**2.1.3 Dielectric friction theories**

The simple description of hydrodynamic friction arising out of viscosity of the solvent becomes inadequate when the motion concerning rotations of polar and charged solutes desired to be explained. A polar molecule rotating in a polar solvent experiences hindrance due to dielectric friction ( DF *ζ* ), in addition to, the mechanical ( *ζ* <sup>M</sup> ) or hydrodynamic friction. In general, the dielectric and mechanical contributions to the friction are not separable as they are linked due to electrohydrodynamic coupling (Hubbard and Onsager, 1977; Hubbard, 1978; Dote et al., 1981; Felderhof, 1983; Alavi et al., 1991c; Kumar and Maroncelli, 2000). Despite this nonseparability, it is common to assume that the total friction experienced by the probe molecule is the sum of mechanical and dielectric friction components, i.e.,

$$
\mathcal{L}\_{Total} = \mathcal{L}\_M + \mathcal{L}\_{DF} \tag{26}
$$

Mechanical friction can be modeled using both hydrodynamic (Debye, 1929) and quasihydrodynamic (Gierer and Wirtz, 1953; Dote et al., 1981) theories, whereas, dielectric friction is modeled using continuum theories.

The earliest research into dielectric effects on molecular rotation took place in the theoretical arena. Initial investigations were closely intertwined with the theories of dielectric dispersion in pure solvents (Titulaer and Deutch, 1974; Bottcher and Bordewijk, 1978; Cole, 1984). Beginning with the first paper to relate the dielectric friction to rotational motion published by Nee and Zwanzig in 1970, a number of studies have made improvements to the Nee-Zwanzig approach (Tjai et al, 1974; Hubbard and Onsager, 1977; Hubbard and Wolynes, 1978; Bordewijk, 1980; McMahon, 1980; Brito and Bordewijk, 1980; Bossis, 1982; Madden and Kivelson, 1982; Felderhof, 1983; Nowak, 1983; van der Zwan and Hynes, 1985; Alavi et al, 1991a,b,c; Alavi and Waldeck, 1993). These have included the electrohydrodynamic treatment which explicitly considers the coupling between the hydrodynamic (viscous) damping and the dielectric friction components.

#### **i. The Nee-Zwanzig theory**

Though not the first, the most influential early treatment of rotational dielectric friction was made by Nee and Zwanzig (NZ) (1970). These authors examined rotational dynamics of the same solute/solvent model in the simple continuum (SC) description i.e., they assumed an Onsager type cavity dipole with dipole moment μ and radius *a* embedded in a dielectric continuum with dispersion ε(ω). Motion was assumed to be in the purely-diffusive (or Smoluchowski) limit. Using a boundary condition value calculation of the average reaction field, Nee and Zwanzig obtained their final result linking the dielectric friction contribution in the spherical cavity as

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

The simple description of hydrodynamic friction arising out of viscosity of the solvent becomes inadequate when the motion concerning rotations of polar and charged solutes desired to be explained. A polar molecule rotating in a polar solvent experiences hindrance due to dielectric friction ( DF *ζ* ), in addition to, the mechanical ( *ζ* <sup>M</sup> ) or hydrodynamic friction. In general, the dielectric and mechanical contributions to the friction are not separable as they are linked due to electrohydrodynamic coupling (Hubbard and Onsager, 1977; Hubbard, 1978; Dote et al., 1981; Felderhof, 1983; Alavi et al., 1991c; Kumar and Maroncelli, 2000). Despite this nonseparability, it is common to assume that the total friction experienced by the probe molecule is the sum of mechanical and dielectric friction

*Total M DF*

Mechanical friction can be modeled using both hydrodynamic (Debye, 1929) and quasihydrodynamic (Gierer and Wirtz, 1953; Dote et al., 1981) theories, whereas, dielectric

The earliest research into dielectric effects on molecular rotation took place in the theoretical arena. Initial investigations were closely intertwined with the theories of dielectric dispersion in pure solvents (Titulaer and Deutch, 1974; Bottcher and Bordewijk, 1978; Cole, 1984). Beginning with the first paper to relate the dielectric friction to rotational motion published by Nee and Zwanzig in 1970, a number of studies have made improvements to the Nee-Zwanzig approach (Tjai et al, 1974; Hubbard and Onsager, 1977; Hubbard and Wolynes, 1978; Bordewijk, 1980; McMahon, 1980; Brito and Bordewijk, 1980; Bossis, 1982; Madden and Kivelson, 1982; Felderhof, 1983; Nowak, 1983; van der Zwan and Hynes, 1985; Alavi et al, 1991a,b,c; Alavi and Waldeck, 1993). These have included the electrohydrodynamic treatment which explicitly considers the coupling between the

Though not the first, the most influential early treatment of rotational dielectric friction was made by Nee and Zwanzig (NZ) (1970). These authors examined rotational dynamics of the same solute/solvent model in the simple continuum (SC) description i.e., they assumed an

Smoluchowski) limit. Using a boundary condition value calculation of the average reaction field, Nee and Zwanzig obtained their final result linking the dielectric friction contribution

μ

 ζζ

ζ

hydrodynamic (viscous) damping and the dielectric friction components.

 η

*V* is calculated using

*V Bk kT* = *<sup>T</sup>* (24)

*VV V* = − *m s* (25)

= + (26)

and radius *a* embedded in a dielectric

). Motion was assumed to be in the purely-diffusive (or

Δ

Δ

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

Δ

and polyalcohols. Hence, for alcohols

**2.1.3 Dielectric friction theories** 

friction is modeled using continuum theories.

