site as a function of temperature and surface coverage. In this case, the most interesting is the dependence of the diffusion coefficient on the *k*-mer length and the surface coverage. Thus, with increasing size of the molecule the diffusion coefficient for noninteracting or attracting *k*-mers increases too, as in the case of repulsive interactions, the diffusion coefficient can either decrease or increase with the molecule length increasing, depending on the degree of coverage [52-54]. For more complex cases for the two-dimensional systems only approximate analytical expressions were obtained. The most well-known analytical approximation is: 1) the theory of Flory-Huggins [48,55-58], which is a generalization of the theory of binary solutions of polymer molecules in a monomolecular solvent for the twodimensional case. The fact that in the framework of lattice gas model the problem of *k*-mer adsorption on homogeneous surfaces is isomorphic to the problem of binary solutions of polymer in a monomolecular solvent, 2) Guggenheim-DiMarzio approximation [59,60], which is based on calculating the number of possible ways of packaging rigid k-mers on lattices with different coordination numbers 3) the approximation based on the extension of the exact solution for a one-dimensional case [52,53] to higher dimensions [49,61], 4) well known quasichemical approximation [62] and mean-field approximation [63], 5) fractional statistical theory (FSTA) for the adsorption of polyatomic molecules, based on Holdan statistics [64], 6) semi-empirical model [61,65], etc. Unfortunately, none of these approximations is universal, and each shows quite good results, depending on the parameters of the model – a flexible or rigid *k*-mer, the length of *k*-mer, the presence or absence of lateral interactions between adsorbed molecules, etc. A brief description of the approximations and compare them with each other can be found in [49,65]. Generally, more recent analytical approximations for the *k*-mers adsorption include earlier ones as special cases. In this regard, let's consider the best of them – FSTA and semi-empirical

In ref. [51] the authors, by comparing experimental data with results obtained by means of analytical approximations and a Monte Carlo simulation, studied the adsorption of linear and flexible polyatomic molecules on honeycomb, square and triangular lattices. Data obtained by the FSTA model in the case of a square lattice are almost identical to the data obtained by the Monte-Carlo simulation. A similar analytical calculation by FSTA for the *k*mers adsorption on triangular lattice gives an inaccurate result, because there are a larger

In order to verify the accuracy of the proposed approach (FSTA) the authors have constructed and analytically calculated the two models of real processes. The first is a model of oxygen adsorption on 5A zeolite and the second one – a model of adsorption of propane on the 13X zeolite. The results of analytical calculations were almost identical to the experimental data. The principal difference FSTA from earlier models of multisite adsorption is that in addition to the size of molecules, it also takes into account their shape and surface geometry. Thereby FSTA can describe the adsorption of both rigid and flexible

Analyzing the results deviations for various approximations from the results of Monte Carlo, it was shown that the most accurate approximation is the semi-empirical model,

number of possible configurations of a single *k*-mer on the surface.

approximation.

molecules.

The dimer model in the framework of the lattice gas model can be described as follows. Let consider the lattice (for simplicity, a square) of the active sites. Each lattice site can be occupied by one of the segments of a dimer, or be empty. In addition, the orientation of the dimer should be specified. The last stage of building the lattice gas model is a complete listing of prohibited configurations. In this case, the system of prohibitions describes the continuity of the dimer. Just as in the simplest of the lattice gas model, different lateral interactions can be considered in the constructed model of dimer adsorption too. The dimer model is actively studied for decades because it is the simplest model of multisite adsorption and is of theoretical and practical interest.

A natural generalization of the dimer adsorption model is an adsorption model of rigid or flexible linear *k*-mers of having no thickness. The only difference from the dimer adsorption model is the assumption that the adsorbed molecule occupies now *k* lattice sites forming a certain configuration. Thus, *k*-mer is called the model of the adsorbate molecule, consisting of *k* equal-sized segments, and the bond length between the segments is equal to the lattice constant and does not change, and not broken in the process of modeling. In the case of dimers, *k* is equal to two. Thus, from a formal point of view the lattice gas model for the dimer adsorption does not differ from the lattice gas model for *k*-mers, so it makes sense to discuss the results obtained in the framework of these models together. Next, referring to the *k*-mers reader should keep in mind the dimers adsorption model, except where otherwise noted.

Depending on whether the same or different properties are segments of the molecule, *k*mers differ by homonuclear and heteronuclear, respectively. Works devoted to the study of *k*-mers can be divided into two groups according to the shape of the molecule (flexibility), the first group is works devoted to the study of flexible *k*-mers [48-51], the second group is works studying the adsorption properties of rigid linear *k*-mers (rigid rods) [49,51,52]. Theoretical analysis of a multisite adsorption of linear molecules in the general case is rather complicated, and the exact solution for *k*-mers found only in the simplest one-dimensional case [52,53]. In these studies were obtained exact expressions for the free energy per active

site as a function of temperature and surface coverage. In this case, the most interesting is the dependence of the diffusion coefficient on the *k*-mer length and the surface coverage. Thus, with increasing size of the molecule the diffusion coefficient for noninteracting or attracting *k*-mers increases too, as in the case of repulsive interactions, the diffusion coefficient can either decrease or increase with the molecule length increasing, depending on the degree of coverage [52-54]. For more complex cases for the two-dimensional systems only approximate analytical expressions were obtained. The most well-known analytical approximation is: 1) the theory of Flory-Huggins [48,55-58], which is a generalization of the theory of binary solutions of polymer molecules in a monomolecular solvent for the twodimensional case. The fact that in the framework of lattice gas model the problem of *k*-mer adsorption on homogeneous surfaces is isomorphic to the problem of binary solutions of polymer in a monomolecular solvent, 2) Guggenheim-DiMarzio approximation [59,60], which is based on calculating the number of possible ways of packaging rigid k-mers on lattices with different coordination numbers 3) the approximation based on the extension of the exact solution for a one-dimensional case [52,53] to higher dimensions [49,61], 4) well known quasichemical approximation [62] and mean-field approximation [63], 5) fractional statistical theory (FSTA) for the adsorption of polyatomic molecules, based on Holdan statistics [64], 6) semi-empirical model [61,65], etc. Unfortunately, none of these approximations is universal, and each shows quite good results, depending on the parameters of the model – a flexible or rigid *k*-mer, the length of *k*-mer, the presence or absence of lateral interactions between adsorbed molecules, etc. A brief description of the approximations and compare them with each other can be found in [49,65]. Generally, more recent analytical approximations for the *k*-mers adsorption include earlier ones as special cases. In this regard, let's consider the best of them – FSTA and semi-empirical approximation.

394 Thermodynamics – Fundamentals and Its Application in Science

such as the classical Ising model and its many generalizations.

and is of theoretical and practical interest.

otherwise noted.

lattice was obtained in the case of the so-called "close-packed limit", i.e. all lattice sites belong to one and only one of the dimers [41-43]. In particular, the entropy per one lattice site was calculated. Interest in the dimer model persists to this day. Exact solutions for the dimer model were recently obtained with close-packed limit in the case of the twodimensional non-orientable surfaces, such as the Möbius strip and Klein bottle [44,45]. The problem of the packing of dimers in the presence of vacancies is much more complicated and largely solved numerically [46,47].This is mainly due to three factors: 1) there are no statistical equivalence between the particle and the vacancy, and 2) any occupied site indicates that at least one of the neighboring sites occupied too and 3) it is impossible to determine exactly whether there will be adsorption on the isolated vacancy. Exact solutions for the dimer model on lattices of dimension greater than two is currently unknown. For three- and more than three dimensional lattice models it is the overall situation characteristic not only for the dimer model, but also for the simpler one-centered model

The dimer model in the framework of the lattice gas model can be described as follows. Let consider the lattice (for simplicity, a square) of the active sites. Each lattice site can be occupied by one of the segments of a dimer, or be empty. In addition, the orientation of the dimer should be specified. The last stage of building the lattice gas model is a complete listing of prohibited configurations. In this case, the system of prohibitions describes the continuity of the dimer. Just as in the simplest of the lattice gas model, different lateral interactions can be considered in the constructed model of dimer adsorption too. The dimer model is actively studied for decades because it is the simplest model of multisite adsorption

A natural generalization of the dimer adsorption model is an adsorption model of rigid or flexible linear *k*-mers of having no thickness. The only difference from the dimer adsorption model is the assumption that the adsorbed molecule occupies now *k* lattice sites forming a certain configuration. Thus, *k*-mer is called the model of the adsorbate molecule, consisting of *k* equal-sized segments, and the bond length between the segments is equal to the lattice constant and does not change, and not broken in the process of modeling. In the case of dimers, *k* is equal to two. Thus, from a formal point of view the lattice gas model for the dimer adsorption does not differ from the lattice gas model for *k*-mers, so it makes sense to discuss the results obtained in the framework of these models together. Next, referring to the *k*-mers reader should keep in mind the dimers adsorption model, except where

Depending on whether the same or different properties are segments of the molecule, *k*mers differ by homonuclear and heteronuclear, respectively. Works devoted to the study of *k*-mers can be divided into two groups according to the shape of the molecule (flexibility), the first group is works devoted to the study of flexible *k*-mers [48-51], the second group is works studying the adsorption properties of rigid linear *k*-mers (rigid rods) [49,51,52]. Theoretical analysis of a multisite adsorption of linear molecules in the general case is rather complicated, and the exact solution for *k*-mers found only in the simplest one-dimensional case [52,53]. In these studies were obtained exact expressions for the free energy per active In ref. [51] the authors, by comparing experimental data with results obtained by means of analytical approximations and a Monte Carlo simulation, studied the adsorption of linear and flexible polyatomic molecules on honeycomb, square and triangular lattices. Data obtained by the FSTA model in the case of a square lattice are almost identical to the data obtained by the Monte-Carlo simulation. A similar analytical calculation by FSTA for the *k*mers adsorption on triangular lattice gives an inaccurate result, because there are a larger number of possible configurations of a single *k*-mer on the surface.

In order to verify the accuracy of the proposed approach (FSTA) the authors have constructed and analytically calculated the two models of real processes. The first is a model of oxygen adsorption on 5A zeolite and the second one – a model of adsorption of propane on the 13X zeolite. The results of analytical calculations were almost identical to the experimental data. The principal difference FSTA from earlier models of multisite adsorption is that in addition to the size of molecules, it also takes into account their shape and surface geometry. Thereby FSTA can describe the adsorption of both rigid and flexible molecules.

Analyzing the results deviations for various approximations from the results of Monte Carlo, it was shown that the most accurate approximation is the semi-empirical model,

developed by Roma et al. Semi-empirical model is a combination of the exact solution for one-dimensional approximation and Guggenheim-DiMarzio approximation [65]. A new theoretical approach is significantly better than other existing approximations and allows fairly simple explaining the experimental data.

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption 397

[74] lattices. It was shown that the system does not belong to the universality class of two-

**Figure 1.** Adsorption isotherms of dimers on square, triangular and honeycomb lattices and

corresponding ordered phases [71,73].

dimensional Ising model.

When comparing the adsorption isotherms of monomers and k-mers, it was found that in the second case the symmetry of the "particle-vacancy" is broken. The isotherms are shifted toward lower coverage with increasing coordination number of the lattice. In other words, for a given value of the chemical potential the surface coverage increases with the lattice coordination number.

As can be seen in most works devoted to the development of new analytical methods, the results are compared not only with the experimental data, but also with the results obtained by the Monte-Carlo, as the reference.

The Monte Carlo method has proved to be a very powerful tool in the study of *k*-mers adsorption. Using this method with different techniques (reweighing, finite-size scaling, and others [66,67]) many important parameters of the phase behavior for different *k*-mers adsorption systems were identified, such as the types of ordered phase structures, the points of phase transitions and critical indexes of phase transitions, etc. [68,69]. The appearance of ordered phases for the model *k*-mers in the presence of lateral interactions has its own specifics, this is due to the presence of orientation in the arrangement of the adsorbed molecules relative to each other.

In ref. [70] the authors, using the transfer-matrix method, investigated the ordered structures of the adsorption layer consisting of interacting dimers adsorbed on a square lattice. Analysis of the changes of the adlayer entropy and the surface diffusion coefficient showed that there is a finite number of ordered phases in case of repulsion lateral interactions between the nearest neighboring molecules.

Later in ref. [71] Ramirez-Pastor et al. using the Monte Carlo method have considered both attractive and repulsive interactions between adsorbed dimers on a square lattice. It was shown that in the case of attractive interactions, the phase diagram is similar to the diagram for a monoatomic gas, but the critical temperature is shifted to higher values. The most interesting case is repulsive interactions when a variety of ordered structures take place. In the case of dimers the symmetry of the "particle-vacancy", typical of monatomic particles, disturbed, that leads to the asymmetry of the adsorption isotherm with respect to the line *θ* = 0.5, on the isotherm two steps take place. When *θ* = 0.5 *c*2 4 structure formed, which is characterized by the alternation of the adsorbed dimer and two adjacent vacancies. When the chemical potential *μ* increases and *θ* close to *θ* = 2/3, adsorbed dimers form parallel zigzag rows (ZZ phase) [71.72]. A similar phase behavior of adsorbed layer of dimers is observed in the case of triangular and honeycomb lattices [73] (Fig. 1). In addition, the scientific group of Ramirez-Pastor, using Monte Carlo method and finite-size scaling techniques, calculated the critical exponents and critical temperatures, and calculated a phase diagram for dimers with repulsive lateral interactions on a square [71] and triangular

[74] lattices. It was shown that the system does not belong to the universality class of twodimensional Ising model.

396 Thermodynamics – Fundamentals and Its Application in Science

fairly simple explaining the experimental data.

coordination number.

by the Monte-Carlo, as the reference.

molecules relative to each other.

interactions between the nearest neighboring molecules.

developed by Roma et al. Semi-empirical model is a combination of the exact solution for one-dimensional approximation and Guggenheim-DiMarzio approximation [65]. A new theoretical approach is significantly better than other existing approximations and allows

When comparing the adsorption isotherms of monomers and k-mers, it was found that in the second case the symmetry of the "particle-vacancy" is broken. The isotherms are shifted toward lower coverage with increasing coordination number of the lattice. In other words, for a given value of the chemical potential the surface coverage increases with the lattice

As can be seen in most works devoted to the development of new analytical methods, the results are compared not only with the experimental data, but also with the results obtained

The Monte Carlo method has proved to be a very powerful tool in the study of *k*-mers adsorption. Using this method with different techniques (reweighing, finite-size scaling, and others [66,67]) many important parameters of the phase behavior for different *k*-mers adsorption systems were identified, such as the types of ordered phase structures, the points of phase transitions and critical indexes of phase transitions, etc. [68,69]. The appearance of ordered phases for the model *k*-mers in the presence of lateral interactions has its own specifics, this is due to the presence of orientation in the arrangement of the adsorbed

In ref. [70] the authors, using the transfer-matrix method, investigated the ordered structures of the adsorption layer consisting of interacting dimers adsorbed on a square lattice. Analysis of the changes of the adlayer entropy and the surface diffusion coefficient showed that there is a finite number of ordered phases in case of repulsion lateral

Later in ref. [71] Ramirez-Pastor et al. using the Monte Carlo method have considered both attractive and repulsive interactions between adsorbed dimers on a square lattice. It was shown that in the case of attractive interactions, the phase diagram is similar to the diagram for a monoatomic gas, but the critical temperature is shifted to higher values. The most interesting case is repulsive interactions when a variety of ordered structures take place. In the case of dimers the symmetry of the "particle-vacancy", typical of monatomic particles, disturbed, that leads to the asymmetry of the adsorption isotherm with respect to the line *θ* = 0.5, on the isotherm two steps take place. When *θ* = 0.5 *c*2 4 structure formed, which is characterized by the alternation of the adsorbed dimer and two adjacent vacancies. When the chemical potential *μ* increases and *θ* close to *θ* = 2/3, adsorbed dimers form parallel zigzag rows (ZZ phase) [71.72]. A similar phase behavior of adsorbed layer of dimers is observed in the case of triangular and honeycomb lattices [73] (Fig. 1). In addition, the scientific group of Ramirez-Pastor, using Monte Carlo method and finite-size scaling techniques, calculated the critical exponents and critical temperatures, and calculated a phase diagram for dimers with repulsive lateral interactions on a square [71] and triangular

**Figure 1.** Adsorption isotherms of dimers on square, triangular and honeycomb lattices and corresponding ordered phases [71,73].

