Bulk correlation length

This formula may also well approximate the strict solution (Eq. 30) for larger values of bulk concentration but in this case its decay length is no longer equal to the bulk correlation length The mean field theory also makes a prediction for the total surface excess z ... 

Fig. 67. Order parameter profiles m(z)/mt, associated with surface-induced disorder. The coordinate z measures the distance from the surface (z = 0). is the bulk correlation length and mu the bulk order parameter. If ease (a) persists up to the first-order transition temperature Tc, this means the surface stays ordered up to 7 c, while case (b) shows surface induced disordering a layer of thickness L gets disordered already at T <71, and as T — Tc the (delocalized) interface at mean position z = L from the surface advances into the bulk, fJ(7 ) -+ oo as T - Tc, and the surface order parameter mj = m(z = 0) then vanishes continuously, mj a (1 - T/Tc). From Dasch et al. (1988).

Figure 2.2 Interfacial profile where > is equal to the order parameter, ifr, scaled to its bulk value, o. and m = z/, the bulk correlation length. The width of the interfacial region centered at u = 0 scales with...

We are quite familiar with the fact that the bulk properties of ordinary coexisting fluid phases determine the structure of the interface between them. For example, the thickness of the interfacial profile is set by the scale of the bulk correlation length. Similarly, the proper-... 

Figure 5.5. The surface enrichment profile (solid line) of deuterated poly(ethylene propylene) at the surface of a blend with normal poly(ethylene propylene), measured by neutron reflectivity. The degree of polymerisation of each component was 2250 and the sample was annealed at 70 °C. The depth is normalised by the bulk correlation length and the dashed line is the prediction of equation (5.1.11). After Norton et al. (1995).

Another way of thinking about this result draws on the notion we introduced in section 5.1 that the concentration perturbation introduced by the surface decays to the bulk over a distance characterised by the bulk correlation length. For a system such as a symmetrical polymer blend, for which the correlation length is a weak function of the concentration, this leads to an exponential adsorption profile... 

Note that all these regimes occur even when the bulk concentration Cb is in the dilute range. When Cb is in the semi-dilute regime, the above discussion is valid for distances smaller than the bulk correlation length. For larger distances, the concentration is the average bulk concentration. There is still however an open question related to the normal extension of the polymers are they localized in the vicinity of the... 

Since we assumed D this means that the central portion of the tube is unaffected, and reaches a concentration d> equal to the bulk value. On the other hand, if Z> the depletion layers occiq)y all the tube volume, and d> < d>B. Thus we are led to the following rule upon increasing the bulk concentration < >, penetration in the pore occurs when the bulk correlation length = a becomes comparable to the pore... 

It can indeed be shownthat for continuous fluids in three or fewer dimensions of space in which the molecules interact via short-ranged forces, such as Lennard-Jones interactions, no density profile exists on the scale of the bulk correlation length in the absence of a macroscopic external field. 

In two dimensions, however, the situation will prove to be very different from what it is in three or more dimensions whereas van der Waals theory predicts o) = 1, which follows at once from the postulated identification a> = v, with v the critical exponent describing the divergence of the bulk correlation length and the exact... 

When one chooses next a more diffuse intrinsic profile such as an exponential profile with a decay length equal to the bulk correlation length K becomes simply ... 

The few available experimental data had not enabled one to discriminate between the vastly diverging points of view advocated in the van der Waals and capillary wave theories. Although no reflectivity measurements have been performed very close to the critical point since the pioneering studies of Webb and his co-workers,new measurements and analyses of the critical parameters (bulk correlation length and surface tension) of the systems originally studied by Webb et al.. have been performed recently and very accurate values for both quantities, which differ appreciably from those used in earlier work, have recently become available. 

It is convenient to estimate the root-mean-square displacement of the interface ()2 in units of the bulk correlation length In the one-phase. region, the correlation length behaves as with t (T... 

From Eqs. (43) and (44), the bare surface tension can be expressed in units of the bulk correlation length... 

For the cyclohexane-methanol system, the relevant data are th bulk correlation length amplitude = 3.24 i, the critical temperature T = 318 K, and the interfacial thickness deduced from the reflectivity data, L = 12.4 (- ) [A] = 4. The corresponding effective ratio is R r = Z.29. ... 

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