Swollen polymer lattice

Solid polymer and gel polymer electrolytes could be viewed as the special variation of the solution-type electrolyte. In the former, the solvents are polar macromolecules that dissolve salts, while, in the latter, only a small portion of high polymer is employed as the mechanical matrix, which is either soaked with or swollen by essentially the same liquid electrolytes. One exception exists molten salt (ionic liquid) electrolytes where no solvent is present and the dissociation of opposite ions is solely achieved by the thermal disintegration of the salt lattice (melting). Polymer electrolyte will be reviewed in section 8 ( Novel Electrolyte Systems ), although lithium ion technology based on gel polymer electrolytes has in fact entered the market and accounted for 4% of lithium ion cells manufactured in 2000. On the other hand, ionic liquid electrolytes will be omitted, due to both the limited literature concerning this topic and the fact that the application of ionic liquid electrolytes in lithium ion devices remains dubious. Since most of the ionic liquid systems are still in a supercooled state at ambient temperature, it is unlikely that the metastable liquid state could be maintained in an actual electrochemical device, wherein electrode materials would serve as effective nucleation sites for crystallization. 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

The self-diffusion coefficients of toluene in polystyrene gels are approximately the same as in solutions of the same volume fraction lymer, according to pulsed field gradient NMR experiments (2fl). Toluene in a 10% cross-linked polystyrene swollen to 0.55 volume fraction polymer has a self-diffusion coefficient about 0.08 times that of bulk liquid toluene. Rates of rotational diffusion (molecular Brownian motion) determined from NMR spin-lattice relaxation times of toluene in 2% cross-linked ((polystytyl)methyl)tri-/t-butylphosphonium ion phase transfer catalysts arc reduced by factors of 3 to 20 compai with bulk liquid toluene (21). Rates of rotational diffusion of a soluble nitroxide in polystyrene gels, determined from ESR linewidths, decrease as the degree of swelling of the polymer decreases (321. 

In the limit when the number N of monomer nnits goes to infinity, there is sharp transition from the swollen to collapsed phase at a critical value of u. This transition is described as a critical phenomena analogous to a tricritical point for magnetic system [2]. For large polymers, the average gyration radius R at the transition behaves as Rjv N <> where the exponent vg is intermediate between the value v for swollen state and the value Vc = i-/D for compact globule on a lattice of fractal dimension D. 

Figure 18. Sections of the u — u — t phase diagram for two different values of t for the 4-simplcx lattice (a) t= 0.2 (b) t= 0.5. Regions marked by AS, DS and DC represent, respectively, the arborbed polymer in swollen state, desorbed polymer in swollen and collapsed (globular) state. The collajee transition between the DC and DS phases is denoted by the dashed line. The special adsorption line is indicated by full line and part of it 1 dotted line. The point where line meets with the adsorption line is a multicritical point. The dotted part of the adsorption line indicates the re on of coexistence of adsorbed S.AVV and the (bulk) globule phase.

As already shown in section 6.2 a polymer chain always remains in a swollen state for all values of ini eraction, (self-) attraction on a 5-simplex lattice. This state is characterized by the fixed point (0.3265..., 0.0279...). Tn this case we therefore have only one combination, i.e. SS of chains. Using these values of fixed points for single chains, the recursion relation for the partition function E of two interacting diains (see [62]) is solved and fixed points E = 0.0279 or E = 0.2713 are found. The fixed point E is found for all values of U3 < U3e(x3). At Us = useixs) fixed point E is found. 

Z2, L and Y in the example above (it would also include P in the case of a coiled polymer). To illustrate the problem we may consider a situation where polymer particles are swollen with a mixture of two compounds, i and j, present as a bulk phase. Here, i and j may represent either Z-j and or any pair of the compounds Z, Y and L. The semi-equilibrium state between the swollen particles (phase (a)) and the mixture of i and j (phase (b) or (c)) according to Flory s lattice theory should read ... 

The solubility of crystalline polymers is rather less than for amorphous polymers. Polyethylene and isotactic polypropylene have much less miscibility in potential solvents than that exhibited by amorphous aliphatic polymers such as polyisobutylene or ethylene-propylene copolymer. This is associated with the necessity of the heat of mixing overcoming the heat of fusion of the crystalline lattice. The aliphatic hydrocarbon solvents, which dissolve amorphous polymers such as polyisobutylene, have small heats of mixing which carmot disrupt the crystalline lattices of polyolefins. The amorphous regions of the semi-crystaUine polyolefins are swollen by aliphatic hydrocarbon solvents and their mechanical properties are lowered, but they do not dissolve. It is only when the temperature approaches the polymer s melting point that dissolution occurs. 

A cross-linked polymer is swollen in three solvents. What is the predicted swollen volume in solvent C Assume that the lattice parameter is 0.300 for all three solvents. All data are taken at 27°C. 

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