**2. Introduction to rotational dynamics**

Understanding solute-solvent interaction has been of great relevance in physico-chemical processes due to the importance of these interactions in determining properties such as chemical reaction yield and kinetics or the ability to isolate one compound from another. Interactions between the solutes and their surrounding solvent molecules are difficult to

*It It* ⊥ ⊥

*I t*( ) and *I t*( ) <sup>⊥</sup> are the fluorescence intensity decays collected with the polarization of the emission polarizer maintained parallel and perpendicular to the polarization of the excitation source, respectively. For a fluorophore in a sample solvent, the fluorescence depolarization is simply due to rotational motion of the excited fluorophore and the decay parameters depend on the size and shape of the fluorophore. For spherical fluorophores, the anisotropy decay is a single exponential with a single rotational correlation time and is

τ

where 0*r* is the initial anisotropy (anisotropy at time t=0 or anisotropy observed in the

anisotropy 0*r* is related to the angle (θ) between the absorption and emission dipoles of the

2

τ

*V kT* η τ

τ

*r*

τ

τ

, temperature ( ) *T* of the solution and the molecular volume ( ) *V* of the

2 3cos (θ) 1 5 2

 <sup>−</sup> <sup>=</sup> 

where the value 0*r* can vary between 0.4 and –0.2 as the angle (θ) varies between <sup>0</sup> 0 and

fluorophore. This is given by Stokes-Einstein relation (Fleming, 1986) as shown below:

*r*

The relation between the steady-state anisotropy (*r*), initial anisotropy ( <sup>0</sup>*r* ), rotational

<sup>0</sup> 1 *<sup>f</sup>*

The Perrin equation is very useful in obtaining the correlation time without the

developed for more complicated shapes of the fluorophore show that a maximum of five

Understanding solute-solvent interaction has been of great relevance in physico-chemical processes due to the importance of these interactions in determining properties such as chemical reaction yield and kinetics or the ability to isolate one compound from another. Interactions between the solutes and their surrounding solvent molecules are difficult to

*r r*

exponentials are enough to explain the fluorescence anisotropy decay (Steiner, 1991).

τ

0

*r*

) and fluorescence lifetime ( *<sup>f</sup>*

measurement of polarization dependent fluorescence decays [ ||

<sup>−</sup> <sup>=</sup> + (4)

(5)

(6)

of the fluorophore is governed by the

= (7)

) is given by Perrin equation as follows

*I t*( ) and *I t*( ) <sup>⊥</sup> ]. The theory

= + (8)

is the rotational correlation time. The initial


*r t*

( ) exp( / ) <sup>0</sup> *rt r t <sup>r</sup>* = −

absence of any depolarizing processes) and *<sup>r</sup>*

<sup>0</sup> 90 respectively. The rotational correlation times *<sup>r</sup>*

where ||

given by (Lakowicz, 2006)

fluorophore under study as

viscosity ( )

η

correlation time ( *<sup>r</sup>*

(Lackowicz, 1983)

where *k* is the Boltzmann constant.

τ

**2. Introduction to rotational dynamics** 

resolve because, unlike in solids, the spatial relationship between the molecules are not fixed on time scales that can be accessed using structural measurements such as X-ray diffraction or multidimensional NMR spectrometry. Intermolecular interactions in the liquid phase are more complex than those in gas phase because of their characteristic strength, the property that gives rise to the liquid phase and at the same time prevents a simple statistical description of collisional interactions from providing adequate insight (Fleming, 1986). Regardless of almost three and a half decades of continuous investigation, the details of solute-solvent interactions, particularly in polar solvent systems, remain to be understood in detail. Most investigations of intermolecular interactions in solution have used a "probe" molecule present at low concentration in neat or binary solvent systems. Typically, a short pulse of light is shone to establish some non-equilibrium condition in the ensemble of probe molecules, with the object of the experiment being to monitor the return to equilibrium. These studies have included fluorescence lifetime, molecular reorientation (Eisenthal, 1975; Shank and Ippen, 1975; von Jena and Lessing, 1979a; Sanders and Wirth, 1983; Templeton et al., 1985; Blanchard and Wirth, 1986; Templeton and Kenney-Wallace, 1986; Blanchard, 1987, 1988, 1989; Blanchard and Cihal, 1988; Hartman et al., 1991; Srivastava and Doraiswamy, 1995; Imeshev and Khundkar, 1995; Dutt, et al., 1995; Chandrashekhar et al., 1995; Levitus et al., 1995; Backer et al., 1996; Biasutti et al., 1996; Horng et al., 1997; Hartman et al., 1997; Laitinen et al., 1997; Singh, 2000; Dutt and Raman, 2001; Gustavsson et al., 2003; Dutt and Ghanty, 2004; Kubinyi et al., 2006), vibrational relaxation (Heilweil et al., 1986, 1987, 1989; Lingle Jr. et al., 1990; Anfinrud et al., 1990; Elsaesser and Kaiser, 1991; Hambir et al., 1993; Jiang and Blanchard, 1994a & b, 1995; McCarthy and Blanchard, 1995, 1996) and timedelayed fluorescence Stokes shift (Shapiro and Winn, 1980; Maroncelli and Fleming, 1987; Huppert et al. 1989, 1990; Chapman et al., 1990; Wagener and Richert, 1991; Fee et al., 1991; Jarzeba et al., 1991; Yip et al., 1993; Fee and Maroncelli, 1994; Inamdar et al., 1995) measurements. Of these, molecular reorientation of molecules in solution has been an important experimental and theoretical concept for probing the nature of liquids and the interactions of solvents with molecules. This has proven to be among the most useful because of the combined generality of the effect and the well-developed theoretical framework for the interpretation of the experimental data (Debye, 1929; Perrin, 1936; Chuang and Eisethal, 1972; Hu and Zwanzig, 1974; Youngren and Acrivos, 1975; Zwanzig and Harrison, 1985). Though, the effect of solute-solvent interactions on the rotational motion of a probe molecule in solution has been extensively studied, these interactions are generally described as friction to probe rotational motion and can be classified into three types. The first category includes short-range repulsive forces, which dominate intermolecular dynamics during molecular collisions. These interactions are present in all liquids and lead to viscous dissipation, which is well described by hydrodynamic theories (Fleming, 1986). The second category includes long-range electrostatic interactions between a charged or dipolar probe and polar solvent molecules. As the solute turns, the induced solvent polarization can lag behind rotation of the probe, creating a torque, which systematically reduces the rate of rotational diffusion. This effect, termed dielectric friction, arises from the same type of correlated motions of solvent molecules, which is responsible for the time dependent Stokes' shift (TDSS) dynamics of fluorescent probes (van der Zwan and Hynes, 1985; Barbara and Jarzeba, 1990; Maroncelli, 1993). The third category includes specific solute-solvent interactions. Hydrogen bonding is probably the most frequently encountered example of this kind. Strong hydrogen bonds will lead to the formation of

