Effective interaction parameters

A more accurate analysis of this problem incorporating renormalization results, is possible [86], but the essential result is the same, namely that stretched, tethered chains interact less strongly with one another than the same chains in bulk. The appropriate comparison is with a bulk-like system of chains in a brush confined by an impenetrable wall a distance RF (the Flory radius of gyration) from the tethering surface. These confined chains, which are incapable of stretching, assume configurations similar to those of free chains. However, the volume fraction here is q> = N(a/d)2 RF N2/5(a/d)5/3, as opposed to cp = N(a/d)2 L (a/d)4/3 in the unconfined, tethered layer. Consequently, the chain-chain interaction parameter becomes x ab N3/2(a/d)5/2 %ab- Thus, tethered chains tend to mix, or at least resist phase separation, more readily than their bulk counterparts because chain stretching lowers the effective concentration within the layer. The effective interaction parameters can be used in further analysis of phase separation processes... 

The effective interaction parameter of two random EPM polymers is given by... 

The miscibility gap will be described more accurately when a meanfield lattice gas approach is choosen [30], The mathematical form of the interaction function in all the above models may bring about a negative value for the effective interaction parameter, g, while all binary interactions by themselves are positive. The complexity of copolymer phase behaviour can be attributed to this peculiarity, like the miscibility-windows in mixtures of a copolymer with another homopolymer [37], or with a second copolymer [38,39]. 

Here, we describe and compare the results of simulations for two multichain systems corresponding to alternating and protein-like HA copolymers [212], The multichain systems consisting of 127-unit copolymers were simulated for the range of the effective interaction parameter / (which is similar to the Flory-Huggins parameter) under solvent conditions when single chains can form strongly collapsed conformations. 

Fig. 46. Effective interaction parameters Xrcp-PS (—) and Xrcp pvp( ) as functions of monomer ratio in the copolymer/for a poly(styrene-d8-co-4-hydroxystyrene) where/is the molar fraction of 4-hydroxystyrene. The dotted line ( = 0.015) is the maximum value for which a strong rcp/homopolymer interface is formed...

Xc Xe/ee Xh/d Xs Xsx XsANS Xthxj/dxpei) merXNP=X 4-1 critical interaction parameter %(TC) 2.1 microstructural contribution to x(hXj/dx,ej) 2.2.3 isotopic contribution to xlhXj/dXpej) 2.2.3 surface energy difference parameter 3.1.2.2 interaction parameter between surface and species X 3.1.2.2 effective interaction parameter as determined by SANS 2.1 interaction parameter between protonated Xj and deuterated (to extent ej) X 2.2.3 ... 

Fig. lO.a The inset shows the postulated variation of the solubility parameter 8 caused by deuterium labeling (symbols and V correspond to labeled and nonlabeled copolymers, respectively) and due to the change in ethyl ethylene fraction x. The cumulative analysis, described in text, yields the absolute 8 value for deuterated dx (A) and protonated hx (V) copolymers as a function of x at a reference temperature Tref=100 °C determined interaction parameters (as in Fig. 9) allow us to determine two sets of differences AS adjusted here to fit independent PVT data [140,141] measured at 83 °C ( ) and at 121 °C (O). b The interaction parameter, yE/EE, arising from the microstructural difference contribution to the overall effective interaction parameter (hxj/dxpej) in Eq. (19) as a function of the average blend composition (xi+Xj)/2 at a reference temperature of 100 °C.%E/ee values are calculated (see text) from coexistence data ( points correspond to [91,143] and O symbols to [136]) for blend pairs, structurally identical but with swapped labeled component. X marks %e/ee yielded directly [134] for a blend with both components protonated. Solid line is the best fit to data... 

Finally the effective interaction parameter (hxj/dxj.ej) in a mixture of a pro-tonated random copolymer hx- and a partly deuterated (to extent ej) random copolymer dxj is written using Eqs. (16a) and (18b) for Xj value close to Xj one ... 

The effective interaction parameter (hXj/dXj.ej) in Eq. (19) is a function of two unknown parameters, %E/EE and %h/d, so these can be extracted [143] from two experimentally determined values of the overall interaction parameter for each pair of structurally identical mixtures with a swapped isotope labeled component, i.e., for dXj/hXj and hxj/dxj. In order to fit to the specific form AT-1 of the interaction parameter used in the approach described above we re-express all effective interaction parameters as [91] x(hXj/dxi,e1)=xcTc/T, where %c= (N1l/2+Njl/2)2/(2N1Nj) and Tc is given by experiment. It turns out [145] that this is a good approximation for studied random polyolefins (see Fig. 9) as the entro-pic term B of Eq. (17a,b) is small (it contributes less than 6% for most of the blends) and the weak ( -dependence of % may be neglected as it is not directly involved here. 

In this section we address the question of accordance between coexistence conditions determined for polymer mixtures in the bulk and confined in thin bilayer films. Macroscopic samples with the size of ca. 1 mm are analyzed by Small Angle Neutron Scattering. It probes the compositional fluctuations away from binodal to yield the effective interaction parameter %SANs(bilayer films are studied by profiling techniques to yield concentrations < q and 2 at binodal. These are described by the composition dependent interaction parameter %(([>). In fact only the section of the relation %(([>) bounded by ( q and <(>2 is relevant as it describes the whole intrinsic profile of Fig. 2. 

The simplest method consists of considering that both SA and SB are classical spins with gA and g effective g-factors and a Je effective interaction parameter. The effective parameters are related to the actual parameters through ... 

In the case of d ° cubic perovskites we make reference to the effective interaction parameters shown in Fig. 2a. These are ... 

These conclusions illustrate the importance of correlations in low-dimensional systems [43-45, 51]. Similar trends are expected to hold for the realistic spd-band model given by Eq. (1). However, it should also be noted that the Hubbard model with the restriction to one orbital per site tends to exaggerate the effects of quantum fluctuations. Extensions of the model by including either several bands or nonlocal interactions should tend to weaken strong fluctuation effects. Moreover, the validity of the Hartree-Fock approximation is considerably improved by using effective interaction parameters that are fitted to experimental bulk results like the bulk magnetic moment /Xb and thus include part of the electron correlations. 

This chapter presents a general theoretical framework for the stndy of polymer solutions in which polymers are associated with each other by strongly attractive forces, such as hydrogen bonding and hydrophobic interaction. The Flory-Huggins free energy is combined with the free energy of association (reversible reaction) to study the mutual interference between phase separation and molecular association. The effective interaction parameters renormalized by the specific interactions are derived as functions of the polymer concentration. 

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