Edwards’ mean field theory

As mentioned above, Edwards has used a self-consistent mean field theory to derive similar (though not identical) results to those obtained from scaling theory (Edwards, 1966 Edwards and Jeffers, 1979). One virtue of this approach is that the relationships were obtained from a single extrapolation formula which encompasses all concentrations from an infinitely dilute solution to bulk polymer. This means that a somewhat more intuitively acceptable picture of polymer solutions emerges, one where there is a gradual change from one concentration regime to another. 

Before discussing Edwards asymptotic solution to Eq. (3.72) with (3.82) - - obtained for the polymer case - - it is convenient to discuss his simpler mean field theory, in which the mean-field is constant in certain regions of space (Edwards (1970a, b)). 

As noted above, the SCF theory of the conductivity deserves further attention. Nevertheless in the case of the simple mean field theory of Section 3.8.5, Edwards (1970b) notes that the conductivity is identically zero for the localized states of E < 0. The details proceed rather simply as before. In the Kubo-Greenwood-Peierls (Kubo (1956) (1957)) expression for theton-ductivity, it is necessary to obtain the average of the product of two Green s functions... 

Now we use this simple mean field theory of Edwards. Given (s) initially, the thermal Green s function for R(j) is given by... 

Gee RFI, Lacevic N, Fried LE (2005) Atomistic simulations of spinodal phase separation preceding poiymer crystaiiization. Nat Mater 5(l) 39-43 Gupta AM, Edwards SF (1993) Mean-field theory of phase transitions in liquid-crystalline polymers. J Chem Phys 98(2) 1588-1596... 

In spin glasses critical behavior near Tj is not expected in the linear term of the susceptibility M/H, but in the nonlinear susceptibility This is known from mean-field theory of spin glasses (Suzuki 1977) where the order parameter is not the magnetization but the quantity = S, [(5,) ] as suggested by Edwards and Anderson in 1975 (see sec. 4). Then, the field conjugate to the order parameter q is in spin glasses (instead of H which is coupled to the order parameter A/ in a... 

Up to now no realistic model of a spin glass has been solved analytically. The simple model proposed by Edwards and Anderson (1975) (sec. 4.1), however, is shown by Monte Carlo simulations to reproduce many experimental findings on spin glasses remarkably well (sec. 4.2). Its mean-field theory, as realized in the Sherrington-Kirkpatrick (SK) model (1975) (sec. 4.3), is now fairly well understood (which has taken about eight years). The solution proposed by Parisi (1979) yields a rich structure in the ordered phases, described by an infinite number of... 

Gupta AM, Edwards SF. Mean-field theory of phase transitions in hquid-crystalhne polymers. J Chem Phys 1993 98 1588-1596. 

An almost identical conclusion was obtained analytically by way of the self-consistent field theory. Edward [23] showed Pcy j 9/5 for d = 3 in good accord with the numerical calculation mentioned above, together with the mean end-to-end distance that scales as... 

Doi first proposed the generalized dynamic equations for the concentrated solution of rod-like polymers. Such constitutive equations can be derived from the molecular theory developed by Doi and Edwards (1986). The basis for the molecular theory is the Smoluchowski equation or Fokker-Planck equation in thermodynamics with the mean field approximation of molecular interaction. 

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 effects of fluctuations in such systems are small because each molecule contains so many monomers. Hence the mean-field, or self-consistent field, theory of such systems is expected to be quite good over a large part of the parameter space. The self-consistent theory was introduced by Edwards [164] and further developed by Helfand and coworkers [165]. It is easily derived as follows. 

The microphase separation of block copolymers is sometimes called order-disorder transition (ODT). The self-consistent-field theory (SCFT) provides a mean-field method to calculate various geometric shapes of microdomains. Edwards first introduced the SCFT into polymer systems on the basis of making path integrals along chain conformations (Edwards 1965). Helfand applied it to the mean-field description of immiscible polymer blends on the basis of the Gaussian-chain model... 

One of the most powerful methods to assess such phenomena theoretically is the self-consistent field (SCF) theory. Originally introduced by Edwards [8] and later Helfand et al. [9], it has evolved into a versatile tool to describe the structure and thermodynamics of spatially inhomogeneous, dense polymer mixtures [ 10-13]. The SCF theory models a dense multi-component polymer mixture by an incompressible system of Gaussian chains with short-ranged binary interactions and solves the statistical mechanics within the mean-field approximation. 

In the present introduction to Mesodyn we assume the reader has had some exposure to statistical thermodynamics and Flory-Huggins theory, but otherwise we do not suppose familiarity with the typical functional mathematical language of colloid and polymer physics. The introduction chapter to this book contains an extensive list of references to morphologies in block copolymer systems, and mean-field calculations, as in Hamley s book [1]. Here we focus on the work done in our own group [2,3]. General introductory books are those of de Gennes [4] and Doi and Edwards [5]. An excellent recent review paper dealing with dynamical field models is available [6]. 

The first attempt to develop a kinetic theory for concentrated systems using a mean-field approach was that of Doi and Edwards. They studied the motion of a polymer molecule... 

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