Athermalization

The energy of interaction between a pair of solvent molecules, a pair of solute molecules, and a solvent-solute pair must be the same so that the criterion that = 0 is met. Such a mixing process is said to be athermal. The solvent and solute molecules must be the same size so that the criterion... 

Solutions can deviate from ideality because they fail to meet either one or both of these criteria. In reference to polymers in solutions of low molecular weight solvents, it is apparent that nonideality is present because of a failure to meet criterion (2), whether the mixing is athermal or not. 

We concluded the last section with the observation that a polymer solution is expected to be nonideal on the grounds of entropy considerations alone. A nonzero value for AH would exacerbate the situation even further. We therefore begin our discussion of this problem by assuming a polymer-solvent system which shows athermal mixing. In the next section we shall extend the theory to include systems for which AH 9 0. The theory we shall examine in the next few sections was developed independently by Flory and Huggins and is known as the Flory-Huggins theory. 

Since the 0 s are fractions, the logarithms in Eq. (8.38) are less than unity and AGj is negative for all concentrations. In the case of athermal mixtures entropy considerations alone are sufficient to account for polymer-solvent miscibility at all concentrations. Exactly the same is true for ideal solutions. As a matter of fact, it is possible to regard the expressions for AS and AGj for ideal solutions as special cases of Eqs. (8.37) and (8.38) for the situation where n happens to equal unity. The following example compares values for ASj for ideal and Flory-Huggins solutions to examine quantitatively the effect of variations in n on the entropy of mixing. 

Evaluate ASj for ideal solutions and for athermal solutions of polymers having n values of 50, 100, and 500 by solving Eqs. (8.28) and (8.38) at regular intervals of mole fraction. Compare these calculated quantities by preparing a suitable plot of the results. 

The quantity x is called the Flory-Huggins interaction parameter It is zero for athermal mixtures, positive for endothermic mixing, and negative for exothermic mixing. These differences in sign originate from Eq. (8.39) and reaction (8.A). 

Athermal mixing is expected in the case of 61 - 62. Since polymers generally decompose before evaporating, the definition 6 = (AUy/V°) is not useful for polymers. There are noncalorimetric methods for identifying athermal solutions, however, so the 6 value of a polymer is equated to that of the solvent for such a system to estimate the CED for the polymer. The fact that a range of 6 values is shown for the polymers in Table 8.2 indicates the margin of uncertainty associated with this approach. 

Those involving solution nonideality. This is the most serious approximation in polymer applications. As we have already seen, the large differences in molecular volume between polymeric solutes and low molecular weight solvents is a source of nonideality even for athermal mixtures. 

It is convenient to begin by backtracking to a discussion of AS for an athermal mixture. We shall consider a dilute solution containing N2 solute molecules, each of which has an excluded volume u. The excluded volume of a particle is that volume for which the center of mass of a second particle is excluded from entering. Although we assume no specific geometry for the molecules at this time, Fig. 8.10 shows how the excluded volume is defined for two spheres of radius a. The two spheres are in surface contact when their centers are separated by a distance 2a. The excluded volume for the pair has the volume (4/3)7r(2a), or eight times the volume of one sphere. This volume is indicated by the broken line in Fig. 8.10. Since this volume is associated with the interaction of two spheres, the excluded volume per sphere is... 

Linking this molecular model to observed bulk fluid PVT-composition behavior requires a calculation of the number of possible configurations (microstmctures) of a mixture. There is no exact method available to solve this combinatorial problem (28). ASOG assumes the athermal (no heat of mixing) FIory-Huggins equation for this purpose (118,170,171). UNIQUAC claims to have a formula that avoids this assumption, although some aspects of athermal mixing are still present in the model. 

The simplest type of solutions which exhibit non-randomness are those in which the non-randomness is attributable solely to geometric factors, i.e. it does not come from non-ideal energetic effects, which are assumed equal to zero. This is the model of an athermal solution, for which... 

Real solutions are rarely completely athermal, even when there is considerable similarity between the nature of the molecules. For cases in which some energy effects must be taken into account, Flory introduced an additional term into the expression for excess Gibbs free energy. Adapting the format of the Scatchard-Hildebrand equation, the additional contribution to the excess Gibbs free energy is assumed to be of the form ... 

