**10. List of symbols**


description of material transport in closed stationary systems. The macroscopic pressure gradient in such systems is determined by the Gibbs-Duhem equation. The only assumption used is that the heat of transport is equivalent to the negative of the chemical potential. In open and non-stationary systems, the macroscopic pressure gradient is calculated using modified material transport equations obtained by non-equilibrium thermodynamics, where the macroscopic pressure gradient is the unknown parameter. In that case, the Soret coefficient is expressed through combined chemical potentials at constant pressure. The resulting thermodynamic expressions allow for the use of statistical mechanics to relate the

This refined thermodynamic theory can be supplemented by microscopic calculations to explain the characteristic features of thermodiffusion in binary molecular solutions and suspensions. The approach yields the correct size dependence in the Soret coefficient and the correct relationship between the roles of electrostatic and Hamaker interactions in the thermodiffusion of colloidal particles. The theory illuminates the role of translational and rotational kinetic energy and the consequent dependence of thermodiffusion on molecular symmetry, as well as the isotopic effect. For non-dilute molecular mixtures, the refined thermodynamic theory explains the change in the direction of thermophoresis with concentration in certain mixtures, and the possibility of phase layering in the system. The concept of a Laplace-like pressure established in the force field of the particle under consideration plays an important role in microscopic calculations. Finally, the refinements make the thermodynamic theory consistent with hydrodynamic theories and with empirical

*a* Energetic parameter characterizing the interaction between the different

gradient in chemical potential to macroscopic parameters of the system.

data.

**10. List of symbols** 

kinds of molecules

*E* Electric field strength

*h* Planck constant

*eJ* Energy flux

*am* Empiric coefficient in Eq. (38) *<sup>i</sup> b* Empiric coefficient in Eq. (38)

*ie* Electric charge of the respective ion

*J* Total material flux in the system

*Li* and *LiQ* Individual molecular kinetic coefficients

*N* Number of components in the mixture

*l* Thickness of a spherical layer around the particle *mi* Molecular mass of the respective component

*<sup>i</sup> J* Component material fluxes

*<sup>N</sup>*<sup>1</sup> *m* Mass of the virtual particle

*k* Boltzmann constant

*ij g* Pair correlation function for respective components

1 2 *I I*, , <sup>3</sup>*I* and Principal values of the tensor of the moment of inertia



Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 367

Anderson, J.L. (1989) Colloid Transport by Interfacial Forces. *Annual Review of Fluid* 

Bringuier, E., Bourdon, A. (2003). Colloid transport in nonuniform temperature. *Physical* 

Bringuier, E., Bourdon, A. (2007). Colloid Thermophoresis as a Non-Proportional Response.

De Groot, S. R. (1952). *Thermodynamics of Irreversible Processes.* North-Holland, Amsterdam,

De Groot, S. R., Mazur, P. (1962). *Non-Equilibrium Thermodynamics.* North-Holland,

Dhont, J. K. G. (2004). Thermodiffusion of interacting colloids. *The Journal of Chemical Physics*.

Dhont, J. K. G. et al, (2007). Thermodiffusion of Charged Colloids: Single-Particle Diffusion.

Duhr, S., Braun, D. (2006). Thermophoretic Depletion Follows Boltzmann Distribution. Physical Review Letters*, Vol.* 96, No. 16 (April 2006) 168301 (4 pages) Duhr, S., Braun, D. (2006). Why molecules move along a temperature gradient. *Proceedings of* 

Fisher, I. Z. (1964). *Statistical Theory of Liquids.* Chicago University Press, Chicago, USA. Ghorayeb, K., Firoozabadi, A. (2000). Molecular, pressure, and thermal diffusion in nonideal multicomponent mixtures. *AIChE Journal*, Vol. 46, No. 5 ( May 2000), 883–891. Giddings, J. C. et al. (1995). Thermophoresis of Metal Particles in a Liquid. *The Journal of* 

Gyarmati, I. (1970). *Non-Equilibrium Thermodynamics.* Springer Verlag, Berlin, Germany. Haase, R. (1969). *Thermodynamics of Irreversible Processes,* Addison-Wesley: Reading,

Hunter, R. J. (1992). *Foundations of Colloid Science*. Vol. 2, Clarendon Press, London, Great

Kirkwood, I. , Boggs, E. (1942). The radial distribution function in liquids. *The Journal of* 

Kirkwood, J. G. (1939). Order and Disorder in Liquid Solutions. *The Journal of Physical* 

Kondepudi, D, Prigogine, I. (1999). *Modern Thermodynamics: From Heat Engines to Dissipative Structures*, ISBN 0471973947, John Wiley and Sons, New York, USA. Landau, L. D., Lifshitz, E. M. (1954). *Mekhanika Sploshnykh Sred* (Fluid Mechanics) (GITTL,

Landau, L. D., Lifshitz, E. M. (1980). *Statistical Physics*, Part 1, English translation, Third

Ning, H., Wiegand, S. (2006). Experimental investigation of the Soret effect in acetone/water

Moscow, USSR) [Translated into English (1959, Pergamon Press, Oxford, Great

Edition, Lifshitz, E. M. and Pitaevskii, L. P., Pergamon Press, Oxford, Great Britain.

and dimethylsulfoxide/water mixtures. *The Journal of Chemical Physics.* Vol. 125,

*The Journal of Non-equilibrium. Thermodynamics. Vol.* 32, No. 3 (July 2007), 221–229,

*Review E*, Vol. 67, No. 1 (January 2003), 011404 (6 pages).

