Interfacial profile

Figure B3.6.2. Local mterface position in a binary polymer blend. After averaging the interfacial profile over small lateral patches, the interface can be described by a single-valued function u r. (Monge representation). Thennal fluctuations of the local interface position are clearly visible. From Wemer et al [49].

For distances z from the surface smaller than f the order parameter amplitude A(z) basically takes already its value At at the transition, while for z = f there is a rather well developed interfacial profile, that resembles the interfacial profile between bulk ordered and disordered phases coexisting at %t [5],see Figs. 12,13. Writing. -j=z/q and using a solution in terms of waves cos (q z), sin (q z) as... 

Fig. 12. Schematic variation of the order parameter profile /(z) of a symmetric (f=l/2) diblock copolymer melt as a function of the distance z from a wall situated at z=0. It is assumed that the wall attracts preferentially species A. Case (a) refers to the case % %v where non-linear effects are still negligible, correlation length and wavelength X are then of the same order of magnitude, and it is also assumed that the surface "field" Hj is so weak that at the surface it only induces an order parameter 0.2 n if mb is the order parameter amplitude that appears for %=%t at the first-order transition in the bulk. Case (b) refers to a case where % is only slightly smaller than %t, such that an ordered "wetting layer" of thickness 1 [Eq. (76)] much larger than the interfacial thickness which is of the same order as [Eq. (74)] is stabilized by the wall, while the bulk is still disordered. The envelope (denoted as m(z) in the figure) of the order parameter profile is then essentially identical to an interfacial profile between the coexisting ordered phase at T=Tt for (zl). The quantitative form of this profile [234] is shown in Fig. 13. From Binder [6]...

D<16 the observed width w is even smaller than the intrinsic width a>oCF > i.e. the confinement effect on the interface is so strong that the intrinsic interfacial profile is squeezed In fact, in these ultrathin films one observes [266] a behavior... 

The proximal radial distribution functions for carbon-oxygen and carbon-(water)hydrogen in the example are shown in Fig. 1.11. The proximal radial distribution function for carbon-oxygen is significantly more structured than the interfacial profile (Fig. 1.9), showing a maximum value of 2. This proximal radial distribution function agrees closely with the carbon-oxygen radial distribution function for methane in water, determined from simulation of a solitary methane molecule in water. While more structured than expected from the... 

We conclude that the proximal radial distribution function (Fig. 1.11) provides an effective deblurring of this interfacial profile (Fig. 1.9), and the deblurred structure is similar to that structure known from small molecule hydration results. The subtle differences of the ( ) for carbon-(water)hydrogen exhibited in Fig. 1.11 suggest how the thermodynamic properties of this interface, fully addressed, can differ from those obtained by simple analogy from a small molecular solute like methane such distinctions should be kept in mind together to form a correct physical understanding of these systems. 

In order to discuss a number of important quantities we consider the interfacial profile for the case of positive adsorption. Such a profile is sketched in fig. 5.6. It represents the polymer concentration dz) as a function of the distance z from the interface. The quantity c(z) is related to the volume fraction concentration profile of segments belonging to free molecules, having no contact with the surface. The excess adsorbed amount r (the amount of... 

The above discussion does not exhaust modem approaches for interpreting surface tensions and interfacial profiles on the basis of van der Waals ideas. Improvements include ... 

Interfacial Profiling. Neutron depth profiling is well suited for measurements across interfacial boundaries. Kvitek et al. ( ) and others (16,17,21,30) have studied profiles of boron implanted and diffused across the interfacial region of Si/Si02. Other NDP experiments (33,34) have been described for interfaces of silicon, silicon dioxide or metal on metal, where diffusion distributions and segregation coefficients were studied. 

Fig. 6. Coarse-grained description of a liquid-gas interface, where the intrinsic profile and local structure of the interface is disregarded, and one rather treats the interface as an elastic membrane at position z h(x, v) ( sharp kink -approximation for the interfacial profile).

We return here to the simple mean field description of second-order phase transitions in terms of Landau s theory, assuming a scalar order parameter cj)(x) and consider the situation T < Tc for H = 0. Then domains with = + / r/u can coexist in thermal equilibrium with domains with —domain with exists in the halfspace with z < 0 and a domain with 4>(x) = +

0 (fig. 35a), the plane z = 0 hence being the interface between the coexisting phases. While this interface is sharp on an atomic scale at T = 0 for an (sing model, with = -1 for sites with z < 0, cpi = +1 for sites with z > 0 (assuming the plane z = 0 in between two lattice planes), we expect near Tc a smooth variation of the (coarse-grained) order parameter field (z), as sketched in fig. 35a. Within Landau s theory (remember 10(jc) 1, v 00 01 < 1) the interfacial profile is described by... 

Thus the thickness of the interfacial profile diverges in the same way as the correlation length does, when T approaches the critical temperature TQ. 

We return here to mean field theory with a scalar order parameter thick film geometry, assuming hard walls or surface against vacuum, respectively) at z = 0 and z = L. Starting again from eq. (14), we may disregard the x and y-coordinates [as in our treatment of the interfacial profile, eqs. (177)—(181)], but now we have to add a perturbation 2Fv(bare) to... 

As a first application for the use of the free energy functional, we will discuss the calculation of interfacial concentration profiles (x) between coexisting unmixed phases and interfacial tension [132-134]. For a symmetric mixture (Na = Nb = N) phase coexistence occurs for p = 0, and since the interfacial profile 4HX) also must be found by minimizing Eq. (47), we look for a solution of (cja = ctb = a)... 

The prefactor q4 in Eq. (166) describes the scattering from sharp interfaces ( Porods law I(q)ocq-4 [182]). In the strong segregation limit, the interfacial profile between unmixed phases (see also Sect. 2.3) is obtained as [132,305-313]... 

Werner A, Schmid F, Muller M and Binder K 1997 Anomalous size-dependence of interfacial profiles between ooexisting phases of polymer mixtures in thin film geometry a Monte-Carlo study J. Chem. Phys. 107 8175... 

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...

The total interfacial free energy per unit area, consists of the sum of /o and the free energy per unit area that comes from the liquid-vapor interface. In equilibrium, one minimizes the total free energy subject to the conservation constraint — i.e., one works at fixed chemical potential. As explained in the discussion of the gas-fiquid interface in Chapter 2, the appropriate bulk free energy to minimize to find the interfacial profile is the grand potential per unit area, gs, which is written ... 

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