The study of the adsorption monolayers consisting of heteronuclear dimers showed that the phase diagram of the system greatly influenced by the quantity of energy of lateral interactions between the different types of segments (it is about two different molecules). In the study of phase diagrams of these films interesting phenomena were found. In particular, the coexistence of three phases and a variety of structural transitions, and ordered linear type structure exists even at high temperatures [75].

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption 399

the surface is of great interest both from the point of view of phase transitions science, as well as from the applied point of view, in particular the appearance of surface conductivity. In the case of the two-dimensional lattice gas model the percolation threshold is so minimum value of the coverage *θ* on an infinite surface, as infinitely large cluster of adsorbed molecules is formed. In ref. [82,83] it is shown that in addition to the dimension of the system and the lattice coordination number on the percolation threshold a size of *k*-mers has a significant effect. Influence of temperature on percolation in adsorption systems with flexible *k*-mers studied in ref. [50] by the Monte Carlo method. There was found an interesting phenomenon – a nonmonotonic behavior of the percolation threshold as function of temperature, i.e. there exists a temperature at which the percolation threshold is a minimum, and the position of the minimum does not depend on the length of *k*-mers.

Thus, all existing works on the *k*-mers adsorption, as in the case of dimers, are concentrated in four main areas: (1) study of the influence of the chemical structure of noninteracting adsorbed molecules on the behavior of their adsorption, (2) study of the influence of surface heterogeneity on the *k*-mers adsorption, (3) description of first order phase transition in the adlayer in the case of mutual attraction of the adsorbate – adsorbate. Only a small number of papers devoted to the study of phase transition "order-disorder", which takes places in the case of repulsive intermolecular interactions [69,70]. In particular, in ref. [69], using Monte Carlo simulation, two important characteristics of the critical properties of repulsive k-mers were identified: (1) the minimum critical temperature have on the value of *k* equal to 2 (dimers) and (2) for *k* > 2 the critical temperature increases monotonically with increasing *k*. Similar results (qualitatively) have been obtained from the analytical calculation of the mean

Additionally it should be noted that long before the above works, the study of the multisite adsorption was engaged by Soviet scientists in Temkin (1938 [84]), and Snagovski(1972 [85- 87]). In ref. [84-87] the adsorption isotherms were analytically obtained in the case of multisite adsorption of two-center (*k* = 2), square (*k* = 4) and hexagonal (*k* = 7) complexes. Today's papers on multicenter adsorption are essentially the development, and sometimes

As one can see, at present the theory of adsorption of *k*-mers are actively developing – adsorption isotherms are calculated, phase diagrams are constructed, percolation thresholds are determined, etc. This is due primarily to a large applied importance of such research. Already, it can be concluded that the behavior of adlayers consisting of molecules that occupy more than one active site of the surface is significantly different from the behavior of systems with monosite adsorption. However, it should be noted that in all these considered

Today there is a small number of papers devoted to theoretical investigation of the behavior of adsorption monolayer consisting of molecules that can have a different orientation with

works the adsorbate molecule can adsorb only one way to the surface – is planar.

field approximation and the principle of minimum free energy.

repeating, of the works of Temkin and Snagovski.

**3. Multisite adsorption of orientable molecules** 

respect to the surface [88-92]. Let's examine them in detail.

Similar calculations were performed for the adsorption of homonuclear dimers on heterogeneous surfaces [1,10,76-79]. In the case where the surface is represented by heterogeneous clusters of active sites of one kind and another, the approximate solution is relatively simple (the solution for multisite adsorption on a homogeneous surface can be used). In describing the adsorption of dimers on heterogeneous surfaces, created at random, the task becomes more complicated. Slightly more than twenty years ago an approximate solution was suggested for this model [1,10,79]. However, the original approach of Nitta et al. could only be applied to surfaces with a discrete distribution of adsorption energy. Later on the basis of this approximation Rudzinski and Everett [1] obtained a solution for a model with continuous distribution of adsorption energy on the surface. However, the usable area of this and other approximations [78] is limited. For example, in [77], the authors investigated an analytical approximation by Monte Carlo method to describe the adsorption of homonuclear dimers on heterogeneous surfaces, created randomly (random heterogeneous surface). The calculation shows that this approximation yields accurate results when the difference between the energies of adsorption on active sites of one and other kind is small.

There are papers devoted to the description of the first order phase transition such as "surface gas – surface liquid" [75,76,80]. The most important conclusion from all these studies is that with increasing molecular size the critical temperature shifts to higher values. On the other hand, increasing of the molecule flexibility leads to decreasing of critical temperature. A very interesting phenomenon was observed in the study of phase transition "surface gas – surfacee liquid" in the adlayer consisting of heteronuclear trimers – namely, the coexistence of three phases (there is a second phase transition "liquid – liquid"). In this case, the phase diagram is asymmetric – there are a shift of the critical density to the unit and an increasing the critical temperature.

The findings in the study of dimers adsorption on heterogeneous surfaces with a sufficient degree of accuracy can be extended to the adsorption systems of *k*-mers [10,78,79]. However, as was shown in ref. [81], the calculation accuracy of the approximations developed for the adsorption of dimers on heterogeneous surfaces decreases with increasing molecular size.

Summarizing the results obtained in these works, one can conclude that the phase behavior of adlayer of dimers on heterogeneous surfaces defined by the following factors: 1) the distribution of the various active sites of adsorption, and 2) the relation between the *k*-mer length and the size of the local heterogeneity, 3) adsorption energy on different active sites.

It should be noted about the theoretical studies of percolation threshold in systems with multisite adsorption. Information on transitions "percolate region" – "nonpercolate area" on the surface is of great interest both from the point of view of phase transitions science, as well as from the applied point of view, in particular the appearance of surface conductivity. In the case of the two-dimensional lattice gas model the percolation threshold is so minimum value of the coverage *θ* on an infinite surface, as infinitely large cluster of adsorbed molecules is formed. In ref. [82,83] it is shown that in addition to the dimension of the system and the lattice coordination number on the percolation threshold a size of *k*-mers has a significant effect. Influence of temperature on percolation in adsorption systems with flexible *k*-mers studied in ref. [50] by the Monte Carlo method. There was found an interesting phenomenon – a nonmonotonic behavior of the percolation threshold as function of temperature, i.e. there exists a temperature at which the percolation threshold is a minimum, and the position of the minimum does not depend on the length of *k*-mers.

398 Thermodynamics – Fundamentals and Its Application in Science

type structure exists even at high temperatures [75].

and an increasing the critical temperature.

The study of the adsorption monolayers consisting of heteronuclear dimers showed that the phase diagram of the system greatly influenced by the quantity of energy of lateral interactions between the different types of segments (it is about two different molecules). In the study of phase diagrams of these films interesting phenomena were found. In particular, the coexistence of three phases and a variety of structural transitions, and ordered linear

Similar calculations were performed for the adsorption of homonuclear dimers on heterogeneous surfaces [1,10,76-79]. In the case where the surface is represented by heterogeneous clusters of active sites of one kind and another, the approximate solution is relatively simple (the solution for multisite adsorption on a homogeneous surface can be used). In describing the adsorption of dimers on heterogeneous surfaces, created at random, the task becomes more complicated. Slightly more than twenty years ago an approximate solution was suggested for this model [1,10,79]. However, the original approach of Nitta et al. could only be applied to surfaces with a discrete distribution of adsorption energy. Later on the basis of this approximation Rudzinski and Everett [1] obtained a solution for a model with continuous distribution of adsorption energy on the surface. However, the usable area of this and other approximations [78] is limited. For example, in [77], the authors investigated an analytical approximation by Monte Carlo method to describe the adsorption of homonuclear dimers on heterogeneous surfaces, created randomly (random heterogeneous surface). The calculation shows that this approximation yields accurate results when the difference between the energies of adsorption on active sites of one and other kind is small.

There are papers devoted to the description of the first order phase transition such as "surface gas – surface liquid" [75,76,80]. The most important conclusion from all these studies is that with increasing molecular size the critical temperature shifts to higher values. On the other hand, increasing of the molecule flexibility leads to decreasing of critical temperature. A very interesting phenomenon was observed in the study of phase transition "surface gas – surfacee liquid" in the adlayer consisting of heteronuclear trimers – namely, the coexistence of three phases (there is a second phase transition "liquid – liquid"). In this case, the phase diagram is asymmetric – there are a shift of the critical density to the unit

The findings in the study of dimers adsorption on heterogeneous surfaces with a sufficient degree of accuracy can be extended to the adsorption systems of *k*-mers [10,78,79]. However, as was shown in ref. [81], the calculation accuracy of the approximations developed for the adsorption of dimers on heterogeneous surfaces decreases with increasing molecular size.

Summarizing the results obtained in these works, one can conclude that the phase behavior of adlayer of dimers on heterogeneous surfaces defined by the following factors: 1) the distribution of the various active sites of adsorption, and 2) the relation between the *k*-mer length and the size of the local heterogeneity, 3) adsorption energy on different active sites.

It should be noted about the theoretical studies of percolation threshold in systems with multisite adsorption. Information on transitions "percolate region" – "nonpercolate area" on Thus, all existing works on the *k*-mers adsorption, as in the case of dimers, are concentrated in four main areas: (1) study of the influence of the chemical structure of noninteracting adsorbed molecules on the behavior of their adsorption, (2) study of the influence of surface heterogeneity on the *k*-mers adsorption, (3) description of first order phase transition in the adlayer in the case of mutual attraction of the adsorbate – adsorbate. Only a small number of papers devoted to the study of phase transition "order-disorder", which takes places in the case of repulsive intermolecular interactions [69,70]. In particular, in ref. [69], using Monte Carlo simulation, two important characteristics of the critical properties of repulsive k-mers were identified: (1) the minimum critical temperature have on the value of *k* equal to 2 (dimers) and (2) for *k* > 2 the critical temperature increases monotonically with increasing *k*. Similar results (qualitatively) have been obtained from the analytical calculation of the mean field approximation and the principle of minimum free energy.

Additionally it should be noted that long before the above works, the study of the multisite adsorption was engaged by Soviet scientists in Temkin (1938 [84]), and Snagovski(1972 [85- 87]). In ref. [84-87] the adsorption isotherms were analytically obtained in the case of multisite adsorption of two-center (*k* = 2), square (*k* = 4) and hexagonal (*k* = 7) complexes. Today's papers on multicenter adsorption are essentially the development, and sometimes repeating, of the works of Temkin and Snagovski.

As one can see, at present the theory of adsorption of *k*-mers are actively developing – adsorption isotherms are calculated, phase diagrams are constructed, percolation thresholds are determined, etc. This is due primarily to a large applied importance of such research. Already, it can be concluded that the behavior of adlayers consisting of molecules that occupy more than one active site of the surface is significantly different from the behavior of systems with monosite adsorption. However, it should be noted that in all these considered works the adsorbate molecule can adsorb only one way to the surface – is planar.

## **3. Multisite adsorption of orientable molecules**

Today there is a small number of papers devoted to theoretical investigation of the behavior of adsorption monolayer consisting of molecules that can have a different orientation with respect to the surface [88-92]. Let's examine them in detail.

The earliest papers on theoretical study of molecular reorientation in the adsorption monolayer were carried out at MSU by Gorshtein and Lopatkin in 1971 [88,89]. They investigated one- and two-dimensional lattice models of diatomic molecules adsorption. It was assumed that the molecule can adsorb in two different ways with respect to the surface: vertically and horizontally. Each type of adsorption had its heat of adsorption, and adsorption energy of vertically oriented molecule was approximately two times smaller. The lateral interactions between adsorbed molecules were not taken into account. The authors derived an exact analytical expression for the adsorption isotherms in one-dimensional case and the approximate equation for two-dimensional lattice. It is shown that for the large values of adsorption heats at low pressures, most of the molecules adsorbed horizontally, and the number of vertically orientated molecules is very small. When coverage increasing the horizontally adsorbed molecules change the orientation, and the number of molecules adsorbed vertically grows fast. In addition, the authors had obtained expressions for isosteric heats of adsorption. Having analyzed calculated thermodynamic functions they concluded that the system exhibits two modes of adsorption: on two neighbor sites in the region of low pressure and on one site at high pressures.

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption 401

sites) or perpendicular (adsorption on one active site) to the surface. In the language of this general model homonuclear dimers adsorption model belongs to the class of models with 1 2 *km m* 2, 1, 2, and heteronuclear dimers adsorption model [90] – to the class with <sup>123</sup> *km m m* 3, 1, 1, 2. The model studied in [91,92] belongs to the class of 1 2 *km m* 2, 1, 1. It should be noted that all these simple classes of models have a single representative and the set of numbers 1 2 , , ,, *<sup>k</sup> km m m* uniquely identifies the type of model.

**Figure 2.** Phase diagrams for dimers on square and triangular lattices. Ordered structures: black circles are site occupied by dimers adsorbed on two sites; gray circles are sites occupied by dimers adsorbed on one site.

The authors of [90] studied the adsorption of heteronuclear dimers (A-B) on a homogeneous surface with a mean-field approximation. In this case the dimer can be adsorbed on the surface in three different ways: horizontal adsorption with two segments at the same time and vertical adsorption with the A segment or vertical adsorption with the B segment. It was assumed that all three types of adsorption differed in the adsorption heat, and the interaction between adsorbed molecules were absent. The authors derived analytical expressions for the adsorption isotherms and isobars. In fact, the physical results obtained in this work are in qualitatively agreement with Gorshteyn and Lopatkin's ones and partially duplicate it.

There is the quite interesting model of spin-1 type, which can describe the adsorption of heteronuclear dimers on different lattices [91,92]. It is assumed in the model that all adsorbed molecules are oriented vertically to the surface only, and the energy of adsorption depends on what segment, A or B, molecule adsorb with. The authors of [91,92] considered the various sets of lateral interactions between dimers, in particular, they took into account not only interactions between nearest neighbors, but also between next-nearest neighbors. Such complex lateral interactions in the system led to the set of ordered surface structures and phase transitions. It had been shown that if the lattice is completely filled the "orderdisorder" transition may occure via a continuous phase transition as well as the first-order phase transition depending on the model parameters. Moreover, the continuous phase transition is nonuniversal. It should be noted that the model does not take into account the possibility of horizontal orientation of the dimer and generally speaking this model can be attributed to the well-known Blume-Emery-Griffiths model [66].

In [93] authors proposed the general lattice gas model describing the adsorption of complex molecules. The model can be formulate by following: the molecule can be adsorbed on the surface by the *k* ways occupying the 1 2 , , , *mm m <sup>k</sup>* active sites located in the corresponding configurations on homogeneous or heterogeneous lattice, respectively. The simplest model of this type is the model of homonuclear dimers adsorption. In this model it is assumed that dimer may have two different orientations in the adlayer – parallel (adsorption on two active sites) or perpendicular (adsorption on one active site) to the surface. In the language of this general model homonuclear dimers adsorption model belongs to the class of models with 1 2 *km m* 2, 1, 2, and heteronuclear dimers adsorption model [90] – to the class with <sup>123</sup> *km m m* 3, 1, 1, 2. The model studied in [91,92] belongs to the class of 1 2 *km m* 2, 1, 1. It should be noted that all these simple classes of models have a single representative and the set of numbers 1 2 , , ,, *<sup>k</sup> km m m* uniquely identifies the type of model.

400 Thermodynamics – Fundamentals and Its Application in Science

region of low pressure and on one site at high pressures.

attributed to the well-known Blume-Emery-Griffiths model [66].

The earliest papers on theoretical study of molecular reorientation in the adsorption monolayer were carried out at MSU by Gorshtein and Lopatkin in 1971 [88,89]. They investigated one- and two-dimensional lattice models of diatomic molecules adsorption. It was assumed that the molecule can adsorb in two different ways with respect to the surface: vertically and horizontally. Each type of adsorption had its heat of adsorption, and adsorption energy of vertically oriented molecule was approximately two times smaller. The lateral interactions between adsorbed molecules were not taken into account. The authors derived an exact analytical expression for the adsorption isotherms in one-dimensional case and the approximate equation for two-dimensional lattice. It is shown that for the large values of adsorption heats at low pressures, most of the molecules adsorbed horizontally, and the number of vertically orientated molecules is very small. When coverage increasing the horizontally adsorbed molecules change the orientation, and the number of molecules adsorbed vertically grows fast. In addition, the authors had obtained expressions for isosteric heats of adsorption. Having analyzed calculated thermodynamic functions they concluded that the system exhibits two modes of adsorption: on two neighbor sites in the

The authors of [90] studied the adsorption of heteronuclear dimers (A-B) on a homogeneous surface with a mean-field approximation. In this case the dimer can be adsorbed on the surface in three different ways: horizontal adsorption with two segments at the same time and vertical adsorption with the A segment or vertical adsorption with the B segment. It was assumed that all three types of adsorption differed in the adsorption heat, and the interaction between adsorbed molecules were absent. The authors derived analytical expressions for the adsorption isotherms and isobars. In fact, the physical results obtained in this work are in qualitatively agreement with Gorshteyn and Lopatkin's ones and partially duplicate it.