Onsager type cavity dipole with dipole moment

ε(ω

**i. The Nee-Zwanzig theory** 

continuum with dispersion

in the spherical cavity as

components, i.e.,

$$
\tau\_{\rm DF}^{\rm NZ} = \frac{\mu^2}{9a^3 kT} \frac{(\varepsilon\_\circ + 2)^2 (\varepsilon\_0 - \varepsilon\_\circ)}{\left(2\varepsilon\_0 + \varepsilon\_\circ\right)^2} \tau\_D \tag{27}
$$

where 0 ε , ε <sup>∞</sup> and τ *<sup>D</sup>* are the zero frequency dielectric constant, high-frequency dielectric constant and Debye relaxation time of the solvent, respectively.

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

$$
\pi\_r^{\alpha \text{obs}} = \pi\_{\text{SED}} + \pi\_{\text{DF}} \tag{28}
$$

Therefore, the observed rotational reorientation time ( *obs r* τ ) is given as the sum of reorientation time calculated using SED hydrodynamic theory and dielectric friction theory.

$$
\sigma\_r^{\text{obs}} = \frac{\eta V \mathcal{f} \mathcal{C}}{kT} + \frac{\mu^2}{9a^3 kT} \frac{(\varepsilon + 2)^2 (\varepsilon\_0 - \varepsilon\_\circ) \tau\_D}{\left(2\varepsilon\_0 + \varepsilon\_\circ\right)^2} \tag{29}
$$

It is clear from the above equation that for a given solute molecule, the dielectric friction contribution would be significant in a solvent of low ε and high τ*<sup>D</sup>*. However, if the solute is 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 ε and large τ*D*.

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

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 solvation time, τ*<sup>s</sup>*, and solute Stokes shift, *S*. According to this theory the dielectric friction is given by (van der Zwan and Hynes, 1985)

$$
\sigma\_{\rm DF} = \frac{\mu^2}{\left(4\mu\right)^2} \frac{S\tau\_s}{6kT} \tag{30}
$$

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

$$S = h\nu\_a - h\nu\_f\tag{31}$$

where *<sup>a</sup> h*ν and *<sup>f</sup> h*ν are the energies of the 0-0 transition for absorption and fluorescence, respectively. The solvation time is approximately related to the solvent longitudinal relaxation time, 0 ( /) τ τε ε *L D* = <sup>∞</sup> and is relatively independent of the solute properties. Hence, τ*L* can be used in place of τ*<sup>s</sup>* in Eqn. (30).

Assuming the separability of the mechanical and dielectric friction components, the rotational reorientation time can be expressed as

$$
\tau\_r^{obs} = \frac{\eta V \mathcal{f} \mathcal{C}}{kT} + \frac{\mu^2}{\left(\Delta\mu\right)^2} \frac{\hbar c}{6kT} \mathcal{L}\_s \tag{32}
$$

Rotational Dynamics of Nonpolar and Dipolar

**3.1b Time-resolved fluorescence measurements** 

polarizers are kept horizontal.

(Lakowicz, 1983)

fluorescence lifetime.

valid for stick boundary condition.

**3.2 Fluorescent probes used in the study** 

Kauffman, 1994)

**Nonpolar probes** 

subslip trend in alcohols.

where ρ

Molecules in Polar and Binary Solvent Mixtures 199

where *HV I* is the fluorescence intensity when the excitation polarizer is kept horizontal and the emission polarizer vertical and *IHH* is the fluorescence intensity when both the

The fluorescence lifetimes of all the probes were measured with time correlated single photon counting technique (TCSPC) using equipment described in detail elsewhere (Selvaraju and Ramamurthy, 2004). If the decay of the fluorescence and the decay of the

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

τ

*<sup>r</sup> r r*

where *r0* is the limiting anisotropy when all the rotational motions are frozen and

<sup>=</sup> < >−

In case of a prolate-ellipsoid model, the parameter *stick f* is given by (Anderton and

2 2 32 2 2 12 2 12

is the ratio of major axis (*a*) to the minor axis (*b*) of the ellipsoid. This expression is

*/*

2( 1) ( 1) 3 [( 2 1) ln{ ( 1) } ( 1) ]

*stick / /*

 *ρ ρ ρ ρρ* + − <sup>=</sup> − +− − −

A variety of the nonpolar fluorescent probe molecules have been studied extensively in the recent past. Most of the nonpolar probes so far studied have the radii of 2.5 Å to 5.6 Å (Inamdar et al., 2006) and a transition towards stick boundary condition is evident with increase in size of the solute. Most of the medium sized neutral nonpolar molecules rotate faster in alcohols compared to alkanes, which is in contrast to that of smaller neutral solutes. It is also noted that the quasihydrodynamic description is adequate for small solutes of 2-3 Å radius in case of GW theory whereas, the DKS model with experimental value in alcohols fail beyond the solute radius of 4.2 Å. Our earlier work on rotational dynamics of exalite probes E392A (*r* = 5.3 Å), and E398 (*r* = 6.0 Å), yielded striking results (Inamdar et al., 2006), in that, these large probes rotated much faster than slip hydrodynamics and followed

The quest to understand the influence of size of solute on rotational dynamics is continued with three nonpolar solutes viz., Exalite 404 (E404), Exalite 417 (E417) and Exalite 428 (E428), which may further fill the gap between the existing data. These probes have an anistropic shape and a dipole emission along their long rod-like backbones. The rod like or cylinder shape is a macromolecular model of great relevance. A number of biopolymers including

anisotropy are represented by single exponential, then the reorientation time

τ

*ρ ρ <sup>f</sup>*

ρ

*<sup>I</sup> <sup>G</sup>*

*HV HH*

*<sup>I</sup>* <sup>=</sup> (36)

τ

(37)

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

τ*<sup>f</sup>* is the

(38)

where the first term represents the mechanical contribution and the second the dielectric contribution.