Rotational Dynamics of Nonpolar and Dipolar

accessible bulk properties of the solvent.

molecule.

Molecules in Polar and Binary Solvent Mixtures 191

size effects in a qualitative way, it could not explain the faster rotation of nonpolar probes in alcohols compared to alkanes. The second relatively new quasihydrodynamic theory was proposed by Dote, Kivelson and Schwartz (DKS) (1981). The DKS theory not only takes into consideration the relative sizes of the solvent and the probe but also the cavities or free spaces created by the solvent around the probe molecule. If the size of the solute is comparable to the free volumes of the solvent, the coupling between the solute and the solvent will become weak which results in reduced friction experienced by the probe

On the other hand, rotational dynamics of small and medium sized polar solutes dissolved in polar solvents experiences more friction than predicted by the hydrodynamic theories. This 'additional friction' is attributed to the solute-solvent hydrogen bonding. The first and the oldest concept of dielectric friction invoked by chemists is the 'solvent-berg' model, in which it is assumed that there is a solute-solvent interaction causing increase in the volume of the solute. Such an enhancement of the volume automatically causes the molecule to rotate slower. However, reservations against such an explanation have also been expressed (Chuang and Eisenthal, 1972; Horng et al., 1997). Objections to this kind of interpretations arise from the fact that in bulk solution, the solvent molecules are expected to exchange (solute-solvent hydrogen bonding dynamics) on a much faster time scale compared to the rotational dynamics. Later, the slower reorientation times of polar molecules in polar solvents have been interpreted using dielectric friction theories (Phillips et al., 1985; Dutt et al., 1990; Alavi et al., 1991b,c; Dutt and Raman, 2001; Gustavsson et al., 2003). Dielectric friction on a rotating solute arises because the polar molecule embedded in a dielectric medium polarizes the surrounding dielectric. As the solute tries to rotate, the polarization of the medium cannot instantaneously keep in phase with the new orientation of the probe molecule and this lag exerts a retarding force on the probe molecule, giving rise to rotational dielectric friction. Although molecular theories of dielectric friction are available, at present these theories are difficult to apply because they require some knowledge of the intermolecular potential or some unavailable properties of the solvent. Continuum theories offer advantages of simplicity and the calculation of molecular friction in terms of easily

The SED theory has been found to describe the rotational dynamics of medium sized molecules fairly well when the coupling between the solute and solvent is purely mechanical or hydrodynamic in nature. It is documented that the SED model correctly predicts the linear dependence of the rotational reorientation times on the solvent viscosity for polar and cationic dyes dissolved in polar and non polar solvents (Chuang and Eisenthal, 1971; Fleming et al., 1976; 1977; Porter et al., 1977; Moog et al., 1982; Spears and Cramer, 1978; Millar et al., 1979; von Jena and Lessing, 1979a, b; 1981; Rice and Kenney-Wallace, 1980; Waldeck and Fleming, 1981; Dutt et al., 1990; Alavi et al., 1991a, b, c; Krishnamurthy et al., 1993; Dutt et al., 1998) that have been interpreted using dielectric fiction theories. The dielectric friction can be modeled using continuum theories of Nee-Zwanzig (NZ) (Nee and Zwanzig, 1970), which treats the solute as a point dipole rotating in a spherical cavity, Alavi-Waldeck (AW) (Alavi and Waldeck, 1991b; 1993) model which is an extension of the NZ theory where the solute is treated as a distribution of charges instead of point dipole and the semiempirical approach of van der Zwan and Hynes (vdZH) (van der Zwan and Hynes, 1985) in which fluorescence Stokes shift of the solute in a given solvent is related to dielectric friction. Conversely, the results of neutral and nonpolar solutes deviate

solute-solvent complexes of well-defined stoichiometry. These new, larger species can persist in solution for fairly long times and will rotate more slowly than the bare solute. Formation and breakage of weak H binds occurring on time scales faster than probe rotation will provide a channel for rotational energy dissipation giving rise to additional friction.

The theoretical interest in the study of rotational reorientation kinetics of molecules in liquids arises from the fact that it provides information about the intermolecular interaction in the condensed phase. However, the theoretical modeling of molecular reorientation in liquids and its correlation with experimental data is still far from satisfactory. Thus far, two kinds of approaches have been employed in understanding the rotational dynamics. In the first approach, binary collision approximation has been used to explain the rotational dynamics. With this approach, kinetic theory model for rotational relaxation has been employed for rough sphere fluids (Widom, 1960; Rider and Fixman, 1972; Chandler, 1974) and for smooth convex bodies (Evans et al., 1982; Evans and Evans, 1984; Evans, 1988). Evans model along with Enskog equation for viscosity has been employed to express rotational reorientation time (τ*<sup>r</sup>*) as a function of the solvent viscosity. However, explaining rotational dynamics from such a molecular point of view is severely constrained on account of multibody interaction in a fluid. For real systems the quantitative predictions can be made about the variation of τ*<sup>r</sup>* with solvent viscosity. The second approach is the macroscopic approach of understanding the rotational dynamics, where the solvent is assumed to be a structureless continuum and the rotational motion of solutes is considered Markovian or diffusional. A considerable degree of success on the rotational dynamics arises from the Stokes-Einstein-Debye (SED) hydrodynamic theory, which forms the basis of understanding molecular rotations of medium sized molecules (few hundred Å3 volumes) in liquids (Einstein, 1906;Debye, 1929; Stokes, 1956), according to which the rotational reorientation time (τ*<sup>r</sup>*) of a solute molecule is proportional to its volume (*V*), bulk viscosity (η) of the solvent and inversely related to its temperature (*T*).