There is the quite interesting model of spin-1 type, which can describe the adsorption of heteronuclear dimers on different lattices [91,92]. It is assumed in the model that all adsorbed molecules are oriented vertically to the surface only, and the energy of adsorption depends on what segment, A or B, molecule adsorb with. The authors of [91,92] considered the various sets of lateral interactions between dimers, in particular, they took into account not only interactions between nearest neighbors, but also between next-nearest neighbors. Such complex lateral interactions in the system led to the set of ordered surface structures and phase transitions. It had been shown that if the lattice is completely filled the "orderdisorder" transition may occure via a continuous phase transition as well as the first-order phase transition depending on the model parameters. Moreover, the continuous phase transition is nonuniversal. It should be noted that the model does not take into account the possibility of horizontal orientation of the dimer and generally speaking this model can be

In [93] authors proposed the general lattice gas model describing the adsorption of complex molecules. The model can be formulate by following: the molecule can be adsorbed on the surface by the *k* ways occupying the 1 2 , , , *mm m <sup>k</sup>* active sites located in the corresponding configurations on homogeneous or heterogeneous lattice, respectively. The simplest model of this type is the model of homonuclear dimers adsorption. In this model it is assumed that dimer may have two different orientations in the adlayer – parallel (adsorption on two active

**Figure 2.** Phase diagrams for dimers on square and triangular lattices. Ordered structures: black circles are site occupied by dimers adsorbed on two sites; gray circles are sites occupied by dimers adsorbed on one site.

As it noted in [90], the results obtained for adsorption of complex molecules can be extended also to the gas mixtures. In other words, the general model can be considered as a special case of gas mixtures adsorption model. In this case the number of mixture components will be characterized by a number *k* and the molecule kind by value *m* . So if one consider the simplest representative of the general model – dimer adsorbed on two or one active site, it would be analogous to a binary mixture of dimers and monomers. These systems will be equivalent to each other if the model of binary mixture has the fixed difference between the chemical potentials of the components. From a physical point of view this means that the adsorbed molecules of one kind, and the behavior of the system is very similar to the adsorption of the gas mixture. Indeed, when the adsorption properties of complex molecules [93-97] was studed, it was discovered the phenomenon of non-monotonic changes in surface coverage with the chemical potential increasing, and a similar phenomenon was observed in the study of adsorption of binary mixtures [98,99].

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption 403

**Figure 3.** Possible configurations of the adsorbed molecules. The small circles represent molecules adsorbed on one active site, and the large ones – molecule adsorbed on the four active sites. The

infinitely strong repulsion between the nearest neighboring molecules in the model (a nearest neighborhood between two adsorbed molecules independently of the adsorption way is prohibited as well) are assumed, which in a first approximation, take into account the complex structure of the molecule. Active sites prohibited for the adsorption, in the case of two different orientations of the adsorbed molecules are indicated on Fig. 3 by the cross. It is seen that the proposed model of multisite adsorption eliminates the effect of varying the orientation of molecules relative to each other on the structure and thermodynamic properties of the adsorbed layer, since all possible orientations of the molecules in the plane parallel to the interface are taken into account simultaneously (Fig. 3). Thus, this model is the simplest model, which allows to study the effect of varying the orientation of complex organic molecules relative to the interface on the structure and properties of the adsorbed layer regardless of other factors. To reduce the dimensionality of the model the value 2 1 *q q* is introduced as the difference between the heats of adsorption of the four-site location and the mono-site one.

> 1 4 4 *i ii i ii H n nc*

where the occupation numbers *<sup>i</sup> c* and *<sup>i</sup> n* are equal to unity for occupied by molecule

The thermodynamic properties of the model have been investigated with standard importance sampling Monte Carlo method [93]. The calculations of the isotherms and the surface coverage as the function of the gas phase pressure (chemical potential) are carried out with the linear size of the lattice 96 *L* and *L* 24, 36, 48, 60 used for finite-size scaling procedures. The linear lattice size was chosen for the adlayer structures to be not perturbed.

(2)

is the chemical

numbers from 0 to 5 correspond to all possible states of the active site (cell).

The effective Hamiltonian of the model under consideration can be written as

adsorbed on one and four sites respectively and zero for empty sites;

potential of the adsorbed particles.

In order to evaluate the influence of surface geometry on the phase behavior of adsorbed monolayer the model of orientable dimers on the square and triangular lattice was studied [100,101]. It was shown that the influence of the coordination number (the number of nearest neighbors) plays an important role in the phase formation process. Namely, in the case of square lattice only two ordered structures consisting of dimers adsorbed only vertically or only horizontally can form, in the case of triangular lattice except for the phases of this type another phase consisting of differently oriented molecules appears. Phase diagrams are presented in Fig.2.

The simplest special case of the general model of adsorption of molecules with different orientations in the adsorbed monolayer is the lattice model of dimers adsorption discussed in detail above. Extending the model of dimers adsorption on molecules with more complex form (cyclic hydrocarbons, aromatic systems, etc.) one can get the lattice model of complex organic molecules adsorption, which takes into account, firstly, the possibility of different orientations with respect to the solid surface and, secondly, the diversity and complicated structure of surface complexes (non-linear shape of the adsorbate molecules). Further, a special case of the generalized model, which allows to study the effect of varying the orientation of the complex organic molecules on the behavior of the adsorbed layer in «pure form» will be considered. The fact is the model of dimers adsorption along with the possibility of different orientations of the molecules with respect to the surface takes into account the ability to the different orientation of molecules relative to each other *a priori*. Indeed, in the case of square lattice, the dimer adsorbed parallel to the surface can have two different orientations which does not allow us, in the framework of this model, to focus on studying the effect of varying the orientation of the molecule with respect to the solid surface on the structure and thermodynamic properties of the adlayer.

As model of solid surface homogeneous square lattice is considered and it is assumed that the molecule can be adsorbed in two different ways: 1) on four active sites (Fig. 3a) and 2) on one active site (Fig. 3b). In the first case the four active sites involved in adsorption process form the square. Thus, for the constructed model one have *k=2, m1=1, m2=4*. In addition, an

diagrams are presented in Fig.2.

As it noted in [90], the results obtained for adsorption of complex molecules can be extended also to the gas mixtures. In other words, the general model can be considered as a special case of gas mixtures adsorption model. In this case the number of mixture components will be characterized by a number *k* and the molecule kind by value *m* . So if one consider the simplest representative of the general model – dimer adsorbed on two or one active site, it would be analogous to a binary mixture of dimers and monomers. These systems will be equivalent to each other if the model of binary mixture has the fixed difference between the chemical potentials of the components. From a physical point of view this means that the adsorbed molecules of one kind, and the behavior of the system is very similar to the adsorption of the gas mixture. Indeed, when the adsorption properties of complex molecules [93-97] was studed, it was discovered the phenomenon of non-monotonic changes in surface coverage with the chemical potential increasing, and a similar

phenomenon was observed in the study of adsorption of binary mixtures [98,99].

In order to evaluate the influence of surface geometry on the phase behavior of adsorbed monolayer the model of orientable dimers on the square and triangular lattice was studied [100,101]. It was shown that the influence of the coordination number (the number of nearest neighbors) plays an important role in the phase formation process. Namely, in the case of square lattice only two ordered structures consisting of dimers adsorbed only vertically or only horizontally can form, in the case of triangular lattice except for the phases of this type another phase consisting of differently oriented molecules appears. Phase

The simplest special case of the general model of adsorption of molecules with different orientations in the adsorbed monolayer is the lattice model of dimers adsorption discussed in detail above. Extending the model of dimers adsorption on molecules with more complex form (cyclic hydrocarbons, aromatic systems, etc.) one can get the lattice model of complex organic molecules adsorption, which takes into account, firstly, the possibility of different orientations with respect to the solid surface and, secondly, the diversity and complicated structure of surface complexes (non-linear shape of the adsorbate molecules). Further, a special case of the generalized model, which allows to study the effect of varying the orientation of the complex organic molecules on the behavior of the adsorbed layer in «pure form» will be considered. The fact is the model of dimers adsorption along with the possibility of different orientations of the molecules with respect to the surface takes into account the ability to the different orientation of molecules relative to each other *a priori*. Indeed, in the case of square lattice, the dimer adsorbed parallel to the surface can have two different orientations which does not allow us, in the framework of this model, to focus on studying the effect of varying the orientation of the molecule with respect to the solid

As model of solid surface homogeneous square lattice is considered and it is assumed that the molecule can be adsorbed in two different ways: 1) on four active sites (Fig. 3a) and 2) on one active site (Fig. 3b). In the first case the four active sites involved in adsorption process form the square. Thus, for the constructed model one have *k=2, m1=1, m2=4*. In addition, an

surface on the structure and thermodynamic properties of the adlayer.

**Figure 3.** Possible configurations of the adsorbed molecules. The small circles represent molecules adsorbed on one active site, and the large ones – molecule adsorbed on the four active sites. The numbers from 0 to 5 correspond to all possible states of the active site (cell).

infinitely strong repulsion between the nearest neighboring molecules in the model (a nearest neighborhood between two adsorbed molecules independently of the adsorption way is prohibited as well) are assumed, which in a first approximation, take into account the complex structure of the molecule. Active sites prohibited for the adsorption, in the case of two different orientations of the adsorbed molecules are indicated on Fig. 3 by the cross. It is seen that the proposed model of multisite adsorption eliminates the effect of varying the orientation of molecules relative to each other on the structure and thermodynamic properties of the adsorbed layer, since all possible orientations of the molecules in the plane parallel to the interface are taken into account simultaneously (Fig. 3). Thus, this model is the simplest model, which allows to study the effect of varying the orientation of complex organic molecules relative to the interface on the structure and properties of the adsorbed layer regardless of other factors. To reduce the dimensionality of the model the value 2 1 *q q* is introduced as the difference between the heats of adsorption of the four-site location and the mono-site one. The effective Hamiltonian of the model under consideration can be written as

$$H = -\frac{\Delta}{4} \sum\_{i} n\_i - \mu \left( \frac{1}{4} \sum\_{i} n\_i + \sum\_{i} c\_i \right) \tag{2}$$

where the occupation numbers *<sup>i</sup> c* and *<sup>i</sup> n* are equal to unity for occupied by molecule adsorbed on one and four sites respectively and zero for empty sites; is the chemical potential of the adsorbed particles.

The thermodynamic properties of the model have been investigated with standard importance sampling Monte Carlo method [93]. The calculations of the isotherms and the surface coverage as the function of the gas phase pressure (chemical potential) are carried out with the linear size of the lattice 96 *L* and *L* 24, 36, 48, 60 used for finite-size scaling procedures. The linear lattice size was chosen for the adlayer structures to be not perturbed. Thermodynamic equilibrium is reached by spin-flip (Glauber) dynamics [102] and diffusion relaxation (Kawasaki dynamics) [103]. To calculate thermodynamic functions successive configurations of the adlayer are generated using Metropolis transition probabilities [104] in the grand canonical ensemble.

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption 405

The adsorption isotherms calculated for different values of the ratio *∆/RT* are shown in Fig. 6. It is seen that for sufficiently large values of *∆/RT* there are three distinct plateaus on the isotherms, which correspond to thermodynamically stable phases of the adsorption monolayer at *ρ=0,125, ρ=0,22(2)* и *ρ=0,5*, where *ρ* is density of the adlayer (the amount of adsorbed molecules per site). Based on the values of *ρ* corresponded to each horizontal plateau one can conclude that the first plateau determines the existence region of the ordered phase *c(4×4)4*, the second plateau – the existence region of the phase *c(3×3)4-1*, and the third – the existence region of the phase *c(2×2)*. In addition, according to the shape of the adsorption isotherms it can be assumed that the ordered structure *c(4×4)4* is formed from disordered lattice gas phase via second-order phase transition, which associated with the origin of a new symmetry element in the system. On the other hand, the phase transitions *c(4×4)4* – *c(3×3)4-1* and *c(3×3)4-1* – *c(2×2)* are the first-order phase transitions which associated with sharp changing in the first derivatives of the free energy of the system, in this case it is expressed by the sharp changing in the number of adsorbed molecules in the system. It is worth to note that along with an abrupt changing in the number of adsorbed molecules in the system, phase transitions *c(4×4)4* – *c(3×3)4-1* and *c(3×3)4-1* – *c(2×2)* are also attended by the

reorientation of the adsorbed molecules with respect to the solid surface.

**Figure 6.** Adsorption isotherms (left) and the dependencies of surface coverage on reduced chemical

The dependencies of surface coverage on reduced chemical potential *Δ/RT* demonstrate the same phase behavior of adsorbed monolayer (Fig. 6). Namely, there are three horizontal plateaus on the surface coverage curves at *θ=0,5, θ=0,55(5)* and *θ=0,5* for sufficiently large values of the ratio *Δ/RT*. These values of surface coverage, as in the case of the adsorption isotherms, coincide with the values of surface coverage for the above mentioned phases *c(4×4)4, c(3×3)4-1* and *c(2×2)*, respectively . However, there is one significant difference between the adsorption isotherms and the dependencies of surface coverage on reduced chemical potential: the isotherm is not only coincides with the curve of surface coverage, but

potential (right) calculated for different values of *Δ/RT*.

Analysis of the ground state (T = 0K) of the model allowed to conclude that in the adlayer due to the infinitely strong repulsive interactions between nearest-neighbor molecules the set of chessboard type ordered structures forms: *c(4×4)4*, *c(3×3)4-1* and *c(2×2)*.The structures are schematically shown in Fig. 4. In addition, the lattice gas phase (LG) with zero coverage exists in the ground state of the system. The corresponding phase diagram of the adsorption monolayer in the ground state of the system shown in Fig. 5.

**Figure 4.** Ordered structure of the adlayer. Gray circle denotes a molecule adsorbed on the four active sites, and the black circle – molecule adsorbed on one active site. The structures are shown in order of their formation with the chemical potential increasing.

**Figure 5.** Phase diagram of the adlayer in the ground state. The solid line separates the stability region of the LG phase and stability region of phase *c(4×4)4*, while the dashed and dotted lines separate the stability region of phases *c(4×4)4* – *c(3×3)4-1* and phases *c(3×3)4-1* – *c(2×2)*, respectively.

The adsorption isotherms calculated for different values of the ratio *∆/RT* are shown in Fig. 6. It is seen that for sufficiently large values of *∆/RT* there are three distinct plateaus on the isotherms, which correspond to thermodynamically stable phases of the adsorption monolayer at *ρ=0,125, ρ=0,22(2)* и *ρ=0,5*, where *ρ* is density of the adlayer (the amount of adsorbed molecules per site). Based on the values of *ρ* corresponded to each horizontal plateau one can conclude that the first plateau determines the existence region of the ordered phase *c(4×4)4*, the second plateau – the existence region of the phase *c(3×3)4-1*, and the third – the existence region of the phase *c(2×2)*. In addition, according to the shape of the adsorption isotherms it can be assumed that the ordered structure *c(4×4)4* is formed from disordered lattice gas phase via second-order phase transition, which associated with the origin of a new symmetry element in the system. On the other hand, the phase transitions *c(4×4)4* – *c(3×3)4-1* and *c(3×3)4-1* – *c(2×2)* are the first-order phase transitions which associated with sharp changing in the first derivatives of the free energy of the system, in this case it is expressed by the sharp changing in the number of adsorbed molecules in the system. It is worth to note that along with an abrupt changing in the number of adsorbed molecules in the system, phase transitions *c(4×4)4* – *c(3×3)4-1* and *c(3×3)4-1* – *c(2×2)* are also attended by the reorientation of the adsorbed molecules with respect to the solid surface.

404 Thermodynamics – Fundamentals and Its Application in Science

monolayer in the ground state of the system shown in Fig. 5.

their formation with the chemical potential increasing.

the grand canonical ensemble.

