Interaction parameters chain

Figure 5 The dependence of limited length of thermodynamically stable chain Lum on interaction parameter chain-solvent p.

Figure 8.3 Volume fraction polymer in equilibrium phases for chains of different length, (a) Theoretical curves drawn for the indicated value of n, with the interaction parameter as the ordinate. Note that x increases downward. (Redrawn from Ref. 6.) (b) Experimental curves for the molecular weights indicated, with temperature as the ordinate. [Reprinted with permission from A. R. Shultz and P. J. Flory, J. Am. Chem. Soc. 74 4760 (1952), copyright 1952 by the American Chemical Society.]...

We saw in Sec. 1.11 that coil dimensions are affected by interactions between chain segments and solvent. Both the coil expansion factor a defined by Eq. (1.63) and the interaction parameter x are pertinent to describing this situation. 

More fundamental treatments of polymer solubihty go back to the lattice theory developed independentiy and almost simultaneously by Flory (13) and Huggins (14) in 1942. By imagining the solvent molecules and polymer chain segments to be distributed on a lattice, they statistically evaluated the entropy of solution. The enthalpy of solution was characterized by the Flory-Huggins interaction parameter, which is related to solubihty parameters by equation 5. For high molecular weight polymers in monomeric solvents, the Flory-Huggins solubihty criterion is X A 0.5. 

Helfand and Tagami [75,76] introduced a model which considered the probability that a chain of polymer 1 has diffused a given distance into polymer 2 when the interactions are characterised by the Flory-Huggins interaction parameter x They predicted that at equilibrium the thickness , d c, of the interface would depend upon the interaction parameter and the mean statistical segment length, b, as follows ... 

Monte Carlo simulations [17, 18], the valence bond approach [19, 20], and g-ology [21-24] indicate that the Peierls instability in half-filled chains survives the presence of electron-electron interactions (at least, for some range of interaction parameters). This holds for a variety of different models, such as the Peierls-Hubbard model with the onsite Coulomb repulsion, or the Pariser-Parr-Pople model, where also long-range Coulomb interactions are taken into account ]2]. As the dimerization persists in the presence of electron-electron interactions, also the soliton concept survives. An important difference with the SSH model is that neu-... 

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... 

Table 5.1 Parameters of the united atom force field for polyethylene used as the atomistic input for the coarse-graining procedure. The Lennard-Jones parameters pertain to CH2-group interaction, since chain ends were not considered in the coarse-graining. 

Fig. 51 Phase diagram for PS-PI diblock copolymer (Mn = 33 kg/mol, 31vol% PS) as function of temperature, T, and polymer volume fraction, cp, for solutions in dioctyl ph-thalate (DOP), di-n-butyl phthalate (DBP), diethyl phthalate (DEP) and M-tetradecane (C14). ( ) ODT (o) OOT ( ) dilute solution critical micelle temperature, cmt. Subscript 1 identifies phase as normal (PS chains reside in minor domains) subscript 2 indicates inverted phases (PS chains located in major domains). Phase boundaries are drawn as guide to eye, except for DOP in which OOT and ODT phase boundaries (solid lines) show previously determined scaling of PS-PI interaction parameter (xodt

The loss of phase complexity in both systems may be attributed to an increase of the PS/PEO and PI/PEO interaction parameters. Because LiClC is selectively located in the PEO domains, the interaction parameters (/ps-peo and xpi-peo ) must increase, leading to variations in domain type and dimension. As the lithium salt increases the polarity (and presumably the solubility parameters) of the PEO domains, the interfacial tensions between PEO and PI, and PEO and PS are elevated. Thus, a reduction in the overall PEO interfacial area is required, which necessitates additional chain stretching. In consequence, the CSC structure becomes dominant when comparing doped and non-doped samples [171] (Figs. 54 and 55b). 

Figure 14. The phase diagram of the gradient copolymer melt with the distribution functions g(x) = l — tanh(ciit(x —fo)) shown in the insert of this figure for ci = 3,/o = 0.5 (solid line), and/o — 0.3 (dashed line), x gives the position of ith monomer from the end of the chain in the units of the linear chain length. % is the Flory-Huggins interaction parameter, N is a polymerization index, and/ is the composition (/ = J0 g(x) dx). The Euler characteristic of the isotropic phase (I) is zero, and that of the hexagonal phase (H) is zero. For the bcc phase (B), XEuier = 4 per unit cell for the double gyroid phase (G), XEuier = -16 per unit cell and for the lamellar phases (LAM), XEuier = 0.

Here, AGeiastic is the contribution due to the elastic retractive forces developed inside the gel and A6mixi g is the result of the spontaneous mixing of the fluid molecules with the polymer chains. The term AGmjXjng is a measure of the compatibility of the polymer with the molecules of the surrounding fluid. This compatibility is usually expressed by the polymer-solvent interaction parameter, xi (Flory, 1953). 

Here k and T have their usual meaning, x is the ratio between the molecular volumes of a polymer chain and a solvent molecule and X is the polymer-solvent interaction parameter. 

With respect to SCF models that focus on the tail properties only (typically densely packed layers of end-grafted chains), the molecularly realistic SCF model exemplified in this review needs many interaction parameters. These parameters are necessary to obtain colloid-chemically stable free-floating bilayers. A historical note of interest is that it was only after the first SCF results [92] showed that it was not necessary to graft the lipid tails to a plane, that MD simulations with head-and-tail properties were performed. In the early MD simulations (i.e. before 1983) the chains were grafted (by a spring) to a plane it was believed that without the grafting constraints the molecules would diffuse away and the membrane would disintegrate. Of course, the MD simulations that include the full head-and-tails problem feature many more interactions than the early ones. 

Chiu, S. W., Clark, M. M., Jakobsson, E., Subramaniam, S. and Scott, H. L. (1999). Optimization of hydrocarbon chain interaction parameters application to the simulation of fluid phase lipid bilayers, J. Phys. Chem. B, 103, 6323-6327. 

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