**11. References** 

*Mechanics.* Vol. 21, 61–99.

ISSN 0340-0204

The Netherlands.

Massachusetts, USA.

Britain.

Britain)].

Amsterdam, The Netherlands.

Vol.120, No. 3 (February 2004) 1632-1641.

*Langmuir*, Vol. 23, No. 4 (November 2007), 1674-1683.

*National Academy of USA*, Vol. 103, 19678-19682.

*Colloid and Interface Science.* Vol. 176, No. 454-458.

*Chemical Physics*., Vol. 10, n.d., 394-402.

No. 22 (December 2006), 221102 (4 pages).

*Chemistry*, Vol. 43, n.d., 97–107

### **11. References**

366 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

*I* Difference in the moment of inertia for the molecules constituting the

*ij r* Interaction potential for the respective molecules

*<sup>i</sup>*<sup>1</sup> *r* Hamaker potential of isolated colloid particle

*<sup>i</sup>* Volume fraction of the respective component

*<sup>i</sup>* Chemical potential of the respective component

*M* Difference in the mass for the molecules constituting the binary mixture

*<sup>N</sup>* Total interaction potential of the atoms or molecules included in the

2 Volume fraction of the second component in binary mixtures

*<sup>i</sup>* Molecular symmetry number for the respective component

Parameter which describes the gradual "switching on" of the

1,2 Boundary values of stable volume fractions in binary systems below the

*<sup>N</sup>*1 Molecular symmetry number for the virtual particle in binary mixture

<sup>0</sup>*<sup>i</sup>* Chemical potential of the ideal gas of the molecules or atoms of the

*P P* 21 Combined chemical potential at the constant pressure for the binary

Combined chemical potential for the respective components

*<sup>e</sup>* Electrostatic contribution to the chemical potential at the constant volume

*<sup>P</sup>* Electrostatic contribution to the chemical potential at the constant pressure

*i* Local pressure distribution around the respective molecule or particle *e* Electrostatic contribution to the local pressure distribution around the

*<sup>c</sup> T T* Ratio of the temperature at the point of measurement to the critical

Chemical potentials of the respective component at the constant pressure

*ij* Energy of interaction between the molecules of the respective components

binary mixture

respective virtual particle

critical temperature

*<sup>D</sup>* Debye length

 *i ik i k k v v*

systems

 \*

intermolecular interaction

respective component

and volume, respectively

for the charged colloid particle

for the charged colloid particle

*ij* Minimal molecular approach distance

charged colloid particle

Electrokinetic potential

temperature

 Macroscopic electrical potential *<sup>e</sup> e* Electrostatic interaction energy

 *ik j*

\*

 

\*

 

*iP iV* ,

2

2 *e*


**14** 

*France* 

**Thermodynamics of Surface Growth with** 

In physics, surface growth classically refers to processes where material reorganize on the substrate onto which it is deposited (like epitaxial growth), but principally to phenomena associated to phase transition, whereby the evolution of the interface separating the phases produces a crystal (Kessler, 1990; Langer, 1980). From a biological perspective, *surface growth* refers to mechanisms tied to accretion and deposition occurring mostly in hard tissues, and is active in the formation of teeth, seashells, horns, nails, or bones (Thompson, 1992). A landmark in this field is Skalak (Skalak et al., 1982, 1997) who describe the growth or atrophy of part of a biological body by the accretion or resorption of biological tissue lying on the surface of the body. Surface growth of biological tissues is a widespread situation, with may be classified as either fixed growth surface (e.g. nails and horns) or moving growing surface (e.g. seashells, antlers). Models for the kinematics of surface growth have been developed in (Skalak et al., 1997), with a clear distinction between cases of fixed and moving growth surfaces, see (Ganghoffer et al., 2010a,b; Garikipati, 2009) for a recent

Following the pioneering mechanical treatments of elastic material surfaces and surface tension by (Gurtin and Murdoch, 1975; Mindlin, 1965), and considering that the boundary of a continuum displays a specific behavior (distinct from the bulk behavior), subsequent contributions in this direction have been developed in the literature (Gurtin and Struthers, 1990; Gurtin, 1995, Leo and Sekerka, 1989) for a thermodynamical approach of the surface stresses in crystals; configurational forces acting on interfaces have been considered e.g. in (Maugin, 1993; Maugin and Trimarco, 1995) – however not considering surface stress -, and (Gurtin, 1995; 2000) considering specific balance laws of configurational forces localized at

Biological evolution has entered into the realm of continuum mechanics in the 1990's, with attempts to incorporate into a continuum description time-dependent phenomena, basically consisting of a variation of material properties, mass and shape of the solid body. One outstanding problem in developmental biology is indeed the understanding of the factors that may promote the generation of biological form, involving the processes of growth (change of mass), remodeling (change of properties), and morphogenesis (shape changes), a

The main focus in this chapter is the setting up of a modeling platform relying on the thermodynamics of surfaces (Linford, 1973) and configurational mechanics (Maugin, 1993)

**1. Introduction** 

exhaustive literature review.

classification suggested by Taber (Taber, 1995).

interfaces.

**Application to Bone Remodeling** 

*LEMTA – ENSEM, 2, Avenue de la Forêt de Haye,* 

Jean-François Ganghoffer

