3485 4 r l l r : l 3 1 ' \ ' \ ' \ ' \ T Figure 4. Temperature dependencies of the normalized rotation relaxation time, 7iR at a given frequency; y = 2R'kTIw/p2. The temperatures TIL&) and T2*(L2) are the transition temperatures in eq 36 for the pores with different sizes, L, > L2; To = I , cI = 1, cb = 80, w i = 0.1, c. = 3. Curves 1 and 1' are obtained at R / & = IO; curves 2 and 2' at R / & = 4.interval around the transition point, T*(L), the dielectric friction and correspondingly the orientational relaxation time change from the value €6") to the value €6")' determined by eq 29b. The temperature and size dependencies of the rotational relaxation time at different values of parameters are shown in Figure 4. It should be emphasized, however, that eqs 36-39 present an example rather than an analysis of a real system. A different temperature dependence will be obtained using the same methodology but assuming different behaviors of A and T .In conclusion, we have studied the role of a boundary in modifying the relaxation behavior of a dipole embedded in a liquid. The formalism presented here which is based on the continuum approach extends previous works by introducing a nonlocal dielectric description of the liquid. Although the relaxation may be nonexponential in time we have calculated the dielectric friction for a given frequency as a function of zo, the distance of the dipole from the boundary. The effect of the boundary has been shown to be small unless the properties of the liquid itself are drastically changed due to the presence of the interface. The nonlocal nature of dielectric function modifies the dielectric friction derived within the local approximation and allows to introduce temperature dependence into the relaxation process through the characteristic length,
AppendixIn order to calculate the dielectric friction in the bulk liquid we will follow the cavity approach.12J6q'8 Consider a dielectric sphere of radius R and dielectric constant el at the center of which a dipole is located. Outside the sphere there is a liquid with dielectric function e(k,w), eq 1. For this model the electrostatic potential, 4, can be written by expanding the solution of the Laplace equation in spherical coordinates. The potential in the liquid, r > R , should vanish at infinite distance, and the homogeneous part of the potential inside the sphere, r < R , has to be analytic at the origin. The nonhomogeneous part of the potential in the sphere arises from the point dipole with the dipole moment p at the center. Thus we have
=('42) The coefficients A, B, and C are obtained from the boundary conditions at r = R which are similar to the conditions 8,9, and 10 used for the plane interface. As a result the field induced at the surface of dielectric sphere, r = R, has the form € I ) 4 R/ A) I I) (A3 1 withf(R/A) given by eq 28.A similar approach for the calculation of the dipole damping in the liquid (with the same boundary conditions) was adopted by van der Zwan and Hynes in ref 12. Our results, however, differ from theirs.A ...