Rotational dynamics of number of nonpolar and polar solutes have been carried out in homologous series of polar and nonpolar solvents. In general, the experimentally measured reorientation times of most of the nonpolar probes could be described by the SED theory with slip boundary condition. In some cases the reorientation times were found to be faster than predicted by the slip boundary condition, a situation termed as subslip behavior. However, for a given probe in a homologous series of solvents (alkanes or alcohols) the normalized reorientation times (i.e., reorientation times at unit viscosity) decrease as the size of the solvent increased. In other words, the reorientation times did not scale linearly with solvent viscosity. This behavior, known as the size effects, could not be explained with SED theory. Another observation, which the SED theory failed to explain, is that the experimentally measured reorientation times of nonpolar probes are faster in alcohols than in alkanes of similar viscosity. To explain the observed size effects two quasihydrodynamics theories have been used. The first one is a relatively old theory proposed by Geirer and Wirtz (GW) (1953), which takes into account both the size of the solute as well as that of the solvent while calculating the boundary condition. This theory visualizes the solvent to be made up of concentric shells of spherical particles surrounding the spherical probe molecule at the center. Each shell moves at a constant angular velocity and the velocity of successive shells decreases with the distance from the surface of the probe molecule, as though the flow between the shells is laminar. As the shell number increases, i.e., at large distances, the angular velocity vanishes. Although, the GW theory is successful in predicting the observed

solute-solvent complexes of well-defined stoichiometry. These new, larger species can persist in solution for fairly long times and will rotate more slowly than the bare solute. Formation and breakage of weak H binds occurring on time scales faster than probe rotation will provide a channel for rotational energy dissipation giving rise to additional friction. The theoretical interest in the study of rotational reorientation kinetics of molecules in liquids arises from the fact that it provides information about the intermolecular interaction in the condensed phase. However, the theoretical modeling of molecular reorientation in liquids and its correlation with experimental data is still far from satisfactory. Thus far, two kinds of approaches have been employed in understanding the rotational dynamics. In the first approach, binary collision approximation has been used to explain the rotational dynamics. With this approach, kinetic theory model for rotational relaxation has been employed for rough sphere fluids (Widom, 1960; Rider and Fixman, 1972; Chandler, 1974) and for smooth convex bodies (Evans et al., 1982; Evans and Evans, 1984; Evans, 1988). Evans model along with Enskog equation for viscosity has been employed to express

rotational dynamics from such a molecular point of view is severely constrained on account of multibody interaction in a fluid. For real systems the quantitative predictions can be

macroscopic approach of understanding the rotational dynamics, where the solvent is assumed to be a structureless continuum and the rotational motion of solutes is considered Markovian or diffusional. A considerable degree of success on the rotational dynamics arises from the Stokes-Einstein-Debye (SED) hydrodynamic theory, which forms the basis of understanding molecular rotations of medium sized molecules (few hundred Å3 volumes) in liquids (Einstein, 1906;Debye, 1929; Stokes, 1956), according to which the rotational

Rotational dynamics of number of nonpolar and polar solutes have been carried out in homologous series of polar and nonpolar solvents. In general, the experimentally measured reorientation times of most of the nonpolar probes could be described by the SED theory with slip boundary condition. In some cases the reorientation times were found to be faster than predicted by the slip boundary condition, a situation termed as subslip behavior. However, for a given probe in a homologous series of solvents (alkanes or alcohols) the normalized reorientation times (i.e., reorientation times at unit viscosity) decrease as the size of the solvent increased. In other words, the reorientation times did not scale linearly with solvent viscosity. This behavior, known as the size effects, could not be explained with SED theory. Another observation, which the SED theory failed to explain, is that the experimentally measured reorientation times of nonpolar probes are faster in alcohols than in alkanes of similar viscosity. To explain the observed size effects two quasihydrodynamics theories have been used. The first one is a relatively old theory proposed by Geirer and Wirtz (GW) (1953), which takes into account both the size of the solute as well as that of the solvent while calculating the boundary condition. This theory visualizes the solvent to be made up of concentric shells of spherical particles surrounding the spherical probe molecule at the center. Each shell moves at a constant angular velocity and the velocity of successive shells decreases with the distance from the surface of the probe molecule, as though the flow between the shells is laminar. As the shell number increases, i.e., at large distances, the angular velocity vanishes. Although, the GW theory is successful in predicting the observed

*<sup>r</sup>*) as a function of the solvent viscosity. However, explaining

*<sup>r</sup>* with solvent viscosity. The second approach is the

*<sup>r</sup>*) of a solute molecule is proportional to its volume (*V*), bulk viscosity

rotational reorientation time (

made about the variation of

τ

reorientation time (

(η τ

τ

) of the solvent and inversely related to its temperature (*T*).

size effects in a qualitative way, it could not explain the faster rotation of nonpolar probes in alcohols compared to alkanes. The second relatively new quasihydrodynamic theory was proposed by Dote, Kivelson and Schwartz (DKS) (1981). The DKS theory not only takes into consideration the relative sizes of the solvent and the probe but also the cavities or free spaces created by the solvent around the probe molecule. If the size of the solute is comparable to the free volumes of the solvent, the coupling between the solute and the solvent will become weak which results in reduced friction experienced by the probe molecule.