In the result of the simulation one expects the times r23 to be proportional to rR and to be proportional to rp, and, indeed, from Fig. 12 the respective power laws are nicely seen in the athermal case of neutral walls, e/k T = 0. Good agreement with predicted results (not shown graphically here) is obtained also for the scahng of with N, and with D [14]. 

Finally, at even lower transformation temperatures, a completely new reaction occurs. Austenite transforms to a new metastable phase called martensite, which is a supersaturated solid solution of carbon in iron and which has a body-centred tetragonal crystal structure. Furthermore, the mechanism of the transformation of austenite to martensite is fundamentally different from that of the formation of pearlite or bainite in particular martensitic transformations do not involve diffusion and are accordingly said to be diffusionless. Martensite is formed from austenite by the slight rearrangement of iron atoms required to transform the f.c.c. crystal structure into the body-centred tetragonal structure the distances involved are considerably less than the interatomic distances. A further characteristic of the martensitic transformation is that it is predominantly athermal, as opposed to the isothermal transformation of austenite to pearlite or bainite. In other words, at a temperature midway between (the temperature at which martensite starts to form) and m, (the temperature at which martensite... 

Objections were raised to other results of these authors derived by viscometric techniques. Thus the viscometric technique led to the erroneous value 119) of 1, instead of 4, as required by symmetry, for the equilibrium constant of the athermal exchange l20> ... 

There is sfill some dispufe about how microwave irradiation accelerates reactions. Besides the generally accepted thermal effects, one beheves that there are some specific (but also thermal) microwave effects, such as the formation of hot spots . There is still some controversy about the existence of non-thermal (athermal) microwave effects. At the present time, new techniques such as coohng while heating are being investigated and the problem of upscahng... 

Ironis very reactive chemically and oxidizes readily. It has four allotropic forms, one of which (a) is magnetic with a Curie transition point of770°C. Ithasadensity of7.6g/cm, ameltingpointof 1536°C, athermal expansion of 12.6 ppm/°C at 25°C, a thermal conductivity of 0.80 W/ cm °C at25°C and an electrical resistivity of 9.71 iohm-cm at 20°C. 

The common white tin (P-Sn) has atetragonal stmcture, a density of7.3 g/cm andalowmelting point of232°C. Itisgenerally chemically inert and oxidationresistant. Ithasathermal expansion of20ppm/°C at 25°C, athermal conductivity of 0.6 W/cm °C at 25°C and an electrical resistivity of 1 l iohm-cm atO°C. 

A novel approach [98], proposed for generating starting configurations of amorphous dense polymeric systems, departs from a continuous vector field and its stream lines. The stream lines of continuous vector fields never intersect. If the backbones of linear polymer chains can be associated with such stream lines, the property of the stream lines partly alleviates the problem of excluded volume, which - due to high density and connectivity - constitutes the major barrier to an efficient packing method of dense polymeric systems. This intrinsic repulsive contact can be compared to an athermal hard-core potential. Considering stream lines immensely simplifies the problem. 

Fig. 3.14. Comparison of the torsion angle density distribution for a MC simulation of an athermal Ci0 phantom chains with Jacobian correction (the horizontal noisy line) and without Jacobian correction (the noisy curve showing two maxima at ca. 100° and 260° and a sharp minimum at 180°)...

Although this athermal bond fluctuation model is clearly not yet a model for any specific polymeric material, it is nevertheless a useful starting point from which a more detailed chemical description can be built. This fact already becomes apparent, when we study suitably rescaled quantities, such that, on this level, a comparison with experiment is already possible. As an example, we can consider the crossover of the self-diffusion constant from Rouse-like behavior for short chains to entangled behavior for longer chains. 

Fig. 5.3. Log-log plot of the self-diffusion constant D of polymer melts vs. chain length N. D is normalized by the diffusion constant of the Rouse limit, DRoUse> which is reached for short chain lengths. N is normalized by Ne, which is estimated from the kink in the log-log plot of the mean-square displacement of inner monomers vs. time [gi (t) vs. t]. Molecular dynamics results [177] and experimental data on PE [178] are compared with the MC results [40] for the athermal bond fluctuation model. From [40]...

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