Ginzburg number

As a result of long-range fluctuations, the local density will vary with position in the classical Landan-Ginzburg theory of fluctuations this introdnces a gradient tenn. A Ginzburg number N is defined (for a... 

With some algebra this condition becomes, in d = 3 dimensions (x = l-x/Xcrii) Rgy, a/n or x /Gi 1, where we have introduced the "Ginzburg number"... 

Certain difficulties remain, however, with this approach. First, such an important feature as a secondary structure did not find its place in this theory. Second, the techniques of sequence design ensuring exact reproduction of the given conformation are well developed only for lattice models of polymers. The existing techniques for continuum models are complex, intricate, and inefficient. Yet another aspect of the problem is the necessity of reaching in some cases beyond the mean field approximation. The first steps in this direction were made in paper [84], where an analog of the Ginzburg number for the theory of heteropolymers was established. 

We will demonstrate below that the Ginzburg number Gi = AT/Tc), which determines the broadness of the energy region near the critical temperature, where fluctuations essentially contribute, is Gi A(Tc/iiq)4 with A 500 in our case. To compare, for clean metals A 100, p,q — fi,., the latter is the electron chemical potential. Thus Gi 1, if Tc is rather high, Tc (f -t- )p,q, and we expect a broad region of temperatures, where fluctuation effects might be important. 

As an essential feature of these theories, crossover behavior is governed by two physical parameters [317] (1) a scaled coupling constant u which reflects the strength and range of the intermolecular forces as represented by 0 and (2) a cutoff wave number A which is assumed to be inversely proportional to a structural length When = 0, one has only one length scale, and one recovers the Ginzburg number with Afa oc (wA)2. 

Considering now results from DH theory, Eq. (18) leads to a very small Ginzburg number [92], apparently confirming rapid crossover. However, the GDH theory leads to a Ginzburg number, which is large, even if compared to values for ordinary non-ionic fluids [94], The same result is obtained when applying Eq. (19) or when the density dependence of the pair correlation function in Eq. (18) [95] is taken into account. It is quite tricky that the predictions for ATq depend sensitively on the approach by which C2g is calculated. Note, the differences of the estimates of C2g become negligible in non-ionic and non-polar fluids because, 13(f) (r) dominates, which vanishes in ionic and polar fluids. 

D. Schwahn, G. Meier, K. Mortensen, and S. Janssen (1994) On the N-scaling of the ginzburg number and the critical amplitudes in various compatible polymer blends. J. Phys. II (France) 4, pp. 837-848 H. Frielinghaus, D. Schwahn, L. Willner, and T. Springer (1997) Thermal composition fluctuations in binary homopolymer mixtures as a function of pressure and temperature. Physica B 241, pp. 1022-1024... 

The lattice gas (Ising model), the simplest model that describes condensation of fluids, has played an important role in the theory of critical phenomena [1] providing crucial tests for most basic theoretical concepts. Recently, accurate numerical results for the crossover from asymptotic (Ising-like) critical behavior to classical (mean-field) behavior have been reported both for two-dimensional [29, 30] and three-dimensional [31] Ising lattices in zero field with a variety of interaction ranges. The Ginzburg number, as defined by Eq. (36), depends on the normalized interaction range R = as... 

For a three-dimensional Ising lattice A = tt, so that the Ginzburg number... 

Macrophase separation can also occur in random copolymers [64] which consist of a random sequence of Q blocks each of which comprises either M segments of type A or M segments of type B. Macrophase separation occurs when xM is of order unity, i.e., independent from the number of blocks, and the A-rich and B-rich phases differ in their composition only by an amoimt of order 1/VQ- The strength of fluctuation effects can be quantified by the Ginzburg number [65]... 

The coupling constant u controls the magnitude of the corrections to the asymptotic power-law behaviour as can be seen from Table 10.8. The Ginzburg number Nq is a measure of the value of the temperature variable t, where the crossover from Ising-like to mean-field critical behaviour occurs. 

In these equations k, a, ct, d ( = Pu in eq 10.29), and g are system-dependent coefficients with g being related to the inverse of the Ginzburg number AIq. Slightly different versions for the crossover function R q) have also been used. In the critical limit 0 one recovers the linear-model parametric equation in Section 10.2.2 with coefficients a and k. In the classical limit q-rcc. Ad becomes an analytic function of AT and Ap. For a comparison of this phenomenological parametric crossover equation with the crossover Landau models the reader is referred to some previous publications. " " ... 

In order to clarify the relation between the phase behavior, interactions between droplets, and the Ginzburg number, we have undertaken further SANS studies of critical phenomenon in a different three-component microemulsion system called WBB, consisting of water, benzene, and BHDC (benzyldimethyl-n-hexadecyl ammonium chloride). This system also has a water-in-oil-type droplet structure at room temperature and decomposes with decreasing temperature. Above the (UCST) phase separation point, critical phenomena have been investigated by Beysens and coworkers [9,10], who obtained the critical indexes, 7 = 1.18 and v = 0.60, and concluded that their data could be interpreted within the 3D-Ising universality. However, Fisher s renormalized critical exponents were not obtained. 

Fig. 2 All the observed temperature dependence of the inverse susceptibility of droplet density fluctuation for the WBB system is shown. The vertical axis indicates the inverse of the renormalized susceptibility and the horizontal the renormalized temperature f = r/Gi. The pure Ising region is characterized by the reduced temperature which is smaller than the Ginzburg number i.e. f < 1...

WBB system) do not require such an interaction. The present results from the WBB system indicate that the droplets scatter rather uniformly and the Ginzburg number is of the same order as that of simple fluids and are therefore consistent with its phase behavior. 

Experimentally determined susceptibility, 1(0) versus T, like that shown in Figure 7.11, can effectively be analyzed by applying a single function describing 1(0) within the whole one-phase regime. Based on an e-expansion model, one may develop a function that describes the experimental data very well [13], as seen in Figure 7.11. The parameters of the crossover Junction are the Ginzburg number, the critical temperature, and the critical exponents. 

A more quantitative description of this problem shows that one can describe this crossover by comparing the reduced temperature distance t to the so-called Ginzburg number" Gi, which for the Flory-Huggins theory of a symmetric polymer mixture reduces to" ... 

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