Mean-field assumption

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

This effective medium or mean field assumption is easy to understand if there is a very large size difference between the newly added particles and any there previously, for example if we think of adding particles to a molecular liquid, we merely treat the liquid as a structureless continuum. However when the dimensions become comparable, the finite volume of the particles present prior to each addition must be considered, i.e. new particles can only replace medium and not particles. The consequence of this crowding is that the concentration change is greater than expressed in Equation (3.52) and it must be corrected to the volume available ... 

While CG-KMC can reach large scales at reasonable computational cost, it can lead to substantial errors at boundaries and interfaces where large gradients exist, and the local mean field assumption is not as accurate. Recent... 

Mathematically speaking, the mean-field assumption consists of mapping the m y. z occupation-number matrix s onto the z-dimensional vectors = (n, Uj,..., n ) and n = (n ,. .., nf) where nj is the total... 

This statement is, of course, in a contradiction with the conventional mean-field assumption that the orthobaric curve forms a line along which e,s,v all increase monotonically on passing from liquid, through the critical point, to the gas. 

This is the equivalent of the mean field assumption in statistical mechanics The n-electron wave function can be written as the antisymmetrized product of one-electron functions o, denotes the spin of the electron ... 

The WSL theory developed by Leibler has been shown to be incorrect because of deviations from the fundamental underlying mean-field assumption. Figure 13.14 shows experimental results for a poly(ethylene-propylene/ethylethylene) (PEP-PEE) diblock copolymer that has been fit to the predictions of the Leibler theory without any adjustable parameters, since the ODT and / were calculated from rheological measurements (Bates et al., 1990). This mean-field theory does not qualitatively describe the behavior of this material. Other experiments have indicated that the RPA approximation (Sttihn and Stickel, 1992) and the Gaussian coil assumption (Bates and Hartney, 1985 Holzer et al., 1991) are inaccurate near the ODT. 

As discussed in this section, the tube-dilation effect, i.e. M J/Me > 1, mainly occurs in the terminal-relaxation region of component two in a binary blend. This effect means that the basic mean-field assumption of the Doi-Edwards theory (Eq. (8.3)) has a dynamic aspect when the molecular-weight distribution of the polymer sample is not narrow. This additional dynamic effect causes the viscoelastic spectrum of a broadly polydisperse sample to be much more complicated to analyze in terms of the tube model, and is the main factor which prevents Eq. (9.19) from being applied... 

The Flory-Huggins theory is in fact nothing more than a two-component polymer version of the simple lattice gas model introduced in section 2. We divide the free energy into an entropic part, which is assumed to take the simplest perfect gas form, while the enthalpic part is estimated using a typical mean-field assumption. 

In an interacting system, each polymer chain will have a distribution function obe5dng equation (4.3.2), each with its own function Ufr) whose form can be obtained if the positions of all the other poljmier chains are known. The resulting system of coupled differential equations, one for each chain in the system, is obviously completely intractable as it stands. However, we can make progress if we make a mean-field assumption - we assume that every polymer chain of the same chemical type experiences an average, mean-field potential U f). This potential has two parts. The first part arises simply from the hard core potential which prevents two segments occupying the same... 

To obtain useful theoretical results for the concentration profile, we need to go beyond these simple scaling arguments. Luckily, at least for the situation of relatively dense, strongly stretched, brushes, we can expect self-consistent field theories to work rather well in such a dense brush the basic mean-field assumption that any polymer chain will interact with its neighbours more than it will with itself should be well obeyed. Niunerical mean-field theories of the kind described in chapter 5 are very well suited to this kind of calculation the earliest results, due to Hirz (these results are still unpublished but some were reproduced by Milner et al. (1988)) showed profiles very different in character from those found for adsorbed chains. Rather than a concave concentration profile, the curves were notably convex, with the concentration dropping rather abruptly to zero on the outside of the brush. In fact it turns out that the profiles are rather well described by a parabolic form (see figure 6.7). It soon turned out that there was a remarkably good analytical solution to the self-consistent mean-field equations which provided an explanation for these parabolic profiles. 

According to the mean-field assumption, the mixing heat... 

The approximation becomes poor for the systems in which concentration fluctuations are large. For instance, in dilute polymer solutions, monomers distribute unevenly inside and outside the region occupied by the polymer chains. The spatial variation of the concentration is so high that the mean field assumption cannot be expected to hold. Also, in the region near the critical point of phase separation, where the concentration fluctuation is large, the mean field picture breaks down. 

The fraction of fully reacted sites in the polymer is plotted versus monomer conversion in Fig. 23, together with experimental resultsand values predicted on the assumption of equal reactivity (mean field assumption). Contrary to the latter, the percolation model nicely predicts the sudden increase in crosslink density at low con-... 

Fig. 23. The fraction of fully reacted monomer units in the polymer (C) as a function of monomer conversion P. Experimental points on continuous curve are for HDDA Dashed-dotted line results from mean-field assumption (equal reactivity of pendent and free double bonds, cf Ref. Dotted line results from the percolation model for the polymerization of a pure divinyl compound in three dimensions...

In the lattice model of polymer solutions, polymer chain is simply represented by a number of consecutively occupied lattice sites, each site corresponding to one chain unit. The rest single sites are assigned to solvents. This simple lattice treatment of polymer solutions allows a very convenient way to calculate thermodynamic properties of flexible and semiflexible polymer solutions from the statistical thermodynamic approach. By the mean-field assumption, the entropy part and the enthalpy part of partition function can be separately calculated. 

How does the Bragg-Williams approximation err If AB interactions are more favorable than AA and BB interactions, then B s will prefer to sit next to /Vs more often than the mean-field assumption predicts. Or, if the selfattractions are stronger than the attractions of A s for B s, then /Vs will tend to cluster together, and B s will cluster together, more than the random mixing assumption predicts. Nevertheless, the Bragg-Williams mean-field expression is often a reasonable first approximation. 

These problems arise from the mean-field assumption used to place the chain segments in the lattice. This picture is more correct in concentrated solutions where the polymer molecules interpenetrate and overlap. It is certainly not a good view of diluted solutions in which the polymer molecules are well separated. In the latter case it is obvious that the concentration of polymer segments is highly non-uniform in the solution. 

It is convenient to partition polymer solutions into three different cases according to their concentration. Dilute solutions involve only a minimum of interaction (overlap) between different polymer molecules. The Flory-Huggins theory does not represent this situation at all well due to its mean-field assumption. The semi-dilute case involves overlapping polymer molecules but still with a considerable separation of the segments of different molecules. 

Mathematically speaking, the mean-field assumption consists of mapping the m. X z occupation-number matrix s onto the 2-dimensional vectors = (n, ri2, - -., ,) and n = (nf, nj, . , n ) where n] is the total number of molecules of species i on lattice plane I regardless of their specific arrangement. Hciicc, wc replace II a) by its mean-field analog llmt (w. w ) where we note in passing that the transformation s — n, n is not bijective in general (see below). 

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