Molecular Relaxation Times from Time-variations of Dispersion and Absorption of Third-order Electric Polarisation in Liquids BY WEADYSEAW JANUSZ BUCHERT ALEXIEWICZ AND STANMAWKIELICH Nonlinear Optics Division Institute of Physics A. Mickiewicz University 60-780 Poznan Poland Received 2nd August 1976 Processes involving an increase in nonlinear 3rd order electric polarisation in dipolar liquids by the action of external electric fields are considered in the approximation of Debye’s rotational dif- fusion model of geometrically spherical molecules. For times comparable with the Debye times of molecular orientation we show that the dispersion and absorption of the above effects are described by the well known Debye factors &(a)and supplementary factors Rlk(m),and that the setting in of steady state polarisation depends on the relaxation times rl z2.We perform a detailed analysis of the case of quadratic changes in electric permittivity in a static electric field. Recently a classical theory has appeared concerning nonlinear processes of mole- cular relaxation in strong electric fields of both low and high frequencies and describing the dispersion and absorption of third-order polarisation of liquid dielectrics in terms of Debye molecular rotational diffusion and Lorentz-Voigt electron dispersion.1*2 For an experimental system in which three electric fields with the frequencies col 02, cog are applied to the dielectric in the same direction i.e. along the z-axis the nonlinear electric polarisation of third order P(2)(co4),is given generally at the fre- quency co4 = col + co2 + co3 by where x is the scalar nonlinear third-order susceptibility of the isotropic dielectric; the dependence of x on the electric field frequencies is considered in ref.(3). The time-variable electric field is given by Ez(t)= 2 Ez(coa)exp(-icoat). a=f(1,2,3) Here by taking into consideration the dependence of the susceptibility tensor x on time we extend the theory to processes of increase and decrease in nonlinear polarisa- tion caused by the imposition and removal of electric fields. In this way we obtain a description of the dynamics of numerous nonlinear processes,such as third-harmcnic generation with the nonlinear susceptibility x(-301 u) co u);t); second-harmonic generation in the presence of a static electric field x(-20 co co 0; t); nonlinear rec- tification of optical and electric frequencies x(-0 co -co 0; t); self-induced changes in electric susceptibility x(-co co co -co; t); and quadratic changes in electric permittivity AE(c~), t)/E2-x(-co co 0 0; t) in a strong static electric field.MOLECULAR RELAXATION TIMES THEORETICAL Consider an isotropic dielectric the N noninteracting axially symmetric molecules of which possess a permanent dipole moment m parallel to the symmetry z-axis the tensor components a, = a, # a, of linear electric polarisability and the tensor components b,,, bxxp= by, of electric hyperpolarisability. We are interested in the time-variations of polarisation due to the switching on of AC electric fields.We calculate these time-variations by determining the probability distribution function of orientation f (6 t) from Debye’s rotational diffusion equation with oJn-the angular part of the spatial derivative operator D -the rotational diffusion coefficient of a molecule with spherical geometry and the potential energy of an axially symmetric molecule in the external field E,(t). Above a = +(a, + 2axx)is the mean linear polarisability of the molecule; y = a, -axxthe anisotropy of its linear polarisability; b = (b,, + 2bx,,)/3 its mean hyperpolarisability; and K = b,, -3bxx,the anisotropy of its hyperpolarisability. The Pl(6)are Legendre polynomials dependent on the polar angle 6. Eqn (3) can be solved by statistical perturbation calculus applying the highly useful method of Racah algebra.5 To this aim we have recourse to the following series expansion in Legendre polynomials and successive powers of 1/RT,the reciprocal of the energy of thermal motion of the molecules The expansion (5) on insertion into eqn (3) yields a set of differential equations of the form -3 2 us(t)&?-’)(t)[1’(Z’ + 1)-Z(l+ 1) -s(s + 1)]r I’ ‘7.(6) l’,s 000 Eqn (6) enables us to find the explicit form of the dynamical coefficients A(?)(t).7 Above denote the Clebsch-Gordan coefficients ; summation over the indices J’ s is rGtricte2 by the triangle ineq~ality.