Mean field statistic

We did not extensively discuss the consequences of lateral interactions of surface species adsorbed in adsorption overlayers. They lead to changes in the effective activation energies mainly because of consequences to the interaction energies in coadsorbed pretransition states. At lower temperatures, it can also lead to surface overlayer pattern formation due to phase separation. Such effects cannot be captured by mean-field statistical methods such as the microkinetics approaches but require treatment by dynamic Monte Carlo techniques as discussed in [25]. 

In order to understand the thermodynamic issues associated with the nanocomposite formation, Vaia et al. have applied the mean-field statistical lattice model and found that conclusions based on the mean field theory agreed nicely with the experimental results [12,13]. The entropy loss associated with confinement of a polymer melt is not prohibited to nanocomposite formation because an entropy gain associated with the layer separation balances the entropy loss of polymer intercalation, resulting in a net entropy change near to zero. Thus, from the theoretical model, the outcome of nanocomposite formation via polymer melt intercalation depends on energetic factors, which may be determined from the surface energies of the polymer and OMLF. 

The various surface hydroxyls formed may structurally and chemically not be fully equivalent, but to facilitate the schematic representation of reactions and of equilibria, one usually considers the chemical reaction of "a" surface hydroxyl group, S-OHL (see the remarks on mean field statistics in Chapter 3.7). 

Surface Complexation Models and Mean Field Statistics... 

The surface complexation models used are only qualitatively correct at the molecular level, even though good quantitative description of titration data and adsorption isotherms and surface charge can be obtained by curve fitting techniques. Titration and adsorption experiments are not sensitive to the detailed structure of the interfacial region (Sposito, 1984) but the equilibrium constants given reflect - in a mean field statistical sense - quantitatively the extent of interaction. 

All functional groups are treated as if they were identical (mean field statistics)... 

Dynamic DFT method is usitally used to model the dynamic behavior of polymer systems and has been implemented in the software package Mesodyn TM from Accelrys [41]. The DFT models the behavior of polymer fluids by combining Gaussian mean-field statistics with a TDGL model for the time evolution of conserved order parameters. However, in contrast to traditional phenomenological free-energy expan-... 

In Fig. 11.3, we made a comparison between the binodals obtained from dynamic Monte Carlo simulations (data points) and from mean-field statistical thermodynamics (solid lines). First, one can see that even with zero mixing interactions B = 0, due to the contribution of Ep, the binodal curve is still located above the liquid-solid coexistence curve (dashed lines). This result implies that the phase separation of polymer blends occurs prior to the crystallization on cooUng. This is exactly the component-selective crystallizability-driven phase separation, as discussed above. Second, one can see that, far away from the liquid-solid coexistence curves (dashed lines), the simulated binodals (data points) are well consistent... 

We made a brief introduction about our current theoretical models of thermodynamics and kinetics of polymer crystallization. We first introduced basic thermodynamic concepts, including the melting point, the phase diagram, the metastable state, and the mesophase. The mean-field statistical thermodynamics based on a... 

The mean field statistical mixture tbnnulae have been mainly developed for dealing with the metal-insulator composites. The simplest statistical mixture fonnulae are developed by Bottcher [85J and extended to orientated ellipsoidal systems by Hsu [861 ... 

Two-Dimensional Hoppers and Mean Field Statistical Theories... 

In order to understand the thermodynamic issue associated with nanocomposite formation, Vaia et applied a mean-field statistical... 

In order to find approximate solutions of the equations for Ci t) and gi,..j t) one can use regular approximate methods of statistical physics, such as the mean-field approximation (MFA) and the cluster variation method (CVM), as well as its simplified version, the cluster field method (CFM) . In both MFA and CFM, the equations for c (<) are separated from those for gi..g t) and take the form... 

As Eq. (3) sh vs, the critical composition (ticn can be controlled by the asymmetry of chain lengths. Particularly interesting is the limit Na = N, Nb = I (which physically is realized by polymer solutions, B representing a solvent of variable quality). Checking the deviations from the mean field predictions, Eq. (3), further contributes to the understanding of the statistical mechanics of mixtures. 

For general rules, a first-order statistical approximation for limiting densities Pi t —> oo) can be obtained by a method akin to the mean-field theory in statistical mechanics (more sophisticated approaches will be introduced in chapter 4). 

The LST is a finitely parameterized model of the action of a given CA rule, >, on probability measures on the space of configurations on an arbitrary lattice. In a very simple manner - which may be thought of as a generalization of the simple mean field theory (MFT) introduced in section 3.1.3. - the LST provides a sequence of approximations of the statistical features of evolving CA patterns. 

Ri does not show any phase transition. This should not be terribly surprising, since, in the deterministic limit, Ri exhibits either class 2 behavior (i.e. periodicity) or class 4 behavior (spatially separated propagating structures with an ill-defined statistical limit). The density p therefore has no well-defined statistical mean for p = 1 and the periodicity and/or propagating structures are rapidly destroyed (and thus p — 0) whenever p < 1. Moreover, from the above mean-field... 

As previously discussed, we expect the scaling to hold if the polydisper-sity, P, remains constant with respect to time. For the well-mixed system the polydispersity reaches about 2 when the average cluster size is approximately 10 particles, and statistically fluctuates about 2 until the mean field approximation and the scaling break down, when the number of clusters remaining in the system is about 100 or so. The polydispersity of the size distribution in the poorly mixed system never reaches a steady value. The ratio which is constant if the scaling holds and mass is conserved,... 

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. 

In what follows, we use simple mean-field theories to predict polymer phase diagrams and then use numerical simulations to study the kinetics of polymer crystallization behaviors and the morphologies of the resulting polymer crystals. More specifically, in the molecular driving forces for the crystallization of statistical copolymers, the distinction of comonomer sequences from monomer sequences can be represented by the absence (presence) of parallel attractions. We also devote considerable attention to the study of the free-energy landscape of single-chain homopolymer crystallites. For readers interested in the computational techniques that we used, we provide a detailed description in the Appendix. ... 

A central issue in statistical thermodynamic modelling is to solve the best model possible for a system with many interacting molecules. If it is essential to include all excluded-volume correlations, i.e. to account for all the possible ways that the molecules in the system instantaneously interact with each other, it is necessary to do computer simulations as discussed above, because there are no exact (analytical) solutions to the many-body problems. The only analytical models that can be solved are of the mean-field type. 

Fattal, D. R. and Ben-Shaul, A. (1994). Mean-field calculations of chain packing and conformational statistics in lipid bilayers comparison with experiments and molecular dynamics studies, Biophys. J., 67, 983-995. 

We have seen that Lagrangian PDF methods allow us to express our closures in terms of SDEs for notional particles. Nevertheless, as discussed in detail in Chapter 7, these SDEs must be simulated numerically and are non-linear and coupled to the mean fields through the model coefficients. The numerical methods used to simulate the SDEs are statistical in nature (i.e., Monte-Carlo simulations). The results will thus be subject to statistical error, the magnitude of which depends on the sample size, and deterministic error or bias (Xu and Pope 1999). The purpose of this section is to present a brief introduction to the problem of particle-field estimation. A more detailed description of the statistical error and bias associated with particular simulation codes is presented in Chapter 7. 

Ideally, one would like to choose Np and M large enough that e is dominated by statistical error (X ), which can then be reduced through the use of multiple independent simulations. In any case, for fixed Np and M, the relative magnitudes of the errors will depend on the method used to estimate the mean fields from the notional-particle data. We will explore this in detail below after introducing the so-called empirical PDF. 

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