Cohen and Turnbull

Cohen and Turnbull [20,21] laid down the foundation for the free volume concept in modeling self-diffusion in simple van der Waals liquids. They considered that the volume in a liquid is composed of two parts, the actual volume occupied by the liquid molecules and the free volume surrounding these molecules opened up by thermal fluctuation. Increasing temperature increases only the free volume and not the occupied volume. The average free volume per molecule, vf, can be defined as... 

Figure 2 The temperature dependence of the self-diffusion coefficient of 2,3-dimethyl-butane predicted by Cohen and Turnbull s free volume model. (From Ref. 25.)...

Numerous models have been proposed to interpret pore diffusion through polymer networks. The most successful and most widely used model has been that of Yasuda and coworkers [191,192], This theory has its roots in the free volume theory of Cohen and Turnbull [193] for the diffusion of hard spheres in a liquid. According to Yasuda and coworkers, the diffusion coefficient is proportional to exp(-Vj/Vf), where Vs is the characteristic volume of the solute and Vf is the free volume within the gel. Since Vf is assumed to be linearly related to the volume fraction of solvent inside the gel, the following expression is derived ... 

Generally, the VTF behaviour of all transport properties may be understood from the free volume concept introduced by Doolittle (1951) and further developed by Cohen and Turnbull (1959). Essentially, any diffusing species is depicted as encaged by the nearest atoms in a cell... 

The jamming effect, i.e., the slowing down of the longitudinal diffusion of a polymer chain by the head-on collision with other chains, can be treated by a model similar to that proposed by Cohen and Turnbull [112] for self diffusion of small molecules in a fluid. This model assumes that if at least one surrounding polymer chain exists within the critical hole ahead of a test chain, both collide, and this prevents the test chain from diffusing longitudinally. With this assumption, we express the longitudinal diffusion coefficient Dp of the test chain as... 

This section closely follows Cohen and Turnbull s original derivation [1], The original paper should be consulted for further details. 

Under suitable assumptions Cohen and Turnbull (1959) have shown for a liquid composed of identical molecules that the total probability P(v ) of finding a free volume exceeding a given value v is represented... 

Cohen and Turnbull [87] generalized somewhat the theoretical concepts of the relationship between diffusion and self-diffusion of liquids modelled by assemblies of rigid spheres and obtained on the basis of the theories of Frenkel and Eyring, Fox and Flory [88] and Williams, Landell and Ferry [89] the equation ... 

In a later work Cohen and Turnbull [90] defined the free volume of liquid as a part of the thermal expansion being freely redistributed throughout the volume without any change in energy. Macedo and Litovitz [91] pointed out that Cohen-TurnbulTs equation (74) ensues from an already known equation obtained independently by Fulcher [92] and Tamman [93] ... 

Mathematical treatment of molten salts that supercool was first carried out by Cohen and Turnbull. The principal idea of the hole theory—that diffusion involves ions that wait for a void to turn up before jumping into it—is maintained. However, Cohen and Turnbull introduced into their model a property called thefree volume, Vf. What is meant by this free volume It is the amount of space in addition to that, Vq, filled by matter in a closely packed liquid. Cohen and Turnbull proposed that the free volume is linearly related to temperature... 

To express the probability that the free volume occasionally opens up to form a hole, Cohen and Turnbull first defined a factor y, which allows for the partial filling of the expanded free volume to the size of a hole. It can vary between 1/2 < y < 1. A value of y = 1 means that the holes are empty, and a value of y = 1/2 that they are half filled. 

Cohen and Turnbull s model is oriented to liquids that form glasses. At the glass transition temperature (i.e., at T= Tq), the diffusion coefficient becomes zero, which is a rational consequence of what is thought to be going on the supercooled liquid finally becomes a glass in which D is effectively zero. 

A difficulty might face the worker who wishes to apply Cohen and Turnbull s theory to transport phenomena in molten salts not only near the glass transition temperature but also above the normal melting point (see Section 5.6.2.2). Experimental evidence shows that the heat of activation of diffusion and of conductance for viscous flows is related to the normal melting point of the substance concerned... 

This result increases the credibility of the Cohen and Turnbull view the meaning ofy = 0.2 is that the hole is 20% full, and this seems consistent with the picture of an ion-sized space filled 20% of the time with an ion. 

A similar theoretical basis for such a relationship can be deduced from the work of Cohen and Turnbull [94], which introduces the concept of a dynamic free volume with a low activation energy that is created and into which solute ions move to and fro. The model is valid when solute movement... 

