Polymeric 294 INDEX

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.

The main parameters used to describe a polymer chain are the polymerization index N, which counts the number of repeat units or monomers along the chain, and the size of one monomer or the distance between two neighboring monomers. The monomer size ranges from a few Angstroms for synthetic polymers to a few nanometers for biopolymers. The simplest theoretical description of flexible chain conformations is achieved with the so-called freely-jointed chain (FJC) model, where a polymer consisting of N + I monomers is represented by N bonds defined by bond vectors r/ with j= Each bond vector has a fixed length r,j = a corresponding to the... 

If one of the species is monomeric oxirane, then J = 1. Likewise, if one of the polymeric species supplied the oxirane, then J > 1. The molecule with the acid group is at degree of polymerization k. The degree of polymerization indexes the number of oxirane residuals within the macromolecule. Though the reaction sequence is simplified, it retains the essence of one molecule reacting with every other molecule. This step-growth mechanism (13) develops the thermoset resin microstructure. 

Figure 4. Representative chain configurations for F-F (a) and F-S (b) polymer classes. MD simulations [100] are obtained for isolated chains and are only meant for illustrative purposes. The polymerization index N is taken as iV = 100. The monomer structure is assumed to be common for these two polymer categories and is depicted in Fig. 3b. The bending energies b and are chosen in the simulations as E /ksT = Es/k T = 0, and Eb/k T = 0, Es/ksT = 200 for F-F and F-S polymer classes, respectively.

From Eq. (5.7), we find an = (0/9vxv0)112 = N112/3 at the critical point where N = (f>0/viv0 is the effective polymerization index. To approach this critical point avoiding the spinodal decomposition in the perpendicular directions, we require... 

In mean field theory, two parameters control the phase behavior of diblock copolymers the volume fraction of the A block /A, and the combined interaction parameter xTak- V. where Xab is the Flory-Huggins parameter that quantifies the interaction between the A and B monomers and N is the polymerization index [30], The block copolymer composition determines the microphase morphology to a great extent. For example, comparable volume fractions of block copolymer components result in lamella structure. Increasing the degree of compositional asymmetry leads to the gyroid, cylindrical, and finally, spherical phases [31]. 

There are several ways of forming surface layers of polymer chains, and various solid/polymer systems have been used. The silica/PDMS system is quite convenient since both end-grafted layers with high grafting densities (i.e., brushes) and irreversibly adsorbed layers (i.e., pseudo-brushes) can be formed with controlled molecular characteristics (polymerization index of the tethered chains and surface density), allowing a detailed investigation of the structure and properties of these two different classes of surface anchored polymer layers. 

Such behavior has been interpreted in terms of a molecular model proposed by Brochard-Wyart and de Gennes [143] and further refined [145,146]. The first version of these models considers a solid surface bearing a few end grafted polymer chains, with a surface density, G, below the onset of the mushroom regime gAT<1, with N the polymerization index of the anchored chains). The melt chains have a polymerization index P. Both N and P are assumed to be much larger than Ne, the average number of monomers needed to form an entanglement. Thus the... 

If species 1 is a solvent non-macromolecule of roughly the same size as the monomers composing the polymeric species, then it is natural to estimate V2/V1 M, polymerization index. Eurther, it is natural also to estimate bfi/bn see Eq. (4.34), p. 77. 

We now speciahze these general formulae to the common case that this mixing free energy is expressed on a per monomer basis. As a notational convenience we consider specifically the case of a polymer species mixed into a small molecule solvent, and use M = V2/V1 as the empirical polymerization index parameter. Then... 

Our discussion here explores active connections between the potential distribution theorem (PDT) and the theory of polymer solutions. In Chapter 4 we have already derived the Flory-Huggins model in broad form, and discussed its basis in a van der Waals model of solution thermodynamics. That derivation highlighted the origins of composition, temperature, and pressure effects on the Flory-Huggins interaction parameter. We recall that this theory is based upon a van der Waals treatment of solutions with the additional assumptions of zero volume of mixing and more technical approximations such as Eq. (4.45), p. 81. Considering a system of a polymer (p) of polymerization index M dissolved in a solvent (s), the Rory-Huggins model is... 

Figure 4 Self diffusion coefficient of the small probe, CH3-CH2-CH2-NBD, versus the PDMS polymerization index (now only unlabelled PDMS is used). Given the low accuracy of the experiment at such high values of the diffusion coefficient (relative uncertainty larger than 10%, contrary to the data on figure 3, which are all slower by more than one decade and thus far easier to measure accurately), it seems that there is no significant change of the friction...

For a mixture of identically D-labelled chains in a matrix of H chains with the same polymerization index, the coherent scattered intensity is given by ... 

Figure 7.Dependence of the critical velocity, V, versus the weight average polymerization index of a) the flowing melt. P b) the surface chains, N c) surface chains.N. in the case where it is identical to that of the melt (N = P).

Fig. 22. (a) Snapshot of an interface between two coexisting phases in a binary polymer blend in the bond fluctuation model (invariant polymerization index // = 91, incompatibihty 17, linear box dimension L 7.5iJe, or number of effective segments N = 32, interaction e/ksT = 0.1, monomer number density po = 1/16.0). (b) Cartoon of the configuration illustrating loops of a chain into the domain of opposite type, fluctuations of the local interface position (capillary waves) and composition fluctuations in the bulk and the shrinking of the chains in the minority phase. Prom Miiller [109]... 

For properly chosen rate parameters it attains values close to one. Figures 3 and 4 display the behavior of the polymerization index (An) and PDI as computed and as predicted by eqs 22, 24, and 25. 

If most of the conversion occurs in this state, the polymerization index becomes... 

Figure 8. Time evolution of the polymerization index ln-([M]o/[M]) during a polymerization of 0.76 M styrene in tert-butylbenzene at 130 °C initiated by the alkoxyamine 6 of Scheme 30 for different alkoxyamine concentrations. The solid lines confirm eq 22.

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