~ In the zero’th approximation of perturbation calculus we obtain from (6) the equation of free rotational diffusion of spherical top molecules with the well known solution WLADYSgAW ALEXIEWICZ ET AL.the zl being rotation relaxation times related to the rotational diffusion coefficient D as follows where zl is Debye's relaxation time. For n > 1 eqn (6) has the solution A(y)(t)= -Dexp(-f/q)21yr + 1) -1(1+ 1) -s(s + 1) S,lt 2 I)(?-I)( t)us( t)exp( + t/zz)dt. ( 10) If integration in (10) is performed from 0 to a,the distribution function thus obtained corresponds to the steady state attained by the liquid dielectric after a sufficiently long time subsequent to the moment at which the external electric fields were switched on or removed. The case in question has been dealt with in full detail in ref. (1)-(3). Standard methods of statistical averaging with the distribution function (5) and dynamical coefficients (10) lead to the result that the increase in polarisation due to the application of the AC electric fields at the moment of time t = 0 is described with accuracy to ES(t) by the temperature-dependent contributions to the tensor of 3rd order susceptibility where MOLECULAR RELAXATION TIMES The latter at k = 0 go over into the Debye factors Rl(Co,bJ = (1 -imabczt)-l dis- cussed in ref.(1)-(4) which define the dispersion and absorption of the effects in steady states (for t = a). APPLICATIONS AND CONCLUSIONS The preceding results provide a coherent description of time-dependent nonlinear relaxation effects in the approximation of Debye rotational diffusion taking into ac- count processes of time-dependent growth of polarisation from the moment of applica- tion of external fields onwards.This is of especial significance in relation to measure- ments of the dielectric and optical properties of molecules by pulse techniques when the times necessary for switching the field on can be comparable with the Debye rotational relaxation times and when difference frequencies between various laser modes can cause nonlinear Debye dispersion in the optical range. Future dielectric studies of the relaxation times of molecules need not be restricted to dispersion and absorption measurements of nonlinear effects but can extend to determinations of the time-variations of dispersion and absorption. For example we have the following formulae for the quadratic variations in electric permittivity in the presence of a strong static electric field (for the sake of clarity we omit the fre- quency-dependence of the molecular parameters assuming electron dispersion to be absent) We have thus resolved each of the temperature-dependent contributions into a part describing the steady state (t = 00) and parts which decrease with z1 and r2and describe WEADYSKAW ALEXIEWICZ ET AL.the increase in polarisation of the medium. The latter are characterised by the oc- currence in addition to factors oscillating with the frequency of the measuring field of non-oscillating factors decreasing in time exponentially and non-exponentially with dispersion given by the function RIk(co). Other effects described by the susceptibility x,also involve these non-oscillating factors e.g.the effects of rectification of dielectric and optical frequencies of the emergence of constant polarisation of isotropic bodies under illumination and of dielectric saturation. The method used by us for the determination of the distribution function from Debye’s kinetic diffusion equation is well adapted to extension by taking intermolecu- lar interactions into a~count~*~ as well as to application within the framework of the more general response function method.lG12 The distribution function determined from eqn (5) can serve as well for calcula- tions of orientation time-autocorrelation tensors in effects of linear and nonlinear light scatteringl3 and for spectral line shape determinations in elastic and inelastic scattering of light by molecules of liquids oriented by the action of an external electric A similar description can be used for pulse electric field-induced effects of birefringence in liquid^.^-^^^^^ B.Kasprowicz-Kielich and S. Kielich Adv. MoZ. ReZaxation Proc. 1975 7 275. B Kasprowicz-Kielich S. Kielich and J. R. Lalanne MoZecuZar Motions in Liquids ed. J. Lascombe (D. Reidel Dordrecht Holland 1974) p. 563. J. Buchert B. 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