On the other hand, rotational dynamics of small and medium sized polar solutes dissolved in polar solvents experiences more friction than predicted by the hydrodynamic theories. This 'additional friction' is attributed to the solute-solvent hydrogen bonding. The first and the oldest concept of dielectric friction invoked by chemists is the 'solvent-berg' model, in which it is assumed that there is a solute-solvent interaction causing increase in the volume of the solute. Such an enhancement of the volume automatically causes the molecule to rotate slower. However, reservations against such an explanation have also been expressed (Chuang and Eisenthal, 1972; Horng et al., 1997). Objections to this kind of interpretations arise from the fact that in bulk solution, the solvent molecules are expected to exchange (solute-solvent hydrogen bonding dynamics) on a much faster time scale compared to the rotational dynamics. Later, the slower reorientation times of polar molecules in polar solvents have been interpreted using dielectric friction theories (Phillips et al., 1985; Dutt et al., 1990; Alavi et al., 1991b,c; Dutt and Raman, 2001; Gustavsson et al., 2003). Dielectric friction on a rotating solute arises because the polar molecule embedded in a dielectric medium polarizes the surrounding dielectric. As the solute tries to rotate, the polarization of the medium cannot instantaneously keep in phase with the new orientation of the probe molecule and this lag exerts a retarding force on the probe molecule, giving rise to rotational dielectric friction. Although molecular theories of dielectric friction are available, at present these theories are difficult to apply because they require some knowledge of the intermolecular potential or some unavailable properties of the solvent. Continuum theories offer advantages of simplicity and the calculation of molecular friction in terms of easily accessible bulk properties of the solvent.

The SED theory has been found to describe the rotational dynamics of medium sized molecules fairly well when the coupling between the solute and solvent is purely mechanical or hydrodynamic in nature. It is documented that the SED model correctly predicts the linear dependence of the rotational reorientation times on the solvent viscosity for polar and cationic dyes dissolved in polar and non polar solvents (Chuang and Eisenthal, 1971; Fleming et al., 1976; 1977; Porter et al., 1977; Moog et al., 1982; Spears and Cramer, 1978; Millar et al., 1979; von Jena and Lessing, 1979a, b; 1981; Rice and Kenney-Wallace, 1980; Waldeck and Fleming, 1981; Dutt et al., 1990; Alavi et al., 1991a, b, c; Krishnamurthy et al., 1993; Dutt et al., 1998) that have been interpreted using dielectric fiction theories. The dielectric friction can be modeled using continuum theories of Nee-Zwanzig (NZ) (Nee and Zwanzig, 1970), which treats the solute as a point dipole rotating in a spherical cavity, Alavi-Waldeck (AW) (Alavi and Waldeck, 1991b; 1993) model which is an extension of the NZ theory where the solute is treated as a distribution of charges instead of point dipole and the semiempirical approach of van der Zwan and Hynes (vdZH) (van der Zwan and Hynes, 1985) in which fluorescence Stokes shift of the solute in a given solvent is related to dielectric friction. Conversely, the results of neutral and nonpolar solutes deviate

Rotational Dynamics of Nonpolar and Dipolar

time [ ] <sup>1</sup> ( 1) *<sup>l</sup>* τ

(Lakowicz, 2006) as

**2.1.1 Hydrodynamic theory** 

The rotational correlation time (

substitution of Eqn. (12) in (14) gives

Eqn. (11) reduces to

given by

where τ

where ζ

Molecules in Polar and Binary Solvent Mixtures 193

<sup>1</sup> ( 1) *dC Dl l C*

The rotational diffusion co-efficient of a solute is given by the Stokes-Einstein model

*kT <sup>D</sup>* ζ

great importance in theoretical as well as experimental studies. A molecule rotating in liquid experiences friction on account of its continuous interaction with its neighbors and the desire to understand has been a motivating force in carrying the experimental

Mechanical friction on a rotating solute in solvent is computed employing hydrodynamic theory by treating the solute as a smooth sphere rotating in a continuum fluid, which is

3

<sup>3</sup> 8 *kT <sup>D</sup>* πη

> ζ

*V kT* η τ

1 6 6 *<sup>r</sup> D kT*

*r*

where *V* is the molecular volume. The most widely used Stokes-Einstein-Debye (SED) hydrodynamic equation for the description of rotational dynamics of spherical molecule is

> *r* 0 *V kT* η τ

approximation embedded in a SED is glossy in error and the shape of the probes is however,

 τ

*<sup>0</sup>* is the rotational reorientation time at zero viscosity. It is known that spherical

characterized by a shear viscosity. If '*a*' is the radius of the molecule and '

ζ = 8π η

*<sup>r</sup>*) is given by

τ

*ll D* <sup>−</sup> = + . In fluorescence depolarization experiments, one measures the

is the friction coefficient and *kT* is the thermal energy. It is this friction, which is of

τ

*<sup>r</sup> <sup>D</sup>* <sup>−</sup> <sup>=</sup> .

=− + (10)

− η

<sup>=</sup> (11)

η

*a* (12)

*<sup>a</sup>* <sup>=</sup> (13)

= = (14)

= (15)

= + (16)

' the viscosity of

and the correlation

1

*dt*

This implies that the correlation function decays exponentially, *<sup>t</sup>*/ *e*

anisotropy decay which is *l=*2 correlation and hence <sup>1</sup> (6 )

measurements of rotational reorientation in liquids.

the liquid, then according to Stokes law (Stokes, 1956)

τ

significantly from the hydrodynamic predictions at higher viscosities (Waldeck et al., 1982; Canonica et al., 1985; Phillips et al., 1985; Courtney et al., 1986; Ben Amotz and Drake, 1988; Roy and Doraiswamy, 1993; Williams et al., 1994; Jiang and Blanchard, 1994; Anderton and Kauffman, 1994; Brocklehurst and Young, 1995; Benzler and Luther, 1997; Dutt et al., 1999; Ito et al., 2000; Inamdar et al., 2006). These probes rotate much faster than predicted by the SED theory with stick boundary condition and are described by either slip boundary condition or by quasihydrodynamic theories. Slip boundary condition (Hu and Zwanzig, 1974) assumes the solute-solvent coupling parameter to be less than unity, contrary to the stick boundary condition. Quasihydrodynamic theories of Gierer and Wirtz (GW) (Gierer and Wirtz, 1953) and Dote, Kivelson and Schwartz (DKS) (Dote, Kivelson and Schwartz, 1981) attempt to improve upon SED theory by taking into consideration not only the size of the solute but also that of the solvent molecule, thereby modifying the boundary conditions. It has been argued (Ben Amotz and Drake, 1988; Roy and Doraiswamy, 1993) that as the size of the solute molecule becomes much larger than the size of the solvent molecule, the observed reorientation times approach the SED theory with the stick boundary condition.