Thermodynamic equilibrium is reached by spin-flip (Glauber) dynamics [102] and diffusion relaxation (Kawasaki dynamics) [103]. To calculate thermodynamic functions successive configurations of the adlayer are generated using Metropolis transition probabilities [104] in

Analysis of the ground state (T = 0K) of the model allowed to conclude that in the adlayer due to the infinitely strong repulsive interactions between nearest-neighbor molecules the set of chessboard type ordered structures forms: *c(4×4)4*, *c(3×3)4-1* and *c(2×2)*.The structures are schematically shown in Fig. 4. In addition, the lattice gas phase (LG) with zero coverage exists in the ground state of the system. The corresponding phase diagram of the adsorption

**Figure 4.** Ordered structure of the adlayer. Gray circle denotes a molecule adsorbed on the four active sites, and the black circle – molecule adsorbed on one active site. The structures are shown in order of

**Figure 5.** Phase diagram of the adlayer in the ground state. The solid line separates the stability region of the LG phase and stability region of phase *c(4×4)4*, while the dashed and dotted lines separate the

stability region of phases *c(4×4)4* – *c(3×3)4-1* and phases *c(3×3)4-1* – *c(2×2)*, respectively.

**Figure 6.** Adsorption isotherms (left) and the dependencies of surface coverage on reduced chemical potential (right) calculated for different values of *Δ/RT*.

The dependencies of surface coverage on reduced chemical potential *Δ/RT* demonstrate the same phase behavior of adsorbed monolayer (Fig. 6). Namely, there are three horizontal plateaus on the surface coverage curves at *θ=0,5, θ=0,55(5)* and *θ=0,5* for sufficiently large values of the ratio *Δ/RT*. These values of surface coverage, as in the case of the adsorption isotherms, coincide with the values of surface coverage for the above mentioned phases *c(4×4)4, c(3×3)4-1* and *c(2×2)*, respectively . However, there is one significant difference between the adsorption isotherms and the dependencies of surface coverage on reduced chemical potential: the isotherm is not only coincides with the curve of surface coverage, but even out of proportion to it. Indeed, the dependencies of surface coverage on the reduced chemical potential are non-monotonic function for sufficiently large values of *Δ/RT*. While the adsorption isotherms in accordance with the general conditions of stability of thermodynamic systems

$$
\begin{split}
\left(\delta\Omega\right)\_{\text{T}\mu} &= \delta\left(\text{G} - \mu\text{N}\right)\_{\text{T}\mu} = \left(\frac{\partial\text{G}\left(\text{T}, \text{p}, \text{N}\right)}{\partial\text{N}} - \mu\right)\delta\text{N} + \frac{1}{2!}\frac{\partial^{2}\text{G}}{\partial\text{N}^{2}}\left(\delta\text{N}\right)^{2} + \\
&+ \frac{1}{3!}\frac{\partial^{3}\text{G}}{\partial\text{N}^{3}}\left(\delta\text{N}\right)^{3} + \frac{1}{4!}\frac{\partial^{4}\text{G}}{\partial\text{N}^{4}}\left(\delta\text{N}\right)^{4} + ... + \frac{1}{m!}\frac{\partial^{m}\text{G}}{\partial\text{N}^{m}}\left(\delta\text{N}\right)^{m} + \sum\_{n=m}^{\infty}\frac{1}{n!}\frac{\partial^{n}\text{G}}{\partial\text{N}^{n}}\left(\delta\text{N}\right)^{n} > 0,
\end{split}
\tag{3}
$$

$$
\mu = \frac{\partial\text{G}\left(\text{T}, p, \text{N}\right)}{\partial\text{N}}, \quad \left(\frac{\partial\mu}{\partial\text{N}}\right)\_{\text{T}} > 0,
$$

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption 407

**Figure 7.** The curves of partial surface coverage calculated at *Δ/RT* = 12. *θ1* – partial coverage of the surface by molecules adsorbed on one active site; *θ4* – partial coverage of the surface by molecules adsorbed on four active sites; *θ* – total surface coverage; ρ – density of the adlayer (adsorption

**Figure 8.** Ordered structure of the trimesic acid adsorption monolayer on Au (111) [21] and the

Analogous sequence of the ordered structures was also observed in other systems, in particular, in p-sexiphenyl [18] and pyridine adsorption monolayers [23] on Au (111). Thus, the phase behavior of monolayer adsorption which is realized in the model under consideration is qualitatively the same as the phase behavior of real adsorption monolayers of complex organic molecules on homogeneous surfaces. Therefore, a detailed study of the

corresponding structures calculated in the framework of the proposed model.

isotherm).

are monotonically increasing functions for all values of *Δ/RT*, where *Ω* – grand potential, *G –* Gibbs free energy, *μ –* chemical potential, *N –* amount of adsorbed molecules, *T –*  temperature and *p –* pressure in gas phase. The same effect is observed in the dimer adsorption models on square and triangular lattices. Apparently, this effect takes a place in any adsorption monolayer of molecules with different orientations with respect to the interface.

In order to uniquely identify the ordered structures appearing in the adsorption monolayer the authors of [93] calculated the curves of partial surface coverage by molecules adsorbed on one and four active sites as functions of the reduced chemical potential *μ/RT* (Fig. 7). It is seen the phase with coverage *θ = 0,5* formed at low values of μ/RT consists only of molecules adsorbed on four active sites, so it is obvious that it has the structure of *c(4×4)4*. Then, with increasing chemical potential (gas phase pressure) or decreasing in temperature of the substrate, the system undergoes the phase transition from phase *c(4×4)4* to the phase with coverage *θ=0,55(5)*. It is clear from Fig. 7 that this phase consists of both molecules adsorbed on four active sites and on one active site. Therefore, this phase has the structure of *c(3×3)4-1*. With further increase of the chemical potential or when the temperature decreases the system undergoes the phase transition from phase *c(3×3)4-1* to the phase *c(2×2)* with *ρ=0,5* and *θ=0,5* which is formed only by molecules adsorbed on one active site.

Recently, a similar phase behavior of adlayer of complex organic molecules was observed experimentally (Fig. 8). In [24] the authors investigated the behavior of the adsorption monolayer of trimesic acid on Au (111) with electrochemical scanning tunneling microscopy. It was shown that with increasing electrode potential the ordered structure of the adlayer consisting only of molecules oriented parallel to the surface changes into the ordered phase, which contains molecules adsorbed both parallel and perpendicular to the electrode surface. Further increasing in the electrode potential leads to the surface phase which is formed only by molecules oriented perpendicular to the surface.

3 4 3 4 3 4

*Tp Tp*

molecules adsorbed on one active site.

 

thermodynamic systems

interface.

even out of proportion to it. Indeed, the dependencies of surface coverage on the reduced chemical potential are non-monotonic function for sufficiently large values of *Δ/RT*. While the adsorption isotherms in accordance with the general conditions of stability of

*GT pN <sup>G</sup> G N N N*

<sup>2</sup> <sup>2</sup> 2

> 

(3)

 

*m n m n m n n m*

(,, ) , 0,

*T*

*N N*

 

11 1 1 ... 0, 3! 4!!!

*N N*

are monotonically increasing functions for all values of *Δ/RT*, where *Ω* – grand potential, *G –* Gibbs free energy, *μ –* chemical potential, *N –* amount of adsorbed molecules, *T –*  temperature and *p –* pressure in gas phase. The same effect is observed in the dimer adsorption models on square and triangular lattices. Apparently, this effect takes a place in any adsorption monolayer of molecules with different orientations with respect to the

In order to uniquely identify the ordered structures appearing in the adsorption monolayer the authors of [93] calculated the curves of partial surface coverage by molecules adsorbed on one and four active sites as functions of the reduced chemical potential *μ/RT* (Fig. 7). It is seen the phase with coverage *θ = 0,5* formed at low values of μ/RT consists only of molecules adsorbed on four active sites, so it is obvious that it has the structure of *c(4×4)4*. Then, with increasing chemical potential (gas phase pressure) or decreasing in temperature of the substrate, the system undergoes the phase transition from phase *c(4×4)4* to the phase with coverage *θ=0,55(5)*. It is clear from Fig. 7 that this phase consists of both molecules adsorbed on four active sites and on one active site. Therefore, this phase has the structure of *c(3×3)4-1*. With further increase of the chemical potential or when the temperature decreases the system undergoes the phase transition from phase *c(3×3)4-1* to the phase *c(2×2)* with *ρ=0,5* and *θ=0,5* which is formed only by

Recently, a similar phase behavior of adlayer of complex organic molecules was observed experimentally (Fig. 8). In [24] the authors investigated the behavior of the adsorption monolayer of trimesic acid on Au (111) with electrochemical scanning tunneling microscopy. It was shown that with increasing electrode potential the ordered structure of the adlayer consisting only of molecules oriented parallel to the surface changes into the ordered phase, which contains molecules adsorbed both parallel and perpendicular to the electrode surface. Further increasing in the electrode potential leads to the surface phase

which is formed only by molecules oriented perpendicular to the surface.

 

*GG G G NN N N NN N N m n*

(,, ) <sup>1</sup> ( ) 2!

*GT pN*

 

**Figure 7.** The curves of partial surface coverage calculated at *Δ/RT* = 12. *θ1* – partial coverage of the surface by molecules adsorbed on one active site; *θ4* – partial coverage of the surface by molecules adsorbed on four active sites; *θ* – total surface coverage; ρ – density of the adlayer (adsorption isotherm).

**Figure 8.** Ordered structure of the trimesic acid adsorption monolayer on Au (111) [21] and the corresponding structures calculated in the framework of the proposed model.

Analogous sequence of the ordered structures was also observed in other systems, in particular, in p-sexiphenyl [18] and pyridine adsorption monolayers [23] on Au (111). Thus, the phase behavior of monolayer adsorption which is realized in the model under consideration is qualitatively the same as the phase behavior of real adsorption monolayers of complex organic molecules on homogeneous surfaces. Therefore, a detailed study of the model and models similar to that is very useful for deeper understanding of the thermodynamics of self-assembled monolayers of complex organic molecules.

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption 409

1. The phenomenon of non-monotonic changes of the surface coverage as function of the chemical potential is a general property of the systems under consideration and is not observed in models of single-site or multisite single-component gas adsorption which

2. There are ordered structures in the adsorption monolayer of complex organic molecules consisting of the molecules with different orientations relative to the solid surface,

3. The transition between the ordered structures of the adsorption monolayer occurs

4. The phase behavior of adsorption monolayer of complex organic molecules strongly depends on two factors: a) the geometry and chemical structure of the adsorbed

It is worth to note that the adsorption of single-component gas consisting of molecules with complex shape can exhibit the features which typical for the adsorption of multicomponent gas mixtures [90]. In particular, the model of adsorption of a gas mixture A (adsorb on *m1* active sites) and B (adsorb on *m2* active sites) is equivalent to the considered models (with *k =* 

Lattice gas model and its various generalizations are one of the most important models of modern statistical thermodynamics and are currently experiencing a new stage of development. On the one hand, it is related to the solution of some fundamental issues concerning the limiting behavior of two-dimensional lattice models, and, on the other hand, it is related to successful attempts to describe with the help of this model complex systems,

Lattice models of multisite adsorption can take into account the number of characteristic features of complex adsorption systems, the most important of which is the possibility of the orientation varying of the adsorbate molecules, both with respect to each other and relative to the solid surface. Analysis of the thermodynamic properties of models of this type showed that despite its relative simplicity they exhibit a fantastic variety of non-trivial physical phenomena. First of all, it refers to the processes of self-assembly and phase transitions of various types, whose study is of considerable interest from the standpoint of the general thermodynamic theory. In addition, referring to the results of the experimental data, it can be argued that discussed in this chapter multisite adsorption model is firstly qualitatively, and in some cases quantitatively, reproduce the behavior of real systems and secondly have considerable predictive power. The results of study of multisite adsorption

1. The adlayer of complex organic and inorganic molecules can form ordered structures consisting of molecules with different orientations simultaneously (both with respect to

*2*) when the chemical potentials of the gas mixture components are the same, μA = μB.

in particular, multisite adsorption of various molecules on the solid surface.

exclude different orientations in the adsorption monolayer.

abruptly through the first-order phase transition.

molecule, and b) the geometry of the surface.

models can be summarized in following conclusions:

each other and relative to the solid surface).

simultaneously.

**4. Conclusion** 

In the framework of the model under consideration, the modern methods of theoretical physics such as the multiple-histogram reweighting and finite-size scaling techniques (the fourth-order cumulant of the order parameter) have been used to estimate the phase diagram (T, μ) of complicated organic molecules adsorbed on the homogenous square lattice (Fig. 9) [93].

**Figure 9.** Phase diagram of adsorption monolayer of complex organic molecules on homogenous square lattice.

The phase diagram shown in Fig. 9 differs from the phase diagrams of dimers adsorption monolayer on square and triangular lattices that: 1) there is the phase consisting of molecules oriented in both ways (parallel and perpendicular to the solid surface) simultaneously, and 2) the critical temperature of this phase (*c(3×3)4-1*) is higher than the critical temperature of the ordered phase formed only by molecules adsorbed parallel to the surface. Indeed, the phase diagram of dimers adsorption on square lattice contains only two regions corresponding to the ordered phases due to the fact that when dimers adsorb on square lattice the ordered phase consisting of both dimers adsorbed parallel and perpendicular to the surface does not appear. In contrast, when dimers adsorb on triangular lattice that ordered phase is formed, but its critical temperature is lower than the critical temperature of the ordered structure formed only by dimers adsorbed parallel to the surface.

Thus, the numerical analysis of the thermodynamic properties of the lattice models which take into account the possibility of varying the orientation of the adsorbate molecules both with respect to each other and with respect to the solid surface showed that:


It is worth to note that the adsorption of single-component gas consisting of molecules with complex shape can exhibit the features which typical for the adsorption of multicomponent gas mixtures [90]. In particular, the model of adsorption of a gas mixture A (adsorb on *m1* active sites) and B (adsorb on *m2* active sites) is equivalent to the considered models (with *k = 2*) when the chemical potentials of the gas mixture components are the same, μA = μB.

## **4. Conclusion**

408 Thermodynamics – Fundamentals and Its Application in Science

lattice (Fig. 9) [93].

square lattice.

model and models similar to that is very useful for deeper understanding of the

In the framework of the model under consideration, the modern methods of theoretical physics such as the multiple-histogram reweighting and finite-size scaling techniques (the fourth-order cumulant of the order parameter) have been used to estimate the phase diagram (T, μ) of complicated organic molecules adsorbed on the homogenous square

**Figure 9.** Phase diagram of adsorption monolayer of complex organic molecules on homogenous

formed only by dimers adsorbed parallel to the surface.

The phase diagram shown in Fig. 9 differs from the phase diagrams of dimers adsorption monolayer on square and triangular lattices that: 1) there is the phase consisting of molecules oriented in both ways (parallel and perpendicular to the solid surface) simultaneously, and 2) the critical temperature of this phase (*c(3×3)4-1*) is higher than the critical temperature of the ordered phase formed only by molecules adsorbed parallel to the surface. Indeed, the phase diagram of dimers adsorption on square lattice contains only two regions corresponding to the ordered phases due to the fact that when dimers adsorb on square lattice the ordered phase consisting of both dimers adsorbed parallel and perpendicular to the surface does not appear. In contrast, when dimers adsorb on triangular lattice that ordered phase is formed, but its critical temperature is lower than the critical temperature of the ordered structure

Thus, the numerical analysis of the thermodynamic properties of the lattice models which take into account the possibility of varying the orientation of the adsorbate molecules both

with respect to each other and with respect to the solid surface showed that:

thermodynamics of self-assembled monolayers of complex organic molecules.

Lattice gas model and its various generalizations are one of the most important models of modern statistical thermodynamics and are currently experiencing a new stage of development. On the one hand, it is related to the solution of some fundamental issues concerning the limiting behavior of two-dimensional lattice models, and, on the other hand, it is related to successful attempts to describe with the help of this model complex systems, in particular, multisite adsorption of various molecules on the solid surface.

Lattice models of multisite adsorption can take into account the number of characteristic features of complex adsorption systems, the most important of which is the possibility of the orientation varying of the adsorbate molecules, both with respect to each other and relative to the solid surface. Analysis of the thermodynamic properties of models of this type showed that despite its relative simplicity they exhibit a fantastic variety of non-trivial physical phenomena. First of all, it refers to the processes of self-assembly and phase transitions of various types, whose study is of considerable interest from the standpoint of the general thermodynamic theory. In addition, referring to the results of the experimental data, it can be argued that discussed in this chapter multisite adsorption model is firstly qualitatively, and in some cases quantitatively, reproduce the behavior of real systems and secondly have considerable predictive power. The results of study of multisite adsorption models can be summarized in following conclusions:

1. The adlayer of complex organic and inorganic molecules can form ordered structures consisting of molecules with different orientations simultaneously (both with respect to each other and relative to the solid surface).

