The correlation length

The above presentation of scaling for thermodynamic properties is direct but not illuminating. A much better picture of what happens in semi-dilute solutions can be obtained if we investigate spatial properties. Consider the solution shown in Fig. III.4. When photographed at a certain time, this looks very much like a network with a certain average mesh size 

We can use neutron scattering, to measure but we can also use a simple idea fu st suggested by H. Benoit. This amounts to adding a small number of inert spheres with diameter D 50-100 A. When D f, we expect the spheres to move easily, with a friction coefficient which is essentially related to the viscosity of the pure solvent. When D , the spheres are trapped— the effective viscosity controlling their friction is closer to the viscosity of the entangled solution.  

Let us now construct the scaling form of in the semi-dilute regime for a good (athermal) solvent. This is based on two requirements  

the mesh size decreases rapidly with concentration. Note an inter- 

We now focus on one particular chain in the semi-dilute solution this could be, for example, one chain labeled by deuteration with all the other chains being normal. We may visualize it as a succession of units or blobs of size (Fig. ni.5). Inside one blob, (from the defmition of the mesh size) the chain does not interact with other chains. Thus, inside one blob we must still have correlations of the excluded volume type. This implies that the nimber of monomers per blob (g) is related to by the law of swollen coils  

The correlation length in the semi-dilute regime is given by 

In dilute solutions can be set equal to The validity of this equality can be readily seen at a polymer concentration close to Ci, when the coilsjust touch but do not interpenetrate. Recalling that Ci N [see equation (4.36)], we have 

In the semi-dilute regime, the mesh size depends only upon the concentration of polymer, at least for polymers of high molecular weight. This seems intuitively reasonable because the mesh size should be determined solely by the total segmental concentration, being independent of chain length if that 

The corresponding analysis for a 0-solvent differs only insofar as C2 N and Rg° N. These relationships lead to m= — 1, i.e. i C2 . 

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For the case of a double-D coil we multiply each matrix element with an element shifted by a constant distance of the same line. This is done in x- and y-direction. The distance between the two elements is the correlation length X for filtering in x-direction and a second correlation length for the movement in y-direction. Thus one gets two new matrices Ax and Ax for the filtering from the left to the right (positiv x-direction) and vice versa (negativ x-direction). 

We present two optical methods for characterizing wire surfaces. These methods allow us to measure the roughness and the correlation length of the surface. It is also possible to identify qualitatively, at a glance, the variations of the roughness along a wire or among its different zones. 

The divergence m the correlation length is characterized by the critical exponent v defined by... 

The correlation length C = T -T diverges with the exponent v. Assuming that when T>T the site... 

The correlation length follows from the above relation, since... 

Based on this equation, one can make a Debye-Bueche plot by plotting [i (q)] versus q and detemiine the slope and the intercept of the curve. The correlation length thus can be calculated as [21]... 

The quantitative analysis of the scattering profile in the high q range can be made by using the approach of Debye et aJ as in equation (B 1.9.52). As we assume tiiat the correlation fiinction y(r) has a simple exponential fomi y(r) = exp(-r/a ), where is the correlation length), the scattered intensity can be expressed as... 

A fingerprint of a continuous phase transition is the divergence of the correlation length at the critical temperature 7", with for... 

All the above scaling relations have one common origin in the behavior of the correlation length of statistical fluctuations, in a finite system [140,141]. Namely, the most specific feature of the second-order transition is the divergence of at the transition point, as is described by Eq. (22). In the finite system, the development of long-wavelength fluctuations is suppressed by the system size limitation can be, at the most, of the same order as L. Taking this into account, we find from Eqs. (22) and (26) that... 

One can easily extend the above analysis to dilute and semi-dilute solutions of EP [65,66] if one recalls [67] from ordinary polymers that the correlation length for a chain of length / in the dilute limit is given by the size R of the chain oc When chains become so long that they start to overlap at I I (X the correlation length of the chain decreases and reflects... 

In the real space the correlation function (6) exhibits exponentially damped oscillations, and the structure is characterized by two lengths the period of the oscillations A, related to the size of oil and water domains, and the correlation length In the microemulsion > A and the water-rich and oil-rich domains are correlated, hence the water-water structure factor assumes a maximum for k = k 7 0. When the concentration of surfac-... 

Finally, we assume that the fields 4>, p, and u vary slowly on the length scale of the lattice constant (the size of the molecules) and introduce continuous approximation for the thermodynamical-potential density. In the lattice model the only interactions between the amphiphiles are the steric repulsions provided by the lattice structure. The lattice structure does not allow for changes of the orientation of surfactant for distances smaller than the lattice constant. To assure similar property within the mesoscopic description, we add to the grand-thermodynamical potential a term propor-tional to (V u) - -(V x u) [15], so that the correlation length for the orientational order is equal to the size of the molecules. 

Ising model harbors a critical point. It can be shown (see [bax82]) that the correlation length = [ln(Ai/A2)] h If H = 0, however, then it can also be shown that limy g+(Ai/Aj) = 1 and, thus, that oo at // = 7 = 0. Since one commonly associates a divergent correlation length with criticality, it is in this sen.se that 7/ = T = 0 may be thought of as a critical point. 

Just as in phase transitions in statistical mechanical systems, observable quantities in PCA systems display singularities obeying simple power laws with universal critical exponents at the transition point. For example, letting ni be the number of sites with correlation length, and t be the correlation time, Kinzel [kinz85b] finds that for p ... 

Figure 6-14. Average domain size vs. inverse deposition temperature Tor different film thicknesses. Error bars represent the mean absolute error and straight lines the best lit for each film thickness. Doited line is the locus of the transition from grains to lamellae. Data for 50-nm films are estimated from the correlation length of the topography fluctuations. Adapted from Ref. [501.

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