Polymer equilibrium concentration

Another situation effecting residual levels existing during PS devolatilization is polymer decomposition or unzipping, which limits the devolatilization temperature to <260 °C. If one tries to go to higher temperature to achieve a more favorable vapor-polymer equilibrium concentration, polymer decomposition begins to dominate. The rate of polymer decomposition can be affected by stabilizing the polymer by the addition of phenolic antioxidants, e.g. 2,6-di-tert-butyl-4-methylphenol [17]. Several Asahi patents indicate the superior... 

While all liquid media are viscoelastic, concentrated polymer solutions are characterized by an especially rich distribution of relaxation processes. In order to consider this phenomenon, the pure shear deformation will be employed. Consider an equilibrium concentrated polymer solution and identify the absolute location of every atom in the sample. Now deform the sample such that the new x-coordinate of each atom is increased by a constant multiple of its y-coordinate. This affine deformation might be difficult in the laboratory, but it can be executed for a computer sample of a concentrated polymer solution. Now allow the sample to evolve in time in contact with a heat bath. After a long time, the sample will reach equilibrium, with the new shape determined by the initial shear deformation. 

Doi, M. and Edwards, S.F., 1978. Dynamics of concentrated polymer systems 1. Brownian motion in equilibrium state, 2. Molecular motion under flow, 3. Constitutive equation and 4. Rheological properties. J. Cheni. Soc., Faraday Trans. 2 74, 1789, 1802, 1818-18.32. 

Depending on the final polymerization conditions, an equilibrium concentration of monomers (ca 8%) and short-chain oligomers (ca 2%) remains (72). Prior to fiber spinning, most of the residual monomer is removed. In the conventional process, the molten polymer is extmded as a strand, solidified, cut into chip, washed to remove residual monomer, and dried. In some newer continuous processes, the excess monomer is removed from the molten polymer by vacuum stripping. 

Lack of termination in a polymerization process has another important consequence. Propagation is represented by the reaction Pn+M -> Pn+1 and the principle of microscopic reversibility demands that the reverse reaction should also proceed, i.e., Pn+1 -> Pn+M. Since there is no termination, the system must eventually attain an equilibrium state in which the equilibrium concentration of the monomer is given by the equation Pn- -M Pn+1 Hence the equilibrium constant, and all other thermodynamic functions characterizing the system monomer-polymer, are determined by simple measurements of the equilibrium concentration of monomer at various temperatures. 

Cationic polymerization of cyclic acetals generally involves equilibrium between monomer and polymer. The equilibrium nature of the cationic polymerization of 2 was ascertained by depolymerization experiments Methylene chloride solutions of the polymer ([P]0 = 1.76 and 1.71 base-mol/1) containing a catalytic amount of boron trifluoride etherate were allowed to stand for several days at 0 °C to give 2 which was in equilibrium with its polymer. The equilibrium concentrations ([M]e = 0.47 and 0.46 mol/1) were in excellent agreement with that found in the polymerization experiments under the same conditions. The thermodynamic parameters for the polymerization of 1 were evaluated from the temperature dependence of the equilibrium monomer concentrations between -20 and 30 °C. 

Another general type of behavior that occurs in polymer manufacture is shown in Figure 3. In many polymer processing operations, it is necessary to remove one or more solvents from the concentrated polymer at moderately low pressures. In such an instance, the phase equilibrium computation can be carried out if the chemical potential of the solvent in the polymer phase can be computed. Conditions of phase equilibrium require that the chemical potential of the solvent in the vapor phase be equal to that of the solvent in the liquid (polymer) phase. Note that the polymer is essentially involatile and is not present in the vapor phase. 

Values of p can be determined, in principle, from any phase equilibrium data. A small table of p 2 values is available in reference (2). However, one of the most straightforward ways of determining pf values is to fit phase equilibrium data for solvent sorption in concentrated polymer solutions. To do this, equations (2) and (13) are combined to solve for p utilizing experimental partial pressure data. 

There are relatively few phase equilibrium data relating to concentrated polymer solutions containing several solvents. Nevertheless, In polymer devolatilization, such cases are often of prime Interest. One of the complicating features of such cases Is that. In many Instances, one of the solvents preferentially solvates the polymer molecules, partially excluding the other solvents from Interaction directly with the polymer molecules. This phenomenon Is known as "gathering". 