2. The phase diagram of such systems is asymmetric. Its shape depends essentially firstly on the size and shape of the adsorbate molecules and secondly on the coordination number and type of heterogeneity of the substrate surface.

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption 411

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2130.

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(2007) 375.

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2959.


From applied point of view a theoretical study of such systems in general is of great interest because of the many possible applications in which used the adsorption monolayers one way or another (self-assembled monolayers adsorbed, thin organic films). The potential application range of such systems is an organic optoelectronics, the development of active elements of chemical and biological sensors, biomedical materials, heterogeneous catalysis. In this paper it is shown that the development of lattice models of these systems and study of its thermodynamic properties allows ones to understand and generalize laws of physical and chemical processes occurring in such systems, and brings the scientists closer to being able to fully control the phase behavior of monolayers of organic and other molecules on solid surfaces.

## **Author details**

Vasiliy Fefelov, Vitaly Gorbunov and Marta Myshlyavtseva *Omsk State Technical University, Russia* 

Alexander Myshlyavtsev *Omsk State Technical University, Russia Institute of Hydrocarbons Processing SB RAS, Russia* 

## **5. References**


[5] J.L. Riccardo, A.J. Ramirez-Pastor, F. Roma, Multilayer Adsorption with Multisite Occupancy: An Improved Isotherm for Surface Characterization // Langmuir 18 (2002) 2130.

410 Thermodynamics – Fundamentals and Its Application in Science

transition.

solid surfaces.

**Author details** 

**5. References** 

163.

Alexander Myshlyavtsev

*Omsk State Technical University, Russia* 

*Omsk State Technical University, Russia* 

Academic Press: London, 1992.

surfaces // Surf. Sci. 198 (1988) 571.

*Institute of Hydrocarbons Processing SB RAS, Russia* 

number and type of heterogeneity of the substrate surface.

coordination number of the lattice (entropy factor).

Vasiliy Fefelov, Vitaly Gorbunov and Marta Myshlyavtseva

2. The phase diagram of such systems is asymmetric. Its shape depends essentially firstly on the size and shape of the adsorbate molecules and secondly on the coordination

3. The critical temperature of the ordered phases of the adlayer on the one hand increases with the size of the molecule and on the other hand decreases with increasing

4. In models that take into account the possibility of the orientation varying of molecules adsorbed on the solid surface one can see the phenomenon of non-monotonic change in the coverage with increasing chemical potential, and the transition between the ordered structures of the monolayer adsorption occurs abruptly, through first order phase

From applied point of view a theoretical study of such systems in general is of great interest because of the many possible applications in which used the adsorption monolayers one way or another (self-assembled monolayers adsorbed, thin organic films). The potential application range of such systems is an organic optoelectronics, the development of active elements of chemical and biological sensors, biomedical materials, heterogeneous catalysis. In this paper it is shown that the development of lattice models of these systems and study of its thermodynamic properties allows ones to understand and generalize laws of physical and chemical processes occurring in such systems, and brings the scientists closer to being able to fully control the phase behavior of monolayers of organic and other molecules on

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**Section 5** 

**Non-Equilibrium Thermodynamics** 


**Non-Equilibrium Thermodynamics** 

416 Thermodynamics – Fundamentals and Its Application in Science

Chimia I chimicheskaya, 9 (2010) 66.

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**Chapter 16** 

© 2012 Ionescu, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

Although the turbulence is often associated with fluid dynamics, it is in fact a basic feature for most systems with few or infinity freedom degrees. It can be defined as chaotic behavior of the systems with few freedom degrees and which are far from the thermodynamic


In hydrodynamics, the transition problem lays back to the end of last century, at the pioneering works of Osborne Reynolds and Lord Rayleigh. Since the beginnings, it was pointed out the fruitful investigation method of considering the linear stability of basic laminar flow until infinitesimal turbulences. Nonlinearity can act in the sense of stabilizing the flow and therefore the primary state is replaced with another stable motion which is considered as secondary flow. This one can be further replaced with a tertiary stable flow, and so on. It is in fact about a *bifurcations sequence*, and Couette-Taylor flow is maybe the

The situation becomes hard to approach if the non-linearity is acting in the sense of increasing the rate of growing the unstable linear modes. Although it was anticipated that the flows can be stable according to the linear theory, in experiments it was concluded that they are unstable. It must be noticed that Reynolds himself understood this possibility, and suggested that for the transition from laminar to turbulence for a pipe flow, "the condition must be of instability at certain size perturbations and stability at smaller perturbations".

**Influence of Simulation Parameters** 

**on the Excitable Media Behaviour –** 

**The Case of Turbulent Mixing** 

Additional information is available at the end of the chapter

equilibrium. In this area two important zones are distinguished:


Adela Ionescu

**1. Introduction** 

http://dx.doi.org/10.5772/51585

characteristic to turbulence;

most widespread example in this sense.

Adela Ionescu

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51585

## **1. Introduction**

Although the turbulence is often associated with fluid dynamics, it is in fact a basic feature for most systems with few or infinity freedom degrees. It can be defined as chaotic behavior of the systems with few freedom degrees and which are far from the thermodynamic equilibrium. In this area two important zones are distinguished:


In hydrodynamics, the transition problem lays back to the end of last century, at the pioneering works of Osborne Reynolds and Lord Rayleigh. Since the beginnings, it was pointed out the fruitful investigation method of considering the linear stability of basic laminar flow until infinitesimal turbulences. Nonlinearity can act in the sense of stabilizing the flow and therefore the primary state is replaced with another stable motion which is considered as secondary flow. This one can be further replaced with a tertiary stable flow, and so on. It is in fact about a *bifurcations sequence*, and Couette-Taylor flow is maybe the most widespread example in this sense.

The situation becomes hard to approach if the non-linearity is acting in the sense of increasing the rate of growing the unstable linear modes. Although it was anticipated that the flows can be stable according to the linear theory, in experiments it was concluded that they are unstable. It must be noticed that Reynolds himself understood this possibility, and suggested that for the transition from laminar to turbulence for a pipe flow, "the condition must be of instability at certain size perturbations and stability at smaller perturbations".

The issue of transition in flows such Poiseuille flow until Reynolds numbers under the critical value, must be due to instabilities at finite amplitude perturbations. For these flows there was nothing relevant found concerning the eventual secondary stable motions, moreover it seems that turbulence issue directly from primary flow, at a fixed Reynolds number. These *strong turbulence problems* are quite difficult to approach and this gives evidence that still is to be added more substance to the original Reynolds' suggestion, after one hundred years of stability studies. For this quite new theory, *unifying with the classic turbulence* is far from being solving, but the recent challenges could open new research directions.

Generally, the statistical idea of a flow is represented by a map:

$$\mathbf{x} = \boldsymbol{\Phi}\_t \begin{pmatrix} \mathbf{X} \end{pmatrix}, \mathbf{X} = \boldsymbol{\Phi}\_{t=0} \begin{pmatrix} \mathbf{X} \end{pmatrix} \tag{1}$$

Influence of Simulation Parameters on the Excitable Media Behaviour – The Case of Turbulent Mixing 421

**X A** (5)

(7)

(8)

(6)

, *d d d d* **X A M N**

2 21 , det *C MN FC MN ij i j ij i j*

where **D** is the deformation tensor, obtained by decomposing the velocity gradient in its

The flow *x=Φt(X)* has *a good mixing* if the mean values *D(lnλ)/D*t and *D(lnη)/Dt* are not decreasing to zero, for any initial position P and any initial orientations **M** and **N**. As the above two quantities are bounded, the deformation efficiency can be naturally quantified. Thus, there is defined (Otino, 1989) the *deformation efficiency in length*, *e<sup>λ</sup> = eλ (X,M,t)* of the

> 1/2 ln / <sup>1</sup>

 **D D**

*e*

*e*

**2. Issues on turbulent mixing. Results and challenges** 

and 3D models of different flow types.

: *D Dt*

Similarly, there is defined the *deformation efficiency in surface*, *, eη= eη (X,N,t)* of the area element d**A**: in the case of an isochoric flow (the jacobian equal 1), there exists the equation

> 1/2 ln / <sup>1</sup>

 **D D**

: *D Dt*

The deformation tensor **F** and the associated tensors **C**, **C**-1, form the fundamental quantities for the analysis of deformation of infinitesimal elements. In most cases the flow *<sup>t</sup> x X* is unknown and has to be obtained by integration from the Eulerian velocity field. If this can be done analytically, then **F** can be obtained by differentiation of the flow with respect to the material coordinates **X.** The flows of interest belong to two classes: i) flows with a special form of *v* and ii) flows with a special form of **F**. The second class is what we are interested for, as it contains the so-called Constant Stretch History Motion –

The central problem exhibited in this chapter is the challenge of unifying the theory of turbulent mixing. This implies few levels: analytical, computational and experimental. The specific literature is rich in works both on analytical and experimental models. There were realized few comparative analysis, both for analytical and computational standpoint, for 2D

Very often, in practice is used the scalar form of (4), namely the following relation:

 

symmetric and non-symmetric part.

material element d**X**, as follows:

(8):

CSHM.

That means *X is mapped in x* after a time t. In the continuum mechanics the relation (1) is named *flow*, and it is a diffeomorphism of class Ck. Moreover, the relation (1) must satisfy the following equation:

$$J = \det\left(D\left(\Phi\_t\left(X\right)\right)\right) = \det\left(\frac{\partial \mathbf{x}\_i}{\partial X\_j}\right) \tag{2}$$

where D denotes the derivation with respect to the reference configuration, in this case **X**. The equation (2) implies two particles, X1 and X2, which occupy the same position **x** at a moment. Non-topological behavior (like break up, for example) *is not allowed*.

With respect to **X** there is defined the basic measure of deformation, the *deformation gradient*, **F,** namely the equation (3):

$$\mathbf{F} = \left(\nabla\_X \Phi\_t\left(\mathbf{X}\right)\right)^T,\\ F\_{ij} = \left(\frac{\partial \mathbf{x}\_i}{\partial X\_j}\right) \tag{3}$$

where *X* denotes differentiation with respect to **X**. According to equation (2), **F** is non singular. The basic measure for the deformation with respect to **x** is the *velocity gradient*.

After defining the basic deformation of a material filament and the corresponding relation for the area of an infinitesimal material surface, there can be defined the basic deformation measures: the *length deformation* λ and *surface deformation* η, with the following relations (Ottino, 1989):

$$\mathcal{Q} = \left(\mathbf{C} : \mathbf{M}\mathbf{M}\right)^{1/2}, \ \eta = \left(\det F\right) \cdot \left(\mathbf{C}^{-1} : \mathbf{N}\mathbf{N}\right)^{1/2} \tag{4}$$

where **C** (=**F**T·**F**) is the *Cauchy-Green deformation tensor,* and the vectors **M**,**N** - the orientation versors in length and surface respectively, are defined by:

$$\mathbf{M} = \frac{d\mathbf{X}}{|d\mathbf{X}|}, \text{ N} = \frac{d\mathbf{A}}{|d\mathbf{A}|} \tag{5}$$

Very often, in practice is used the scalar form of (4), namely the following relation:

420 Thermodynamics – Fundamentals and Its Application in Science

Generally, the statistical idea of a flow is represented by a map:

directions.

the following equation:

**F,** namely the equation (3):

(Ottino, 1989):

The issue of transition in flows such Poiseuille flow until Reynolds numbers under the critical value, must be due to instabilities at finite amplitude perturbations. For these flows there was nothing relevant found concerning the eventual secondary stable motions, moreover it seems that turbulence issue directly from primary flow, at a fixed Reynolds number. These *strong turbulence problems* are quite difficult to approach and this gives evidence that still is to be added more substance to the original Reynolds' suggestion, after one hundred years of stability studies. For this quite new theory, *unifying with the classic turbulence* is far from being solving, but the recent challenges could open new research

That means *X is mapped in x* after a time t. In the continuum mechanics the relation (1) is named *flow*, and it is a diffeomorphism of class Ck. Moreover, the relation (1) must satisfy

> det det *<sup>i</sup> t*

where D denotes the derivation with respect to the reference configuration, in this case **X**. The equation (2) implies two particles, X1 and X2, which occupy the same position **x** at a

With respect to **X** there is defined the basic measure of deformation, the *deformation gradient*,

*<sup>T</sup> <sup>i</sup>*

*<sup>x</sup> <sup>F</sup> X* 

moment. Non-topological behavior (like break up, for example) *is not allowed*.

,

*X t ij*

where *X* denotes differentiation with respect to **X**. According to equation (2), **F** is non singular. The basic measure for the deformation with respect to **x** is the *velocity gradient*.

After defining the basic deformation of a material filament and the corresponding relation for the area of an infinitesimal material surface, there can be defined the basic deformation measures: the *length deformation* λ and *surface deformation* η, with the following relations

1/2 1/2 <sup>1</sup>

where **C** (=**F**T·**F**) is the *Cauchy-Green deformation tensor,* and the vectors **M**,**N** - the orientation

 

versors in length and surface respectively, are defined by:

*<sup>x</sup> J DX <sup>X</sup>*

<sup>0</sup> , *t t x XX X* (1)

(2)

*j*

*j*

: , det : *F* **C MM C NN** (4)

**F X** (3)

$$\mathcal{X}^2 = \mathbb{C}\_{ij} \cdot \mathbf{M}\_i \cdot \mathbf{N}\_{j'} \ \eta^2 = \left(\det F\right) \cdot \mathbb{C}\_{ij}^{-1} \cdot M\_i \cdot \mathbf{N}\_j \tag{6}$$

where **D** is the deformation tensor, obtained by decomposing the velocity gradient in its symmetric and non-symmetric part.

The flow *x=Φt(X)* has *a good mixing* if the mean values *D(lnλ)/D*t and *D(lnη)/Dt* are not decreasing to zero, for any initial position P and any initial orientations **M** and **N**. As the above two quantities are bounded, the deformation efficiency can be naturally quantified. Thus, there is defined (Otino, 1989) the *deformation efficiency in length*, *e<sup>λ</sup> = eλ (X,M,t)* of the material element d**X**, as follows:

$$e\_{\lambda} = \frac{D\left(\ln \lambda\right) / Dt}{\left(\mathbf{D} : \mathbf{D}\right)^{1/2}} \le 1 \tag{7}$$

Similarly, there is defined the *deformation efficiency in surface*, *, eη= eη (X,N,t)* of the area element d**A**: in the case of an isochoric flow (the jacobian equal 1), there exists the equation (8):

$$\varepsilon\_{\eta} = \frac{D\left(\ln \eta\right) / Dt}{\left(\mathbf{D} : \mathbf{D}\right)^{1/2}} \le 1 \tag{8}$$

The deformation tensor **F** and the associated tensors **C**, **C**-1, form the fundamental quantities for the analysis of deformation of infinitesimal elements. In most cases the flow *<sup>t</sup> x X* is unknown and has to be obtained by integration from the Eulerian velocity field. If this can be done analytically, then **F** can be obtained by differentiation of the flow with respect to the material coordinates **X.** The flows of interest belong to two classes: i) flows with a special form of *v* and ii) flows with a special form of **F**. The second class is what we are interested for, as it contains the so-called Constant Stretch History Motion – CSHM.

#### **2. Issues on turbulent mixing. Results and challenges**

The central problem exhibited in this chapter is the challenge of unifying the theory of turbulent mixing. This implies few levels: analytical, computational and experimental. The specific literature is rich in works both on analytical and experimental models. There were realized few comparative analysis, both for analytical and computational standpoint, for 2D and 3D models of different flow types.

## **2.1. Recent issues in the literature**

The recent discussions and works (Dimotakis, 1983) on the empirical evidence and theoretical ansatz on turbulence support the notion that fully–developed turbulence requires a minimum Reynolds of order of 104 to be sustained. This value must be viewed as a necessary, but not sufficient condition for the flow to be fully developed. Presently available evidence suggests that both the fact that the phenomenon occures and the range of values of Reynolds number where it occures are universal, i.e. independent of the flow geometry.

