Capillary-wave divergences

According to the capillary wave approximation , a rough interface is characterized by a width that diverges logarithmically with its transverse dimensions ... 

Notice that this capillary wave term alone cannot represent the whole correlation because it logarithmically diverges as Ko(es) — log(es) for s -> 0. The singularity 1/p can also be derived from the condition of existence of an inhomogeneous solution to the LMBW equation (2). Indeed, the matrix equation (40) has non-tiivial solutions p 0 at ext -> 0 if and only if... 

Capillary waves do not only broaden the width of the interface but they can also destroy the orientational order in highly swollen lamellar phases (see Fig. 1 for a phase diagram extracted from Monte Carlo Simulations). Those phases occur in mixtures of diblock-copolymers and homopolymers. The addition of homopolymers swells the distance between the lamellae, and the self-consistent field theory predicts that this distance diverges at Lifshitz points. However, general considerations show that mean-field approximations are bound to break down in the vicinity of lifshitz points [61]. (The upper critical dimension is du = 8). This can be quantified by a Ginzburg criterion. Fluctuations are important if... 

The capillary-wave argument above leads to a divergence of the surface thickness as (-In g)i, and any divergence of p (z) wfll be no... 

The divergence of is due to the low-wave-number modes. In two dimensions (one-dimensional interface), at g = 0 and L = oc, the contribution to made by all capillary waves of wave number greater than any prescribed q is kThmq. In three dimensions, again at g = 0 and L = x, it... 

In summary, assuming the equilibrium structure of the fluid interface to result from averaging capillary wave excitations on an intrinsic interface, it is found that while the external field does not affect the divergence of the interfacial thickness in the critical region of fluids in three or more dimensions (except, of course, extremely close to the critical point ), its effect is dramatic in two dimensions, where the critical behavior is found to be non-universal, i.e., depending on the external field. Consequently, the relation p = (d-Do>, which links the critical exponents of surface tension and interfacial thickness to the dimension of space and which is most probably correct in d > 3, appears to be incorrect in d = 2, since there co, unlike p, is strongly field-dependent. ... 

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. 

Fig. 4. Theoretical surface divergence for a 15 Hz capillary-gravity wave. Curves are shown for an ideal inviscid flow, a real flow with zero viscoelasticity, and a surfactant flow with a finite viscoelasticity. Each curve is along the phase of each wave, with the horizontal position at zero corresponding to a wave crest. The wave amplitude wavelength ratio is 1 20...

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