Based on the above description, we have chosen two kinds of solutes categorized as nonpolar and polar to study their rotational reorientation dynamics in nonpolar, polar and binary mixtures of solvents. In the first case, where the nonpolar probes embedded in polar or nonpolar solvents to examine the influence of solute to solvent size ratio and the shape of the solute on the friction experienced by the probe molecule which in turn enables to test the validity of hydrodynamic and quasihydrodynamic theories. The friction experienced by these probes is purely hydrodynamic or mechanical in nature since it is dominated by shortrange repulsive forces. Polar probes used in charged polar solvents with an intention of understanding how the long-range electrostatic interactions between the solute and the solvent, which are charge-dipole or dipole-dipole in nature, influence the rotational dynamics of the probe molecules. Dielectric friction on a rotating solute arises because of the polar molecule entrenched in a dielectric medium polarizes the surrounding dielectric. As the solute tries to rotate, the polarization of the medium cannot instantaneously keep in phase with the new orientation of the probe molecule and this lag exerts a retarding force on the probe molecule, giving rise to rotational dielectric friction.

### **2.1 Theoretical background**

Among the many proposed models for the study of rotational motion, the most commonly employed is the rotational diffusion model outlined by Debye (Debye, 1929), in which the reorientation is assumed to occur in small angular steps. On account of high frequency collisions, a molecule can rotate through a very small angle before undergoing another reorienting collision. The rotational diffusion equation solved to obtain the rotational correlation time τ*r* of the density function ρ(, ) θ φis given by (Lackowicz, 2006)

$$\frac{\partial \rho}{\partial t} = D \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left[ \frac{1}{\sin \theta} \frac{\partial \rho}{\partial \theta} + \frac{1}{\sin^2 \theta} \frac{\partial^2 \rho}{\partial \phi^2} \right] \tag{9}$$

where *D* is the rotational diffusion coefficient. For spherical particles ρ satisfies the form 1 , () ( , ) *C tYl m* θ φ in isotropic liquids, where , (, ) *Yl m* θ φ are the Legendre polynomials and the coefficient 1 *C t*( ) is essentially the same as the correlation function. Substitution of 1 , () ( , ) ρ = *C tYl m* θ φgives an ordinary differential equation for *C* as

significantly from the hydrodynamic predictions at higher viscosities (Waldeck et al., 1982; Canonica et al., 1985; Phillips et al., 1985; Courtney et al., 1986; Ben Amotz and Drake, 1988; Roy and Doraiswamy, 1993; Williams et al., 1994; Jiang and Blanchard, 1994; Anderton and Kauffman, 1994; Brocklehurst and Young, 1995; Benzler and Luther, 1997; Dutt et al., 1999; Ito et al., 2000; Inamdar et al., 2006). These probes rotate much faster than predicted by the SED theory with stick boundary condition and are described by either slip boundary condition or by quasihydrodynamic theories. Slip boundary condition (Hu and Zwanzig, 1974) assumes the solute-solvent coupling parameter to be less than unity, contrary to the stick boundary condition. Quasihydrodynamic theories of Gierer and Wirtz (GW) (Gierer and Wirtz, 1953) and Dote, Kivelson and Schwartz (DKS) (Dote, Kivelson and Schwartz, 1981) attempt to improve upon SED theory by taking into consideration not only the size of the solute but also that of the solvent molecule, thereby modifying the boundary conditions. It has been argued (Ben Amotz and Drake, 1988; Roy and Doraiswamy, 1993) that as the size of the solute molecule becomes much larger than the size of the solvent molecule, the observed reorientation times approach the SED theory with the stick boundary condition. Based on the above description, we have chosen two kinds of solutes categorized as nonpolar and polar to study their rotational reorientation dynamics in nonpolar, polar and binary mixtures of solvents. In the first case, where the nonpolar probes embedded in polar or nonpolar solvents to examine the influence of solute to solvent size ratio and the shape of the solute on the friction experienced by the probe molecule which in turn enables to test the validity of hydrodynamic and quasihydrodynamic theories. The friction experienced by these probes is purely hydrodynamic or mechanical in nature since it is dominated by shortrange repulsive forces. Polar probes used in charged polar solvents with an intention of understanding how the long-range electrostatic interactions between the solute and the solvent, which are charge-dipole or dipole-dipole in nature, influence the rotational dynamics of the probe molecules. Dielectric friction on a rotating solute arises because of the polar molecule entrenched in a dielectric medium polarizes the surrounding dielectric. As the solute tries to rotate, the polarization of the medium cannot instantaneously keep in phase with the new orientation of the probe molecule and this lag exerts a retarding force on

the probe molecule, giving rise to rotational dielectric friction.