Influence of Simulation Parameters on the Excitable Media Behaviour – The Case of Turbulent Mixing 423

almost Reynolds – number independent normalized variance of the jet-fluid concentration on the jet axis, with a strong Reynolds – number dependence found in liquid – phase jets, in

To summarize, recent data on turbulent mixing suppport the notion that the fully developed turbulent flow requires a minimum Reynolds number of 104, or a Taylor Reynolds number of ReT≈102 to be sustained. Conversely, turbulent flow below this Reynolds number cannot be regarded as fully – developed and can be expected to be

The manifestation of the transition to this state may depend on the particular flow geometry, e.g. the appearance of streamwise vortices and three – dimensionality in shear layers. Neverless, the fact that such a transition occurs, as well as the approximate Reynolds number where it is expected, appears to be a universal property of turbulence. It is observed

**2.2. Recent results from experimental and computational / analytical standpoint** 

Recently, it was realized (Ionescu, 2002, 2010) the analysis of the length and surface deformations efficiency for a mathematical 3D model associated to a vortexation

The mathematical study was done in association with the experiments realized in a special vortexation tube, closed at one end. Locally, there is produced a high intensity annular vorticity zone, which is acting like a tornado. The small scale at which the turbulence issues allows retaining the solid particles, mixing the textile fibbers or breaking up the multi-

The special vortex tube used for achieving the breaking up of filaments is a modified version at low pressure of a Ranque-Hilsch tube (Savulescu, 1998). Completely closing an end of the tube, there is obtained in this region a high intensity swirl. The flow in the tube is generally like a swirl, with a rate tangential velocity / axial velocity maxim near the closed

The approaching aerodynamic circuit is made from: a pressure source, the box of tangential inputs, the diffusion zone, the tube where there is produced the swirl with additional

It must be noticed that this torrential flow is concentrating and intensifying the vorticity, in contrast with the usual cyclone-type flow or other flows generators. If in the installation is introduced a pollutant, there issues a turbulent mixing in the annular vorticity zone. The spatial and temporal scales revealed the existence of different domains, starting with laboratory ones and until dissipative domains or others - corresponding to fine –structure wave numbers. Thus, the applications area is very large, including collecting, aggregating,

in a wide variety of flows and turbulent flow phenomena. (Dimotakis, 1983)

phenomena. The biological material used is the aquatic algae *Spirulina Platensis.*

the same Reynolds – number range.

qualitatively different (Dimotakis 1983).

cellular filaments of aquatic algae.

end, where is also created the annular vortex structure.

pollutants inputs, and the closing end with a rotating end.

separating and fragmenting the various pollutants.

On the other hand, how sharp this transition is *does* appear to depend on the details of the flow. In particular, it is considerable sharp, as a function of Reynolds number, in the (Couette- Taylor) flow between concentric rotative cylinders. It is less well – defined for a shear layer and, among the flows considered, the least well – defined for turbulent jets. Perhaps an explanation for this variation lies in the definition of the Reynolds number itself and the manner in which the various factors that enter are specified for each flow. In the case of the Couette-Taylor flow, for example, both the the velocity *U a CT* and the spatial scale *CT b a* ., where Ω is the differential rotation rate, with a and b - the inner and outer cylinder radii, are well-defined by the flow-boundary conditions.

In the case of a zero streamwise pressure gradient shear layer, the velocity 1 *U UU U <sup>S</sup>* 1 2 is a constant, reasonably well specified by the flow boundary conditions at a particular station. The length scale 1 1 *s s x* must be regarded as a stochastic variable in a given flow with a relatively large variance. The Reynolds number for the shear layer is then the product of a well-defined variable and a less well-defined, stochastic, variable.

In the case of a turbulent jet, both the local velocity Uj and the scale δj must be regarded as stochastic flow variables, each with its own large variance. The Reynolds number for the jet is then the product of two stochastic variables, and, as a consequence, its local, instantaneous value is the least well-defined of the three.

As regards fully-developed turbulent flow, the presently available evidence does not support the notion of Reynolds – number – independent mixing dynamics, at least in the case of gas – phase shear layers for which the investigations span a large enough range. In the case of gas – phase turbulent jets, presently available evidence admits a flame length stoichiometric coefficient tending to a Reynolds number – independent behaviour. It must be noticed, however, that the range of Reynolds numbers spanned by experiments may not be large enough to provide us with a definitive statement, at least as evidenced by the range required in the case of shear layers.

In comparing shear layer with turbulent - jet mixing behavior, the more important conclusion may be that they appear to respond in the opposite way to Schmidt number effects, i.e. gas- vs liquid – phase behavior. Specifically, there are high Schmidt number (liquid – phase) shear layers that exhibit a low Reynolds number dependence in chemical product formation, if any. In contrast, there are gas- phase turbulent jets that exhibit an almost Reynolds – number independent normalized variance of the jet-fluid concentration on the jet axis, with a strong Reynolds – number dependence found in liquid – phase jets, in the same Reynolds – number range.

422 Thermodynamics – Fundamentals and Its Application in Science

The recent discussions and works (Dimotakis, 1983) on the empirical evidence and theoretical ansatz on turbulence support the notion that fully–developed turbulence requires a minimum Reynolds of order of 104 to be sustained. This value must be viewed as a necessary, but not sufficient condition for the flow to be fully developed. Presently available evidence suggests that both the fact that the phenomenon occures and the range of values of Reynolds number where it occures are universal, i.e. independent of the flow

On the other hand, how sharp this transition is *does* appear to depend on the details of the flow. In particular, it is considerable sharp, as a function of Reynolds number, in the (Couette- Taylor) flow between concentric rotative cylinders. It is less well – defined for a shear layer and, among the flows considered, the least well – defined for turbulent jets. Perhaps an explanation for this variation lies in the definition of the Reynolds number itself and the manner in which the various factors that enter are specified for each flow. In the case of the Couette-Taylor flow, for example, both the the velocity *U a CT* and the

In the case of a zero streamwise pressure gradient shear layer, the velocity

*U UU U <sup>S</sup>* 1 2 is a constant, reasonably well specified by the flow boundary conditions

in a given flow with a relatively large variance. The Reynolds number for the shear layer is then the product of a well-defined variable and a less well-defined, stochastic, variable.

In the case of a turbulent jet, both the local velocity Uj and the scale δj must be regarded as stochastic flow variables, each with its own large variance. The Reynolds number for the jet is then the product of two stochastic variables, and, as a consequence, its local,

As regards fully-developed turbulent flow, the presently available evidence does not support the notion of Reynolds – number – independent mixing dynamics, at least in the case of gas – phase shear layers for which the investigations span a large enough range. In the case of gas – phase turbulent jets, presently available evidence admits a flame length stoichiometric coefficient tending to a Reynolds number – independent behaviour. It must be noticed, however, that the range of Reynolds numbers spanned by experiments may not be large enough to provide us with a definitive statement, at least as evidenced by the range

In comparing shear layer with turbulent - jet mixing behavior, the more important conclusion may be that they appear to respond in the opposite way to Schmidt number effects, i.e. gas- vs liquid – phase behavior. Specifically, there are high Schmidt number (liquid – phase) shear layers that exhibit a low Reynolds number dependence in chemical product formation, if any. In contrast, there are gas- phase turbulent jets that exhibit an

and outer cylinder radii, are well-defined by the flow-boundary conditions.

 

*b a* ., where Ω is the differential rotation rate, with a and b - the inner

*x* must be regarded as a stochastic variable

**2.1. Recent issues in the literature** 

geometry.

spatial scale *CT*

1

required in the case of shear layers.

at a particular station. The length scale 1 1 *s s*

instantaneous value is the least well-defined of the three.

To summarize, recent data on turbulent mixing suppport the notion that the fully developed turbulent flow requires a minimum Reynolds number of 104, or a Taylor Reynolds number of ReT≈102 to be sustained. Conversely, turbulent flow below this Reynolds number cannot be regarded as fully – developed and can be expected to be qualitatively different (Dimotakis 1983).

The manifestation of the transition to this state may depend on the particular flow geometry, e.g. the appearance of streamwise vortices and three – dimensionality in shear layers. Neverless, the fact that such a transition occurs, as well as the approximate Reynolds number where it is expected, appears to be a universal property of turbulence. It is observed in a wide variety of flows and turbulent flow phenomena. (Dimotakis, 1983)

## **2.2. Recent results from experimental and computational / analytical standpoint**

Recently, it was realized (Ionescu, 2002, 2010) the analysis of the length and surface deformations efficiency for a mathematical 3D model associated to a vortexation phenomena. The biological material used is the aquatic algae *Spirulina Platensis.*

The mathematical study was done in association with the experiments realized in a special vortexation tube, closed at one end. Locally, there is produced a high intensity annular vorticity zone, which is acting like a tornado. The small scale at which the turbulence issues allows retaining the solid particles, mixing the textile fibbers or breaking up the multicellular filaments of aquatic algae.

The special vortex tube used for achieving the breaking up of filaments is a modified version at low pressure of a Ranque-Hilsch tube (Savulescu, 1998). Completely closing an end of the tube, there is obtained in this region a high intensity swirl. The flow in the tube is generally like a swirl, with a rate tangential velocity / axial velocity maxim near the closed end, where is also created the annular vortex structure.

The approaching aerodynamic circuit is made from: a pressure source, the box of tangential inputs, the diffusion zone, the tube where there is produced the swirl with additional pollutants inputs, and the closing end with a rotating end.

It must be noticed that this torrential flow is concentrating and intensifying the vorticity, in contrast with the usual cyclone-type flow or other flows generators. If in the installation is introduced a pollutant, there issues a turbulent mixing in the annular vorticity zone. The spatial and temporal scales revealed the existence of different domains, starting with laboratory ones and until dissipative domains or others - corresponding to fine –structure wave numbers. Thus, the applications area is very large, including collecting, aggregating, separating and fragmenting the various pollutants.

From physical standpoint (Savulescu, 1998), the vortex produced in the installation implies four mechanisms:

Influence of Simulation Parameters on the Excitable Media Behaviour – The Case of Turbulent Mixing 425

Following the opinion of the specialists in cellular biology, this new technological method of

processing the flows is more efficient than the classical centrifugation method.

**Figure 1.** The fragmentation degree variation of *Spirulina Platensis*

the agglomeration of short fibers (*aerodynamic spinning*);

as pH, the temperature, the humidity, etc).

modeling the global swirling streamlines;

 local modeling of the concentrated vorticity structure; introducing the elements of chaotic turbulence.

the widespread isochoric two-dimensional flow (Ottino, 1989):

implies the following three stages:

Three specific applications were performed as fluid waste management:

the breakout of cell membranes of the phytoplankton from polluted waters and the

The mathematical modeling and analytically testing of the above experiment confirmed the experimental study. Concerning the phenomenon scale, there were taken into account medium helicoidally streamlines with approximating 10m width. *There were not gone further to molecular level.* Also, it is important to notice that for this moment of the analytical testing of the model, there has not been studied the influence of strictly biological parameters (such

The complex multiphase flow necessarily implies a theoretical approach for discovering the ways of optimization, developing and control of the installation. Numeric simulation of 3D multiphase flows is currently in study. In the mathematical framework, the flow complexity

The mathematical model associated to the vortex phenomena is, basically, the 3D version of

providing of a cell content solution *with important bio-stimulating features*.

the retention of particles under 5µm without any material filter;


The convection keeps in the transport and deposit of the powder pollutant when the graining diameter is greater than 5m. However, the graining spectra domain under 5m undergoes a turbulent diffusion and stratification effects which are generally out of control. For revealing the local concentration of the streamlines, 2D simplified models were tested. The multiphase 3D flows simulation is still in study.

The turbulent mixing in multiphase flow reveals the following experiment components:


The installation was realized in two versions: a small scale vortex tube (10-20mm diameter) and a large scale one (100-300mm). They correspond generally at two particles processing classes, although in most cases the classes can be superposed.

The first category refers to collecting and separating various powders, from gaseous emissions to ceramic powders. The parameters which must be taken into account are the graining spectra, the atmosphere nature and concentration. The vorticity concentration can be used for processing the deposit of different particles in powder or cement form by an adequate closing lid.

The second category contains the processing on small scale, including deformation and breaking up mechanisms for various particles in a host fluid. It is studied the vorticity concentration near the closing lid.

During such an experiment (Ionescu, 2002), it was processed the aquatic algae *Spirulina Platensis* in the host fluid water. After the processing, the long chains of cellular filaments were fragmented, producing isolated cell units or – sometimes – there was recorded the breaking up of some cellular membranes (with less than 100 Angstrom thickness). The initial and final observations (after the vortexation) are exhibited in figure 1 below. It has been

used the non-dimensional parameter *<sup>a</sup>* <sup>3</sup> *t Q D* , where t represents the time (in seconds), Q

the installation debit (m3/sec) and D the diameter (m3). As it can be seen from the picture, the fragmentation degree starts to increase as τa grows. There can also be observed the algae form before and after the fragmentation.

Following the opinion of the specialists in cellular biology, this new technological method of processing the flows is more efficient than the classical centrifugation method.

**Figure 1.** The fragmentation degree variation of *Spirulina Platensis*

424 Thermodynamics – Fundamentals and Its Application in Science

The multiphase 3D flows simulation is still in study.

four mechanisms:

of the air velocity.

adequate closing lid.

concentration near the closing lid.

used the non-dimensional parameter *<sup>a</sup>* <sup>3</sup>

form before and after the fragmentation.

From physical standpoint (Savulescu, 1998), the vortex produced in the installation implies


The convection keeps in the transport and deposit of the powder pollutant when the graining diameter is greater than 5m. However, the graining spectra domain under 5m undergoes a turbulent diffusion and stratification effects which are generally out of control. For revealing the local concentration of the streamlines, 2D simplified models were tested.

The turbulent mixing in multiphase flow reveals the following experiment components:



The installation was realized in two versions: a small scale vortex tube (10-20mm diameter) and a large scale one (100-300mm). They correspond generally at two particles processing

The first category refers to collecting and separating various powders, from gaseous emissions to ceramic powders. The parameters which must be taken into account are the graining spectra, the atmosphere nature and concentration. The vorticity concentration can be used for processing the deposit of different particles in powder or cement form by an

The second category contains the processing on small scale, including deformation and breaking up mechanisms for various particles in a host fluid. It is studied the vorticity

During such an experiment (Ionescu, 2002), it was processed the aquatic algae *Spirulina Platensis* in the host fluid water. After the processing, the long chains of cellular filaments were fragmented, producing isolated cell units or – sometimes – there was recorded the breaking up of some cellular membranes (with less than 100 Angstrom thickness). The initial and final observations (after the vortexation) are exhibited in figure 1 below. It has been

> *t Q D*

the installation debit (m3/sec) and D the diameter (m3). As it can be seen from the picture, the fragmentation degree starts to increase as τa grows. There can also be observed the algae

, where t represents the time (in seconds), Q

lid near the tube walls, with the pressure source near the tube centre line;



aggregation state or remains fragmented, in the host fluid;

classes, although in most cases the classes can be superposed.

Three specific applications were performed as fluid waste management:


The mathematical modeling and analytically testing of the above experiment confirmed the experimental study. Concerning the phenomenon scale, there were taken into account medium helicoidally streamlines with approximating 10m width. *There were not gone further to molecular level.* Also, it is important to notice that for this moment of the analytical testing of the model, there has not been studied the influence of strictly biological parameters (such as pH, the temperature, the humidity, etc).

The complex multiphase flow necessarily implies a theoretical approach for discovering the ways of optimization, developing and control of the installation. Numeric simulation of 3D multiphase flows is currently in study. In the mathematical framework, the flow complexity implies the following three stages:


The mathematical model associated to the vortex phenomena is, basically, the 3D version of the widespread isochoric two-dimensional flow (Ottino, 1989):

$$\begin{cases} \mathbf{x}\_1 = \mathbf{G} \cdot \mathbf{x}\_2 \\ \vdots \\ \mathbf{x}\_2 = \mathbf{K} \cdot \mathbf{G} \cdot \mathbf{x}\_{1'} \quad -1 \text{( $K \text{(1,G)}$ )} \mathbf{0} \end{cases} \tag{9}$$

Influence of Simulation Parameters on the Excitable Media Behaviour – The Case of Turbulent Mixing 427

of filaments' break up. It was called *rare event* the event of very sudden breaking up of the algae filaments, this corresponding from mathematical standpoint to the break out of the program (because of impossibility of maintaining the required accuracy). Thus, a very

Sumarising these basic stages in the behavior analysis of the turbulent mixing flow, is important to notice that the filaments breaking up is due to alternating loadings that the filament undergoes, in a space-time with *random events*, available to the break phenomena.