In addition polymerizations which proceed exclusively by addition of monomer to the end of a growing chain, the mobility of the monomer, being affected only to a comparatively small degree by surrounding polymer molecules, should be adequate to maintain an equilibrium concentration of monomers in the vicinity of active centers. Even in... 

The diffusion constant of a primary radical must be of the order of 10 cm.2 sec.- the radius r is about 5X10 cm., and as we have seen 1 10 " per second. Hence ]ag l0 radicals per cc. But the radicals are being generated at a rate of 10 cc. sec. hence the average lifetime of a radical from generation to capture by a polymer particle will be only 10 sec. " The rate of termination by reaction between two radicals in the aqueous phase at the calculated equilibrium concentration, 10 radicals per cc., will be given by... 

The addition of heat shifts the equilibrium concentrations away from the products and back towards the reactants, the monomers. This is one reason why processing these types of polymers is often more difficult than processing products of chain growth mechanisms. The thermal degradation process can be dramatically accelerated by the presence of the low molecular weight condensation products such as water. Polyester, as an example, can depolymerize rapidly if processed in the presence of absorbed or entrained water. 

Table 1 gives details of the adsorbed amounts for the six polymer fractions obtained at an equilibrium concentration of 2000 ppm. Based on the full adsorption isotherm 02) these values correspond... 

Our model predicts destabilization of colloidal dispersions at low polymer concentration and restabilisation in (very) concentrated polymer solutions. This restabilisation is not a kinetic effect, but is governed by equilibrium thermodynamics, the dispersed phase being the situation of lowest free energy at high polymer concentration. Restabilisation is a consequence of the fact that the depletion thickness is, in concentrated polymer solutions, (much) lower than the radius of gyration, leading to a weaker attraction. 

Chemical equilibrium. The polymer concentrations in the ideal gas phase of water are related to the association equilibrium constants according to the expression ... 

By using a PES with a different thickness, one can conveniently change the AV ratio. This approach permits some control over the time required to reach equilibrium concentrations. Bartkow et al. (2004) has reported an excellent example of the impact of ratio or thickness on the time to equilibrium. These investigators showed that a 200 pm thick PE sheet took twice as long to reach equilibrium in air as a 100 pm thick PE sheet. In theory, changing membrane thickness will not affect polymer diffusivity and equilibrium membrane-water partition coefficients (I mwS) or solubility coefficients ( p). However, in practice different values of (membrane-air partition coefficient) and membrane... 

Because the onset of monomer-polymer equilibrium can occur before the filaments achieve their own equilibrium concentration behavior, these filaments will undergo polymer length redistribution. This is a slow process in vitro that in many respects resembles crystallization (See Ostwald Ripening). 

The summations are made over all polymer lengths from one to a maximal length m and typically represents the polymer weight concentration i.e., the total amount of monomer present as polymeric species. Thus the two summations are alternate expressions of [P], the approximate concentration of polymer ends. At equilibrium, dX/dt = 0, and the dissociation constant (K = kofilkon) reduces to [X], which is the concentration of monomer that coexists with polymer at equilibrium. We chose to use the infinity sign as a subscript to distinguish this equilibrium concentration of monomer (i.e., [M]oo) from that concentration [M] that may be present at any other extent of reaction. 

Polymers prepared at 250-270 °C contain an equilibrium concentration of up to 10% of cyclic monomer and partly cyclic higher oligomers after cooling, the monomer and oligomers can be recovered by extraction with water or lower al-... 

Figure 2. Equilibrium concentrations of cyclic dimer (9), trimer (O), tetramer (d), and linear polymers (9) as a function of total monomer unit concentration...

When gas concentrations are calculated for these polymers, the combined effects of low diffusivity and higher gas generation rate result in the conclusion that gas diffusion out of the polymer has a small effect on the gas concentration in the polymer for samples in the 0.033 to 0.100-inch thickness range. As a further result, the calculated gas concentration for these samples increases linearly with time in the manner observed for the thicker polystyrene samples (Figure 19). The calculated gas concentration will start to approach an equilibrium concentration at some greater time, when the rate of gas diffusion out equals the rate of gas generation, but this point is evidently not reached within the time scale of the present experiments. 

Necklace models represent the chain as a connected sequence ctf segments, preserving in some sense the correlation between the spatial relationships among segments and their positions along the chain contour. Simplified versions laid the basis for the kinetic theory of rubber elasticity and were used to evaluate configurational entropy in concentrated polymer solutions. A refined version, the rotational isomeric model, is used to calculate the equilibrium configurational... 

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