Mean-field calculation

This Blume-Eiiiery-GrifSths (BEG) model [74] has been studied both by mean field calculations as well as by simulations. There is no pronounced difference between the amphiphile molecules S= 0, the oil or the water. Indeed, the model was first suggested in a quite different context. An extension of the model by Schick and Shih [75] includes an additional interaction of tlie fomi... 

FIG. 13 Phase diagram of a vector lattice model for a balanced ternary amphiphilic system in the temperature vs surfactant concentration plane. W -I- O denotes a region of coexistence between oil- and water-rich phases, D a disordered phase, Lj an ordered phase which consists of alternating oil, amphiphile, water, and again amphi-phile sheets, and L/r an incommensurate lamellar phase (not present in mean field calculations). The data points are based on simulations at various system sizes on an fee lattice. (From Matsen and Sullivan [182]. Copyright 1994 APS.)... 

A phase diagram of the symmetric PS-fc-PI blended with PS homopolymer of shorter chain lengths was constructed by Bodycomb et al. [ 174]. The effect of blend composition on the ODT is shown in Fig. 56 along with the results of mean-field calculations. In analogy to MFT the addition of homopolymer decreases the ODT temperature for the nearly symmetric diblock copolymer. 

Fig. 56 Phase diagram of blend of PS-fi-PI with PS. T0dt. o TDMt, Toot- Vertical lines separating microdomain structures are obtained from total volume fraction PS in system. Dashed line results of mean-field calculation for ODT. The OOT line which exists at volume fractions ps 5 ub was obtained during a heating process. From [174]. Copyright 2000 American Chemical Society...

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. 

Scaling theories are restricted to long polymer chains in good solvents and do not include finite chain effects and polymer-solvent interactions. These models should be complemented by more detailed mean-field calculations and molecular simulations. 

Relativistic mean-field calculation as tabulated in Ref. [57] for a compression modulus K = 240 MeV ( RMF240 )... 

Figure 6. Mass-radius relation of different compact star configurations. The left panels correspond to calculations with parameter set RKH for the quark matter phase and the right panels to parameter set HK, respectively. From the upper panel downwards the hadronic phase is described by a BHF calculation without hyperons [55], a relativistic mean field calculation [57] and a chiral SU(3) model [53].

In the EA model the spin-spin interaction is only of the nearest-neighbor type. The Sherrington-Kirpatrick (SK) model [79] is the infinite-range version of the EA model. It is most useful as a basis for mean-field calculations. One such solution is the replica symmetry breaking theory of Parisi [80-82]. 

Schaink, H.M., Smit, J.A.M. (1997). Mean field calculation of polymer segment depletion and depletion-induced demixing in ternary systems of globular proteins and flexible polymers in a common solvent. Journal of Physical Chemistry, 107, 1004-1015. 

Fig. 6.52 Interfarial excess in thin films of blends of a dPS-P2VP diblock (A PS = 391, jVP2Vp = 68) with PS homopolymer (NK = 6440) (Dai et al. 1992). Since the homopolymer is much longer than the diblock, the diblock forms a dry brush. The circles are the results from forward recoil spectrometry, the lines correspond to theoretical calculations. The dashed line was computed using the theory of Leibler (1988), and the solid line is from the self-consistent mean field calculation of Shull and Kramer (1990).

Mean-field calculations [19] on the general model give a phase diagram similar to fig. (3), with a QCP at xc and an unusual time-reversal violating phase for the pseudogap region. 

The name, DLYO, originates from the first letter in the surname of the four authors (Derjaguin, Landau, Verwey and Overbeek) from two different groups, which originally published these ideas. The theory is based on the competition between two contributions, a repulsive electric double layer and an attractive van der Waals force [4,5]. The interaction in the electric double layer was originally obtained from mean field calculations via the Poisson-Boltzmann equation [Eq. (4)]. However, the interaction can also be determined by MC simulations (Sec. II. B) and by approximate integral equations like HNC (Sec. II. C). This chapter will focus on the first two possibilities. 

FIG. 16. The osmotic pressure (solid line) as a function of the salt concentration determined by mean field calculations. The dashed line is the contribution to the osmotic pressure from the kinetic terms and the dot-dashed line is the asymmetric term [60]. 

P0Sm =Pm— buik, may become attractive. The electrostatic ion-ion correlation is normally attractive. This effect arises because an ion influences the distribution of other ions, which results in a different osmotic pressure than in a mean field calculation. There exists a region where attractive and repulsive correlation may be balanced. This region is under ordinary conditions, however, smaller for ions of higher valency. Therefore, the deviation from PB is often reported for divalent or trivalent ions [69,70]. 

An attentive reader will note that this Ec is (slightly) different from the optimal energy derived from the Flory approach (Eq. 14). This is not a surprise the Flory (mean field) calculation is good to compute sizes, but less good to compute energies [15]. 

Here a = a t - 1), t = T/T°P, and a, b, c are either phenomenological parameters or else they can be calculated from mean-field calculations [48] or better yet, by using the Hubbard-Stratonovich transformation to convert the partition function into a functional integral [39]. In the latter case, one obtains around t = 1 the values... 

Fig. 21 Plots obtained by mean-field calculations for an EHFMI [24]. Calculations are performed for a two-dimensional 16x16 square lattice with open boundary conditions. Parameters used are U = St and t = —0.2t t denotes the second nearest neighbor transfer integrals tjk)- The number of doped holes is 8 half of them are centers of merons and the rest are centers of antimerons. (a) Plot for spin configuration. Centers of spin vortices are indicated as M for a meron (winding number -H spin vortex) and A for an antimeron (winding number —1 spin vortex), respectively, (b) Plot for current density j (short black arrows) and V x (long orange arrows). M and A here indicate centers of counterclockwise and clockwise loop currents, respectively (c) Plot for D(x), which connects j(x) and V/(x) as j(x) = D(x) V/(x) (d) Plot for 2j (thick orange line arrows are not attached but directions are the same as those of the black arrows) and 2Z)(x) V/(x) (black arrows)...

This estimate has been used to normalize the interface tension in Fig. 21. The collapse of the data for different chain lengths onto a common curve shows that the interface tension indeed only depends on the combination xN and the data are well described by numerical mean field calculations [107] and analytic predictions by Semenov [110]. 

Translational and Librational Amplitudes from Mean Field Calculations... 

Very recent Monte Carlo simulations and self consistent mean field calculations [223] have shown that wetting properties might be reflected in the phase diagram of a blend confined between symmetric selective surfaces Close to Tw a convex curvature is exhibited by the phase diagram on the side poor in preferentially adsorbed polymer. Also the temperature dependence of Ap changes around the wetting point Tsv. 

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