*r* of the density function

*D t* ρ

in isotropic liquids, where , (, ) *Yl m*

where *D* is the rotational diffusion coefficient. For spherical particles

Among the many proposed models for the study of rotational motion, the most commonly employed is the rotational diffusion model outlined by Debye (Debye, 1929), in which the reorientation is assumed to occur in small angular steps. On account of high frequency collisions, a molecule can rotate through a very small angle before undergoing another reorienting collision. The rotational diffusion equation solved to obtain the rotational

> 11 1 sin sin sin

∂ ∂∂ ∂ = + ∂ ∂∂ <sup>∂</sup>

 θθ

> θ φ

coefficient 1 *C t*( ) is essentially the same as the correlation function. Substitution of

gives an ordinary differential equation for *C* as

is given by (Lackowicz, 2006)

(9)

satisfies the form

ρ

are the Legendre polynomials and the

2 2 2

ρρ

θ φ

ρ(, ) θ φ

θθ

**2.1 Theoretical background** 

τ

correlation time

1 , () ( , ) *C tYl m* θ φ

= *C tYl m*

ρ

1 , () ( , )

θ φ

$$\frac{dC\_1}{dt} = -Dl(l+1)C\_1\tag{10}$$

This implies that the correlation function decays exponentially, *<sup>t</sup>*/ *e* − η and the correlation time [ ] <sup>1</sup> ( 1) *<sup>l</sup>* τ *ll D* <sup>−</sup> = + . In fluorescence depolarization experiments, one measures the anisotropy decay which is *l=*2 correlation and hence <sup>1</sup> (6 ) τ*<sup>r</sup> <sup>D</sup>* <sup>−</sup> <sup>=</sup> .

The rotational diffusion co-efficient of a solute is given by the Stokes-Einstein model (Lakowicz, 2006) as

$$D = \frac{kT}{\zeta} \tag{11}$$

where ζ is the friction coefficient and *kT* is the thermal energy. It is this friction, which is of great importance in theoretical as well as experimental studies. A molecule rotating in liquid experiences friction on account of its continuous interaction with its neighbors and the desire to understand has been a motivating force in carrying the experimental measurements of rotational reorientation in liquids.

#### **2.1.1 Hydrodynamic theory**

Mechanical friction on a rotating solute in solvent is computed employing hydrodynamic theory by treating the solute as a smooth sphere rotating in a continuum fluid, which is characterized by a shear viscosity. If '*a*' is the radius of the molecule and 'η' the viscosity of the liquid, then according to Stokes law (Stokes, 1956)

$$
\zeta = 8\pi a^3 \eta \tag{12}
$$

Eqn. (11) reduces to

$$D = \frac{kT}{8\pi\eta a^3} \tag{13}$$

The rotational correlation time (τ*<sup>r</sup>*) is given by

$$
\sigma\_r = \frac{1}{6D} = \frac{\mathcal{L}}{6kT} \tag{14}
$$

substitution of Eqn. (12) in (14) gives

$$
\sigma\_r = \frac{\eta V}{kT} \tag{15}
$$

where *V* is the molecular volume. The most widely used Stokes-Einstein-Debye (SED) hydrodynamic equation for the description of rotational dynamics of spherical molecule is given by

$$
\pi\_r = \frac{\eta V}{kT} + \pi\_0 \tag{16}
$$

where τ*<sup>0</sup>* is the rotational reorientation time at zero viscosity. It is known that spherical approximation embedded in a SED is glossy in error and the shape of the probes is however,

Rotational Dynamics of Nonpolar and Dipolar

*C*

**ii. The Dote-Kivelson-Schwartz theory (DKS)** 

where

given by

Schwartz, 1981)

the solute molecule, with

prediction for the sphere of same volume.

compressibility *kT* of the liquid by

where γ φ

and φ

Molecules in Polar and Binary Solvent Mixtures 195

<sup>−</sup>

( ) ( )

*V V V V* <sup>−</sup> = + + +

*Vs* and *Vp* are the volumes of the solvent and probe, respectively. The expression for *CGW* is

*C C GW* = σ

*C* in Eqn. (17) should be replaced with *CGW* obtained from Eqn. (21) for calculating the reorientation times with GW theory. When the ratio *Vs*/*Vp* is very small *CGW* reduces to

Although, the GW theory is successful in predicting the observed size effects in a qualitative way, it could not explain the faster rotation of nonpolar probes in alcohols compared alkanes. Hence, the second relatively new quasihydrodynamic theory, was proposed by Dote, Kivelson and Schwartz (DKS) in 1981. This theory not only takes into consideration the relative sizes of the solvent and the probe but also the cavities or free spaces created by the solvent around the probe molecule. If the size of the solute is comparable to the free volumes of the solvents, the coupling between the solute and the solvent will become weak which results in reduced friction experienced by the probe molecule. According to DKS theory the solute-solvent coupling parameter, *CDKS* is given by (Dote, Kivelson and

> <sup>1</sup> *CDKS* (1 / ) γ φ

> > *p s V V V V*

Δ

γ

/ is the ratio of the free volume available for the solvent to the effective size of

2/3 4 1 *<sup>p</sup>*

 = +

is the ratio of the reorientation time predicted by slip hydrodynamics to the stick

Δ

solvent molecule and some discretion must be applied while calculating this term (Dutt et al., 1988; Anderton and Kauffman, 1994; Dutt and Rama Krishna, 2000). Δ*V* is empirically related to the solvent viscosity, the Hilderbrand-Batchinsky parameter *B* and the isothermal

*s p s p*

<sup>0</sup> 4 3 1/3 1/3 6( / ) 1

12 / 14 /

1/3

= +

σ

*s p*

*V V*

unity and the SED equation with stick boundary condition is obtained.

<sup>1</sup> <sup>1</sup> 3 <sup>0</sup> 1 6 *<sup>s</sup> p <sup>V</sup> <sup>C</sup> V*

(19)

(20)

1

0 (21)

<sup>−</sup> = + (22)

, (23)

*V* is the smallest volume of free space per

more important. In reality, the exact shape of the solute molecule is need not be a spherical and there is a necessary to include a parameter, which should describe the exact shape of nonspherical probes. Hence, the equation for nonspherical molecule proposed by Perrin (Perrin, 1936) is given as follows

$$
\sigma\_r = \frac{\eta V}{kT} (f \text{C}) \tag{17}
$$

where *f* is shape factor and is well specified, *C* is the boundary condition parameter dependent strongly on solute, solvent and concentration. The shapes of the solute molecules are usually incorporated into the model by treating them as either symmetric or asymmetric ellipsoids. For nonspherical molecules, *f* >1 and the magnitude of deviation of *f* from unity describes the degree of the nonspherical nature of the solute molecule. *C*, signifies the extent of coupling between the solute and the solvent and is known as the boundary condition parameter (Barbara and Jarzeba, 1990). In the two limiting cases of hydrodynamic stick and slip for a nonspherical molecule, the value of *C* follows the inequality, 0< *C* ≤ 1 and the exact value of *C* is determined by the axial ratio of the probe.

