Thermodynamic properties derived from the free volume model of liquids
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T H E Free Volume Model, developed by Turnbull and Cohen, 1'~ has been recognized for some time as a valuable model for the description of the transport properties of dense liquids. It has also been applied with success to the problem of describing solidification phenomena, particularly the liquid-glass transition, s The model itself is appealing because of the clearness of its physical concepts and the relative simplicity of its mathematical formalism. For these reasons it is natural to hope that the Free Volume Model could provide reasonably accurate predictions of the thermodynamic properties of liquids near the solidification point. It would then more completely describe the liquid state in a thermodynamic region which is v e r y difficult to understand theoretically. A f i r s t step in this direction was taken by Turnbull 4 when he suggested the form that the Free Volume Model partition function should take Z = k-~Nfe-k~T dSNx ~ {(V/IX s) exp -- [dp(r)IkT]} N.
ill Here k is the thermal wavelength and ~(r) is the Lennard-ffones Intermolecular potentialfunction for an average cell of the liquid
qb(r) = 4e is~r) '2 - (a/r)6].
[2]
Although R in Eq. [1] is the separation of any two molecules, r is the diameter of an average cell or, viewed differently, is the average intermolecular separation. It Is shown that r is related to the average specific volume of the liquid. The free volume, v/, has been defined in different ways b y d i f f e r e n t authors, s,e and Turnbull and Cohen have modified their definition of vf as their model evolved. ~ In the present work, the free volume is defined as the difference between the specific volume per molecule, v = V / N and the actual volume of the m01eRONALDI. MILLERis a Physicist with Nuclear & Space Physics Group, BoeingAerospaceCompany,Huntsville, Ala. 35807. Manuscript submitted June 28, 1973. METALLURGICALTRANSACTIONS
cule, v o = -~ ~d 3, where V is the total volume and d is the molecular diameter.
DERIVATION OF THE EQUATION OF STATE With all terms in [i] defined explicitly,the derivation of the equation of state proceeds according to the formula given by statisticalmechanics 8
P = kT[O(lnZ)/aV]T
[3]
Substitution of ill in [3] yields immediately P V = NkT/(I - vo/v) - NV(a@/gV)T
[4]
Since the terms on the right-hand side of [4] which involve the total volume are ratios, they can be written in t e r m s of the specific volume and the molecular volume co. The derivative of the tntermolecular potential may then be expressed as
(adp/aV)T
=
(a~)/ar) ( a r / W ) T
[5]
The first factor in [5] is easily obtained from [2]. In order to evaluate the second factor, a relationship between the inter-molecular separation, r, and the specific volume must be assumed. This relationship will depend on the shape of the cell in which a particular molecule finds itself, where the cell is defined as the region of space to which the molecule is confined by its nearest neighbors. The cell shape can be expected to vary from molecule to molecule in a random manner, and to change wi
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