Liquid free volume

Free-volume models follow the idea that on cooling the supercooled liquid, free volume progressively reduces,and hence mobility becomes impossible. The mobility of an atom (or more generally a structural unit) is supposed to be possible only when a free volume which can contain the atom exists next to it. We shall consider in the following the average free volume per atom. Let us call v the minimum volume of a cavity that can contain the atom or the mobile structural unit. Diffusion of the atom will depend on the free volume size Vf per atom. Hence either... 

For the evaporation process we mentioned above, the thermodynamic probability of the gas phase is given by the number of places a molecule can occupy in the vapor. This, in turn, is proportional to the volume of the gas (subscript g) 12- oc V In the last chapter we discussed the free volume in a liquid. The total free volume in a liquid is a measure of places for molecules to occupy in the liquid. The thermodynamic probability of a liquid (subscript 1) is thus V, oc V, frgg. Based on these ideas, the entropy of the evaporation process can be written as... 

Neither the volume occupied by a mole of gas at the boiling point nor the free volume of a liquid vary too widely from substance to substance. Taking the former to be about 30,000 ml and the latter to be about 3 ml gives... 

There are two ways in which the volume occupied by a sample can influence the Gibbs free energy of the system. One of these involves the average distance of separation between the molecules and therefore influences G through the energetics of molecular interactions. The second volume effect on G arises from the contribution of free-volume considerations. In Chap. 2 we described the molecular texture of the liquid state in terms of a model which allowed for vacancies or holes. The number and size of the holes influence G through entropy considerations. Each of these volume effects varies differently with changing temperature and each behaves differently on opposite sides of Tg. We shall call free volume that volume which makes the second type of contribution to G. 

Tetrakis(cumylphenoxy)phthalocyanine (5) a PC derivative, having liquid—crystalline-like substituents (38—43) was studied because the cross-section area of the substituents is much larger than that of a normal alkyl chain, and therefore, the requirement of minimized free volume in the assembly may be easier to accomplish. 

The permachor method is an empirical method for predicting the permeabiUties of oxygen, nitrogen, and carbon dioxide in polymers (29). In this method a numerical value is assigned to each constituent part of the polymer. An average number is derived for the polymer, and a simple equation converts the value into a permeabiUty. This method has been shown to be related to the cohesive energy density and the free volume of the polymer (2). The model has been modified to liquid permeation with some success. 

Matthews-Akgerman The free-volume approach of Hildebrand was shown to be valid for binary, dilute liquid paraffin mixtures (as well as self-diffusion), consisting of solutes from Cg to Cig and solvents of Cg and C o- The term they referred to as the diffusion volume was simply correlated with the critical volume, as = 0.308 V. We can infer from Table 5-15 that this is approximately related to the volume at the melting point as = 0.945 V, . Their correlation was vahd for diffusion of linear alkanes at temperatures up to 300°C and pressures up to 3.45 MPa. Matthews et al. and Erkey and Akger-man completea similar studies of diffusion of alkanes, restricted to /1-hexadecane and /i-octane, respectively, as the solvents. 

Fig. 22.8. (a) The volume change when a simple melt (like a liquid metal) crystallises defines the melting point, T (b) the spread of molecular weights blurs the melting point when polymers crystallise ( ) when a polymer solidifies to a gloss the melting point disappears completely, but a new temperature at which the free volume disappears (the gloss temperature, TJ con be defined and measured. 

As the temperature is decreased, free-volume is lost. If the molecular shape or cross-linking prevent crystallisation, then the liquid structure is retained, and free-volume is not all lost immediately (Fig. 22.8c). As with the melt, flow can still occur, though naturally it is more difficult, so the viscosity increases. As the polymer is cooled further, more free volume is lost. There comes a point at which the volume, though sufficient to contain the molecules, is too small to allow them to move and rearrange. All the free volume is gone, and the curve of specific volume flattens out (Fig. 22.8c). This is the glass transition temperature, T . Below this temperature the polymer is a glass. 

L. L. Blyler and T. K. Kwei [39] proposed the direct opposite (to 4). In their reasoning, they proceeded from the known and generally acceptable Doolittle equation, which puts liquid viscosity in exponential dependence on the inverse value of the free volume of the latter. According to [39], gas has a volume of its own, the value of which it contributes to the free volume of the polymer when it dissolves therein as a result, viscosity falls. The theoretical formula obtained by the authors was experimentally confirmed in the same work. The authors measured pressure values at the entrance of cylindrical capillaries, through which melts of both pure polyethylene, and polyethylene with gas dissolved in it, extruded at a constant rate. 

Experimental data on nitrogen obtained from spin-lattice relaxation time (Ti) in [71] also show that tj is monotonically reduced with condensation. Furthermore, when a gas turns into a liquid or when a liquid changes to the solid state, no breaks occur (Fig. 1.17). The change in density within the temperature interval under analysis is also shown in Fig. 1.17 for comparison. It cannot be ruled out that condensation of the medium results in increase in rotational relaxation rate primarily due to decrease in free volume. In the rigid sphere model used in [72] for nitrogen, this phenomenon is taken into account by introducing the factor g(ri) into the angular momentum relaxation rate... 