The equilibrium distribution p (uj) = hard sphere approximation), or e(u,) was linear in (u< — Vf) in the region of interest. Either assumption leads to the result (passing to the continuum limit for Uj)... 

The behavior of the WLF parameters can be explained using the free-volume concept. The two WLF parameters are described in the following form according to the free-volume theory developed by Cohen and Turnbull [105,106] ... 

In equation (3.01), A and B are empirical constants, Vocc is the volume occupied by the constituent particles and v/ is the free volume. In equation (3.02), r]a, C and To are constants. VTF equation implies that viscosities of glass forming supercooled liquids are non-Arrhenius and To is the temperature which linearizes the data of the non-Arrhenius plot. Cohen and Turnbull (Cohen and Turnbull, 1959 Turnbull and Cohen, 1961,... 

The free-volume models reviewed here and in a later section are based on Cohen and Turnbull s theory (18) for diffusion in a hard-sphere liquid. These investigators argue that the total free volume is a sum of two contributions. One arises from molecular vibrations and cannot be redistributed without a large energy change, and the second is in the form of discontinuous voids. Diffusion in such a liquid is not due to a thermal activation process, as it is taken to be in the molecular models, but is assumed to result from a redistribution of free-volume voids caused by random fluctuations in local density. 

The free-volume model proposed by Vrentas and Duda (67-69) is based on the models of Cohen and Turnbull and of Fujita, while utilizing Bearman s (7j0) relation between the mutual diffusion coefficient and the friction coefficient as well as the entanglement theory of Bueche (71) and Flory s (72) thermodynamic theory. The formulation of Vrentas and Duda relaxes the assumptions deemed responsible for the deficiencies of Fujita s model. Among the latter is the assumption that the molecular weight of that part of the polymer chain involved in a unit "jump" of a penetrant molecule is equal to the... 

The Cohen and Turnbull free-volume model [30] assumes a liquid composed of hard-sphere molecules and voids in which diffusion occurs whenever a void larger than some minimum volume V forms in the body of the liquid and a molecule jumps into it. The equation for the diffusion coefficient is... 

Cohen and Turnbull s critical free-volume fluctuations picture of selfdiffusion in dense liquids is similar to the vacancy model of self-diffusion in crystals. However, in crystals individual vacancies exist and retain their identity over long periods of time, whereas in liquids the corresponding voids are ephemeral. The free volume is distributed statistically so that at any given instance there is a certain concentration of molecule-sized voids in the liquid. However, each such void is short-lived, being created and dying in continual free-volume fluctuations. The Frenkel hole theory of liquids ignores this ephemeral, statistical character of the free volume. 

Our result for v>v is essentially identical to that derived earlier by Cohen and Turnbull for the most probable distribution of free volume X, P x) = y/Vfexp( — yx/Vf), where Vy is the free volume averaged over all cells, Vy —pvy, and y is a numerical factor between and 1 introduced to correct for overlap of the volume between neighboring cells. Comparing the exponent yx/vy =(y/p)x/vy whh the exponent in (6.10b), (v-v )/vy, we see that the two distributions are identical if x is taken as (c —u ) and y is taken as p, which would be close to j in the temperature region considered in Ref. 88. 

The original work on the free-volume model by Cohen and Turnbull - showed that the fluidity obeyed the Doolittle equation (2.8). We show that the percolation ideas developed in this paper give rise to the same equation for the fluidity. 

Time-temperature superpositioning was originally derived from free volume models, which assume that the rates of molecular motions are governed by the available unoccupied space. Early studies of molecular liquids led to the Doolittle equation, relating the viscosity to the fractional free volume, f =V /(V - Vo), where V is the specific volume and Vo is the occupied volume normalized by the mass) (Doolittle and Doolittle, 1957 Cohen and Turnbull,... 

A well-known and simple theory for describing molecular transport in a liquid is the free-volume theory of Cohen and Turnbull [1959, 1970]. Employing statistical mechanics, these authors showed that the most probable size distribution of the free volume per molecule in a hard sphere liquid may be described by an exponential decreasing function. It was assumed that diffusion of the hard-spheres can only take place when, due to thermal fluctuations, holes are formed whose size is greater than a critical volume. When applying this theory to a structural relaxation process in a liquid, its (circular) frequency o) = r = 2jtv is expressed by... 

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