Interfacial density profile

Fig. ni-7. (a) Interfacial density profile for an argonlike liquid-vapor interface (density in reduced units) z is the distance normal to the surface, (b) Variations of P-p of Eq. ni-40 (in reduced units) across the interface. [From the thesis of J. P. R. B. Walton (see Ref. 66).]... 

The relationship between (r) for water oxygen atoms (Fig. 1.11) and the oxygen atom interfacial density profile (Fig. 1.9) can be established by superposing these proximal radial distribution functions to model the conditional densities as... 

R-t,R) = interaction potential between an adsorbate molecule at r and the entire sohd surrounding the pore, while (f)FF(r,R) represents the corresponding potential if the bulk fluid was instead present in place of the surrounding sohd. Both these potentials incorporate the interfacial density profile. Using an analysis similar to that described in section 2.1, the equations for estimating the critical thickness are given by... 

Before leaving the subject of interfacial behavior in polymers, it is instructive to consider the interfacial tension, and resulting interfacial density profiles. Making effective use of the Flory interaction parameter x, Helfand and Tagami (1972), Gaines (1972), Wu (1974), and others estimated the interfacial surface tension between incompatible polymer pairs (see Table 13.1). Also shown in Table 13.1 are theoretically estimated values of x-(See Section 4.7 and especially Sections 4.7.3 and 9.6 for related discussion.) Helfand and Tagami found that the characteristic thickness of the interface is proportional to x — y for small /. For a polystyrene/poly(methyl methacrylate) system, the value of / leads to an estimated interfacial thickness of 50 A. This value is much less than that estimated by Voyutskii and Vakula... 

Figure 7. The interfacial density profiles for the system hexane propanol at Xp =0.80 and 298.15 K, computed with the PR equation of state. The solid lines denote the profiles at p 2=0.01, while the dashed lines represent the results at Pi2=0.10.

In Figure 7 two different interfacial density profiles are shown, of which the first is obtained with p 2=0.01 (solid curve), while the second is obtained with Pi2=0.10... 

Figure 7.1 Interfacial density profile AplAp( zb 00) as a function of z/. —, the mean-field (van der Waals) approximation given by eq 7.21 . , calculated in renormalization-group theory and given by eq 7.70.

Abraham, F. F. 1978. The interfacial density profile of a Lennard-Jones fluid in contact with a (100) Lennard-Jones wall and its relationship to idealized fluid/wall systems A Monte Carlo simulation. J. Ghent. Phys. 68 3713. 

Just as for the interfacial density profile itself, two very different points of view on the nature of interfacial correlations have been advocated, the first relying on local thermodynamic concepts in the spirit of the van der Waals theory, the second relying on capillary wave concepts. 

For a planar interface, it is useful to consider the singleparticle density averaged over the spatial directions parallel to the interface - this is the interfacial density profile. Taking the z direction to be perpendicular to the interface gives... 

Another feature seen in the fine-scaled interfacial density profiles is that the crystal layers tend to relax outward into the fluid region, so that the spacing of the peaks in the density oscillations increases. This phenomenon has also been observed in recent and MD simulations of the singlecomponent hard sphere interface, where it appears to be related to the preference of the fluid to order at the interface in a way that is more consistent with a [111] face. 

Second-Order Integral Equations for Associating Fluids As mentioned above in Sec. II A, the second-order theory consists of simultaneous evaluation of the one-particle (density profile) and two-particle distribution functions. Consequently, the theory yields a much more detailed description of the interfacial phenomena. In the case of confined simple fluids, the PY2 and HNC2 approaches are able to describe surface phase transitions, such as wetting and layering transitions, in particular see, e.g.. Ref. 84. 

FIG. 20 (a) Density profiles p(z) vs z for e = —2 and four average bulk densities (f> as indicated, (b) Surface excess vs density in the bulk for four choices of e. (c) Profiles for the diagonal components of the pressure tensor and of the total pressure for (p = l.O and e = —2. Insert in (c) shows the difference between P, and Px to show that isotropic behavior in the bulk of the film is nicely obtained, (d) Interfacial tension between the polymer film and the repulsive wall vs bulk density for all four choices of e. Curve is only a guide for the eye [18]. 

When the two phases separate the distribution of the solvent molecules is inhomogeneous at the interface this gives rise to an additional contribution to the free energy, which Henderson and Schmickler treated in the square gradient approximation [36]. Using simple trial functions, they calculated the density profiles at the interface for a number of system parameters. The results show the same qualitative behavior as those obtained by Monte Carlo simulations for the lattice gas the lower the interfacial tension, the wider is the interfacial region in which the two solvents mix (see Table 3). 

This contribution involves the positive-ion and electron density profiles of the metal, and the former is often assumed not to change with charging of the interface. In 1983 and 1984, several workers30-32,79 showed how certain features of the interfacial capacity curves should depend on the metal. 

Once the free energy of an inhomogeneous system is given, one can calculate by standard methods the properties of the interface—for example, the interfacial tension or the density profile perpendicular the interface [285]. Weiss and Schroer compared the various approximations within square-gradient theory discussed earlier in Section IV.F for studying the interfacial properties for pure DH and FL theory [241, 242], In theories based on local density approximations the interfacial thickness and the interfacial tension were found to differ by up to a factor of four in the various approximations. This contrasts with nonionic fluids, where the density profiles and interfacial... 

Figure 1.2 Number density profiles for the carbon atoms, water oxygen atoms, and methanol carbon atoms for the 10% methanol case. The alkane carbon density is on the left, and the high-density profile is for water oxygen on the right. Distances along z perpendicular to the interfacial plane are given in A. Densities are given in arbitrary units.

This system displays (Fig. 1.9) a traditional interfacial oxygen density profile that has been the object of measurement (Pratt and Pohorille, 2002), monotonic with a width 2-3 times the molecular diameter of a water molecule. This widthis somewhat larger than that of water-alkane liquid-liquid interfaces, though it is still not a broad interface. The enhanced widthis probably associated withroughness of the stationary alkyl layer the carbon density profile is shown in Fig. 1.9 as well. 

The molecular-level stmcture of the electrode/electrolyte interface was studied using two- and three- phase systems, including membrane/vapor, membrane/vapor/catalyst and membraneAfapor/ graphite systems. The simulations of a membraneAfapor interface show a region of dehydration near the interface. The interfacial thickness measured from the water density profile was found to decrease in width with increasing humidity. Hydronium ions displayed a preferential orientation at the interface, with the oxygen exposed to the vapor phase. 

Monte Carlo and molecular dynamics calculations of the density profile of model system of benzene-water [70], 1,2-dichloroethane-water [71], and decane-water [72] interfaces show that the thickness of the transition region at the interface is molecu-larly sharp, typically within 0.5 nm, rather than diffuse (Fig. 4). A similar sharp density profile has been reported also at several liquid-vapor interfaces [73, 74]. The sharpness of interfaces thus seems to be a general characteristic of the boundary between two stable phases and it is likely that the presence of supporting electrolytes would not significantly alter the thickness of the transition region at an ITIES. The interfacial mixed solvent layer [54, 55], if any, would probably have a thickness comparable with this thin inner layer. 

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