Scaling parameters solvent quality

Fig. 42 Ratio R p/Rgi as a function of the interaction parameter x in a semi-logarithmic scale for the 127-unit alternating and protein-like chains in the corresponding multichain systems. The parameter x is similar to the Flory-Huggins interaction parameter X and characterizes solvent quality in an integral manner. Sufficiently large values of x (X > 1) correspond to a poor solvent. Solvent quality becomes poorer with decreasing temperature or with increasing x. Adapted from [212]...

It is known that the coil-globule transition in flexible polymers is well explained by the theory of the type discussed [22]. Note that the chain length and the solvent quality come into the theory in the following combined form x = BN1/2/l3, which is the only dimensionless parameter governing the transition. The presence of the master curve (see Fig. 3.5 below) implies that the phase behavior of the thermodynamic limit with N —> oo is readily discussed from the measurement of shorter chains via finite-size scaling. 

The grouping in the parenthesis of Equation 10.10 can be related to the characteristic ratio and is nearly independent of the polymer molecular weight the dependence of intrinsic viscosity on solvent quality is therefore proportional to the product aM. In theta solvents, a is unity (the intrinsic viscosity scales with and in good solvents a is proportional to (the intrinsic viscosity scales with M ). Comparison with Equation 10.1 suggests that the Mark-Houwink parameter should lie in the range 0.5 expansion factor if theta conditions for the polymer solution are known. 

Scaling theory does not give the prefactors ( amplitudes ) of the power laws. For neutral polymers in solution the Renormalization Group Theory (RNG) has been applied to describe the region from dilute to semi-dilute behaviour with a prediction of the prefactors [184, 185]. As a result, the reduced diffusion coefficient D(Cp)/D(Cp = 0) should be a universal function of the reduced concentration A2CpM , where A2 is the second viiial coefficient. For polyelectrolytes the excluded volume parameter z enters this universal relation which depends on the solvent quality. The RNG developed by Wang and Bloomfield [186] for polyions has been applied to explain the concentration dependence of the polyion diffusion coefficient. 

In contrast to -conditions a large number of NSE results have been published for polymers in dilute good solvents [16,110,115-120]. For this case the theoretical coherent dynamic structure factor of the Zimm model is not available. However, the experimental spectra are quite well described by that derived for -conditions. For example, see Fig. 42a and 42b, where the spectra S(Q, t)/S(Q,0) for the system PS/d-toluene at 373 K are shown as a function of time t and of the scaling variable (Oz(Q)t)2/3. As in Fig. 40a, the solid lines in Fig. 42a result from a common fit with a single adjustable parameter. No contribution of Rouse dynamics, leading to a dynamic structure factor of combined Rouse-Zimm relaxation (see Table 1), can be detected in the spectra. Obviously, the line shape of the spectra is not influenced by the quality of the solvent. As before, the characteristic frequencies 2(Q) follow the Q3-power law, which is... 

To scale up a chemical process to pilot or commercial-scale operations, a significant laboratory effort is required to define the operating ranges of the critical process parameters. A critical process parameter is any process variable that may potentially affect the product quality or yield. This information is required to prepare a Process Risk Analysis, which is an FDA prerequisite for process validation. Process parameters that are often evaluated as part of the risk analysis include reaction temperature, solvent systems, reaction time, raw material and reagent ratios, rate and orders of addition, agitation, and reaction concentration. If catalysts are employed as part of the process, additional laboratory evaluation may also be required to further define the process limits. Experimental design is often used for the evaluation of critical process parameters to minimize the total laboratory effort (4). 

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