It is observed that the experimentally measured rotational reorientation times of number of the nonpolar solutes (Waldeck et al., 1982; Canonica et al., 1985; Phillips et al., 1985; Courtney et al., 1986; Ben Amotz and Drake, 1988; Roy and Doraiswamy, 1993; Williams et al., 1994; Jiang and Blanchard, 1994; Anderton and Kauffman, 1994; Brocklehurst and Young, 1995; Benzler and Luther, 1997; Dutt et al., 1999; Ito et al., 2000; Inamdar et al., 2006) could be described by the SED theory with slip boundary condition (subslip behavior). For a homologous series of solvents such as alcohols or alkanes, the normalized reorientation times decreased as the size of the solvent is increased. In other words, the reorientation times did not scale linearly with solvent viscosity.

### **2.1.2 Quasihydrodynamic theories**

While the SED hydrodynamic theory takes only the size of the solute molecule into account leaving solvent size aside, one needs to consider the size of the solute as well as solvent molecules. Quasihydrodynamic theories consider these and modify the boundary condition accordingly. To explain such observation of size effects, two quasihydrodynamic theories by Gierer and Wirtz (GW) and Dote, Kivelson and Schwartz (DKS) have been used.

#### **i. Gierer and Wirtz theory (GW)**

The first and the relatively old theory proposed by Girer and Wirtz (GW) in 1953, takes into account both the size of the solute as well as that of the solvent while calculating the boundary condition. It visualizes the solvent to be made up of concentric shells of spherical particles surrounding the spherical probe molecule at the center. Each shell moves at a constant angular velocity and the velocity of successive shells decreases with the distance from the surface of the probe molecule, as though the flow between the shells is laminar. As the shell number increase, i.e., at large distances, the angular velocity vanishes. The angular velocity ω1 of the first solvation shell is related to the angular velocity ω<sup>0</sup> of the probe molecule by means of a sticking factor σ.

$$
\alpha \alpha\_1 = \sigma \alpha \rho\_0 \tag{18}
$$

When 1 σ = , it gives the stick boundary condition and σ is related to the ratio of the solute to solvent size, as

$$\sigma = \left[ 1 + 6 \left( \frac{V\_s}{V\_p} \right)^{\frac{1}{3}} \mathbb{C}\_0 \right]^{-1} \tag{19}$$

where

194 Hydrodynamics – Advanced Topics

more important. In reality, the exact shape of the solute molecule is need not be a spherical and there is a necessary to include a parameter, which should describe the exact shape of nonspherical probes. Hence, the equation for nonspherical molecule proposed by Perrin

> ( ) *<sup>r</sup> <sup>V</sup> fC kT* η τ

where *f* is shape factor and is well specified, *C* is the boundary condition parameter dependent strongly on solute, solvent and concentration. The shapes of the solute molecules are usually incorporated into the model by treating them as either symmetric or asymmetric ellipsoids. For nonspherical molecules, *f* >1 and the magnitude of deviation of *f* from unity describes the degree of the nonspherical nature of the solute molecule. *C*, signifies the extent of coupling between the solute and the solvent and is known as the boundary condition parameter (Barbara and Jarzeba, 1990). In the two limiting cases of hydrodynamic stick and slip for a nonspherical molecule, the value of *C* follows the inequality, 0< *C* ≤ 1 and the exact

It is observed that the experimentally measured rotational reorientation times of number of the nonpolar solutes (Waldeck et al., 1982; Canonica et al., 1985; Phillips et al., 1985; Courtney et al., 1986; Ben Amotz and Drake, 1988; Roy and Doraiswamy, 1993; Williams et al., 1994; Jiang and Blanchard, 1994; Anderton and Kauffman, 1994; Brocklehurst and Young, 1995; Benzler and Luther, 1997; Dutt et al., 1999; Ito et al., 2000; Inamdar et al., 2006) could be described by the SED theory with slip boundary condition (subslip behavior). For a homologous series of solvents such as alcohols or alkanes, the normalized reorientation times decreased as the size of the solvent is increased. In other words, the reorientation

While the SED hydrodynamic theory takes only the size of the solute molecule into account leaving solvent size aside, one needs to consider the size of the solute as well as solvent molecules. Quasihydrodynamic theories consider these and modify the boundary condition accordingly. To explain such observation of size effects, two quasihydrodynamic theories by

The first and the relatively old theory proposed by Girer and Wirtz (GW) in 1953, takes into account both the size of the solute as well as that of the solvent while calculating the boundary condition. It visualizes the solvent to be made up of concentric shells of spherical particles surrounding the spherical probe molecule at the center. Each shell moves at a constant angular velocity and the velocity of successive shells decreases with the distance from the surface of the probe molecule, as though the flow between the shells is laminar. As the shell number increase, i.e., at large distances, the angular velocity vanishes. The angular

1 of the first solvation shell is related to the angular velocity

ω σω

σ.

= , it gives the stick boundary condition and

Gierer and Wirtz (GW) and Dote, Kivelson and Schwartz (DKS) have been used.

= (17)

ω

is related to the ratio of the solute

1 0 = (18)

σ

<sup>0</sup> of the probe

(Perrin, 1936) is given as follows

value of *C* is determined by the axial ratio of the probe.

times did not scale linearly with solvent viscosity.