Associating known streamlines to the medium flow (that means helical flow in a

In fact, there were found four types of processes, all of which were matched by the

2. Linear-negative processes, which correspond to alternate tasks of stretching and folding

3. Mixing phenomena where there issue smaller or larger deviations or strong discontinuities; these concern the situations when some pieces are suddenly coming off from the whole filament, followed by the restarting of vortexation for the rest of

4. Rare events – these correspond to the turbulent mixing and represent the sudden break

Crossing over from 2D to 3D case, it is easy to deduce the requirement of a special analysis of the influence of parameters on the behavior of this complex mixing flow. This would include more paramater analysis types for some perturbation models in 2D and 3D case, but

The analysis recently has been continued with more computational simulations, for 2D model, both in periodic and – non – periodic case, and for 3D model, too. A lot of

also another mathematical analysis types, for example spectral analysis.

 the statistic increment in time of the breaking cases, according to the experiments; the relative singularity of the events which could produce the breaking, fact which is confirmed by the quite long duration of the experiments which led to the filaments break up; in very rare cases, the cellular membrane could be broken, and the cellular

 Determining the linear and surface deformations from continuum fluid mechanics; Associating a sequence of random values to the (vectorial) length and surface

important fact happened: *the mathematical simulations matched the experimental analysis.*

 *is significant*, since it brings more cases

computational context that the *surface deformation e*

The modeling has been the following basic stages:

1. Processes with relative linear behavior;

of filaments and are the most;

up of spirulina filaments.

content collected.

The validity of the model is confirmed by two aspects:

**2.3. New influence of simulation parameters** 

cylinder);

orientations.

simulations

algae;

Namely, the following vortex flow model is in study:

$$\begin{cases} \mathbf{x}\_1 = \mathbf{G} \cdot \mathbf{x}\_2 \\ \mathbf{x}\_2 = \mathbf{K} \cdot \mathbf{G} \cdot \mathbf{x}\_1 \\ \vdots \\ \mathbf{x}\_3 = c \end{cases}, -1 \\ \{K/1, c = const.} \tag{10}$$

In the first stage, the flow model (10) was studied from the analytical standpoint. Namely, the solution of the Cauchy problem associated to (10) was found. In order to analyze the length and surface deformations of algae filaments in certain vortex conditions, the Cauchy-Green deformation tensor was calculated. There were obtained quite complex formulas for *e e*, (Ionescu 2002, 2010).

The second stage has been a computational / simulation standpoint. It has been realized a computational analysis of the length and surface deformation efficiency, in some specific vortexation conditions. With the numeric soft MAPLE11, the analysis has been two parts. Firstly, the following Cauchy problems has been solved:

$$\begin{aligned} e\_\lambda \left( t \right) &= 0, \ x \left( 0 \right) = 0 \\ e\_\eta \left( t \right) &= 0, \ x \left( 0 \right) = 0 \end{aligned} \tag{11}$$

using a *discrete* numeric calculus procedure (Abell, 2005). The output of the procedure is a listing of the form , , 0..25 *i i t xt i* . In the second part there have been realized discrete time plots, in 25 time units, following the listings obtained in the first part. The plots represent in fact *the image of the length and surface deformations, in the established scale time*.

Very few *irrational* value sets were chosen for the length, respectively surface versors, taking into account the versor condition:

$$
\sum M\_i = 1,\\
\sum N\_j = 1 \tag{12}
$$

The studied cases are in fact represented by the events associated to different values of length and surface orientation versors: <sup>123</sup> *MMM* , , and <sup>123</sup> *NNN* , , for length and surface respectively. The events were very few, about 60. Their statistical interpretation is synthesized in (Ionescu 2010), including also the two-dimensional case.

The graphical events are illustrating the analysis of the deformations for Spirulina Platensis, in 25 time units' vortexation. According to (Ionescu 2001, Ottino 1989), the algae filaments represent *Lagrange markers*. The special spiral form of the algae gives the answer to the computational context that the *surface deformation e is significant*, since it brings more cases of filaments' break up. It was called *rare event* the event of very sudden breaking up of the algae filaments, this corresponding from mathematical standpoint to the break out of the program (because of impossibility of maintaining the required accuracy). Thus, a very important fact happened: *the mathematical simulations matched the experimental analysis.*

Sumarising these basic stages in the behavior analysis of the turbulent mixing flow, is important to notice that the filaments breaking up is due to alternating loadings that the filament undergoes, in a space-time with *random events*, available to the break phenomena. The modeling has been the following basic stages:


In fact, there were found four types of processes, all of which were matched by the simulations

1. Processes with relative linear behavior;

426 Thermodynamics – Fundamentals and Its Application in Science

Namely, the following vortex flow model is in study:

*e e*, 

(Ionescu 2002, 2010).

into account the versor condition:

.

 

.

 

> . 3

 

Firstly, the following Cauchy problems has been solved:

*x c*

1 2 .

*x Gx*

2 1

1 2 .

*x Gx*

2 1 , 1 1, 0

*x K G x K c const*

In the first stage, the flow model (10) was studied from the analytical standpoint. Namely, the solution of the Cauchy problem associated to (10) was found. In order to analyze the length and surface deformations of algae filaments in certain vortex conditions, the Cauchy-Green deformation tensor was calculated. There were obtained quite complex formulas for

The second stage has been a computational / simulation standpoint. It has been realized a computational analysis of the length and surface deformation efficiency, in some specific vortexation conditions. With the numeric soft MAPLE11, the analysis has been two parts.

> 

using a *discrete* numeric calculus procedure (Abell, 2005). The output of the procedure is a listing of the form , , 0..25 *i i t xt i* . In the second part there have been realized discrete time plots, in 25 time units, following the listings obtained in the first part. The plots represent in fact *the image of the length and surface deformations, in the established scale time*.

Very few *irrational* value sets were chosen for the length, respectively surface versors, taking

The studied cases are in fact represented by the events associated to different values of length and surface orientation versors: <sup>123</sup> *MMM* , , and <sup>123</sup> *NNN* , , for length and surface respectively. The events were very few, about 60. Their statistical interpretation is

The graphical events are illustrating the analysis of the deformations for Spirulina Platensis, in 25 time units' vortexation. According to (Ionescu 2001, Ottino 1989), the algae filaments represent *Lagrange markers*. The special spiral form of the algae gives the answer to the

*et x et x*

synthesized in (Ionescu 2010), including also the two-dimensional case.

0, 0 0 0, 0 0

(11)

1, 1 *M N i j* (12)

, 1 1, .

(9)

(10)

*x KGx K G*


The validity of the model is confirmed by two aspects:


Crossing over from 2D to 3D case, it is easy to deduce the requirement of a special analysis of the influence of parameters on the behavior of this complex mixing flow. This would include more paramater analysis types for some perturbation models in 2D and 3D case, but also another mathematical analysis types, for example spectral analysis.

## **2.3. New influence of simulation parameters**

The analysis recently has been continued with more computational simulations, for 2D model, both in periodic and – non – periodic case, and for 3D model, too. A lot of

comparisons between periodic and non- periodic case, 2D and 3D case were realized (Ionescu 2008, 2009). In the same time, the computational appliances were varied. If innitialy, the model was studied from the standpoint of mixing efficiency, in the works that come after, new appliances of the MAPLE11 soft were tested (Abell 2005, Ionescu 2011), in order to collect more statistical data for the turbulent mixing theory. In what follows, few of these appliances are described.

Influence of Simulation Parameters on the Excitable Media Behaviour – The Case of Turbulent Mixing 429

 Given a set or list of initial conditions (see below), and a system of first order differential equations or a single higher order differential equation, **DEplot** plots solution curves, *by numerical methods*. This means that the initial conditions of the problem must be given in standard form, that is, the function values and all derivatives up to one less than the differential order of the differential equation at the

 A system of two first order differential equations produces a direction field plot, provided the system is determined to be autonomous. In addition, a single first order differential equation produces a direction field (as it can always be mapped to a system of two first order autonomous differential equations). A system is determined to be autonomous when all terms and factors, other than the differential, are free of the independent variable. For systems not meeting these criteria, no direction field is produced (only solution curves are possible in such instances). There can be **only one**

 The default method of integration is *method=rkf45.* Other methods can be specified in the optional equations. Note that because numerical methods are used to generate plots, the output is subject to the characteristics of the numerical method in use. In particular,

 The direction field presented consists of either a grid of arrows or a set of randomly generated arrows. In either case, the arrows are tangential to solution curves. For each grid point, the arrow centered at **(x,y)** has slope **dy/dx**. This slope is computed using **(dy/dt)/(dx/dt)**, where these two derivatives are specified in the first argument to **DEplot**. The curved arrow types (**curves** and **comet**) require additional data for the curvature of the direction field, which is computed by moving an epsilon in the direction of the slope **dy/dx**, and computing **dy/dx**, then moving an epsilon in the direction opposite the slope, and computing **dy/dx**. This data is then sufficient to draw a small portion of the direction field lines local to the point, which is then used to draw

By default, the two dependent variables are plotted, unless otherwise specified in the

 The **deqns** parameter can be given as a procedure, but must conform to the specification as given in **dsolve/numeric**, and the **number** option must be included

where **N** represents the number of first order equations, **ivar** is the independent variable, **Y** is a vector of length **N**, and **YP** is a vector of derivatives which is updated by the procedure

before the initial conditions. In this instance, **deqns** must be of the form:

unusual output may occur when dealing with asymptotes.

Description:

same point.

independent variable

the curved arrows.

proc( N, ivar, Y, YP )

(for the equivalent first order system), also of length **N.** 

 YP[1] := f1(ivar,Y); YP[2] := f2(ivar,Y);

**scene** option.

...

 ... end proc

## *2.3.1. Few MAPLE11 apliances used in simulations*

For the purpose of this chapter, there are presented only the recent used appliances of MAPLE11 soft. There have been used two graphical appliances of this soft, namely "DePlot" and "Phaseportrait" tools, from "DETools" package. Both of them are based on numeric methods in producing trajectories of differential equations systems. The most used numeric method is *Fehlberg fourth-fifth order Runge-Kutta* method - the so-called "rkf45" method - with degree four interpolant (Abell 2005).

*DETools[DePlot]* – this tool plots solutions to a system of differential equations. The calling sequences are as follows:

DEplot(deqns, vars, trange, options)

DEplot(deqns, vars, trange, inits, options)

DEplot(deqns, vars, trange, xrange, yrange, options)

DEplot(deqns, vars, trange, inits, xrange, yrange, options)

DEplot(dproc, vars, trange, number, xrange, yrange, options)

#### Parameters:

deqns - list or set of first order ordinary differential equations, or a single differential equation of any order;

dproc - a Maple procedure representation for first order ordinary differential equations, or a single differential equation of any order;

vars - dependent variable, or list or set of dependent variables;

trange - range of the independent variable;

number - equation of the form 'number'=integer indicating the number of differential equations when **deqns** is given as a function (dproc) instead of expressions;

inits - set or list of lists; initial conditions for solution curves;

xrange - range of the first dependent variable;

yrange - range of the second dependent variable;

options - (optional) equations of the form keyword=value;

Description:

428 Thermodynamics – Fundamentals and Its Application in Science

*2.3.1. Few MAPLE11 apliances used in simulations* 

these appliances are described.

degree four interpolant (Abell 2005).

DEplot(deqns, vars, trange, options)

DEplot(deqns, vars, trange, inits, options)

or a single differential equation of any order;

trange - range of the independent variable;

xrange - range of the first dependent variable;

yrange - range of the second dependent variable;

DEplot(deqns, vars, trange, xrange, yrange, options)

DEplot(deqns, vars, trange, inits, xrange, yrange, options)

DEplot(dproc, vars, trange, number, xrange, yrange, options)

vars - dependent variable, or list or set of dependent variables;

inits - set or list of lists; initial conditions for solution curves;

options - (optional) equations of the form keyword=value;

sequences are as follows:

equation of any order;

Parameters:

comparisons between periodic and non- periodic case, 2D and 3D case were realized (Ionescu 2008, 2009). In the same time, the computational appliances were varied. If innitialy, the model was studied from the standpoint of mixing efficiency, in the works that come after, new appliances of the MAPLE11 soft were tested (Abell 2005, Ionescu 2011), in order to collect more statistical data for the turbulent mixing theory. In what follows, few of

For the purpose of this chapter, there are presented only the recent used appliances of MAPLE11 soft. There have been used two graphical appliances of this soft, namely "DePlot" and "Phaseportrait" tools, from "DETools" package. Both of them are based on numeric methods in producing trajectories of differential equations systems. The most used numeric method is *Fehlberg fourth-fifth order Runge-Kutta* method - the so-called "rkf45" method - with

*DETools[DePlot]* – this tool plots solutions to a system of differential equations. The calling

deqns - list or set of first order ordinary differential equations, or a single differential

dproc - a Maple procedure representation for first order ordinary differential equations,

number - equation of the form 'number'=integer indicating the number of differential

equations when **deqns** is given as a function (dproc) instead of expressions;


```
 ... 
 YP[1] := f1(ivar,Y); 
 YP[2] := f2(ivar,Y); 
 ... 
 end proc
```
where **N** represents the number of first order equations, **ivar** is the independent variable, **Y** is a vector of length **N**, and **YP** is a vector of derivatives which is updated by the procedure (for the equivalent first order system), also of length **N.** 

 The **inits** parameter must be specified as: [ [x(t0)=x0,y(t0)=y0], [x(t1)=x1,y(t1)=y1], ... ]

```
 [ [y(t0)=y0], [y(t1)=y1], ... ]
```
[ y(t0)=y0, y(t1)=y1, ... ]

where, in the above, sets can be used in place of lists, or

[ [t0,x0,y0], [t1,x1,y1], ... ]

{ [t0,x0,y0], [t1,x1,y1], ... }

[ [t0,x0], [t1,x1], ... ]

where the above is a list or set of lists, each sublist specifying one group of initial conditions.

Influence of Simulation Parameters on the Excitable Media Behaviour – The Case of Turbulent Mixing 431

where the above is a list (or set) of lists, each sublist specifying one group of initial

In what follows there is presented the comparative analysis for 2D and 3D mixing flow

2 1 , 1 1, 0

2 1 21

*x K G x K c const*

For all models it is used the same set of parameter values (*G, KG*) as used in (Ionescu 2008),

Each simulation case is labelled on the figure. The time units number was succesively increased, in order to better analyse the solution behavior for each model and corresponding case. The "stepsize" option for the "rkf45" numeric method, on which the "Phaseportrait" procedure is based, is implicitely assigned to 0.05. Also, it must be noticed that the above choice of parameters (one positive and another negative) is optimal for analyzing the

*x KGx G x x*

, 1 1, .

(13)

*x KGx K G*

**inits** should be specified as

conditions.

[ [x(t0)=x0,y(t0)=y0], [x(t1)=x1,y(t1)=y1], ... ]

models. There are followed two main leves of comparison:

i. The 2D (non-periodic) mixing flow, namely the differential system (9):

1 2 .

1 21

*x Gx x*

*x Gx*

.

 

in comparison with a "perturbed" version of it, the system (13):

.

.

 

> . 3

 

direction field associated to the models' solutions.

*x c*

.

1 2 .

*x Gx*

2 1

*2.3.2. Comparative computational analysis* 

ii. The 3D mixing flow model (10)

containing three simulation cases: - case1: *G KG* 0.25, 0.035 ; - case2: *G KG* 0.755, 0.65 ; - case3: *G KG* 0.85, 0.25

The **xrange** and **yrange** parameters must be specified as follows.

```
 x(t) = x1..x2, y(t) = y1..y2 or
```
x = x1..x2, y = y1..y2

More details about the parameters can be found in (Abell 2005).

*DETools[Phaseportrait]*. This tool has the following parameters:

**deqns** - list or set of first order ordinary differential equations, or a single differential equation of any order;

**vars** - dependent variable, or list or set of dependent variables;

**trange** - range of the independent variable;

**inits** - set or list of lists; initial conditions for solution curves;

**options** - (optional) equations of the form keyword=value;

Description


**inits** should be specified as

430 Thermodynamics – Fundamentals and Its Application in Science

where, in the above, sets can be used in place of lists, or

The **xrange** and **yrange** parameters must be specified as follows.

More details about the parameters can be found in (Abell 2005). *DETools[Phaseportrait]*. This tool has the following parameters:

**trange** - range of the independent variable;

**vars** - dependent variable, or list or set of dependent variables;

**inits** - set or list of lists; initial conditions for solution curves;

**options** - (optional) equations of the form keyword=value;

instances). There can be ONLY one independent variable.

where the above is a list or set of lists, each sublist specifying one group of initial conditions.