Any polymer contains some inner free space free volume distributed in a dynamic manner between its molecular chains (see Section 23.2). When it is exposed to a fluid (liquid or gas) the physical possibility exists for fluid absorption by the polymer, if the fluid molecules or atoms are small enough to fit into local regions of this distributed space during kinetic movements. As this happens, subsequent kinetic chain motion must allow for the newly absorbed fluid molecules and, hence, the polymer s overall volume will adjust accordingly this action will coincide with the formation of more free space around these fluid molecules—so the polymer will swell a little. This process will be continued until an equilibrium is reached ( equilibrium swelling ), by which time the extent of swelling can be considerable. The amount of fluid taken up and the rate at which this happens are both important, and are discussed in this and following sections. 

Glass transition temperature (Tg) Swelling will be reduced in an elastomer with a high Tg, that is, in an elastomer with relatively little free volume available to absorb liquid. 

According to free-volume interpretations, the rate of molecular motions is governed entirely by the available unoccupied space ( free volume ). Early studies of molecular liquids led to the Doolittle equation, relating the viscosity to the fractional free volume, / [23,24]... 

A more rigorous free-volume treatment is due to Cohen and Grest (CG) [34,35], according to which the material is comprised of liquid and solid-like cells. The former have free volume, but mobility requires continuity of the local empty space. The temperature dependence of the relaxation times according to the CG model is... 

Generally, the values of the scaling exponent are smaller for polymers than for molecular liquids, for which 3.2 < y < 8.5. A larger y, or steeper repulsive potential, implies greater influence of jamming on the dynamics. The smaller exponent found for polymers in comparison with small-molecule liquids means that volume effects are weaker for polymers, which is ironic given their central role in the historical development of free-volume models. The reason why y is smaller... 

To evaluate the time-dependent function, X(t), a simple model of diffusion is proposed. Starting from Langmuir adsorption theory, we consider that liquid molecules having diffused into the elastomer are localized on discrete sites (which might be free volume domains). In these conditions, we can deduce the rate of occupation of these sites by TCP with time. Only the filhng of the first layer of the sites situated below the liquid/solid interface at a distance of the order of the length of intermolecular interaction, i.e., a few nanometers, needs to be considered to estimate X(t). 

The free volume is considered to represent the difference between the actual volume of the liquid (or the amorphous polymer) and the minimum volume which it would occupy if its molecules were packed firmly in contact with each other. Incompressible molecules with rigid dimensions are implied in this definition of a free volume. The unrealistic nature of this implication undermines precise determination, or even an exact definition, of the free volume. The concept has proved useful nevertheless. 

The next most important mechanism affecting the surface tension at a single component simple fluid gas-liquid interface is, we believe, associated with the nonlocality of the repulsive interaetions. To account for this mechanism, observe that it enters by way of the exeluded volume effeet. In the coarse-grained GvdW(S) theory above, the free volume faetor /(r) is given by... 

In polymer electrolytes (even prevailingly crystalline), most of ions are transported via the mobile amorphous regions. The ion conduction should therefore be related to viscoelastic properties of the polymeric host and described by models analogous to that for ion transport in liquids. These include either the free volume model or the configurational entropy model . The former is based on the assumption that thermal fluctuations of the polymer skeleton open occasionally free volumes into which the ionic (or other) species can migrate. For classical liquid electrolytes, the free volume per molecule, vf, is defined as ... 

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... 

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 ... 

In the molten state polymers are viscoelastic that is they exhibit properties that are a combination of viscous and elastic components. The viscoelastic properties of molten polymers are non-Newtonian, i.e., their measured properties change as a function of the rate at which they are probed. (We discussed the non-Newtonian behavior of molten polymers in Chapter 6.) Thus, if we wait long enough, a lump of molten polyethylene will spread out under its own weight, i.e., it behaves as a viscous liquid under conditions of slow flow. However, if we take the same lump of molten polymer and throw it against a solid surface it will bounce, i.e., it behaves as an elastic solid under conditions of high speed deformation. As a molten polymer cools, the thermal agitation of its molecules decreases, which reduces its free volume. The net result is an increase in its viscosity, while the elastic component of its behavior becomes more prominent. At some temperature it ceases to behave primarily as a viscous liquid and takes on the properties of a rubbery amorphous solid. There is no well defined demarcation between a polymer in its molten and rubbery amorphous states. 

Pratt, L. R., Hummer, G., and Garde, S. (1999). Theories of hydrophobic effects and the description of free volume in complex liquids. In New Approaches to Problems in Liquid State Theory (C. Caccamo, J.-P., Hansen, and G. Stell, eds.), vol. 529, pp. 407-420. Kluwer, Netherlands. NATO Science Series. 

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