**2.1.2 Quasihydrodynamic theories** 

**i. Gierer and Wirtz theory (GW)** 

molecule by means of a sticking factor

velocity

When 1 σ

to solvent size, as

ω

$$\mathbf{C}\_{0} = \left[ \frac{6\left(V\_{s} \;/\; V\_{p}\right)^{1/3}}{\left[1 + 2\left(V\_{s} \;/\; V\_{p}\right)^{1/3}\right]^{4}} + \frac{1}{\left[1 + 4\left(V\_{s} \;/\; V\_{p}\right)^{1/3}\right]^{3}} \right]^{-1} \tag{20}$$

*Vs* and *Vp* are the volumes of the solvent and probe, respectively. The expression for *CGW* is given by

$$\mathbf{C}\_{GN} = \sigma \mathbf{C}\_0 \tag{21}$$

*C* in Eqn. (17) should be replaced with *CGW* obtained from Eqn. (21) for calculating the reorientation times with GW theory. When the ratio *Vs*/*Vp* is very small *CGW* reduces to unity and the SED equation with stick boundary condition is obtained.

#### **ii. The Dote-Kivelson-Schwartz theory (DKS)**

Although, the GW theory is successful in predicting the observed size effects in a qualitative way, it could not explain the faster rotation of nonpolar probes in alcohols compared alkanes. Hence, the second relatively new quasihydrodynamic theory, was proposed by Dote, Kivelson and Schwartz (DKS) in 1981. This theory not only takes into consideration the relative sizes of the solvent and the probe but also the cavities or free spaces created by the solvent around the probe molecule. If the size of the solute is comparable to the free volumes of the solvents, the coupling between the solute and the solvent will become weak which results in reduced friction experienced by the probe molecule. According to DKS theory the solute-solvent coupling parameter, *CDKS* is given by (Dote, Kivelson and Schwartz, 1981)

$$C\_{DKS} = \left(1 + \mathcal{Y} / \phi\right)^{-1} \tag{22}$$

where γ φ/ is the ratio of the free volume available for the solvent to the effective size of the solute molecule, with

$$\mathcal{Y} = \frac{\Delta V}{V\_p} \left[ 4 \left( \frac{V\_p}{V\_s} \right)^{2/3} + 1 \right] \tag{23}$$

and φ is the ratio of the reorientation time predicted by slip hydrodynamics to the stick prediction for the sphere of same volume. Δ*V* is the smallest volume of free space per solvent molecule and some discretion must be applied while calculating this term (Dutt et al., 1988; Anderton and Kauffman, 1994; Dutt and Rama Krishna, 2000). Δ*V* is empirically related to the solvent viscosity, the Hilderbrand-Batchinsky parameter *B* and the isothermal compressibility *kT* of the liquid by

Rotational Dynamics of Nonpolar and Dipolar

τ

where 0 ε , ε<sup>∞</sup> and

and large

solvation time,

Δμ

 and *<sup>f</sup> h*ν

relaxation time, 0 ( /) τ τε

*L* can be used in place of

rotational reorientation time can be expressed as

 ε

τ

*obs*

τ

where

where *<sup>a</sup> h*ν

Hence, τ

and

τ

τ*D*.

Molecules in Polar and Binary Solvent Mixtures 197

3 2 0 ( 2) ( )

ε ε

0

 τ

> *r* τ

∞ ∞

and high

τ

<sup>=</sup> (30)

*<sup>f</sup>* (31)

= + (28)

*<sup>D</sup>* are the zero frequency dielectric constant, high-frequency dielectric

0

 ε ετ

+

+ −

ε

*<sup>s</sup>*, and solute Stokes shift, *S*. According to this theory the dielectric friction is

(27)

) is given as the sum of

(29)

*<sup>D</sup>*. However, if the solute is

ε

 εε

∞ ∞ ∞

2 2

+ − <sup>=</sup> <sup>+</sup>

ε

*NZ*

constant and Debye relaxation time of the solvent, respectively.

Therefore, the observed rotational reorientation time ( *obs*

τ

contribution would be significant in a solvent of low

**ii. The van der Zwan-Hynes theory (vdZH)** 

given by (van der Zwan and Hynes, 1985)

*VfC*

= +

η

τ

μ

*obs*

*kT a kT*

μ

ττ

9 (2 )

If one assumes that the mechanical and dielectric components of friction are separable, then

*r SED DF*

reorientation time calculated using SED hydrodynamic theory and dielectric friction theory.

It is clear from the above equation that for a given solute molecule, the dielectric friction

large, the contribution due to dielectric friction becomes small and the relative contribution to the overall reorientation time further diminishes due to a step increase in the hydrodynamic contribution. Hence, most pronounced contribution due to dielectric friction could be seen in small molecules with large dipole moments especially in solvents of low

A semiempirical method for finding dielectric friction proposed by van der Zwan and Hynes (1985), an improvement over the Nee and Zwanzig model, provides a prescription for determining the dielectric friction from the measurements of response of the solute in the solvent of interest. It relates dielectric friction experienced by a solute in a solvent to

> 2 <sup>2</sup> ( ) 6 *<sup>s</sup> DF*

μ

Δμ

*Sh h* = − ν*a*

respectively. The solvation time is approximately related to the solvent longitudinal

Assuming the separability of the mechanical and dielectric friction components, the

*r s VfC hc kT kT*

2 <sup>2</sup> ( ) 6

μ

Δμ

*<sup>s</sup>* in Eqn. (30).

η

τ

*S kT*

τ

is the difference in dipole moment of the solute in the ground and excited states

ν

are the energies of the 0-0 transition for absorption and fluorescence,

*L D* = <sup>∞</sup> and is relatively independent of the solute properties.

Δν

 τ

= + (32)

2 2

ε

3 2 0 ( 2) ( )

ε ε

9 (2 ) *obs <sup>D</sup> <sup>r</sup>*

 τ

*DF <sup>D</sup> a kT*

$$
\Delta V = B k\_T \eta k T \tag{24}
$$

Since the Frenkel hole theory and the Hilderbrand treatment of solvent viscosity were developed for regular solutions (Anderton and Kauffman, 1994), Equation (24) may not be a valid measure of the free space per solvent molecule for associative solvents like alcohols and polyalcohols. Hence, for alcohols Δ*V* is calculated using

$$
\Delta V = V\_m - V\_s \tag{25}
$$

where *Vm* is the solvent molar volume divided by the Avogadro number.