**deqns** - list or set of first order ordinary differential equations, or a single differential

 Given a list (or set) of initial conditions (see below), and a system of first order differential equations or a single higher order differential equation, **phaseportrait** plots solution curves, by numerical methods. This means that the initial conditions of the problem must be given in standard form, that is, the function values and all derivatives up to one less than the differential order of the differential equation at the same point. A system of two first order differential equations also produces a direction field plot, provided the system is determined to be autonomous. In addition, a single first order differential equation also produces a direction field (as it can always be mapped to a system of two first order autonomous differential equations). For systems not meeting these criteria, no direction field is produced (only solution curves are possible in such

All of the properties and options available in **phaseportrait** are also found in **Deplot .**

 The **inits** parameter must be specified as: [ [x(t0)=x0,y(t0)=y0], [x(t1)=x1,y(t1)=y1], ... ]

[ [y(t0)=y0], [y(t1)=y1], ... ]

[ y(t0)=y0, y(t1)=y1, ... ]

 [ [t0,x0,y0], [t1,x1,y1], ... ] { [t0,x0,y0], [t1,x1,y1], ... }

x(t) = x1..x2, y(t) = y1..y2 or

[ [t0,x0], [t1,x1], ... ]

x = x1..x2, y = y1..y2

equation of any order;

Description

```
[ [x(t0)=x0,y(t0)=y0], [x(t1)=x1,y(t1)=y1], ... ]
```
where the above is a list (or set) of lists, each sublist specifying one group of initial conditions.

#### *2.3.2. Comparative computational analysis*

In what follows there is presented the comparative analysis for 2D and 3D mixing flow models. There are followed two main leves of comparison:

i. The 2D (non-periodic) mixing flow, namely the differential system (9):

$$\begin{cases} \mathbf{x}\_1 = \mathbf{G} \cdot \mathbf{x}\_2 \\\\ \mathbf{x}\_2 = K \cdot \mathbf{G} \cdot \mathbf{x}\_{1'} \ -1 \langle K \langle 1 \rangle G \rangle 0 \end{cases}$$

in comparison with a "perturbed" version of it, the system (13):

$$\begin{cases} \mathbf{x}\_1 = \mathbf{G} \cdot \mathbf{x}\_2 + \mathbf{x}\_1 \\\\ \mathbf{x}\_2 = \mathbf{K} \cdot \mathbf{G} \cdot \mathbf{x}\_1 + \mathbf{G} \cdot \left(\mathbf{x}\_2 - \mathbf{x}\_1\right) \end{cases} \tag{13}$$

ii. The 3D mixing flow model (10)

$$\begin{cases} \mathbf{x}\_1 = \mathbf{G} \cdot \mathbf{x}\_2 \\ \mathbf{x}\_2 = \mathbf{K} \cdot \mathbf{G} \cdot \mathbf{x}\_1 \\ \vdots \\ \mathbf{x}\_3 = \mathbf{c} \end{cases}, -1 / K / 1, c = \text{const.} $$

For all models it is used the same set of parameter values (*G, KG*) as used in (Ionescu 2008), containing three simulation cases:


Each simulation case is labelled on the figure. The time units number was succesively increased, in order to better analyse the solution behavior for each model and corresponding case. The "stepsize" option for the "rkf45" numeric method, on which the "Phaseportrait" procedure is based, is implicitely assigned to 0.05. Also, it must be noticed that the above choice of parameters (one positive and another negative) is optimal for analyzing the direction field associated to the models' solutions.

The "scene" parameter of the graphic procedure was set to {x(t), y(t)], the same in 2D and in 3D case.

Influence of Simulation Parameters on the Excitable Media Behaviour – The Case of Turbulent Mixing 433

**Figure 4.** Case2 of simulation for the model (9). The time units number t=0..40

**Figure 5.** Case2 of simulation for the model (9) with t=0..100. The trajectory tends to "aggregate"

towards the origin.

**Figure 2.** Case1 of simulation for the model (9). Time units number: t= 0..40

**Figure 3.** Case1 of simulation for the model (9) with t=0..100. The movement is realized on the same trajectory

**Figure 4.** Case2 of simulation for the model (9). The time units number t=0..40

**Figure 2.** Case1 of simulation for the model (9). Time units number: t= 0..40

**Figure 3.** Case1 of simulation for the model (9) with t=0..100. The movement is realized on the same

case.

trajectory

The "scene" parameter of the graphic procedure was set to {x(t), y(t)], the same in 2D and in 3D

**Figure 5.** Case2 of simulation for the model (9) with t=0..100. The trajectory tends to "aggregate" towards the origin.

**Figure 8.** Case1 of simulation for the model (13). The trajectory changes, it seems to be no longer

**Figure 9.** Case2 of simulation for the model (13). The time units number t=0..40. The trajectory changes

periodic, but tends to infinity

again and becomes negative

**Figure 6.** Case3 of simulation for the model (9). Time units number t=0.40

**Figure 7.** Case3 of simulation for the model (9) with t=0..100. The trajectory multiplies but does not draw near the origin

434 Thermodynamics – Fundamentals and Its Application in Science

**Figure 6.** Case3 of simulation for the model (9). Time units number t=0.40

**Figure 7.** Case3 of simulation for the model (9) with t=0..100. The trajectory multiplies but does not

draw near the origin

**Figure 8.** Case1 of simulation for the model (13). The trajectory changes, it seems to be no longer periodic, but tends to infinity

**Figure 9.** Case2 of simulation for the model (13). The time units number t=0..40. The trajectory changes again and becomes negative

**Figure 12.** Case3 of simulation for the model (13), t=0..40. The trajectory becomes positive

**Figure 13.** Case3 of simulation for the model (13) with t=0..100

**Figure 10.** Case2 of simulation for the model (13), with t=0..100.

**Figure 11.** Case2 of simulation for the model (13) with t=0..200

**Figure 12.** Case3 of simulation for the model (13), t=0..40. The trajectory becomes positive

**Figure 10.** Case2 of simulation for the model (13), with t=0..100.

**Figure 11.** Case2 of simulation for the model (13) with t=0..200

**Figure 13.** Case3 of simulation for the model (13) with t=0..100

**Figure 16.** Case1 of simulation for the model (10) with t=0..50

**Figure 17.** Case2 of simulation for the model (10), t=0..25. A periodic behaviour of the trajectory

**Figure 14.** Case3 of simulation for the model (13) with t=0..200. The program does not show the field arrows anymore.

**Figure 15.** Case1 of simulation for the model (10). Time units number t=0..25

**Figure 16.** Case1 of simulation for the model (10) with t=0..50

438 Thermodynamics – Fundamentals and Its Application in Science

arrows anymore.

**Figure 14.** Case3 of simulation for the model (13) with t=0..200. The program does not show the field

**Figure 15.** Case1 of simulation for the model (10). Time units number t=0..25

**Figure 17.** Case2 of simulation for the model (10), t=0..25. A periodic behaviour of the trajectory

**Figure 20.** Case3 of simulation for the model (10), time t=0..75. The "circular loops" are fewer than in

the case2, for the same simulation time.

**Figure 21.** Case3 of simulation for the model (10) with t=0..100.

**Figure 18.** Case2 of simulation for the model (10), t=0..75

**Figure 19.** Case3 of simulation for the model (10), time t=0..25

440 Thermodynamics – Fundamentals and Its Application in Science

**Figure 18.** Case2 of simulation for the model (10), t=0..75

**Figure 19.** Case3 of simulation for the model (10), time t=0..25

**Figure 20.** Case3 of simulation for the model (10), time t=0..75. The "circular loops" are fewer than in the case2, for the same simulation time.

**Figure 21.** Case3 of simulation for the model (10) with t=0..100.

## **3. Remarks on the simulations and Maple appliances importance**

Looking above to the figures, it is important to notice some special features.

 It must be pointed out the advance of the "phaseportrait" appliance of Maple soft, and generally of the "plots" package. The possibility of increasing step by step the simulation time units offers important features of the analysis and consequently of the model behaviour. For example in fig.3, when time increases, the trajectory "moves" on the same line. Comparing to this, in fig.4 the trajectory has some "tracks" which overcrowd towards the origin. In case 3, for the same simulation time, the trajectory only multiplies and does not came toward the origin. Also, figures 8-10 illustrate in a good manner the influence of time increasing on the trajectory behaviour: this can be seen from the values variation of x(t) and y(t) on the axes.

Influence of Simulation Parameters on the Excitable Media Behaviour – The Case of Turbulent Mixing 443

Looking at the above pictures, we see that each simulation case brings significant

 For the modified (perturbed) 2D model, just from the case1 of simulation we see that the trajectory has a great change. It is not periodic anymore like in the initial 2D model, but is more evident its trend to infinity, just from the beginning of the simulation. The

 Concerning the 3D model, it is obvious that the field arrows could not appear, since there is a 3D simulation. Instead, the "scene" parameter was chosen to [x(t), y(t)], like in 2D case model. This is optimal for studying the trajectory behaviour, since it is like we "look" at the trajectory from above. The scene parameter is important in analysis, and a further aim would be to change it in the simulations and make some other comparative

Concerning the general context of random events that go with the mixing flow phenomena,

 It is obvious that the above analysis confirms the fact that the mixing flow model is *a far from equilibrium model.* This is confirmed by the number of parameter / time units analysis. Using different appliances – "mixing efficiency" and computational / graphical appliances, the conclusion is the same: *the model becomes far from equilibrium, in certain simulation conditions*. It must be noticed that the "units" of time can be of any type, sufficiently small or large. It depends to the target of the analysis. Enlarging the set of parameter values would bring another important data for completing the panel of repetitive events. This would be a next aim. All this space-time context present in all these analysis, consolidates the basic statement that the *turbulent mixing flows must be approached as chaotic systems*. This is in fact regaining the idea of a system / model high sensitive to initial conditions (Ionescu

 Testing step by step each of the three models above was possible due to the flexible structure of the MAPLE11 graphical / computational appliances. In the same time this shows that these repetitive simulations are relatively easy to perform. Maybe useful

 Also the "step by step" analysis guides the reader by successive steps, to the so- called "far from equilibrium model", by the possibility of sensitive modifying the model

 The issues of *repetitive phenomena*, both in 2D and 3D case, give rises to achieve some appliances of chaotic dynamical systems, whose numeric models would give new

research directions on the behavior in excitable media. This would be a next aim

parameters have a great influence in this sense

the above analysis just brings at that some important conclusions:

comments would produce if sharing some of these files.

parameters and the time at each step of simulation.

differences:

graphical analysis

2009).

**Author details** 

*University of Craiova, Romania* 

Adela Ionescu


## **4. Conclusion**

Looking on the above computational analysis, some conclusions issue: regarding the above comparative analysis, on one hand, and, concerning the general context of *random events* that go with the mixing flow phenomena, on the other hand.

Looking at the above pictures, we see that each simulation case brings significant differences:


Concerning the general context of random events that go with the mixing flow phenomena, the above analysis just brings at that some important conclusions:


## **Author details**

442 Thermodynamics – Fundamentals and Its Application in Science

**3. Remarks on the simulations and Maple appliances importance** 

 It must be pointed out the advance of the "phaseportrait" appliance of Maple soft, and generally of the "plots" package. The possibility of increasing step by step the simulation time units offers important features of the analysis and consequently of the model behaviour. For example in fig.3, when time increases, the trajectory "moves" on the same line. Comparing to this, in fig.4 the trajectory has some "tracks" which overcrowd towards the origin. In case 3, for the same simulation time, the trajectory only multiplies and does not came toward the origin. Also, figures 8-10 illustrate in a good manner the influence of time increasing on the trajectory behaviour: this can be

 All the simulations for 2D case model have been realized in the same initial conditions: (x(t), y(t)) = (1,1). This gives more accuracy for the graphical comparisons. Applying this procedure step by step gives the possibility to have a total parameter control at every stage of simulation. In 3D case the initial conditions were a little modified: (x(t, y(t), z(t))=(1,0,0). It was taken into account the fact that on the z axis is represented the

 Also, the "scene" parameter was the same: [x(t), y(t)] both in 2D and 3D case. That means in 3D case, the trajectory is studied by watching it "from the z axis". The field arrows are an important appliance of this plot tool, as they are very suggestive concerning the trajectory features. In this context it is extremely important to notice the figure 13. In this model (simulation case3), when increasing the time, the field arrows disappear, the program doesn't show them anymore. This can be glossed like a *"lose of equilibrium" for the 2D model,* in certain parameter context, and this fact can be easily

 In figures 8-13 it must be noticed a similarity with the pictures obtained in (Ionescu, 2008), where the origin becomes an *unstable focus*. It is observed here that when increasing the time, e.g. in Fig. 10, the trajectory does not go periodic, it changes only

 A special attention must be paid for the figure 17, 3D simulation case. For t=0..75 units, and looking to the trajectory from above, from the z axis, it is obvious that the *vorticity of the flow appears*. It is very important to notice that if in (Ionescu 2002, 2010), when searching for the mixing efficiency, there were necessary small time units for get the special events of losing the equilibrium of the model, in this case, searching for the phaseportrait needs some more time units, in order to get visible vorticity of the flow. The vorticity allure is also visible in the figures 18-20, with some changes in the loops

Looking on the above computational analysis, some conclusions issue: regarding the above comparative analysis, on one hand, and, concerning the general context of *random events* that

velocity and is natural that the mixing process starts from a zero velocity

Looking above to the figures, it is important to notice some special features.

seen from the values variation of x(t) and y(t) on the axes.

observed from the great values for x(t) and y(t) on the axis

number and their nearness to the origin.

go with the mixing flow phenomena, on the other hand.

the positivity.

**4. Conclusion** 

Adela Ionescu *University of Craiova, Romania* 

## **Acknowledgement**

The basic part of my work in turbulent mixing was done in the period of my Ph. D. thesis at the Polytechnic University of Bucarest Romania. Fruitful experiments and special discussions were done togheter with the scientific team of the Institute of Applied Ecology, Bucarest. I am grateful to them.

**Chapter 17** 

. *QC*

© 2012 Ngouateu Wouagfack and Tchinda, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

*QL* is the heat input rate from the cooling

properly cited.

**ECOP Criterion for Irreversible** 

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51547

**1. Introduction** 

performance criterion.

and .

of .

**2. Thermodynamics analysis** 

heat sinks at temperatures *CT* and *AT* and

Paiguy Armand Ngouateu Wouagfack and Réné Tchinda

**Three-Heat-Source Absorption Refrigerators** 

In this chapter, the optimization analysis based on the new thermo-ecological criterion (ECOP) first performed by Ust et al. [1] for the heat engines is extended to an irreversible three-heat-source absorption refrigerator. The thermo-ecological objective function ECOP is optimized with respect to the temperatures of the working fluid. The maximum ECOP and the corresponding optimal temperatures of the working fluid, coefficient of performance, specific cooling load, specific entropy generation rate and heat-transfer surface areas in the exchangers are then derived analytically. Comparative analysis with the COP criterion is carried out to prove the utility of the ecological coefficient of

The main components of an absorption refrigeration system are a generator, an absorber, a condenser and an evaporator as shown schematically in Fig. 1 [2]. In the shown model, . *QH* is the rate of absorbed heat from the heat source at temperature *HT* to generator,

*QA* are, respectively, the heat rejection rates from the condenser and absorber to the

.

*QH* heat rate input. The weak solution at state 2 passes through the expansion valve into the absorber with a pressure reduction (2–2'). In the condenser, the working fluid at

and reproduction in any medium, provided the original work is properly cited.

space at temperature *<sup>L</sup> T* to the evaporator. In absorption refrigeration systems, usually NH3/H2O and LiBr/H2O are used as the working substances, and these substances abide by ozone depletion regulations, since they do not consist of chlorouorocarbons. In Fig. 1, the liquid rich solution at state 1 is pressurized to state 1' with a pump. In the generator, the working fluid is concentrated to state 3 by evaporating the working medium by means

But my entire acknowledgement points to Ph. D. Prof. Eng. Stefan Savulescu, from Politechnic University of Bucarest. Without the special vortexation installation constructed by him, the mixing flow experiments could not be possible. Due to his great experience in the turbulence field, and his fundamental observations, advices and suggestions, my work in turbulent mixing have an important sense and the great challenge of unifying the turbulence theory takes shape.

## **5. References**

