Mean field theory free energy

This is the well known equal areas mle derived by Maxwell [3], who enthusiastically publicized van der Waal s equation (see figure A2.3.3. The critical exponents for van der Waals equation are typical mean-field exponents a 0, p = 1/2, y = 1 and 8 = 3. This follows from the assumption, connnon to van der Waals equation and other mean-field theories, that the critical point is an analytic point about which the free energy and other themiodynamic properties can be expanded in a Taylor series. 

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

The assumption that the free energy is analytic at the critical point leads to classical exponents. Deviations from this require tiiat this assumption be abandoned. In mean-field theory. 

Theoretically, several aspects of the Thommes-Findenegg experiment can be analyzed at the mean-field level [157]. A key quantity of a mean-field theory of confined fluids is the (Helmholtz) free energy, given by... 

The validity of mean field theory for N —y oo has striking consequences for the initial stages of phase separation. " In a metastable state slightly inside the coexistence curve, the nucleation free energy barrier is due to spherical droplets with a radius R The free energy excess of a droplet is written in terms of bulk and surface terms " "... 

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. ... 

To demonstrate that equation (3) describes a mean-field theory of gas-liquid transitions it will be shown how it can be obtained by minimizing a Landau free-energy function. This objective is achieved by working backwards. 

Semianalytical mean-field theories of block copolymer micellization were formulated by Noolandi et al. [ 197] and by Leibler et al. [198]. In the approach of Noolandi et al., the micellar characteristics were obtained through a minimization of the Gibbs free energy for an isolated micelle. This was applied to PS-PB micelles, and the obtained theoretical values were in good agreement with the experimental ones. 

Fig. 4 a Mean-field result (solid line) for the rescaled brush free energy per polymer as a function of the inverse interaction parameter 1/F- The infinite stretching resnlt is indicated by a horizontal dotted line, the broken straight line denotes the infinite stretching result with the leading correction dne to the finite end-point distribntion entropy. b Rescaled lateral pressnre within mean-field theory (solid line) compared with the asymptotic infinite-stretching result (dotted line)... 

A simple mean field theory for micelle formation by a diblock copolymer in a homopolymeric solvent was developed by Leibler et al. (1983). This model enables the calculation of the size and number of chains in a micelle and its free energy of formation. The fraction of copolymer chains aggregating into micelles can also be obtained. A cmc was found for low copolymer contents even for weak incompatibilities between components. Leibler et al. (1983) emphasize that fora finite aggregation number p, the cmc is a region rather than a well-defined concentration and some arbitrariness is involved in its definition. 

In Eqs. (19)-(21), always the sign has to be chosen which yields the absolute minimum of the free energy. As usual, mean field theory yields metastable branches, and the phase separation from one state 4>coex (D) to the other state I coex (D) in this treatment shows up as an intersection of two branches for AF/kBT when plotted vs Ap or the conjugate variable A=b- [Pg.14]

For this problem already the simple mean field approximation becomes rather involved [197,213]. Therefore, we describe here only an approach, which is even more simplified, appropriate for wavenumbers q near the characteristic wavenumber q, but strictly correct neither for q—>0 nor for large q the spirit of our approach is similar to the long wavelength approximation encountered in the mean field theory of blends, Eq. (7). That is, we write the effective free energy functional as an expansion in powers of t t and include terms (Vv /)2 as well as (V2 /)2, as in the related problem of lamellar phases of microemulsions [232,233],namely [234]... 

Before setting out on the exact mean field theory solution to the one-dimensional colloid problem, I wish to emphasize that the existence of the reversible phase transition in the n-butylammonium vermiculite system provides decisive evidence in favor of our model. The calculations presented in this chapter are deeply rooted in their agreement with the experimental facts on the best-studied system of plate macroions, the n-butylammonium vermiculite system [3], We now proceed to construct the exact mean field theory solution to the problem in terms of adiabatic pah-potentials of both the Helmholtz and Gibbs free energies. It is the one-dimensional nature of the problem that renders the exact solution possible. 

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