Free Volume Theory of Viscosity

The Doolittle free volume equationη=A exp (Bv0/vf), (1)which was first proposed for the viscosity of n-alkanes, is widely accepted as the viscosity equation for various kinds of liquids and polymeric substances. However, Bueche modified it according to a normal coordinate computation for the thermal motion of a simple array of molecules connected by Hookian springs, and derived the following equationη=A exp{B(v0/vf)2}. (2)The purpose of the present study is to clarify the theoretical background for the free volume representation of the liquid viscosity from the view-point of the thermal fluctuation theory.The probability that the thermal fluctuations of energy and volume will amount to ΔE and ΔV respectively in a small, but macroscopic system which is a part of a large macroscopic system, is denoted by prob. (ΔE, ΔV). By a simple thermodynamic computation, it is given byProb. (ΔE, ΔV)∝exp[-1/2kT0{1/CvT(ΔE)2+(p2/CvT-2αp/CvκT+Cp/CvVκT)(ΔV)2+(2p/CvT-2α/CvκT)(ΔE)(ΔV)}]. (3)in which Cv and Cp are the specific heats at constant volume and constant pressure respectively, α the thermal expansion coefficient, and κT the isothermal compressibility. Then, the probability that the volume fluctuation is larger than a certain critical value ΔV* and the energy fluctuation is within the range of average root mean square isProb. (ΔV>ΔV*, ΔE_??_√(ΔE)2)≅1/√2πγ√(ΔV)2e/ΔV*-γ/2(ΔV*)2/(ΔV)2, (4)where γ is the ratio of the specific heat.We consider a liquid system which is composed of N rigid sphere molecules in the potential field of square-well type, χ(V)=-(V0/V)χ0, where V0 is an occupied volume and χ0=-χ(V0). According to a simple statistical mechanical computation the mean square value (ΔV)2 will run as:(ΔV)2=-kT(∂V/∂p)T≅1/N(Vf)2. (5)in which Vf is the free volume, V-V0. After further computation the following viscosity-free volume equation is obtained:γ/2(Δv*/vf)2 γ/2(vv/vf)2η=Ae =Ae, (6)where the critical volume required for the shift of a molecule, Δv*, is assumed to be nearly equal to the occupied volume per molecule, v0.Equation (6) shows that logη is proportional to the square of the reciprocal of the free volume fraction, and this relation is the same as that obtained from Bueche's normal coordinate theory. The value of the parameter B in equation (2) is (γ/2) in the author's equation, and differs from the value of Bueche, (9/π).In conclusion, the modified Doolittle equation, η=A exp{B(v0/vf)2}derived from the thermal fluctuation theory is in accordance with Bueche's result obtained from the normal coordinate theory, and against Cohen and Turnbull's theory.