Density profile, across interface

Figure LI Density profile across the interface, defining thickness of the interphase.

The density profile across the interface follows an exponential decay (see Figure 1.1). The intercepts of the steepest tangential line with the horizontal lines defining the volume fraction of either one of the two polymeric ingredients, (p = 0 and 1, define the thickness of the interphase, Al [Helfand and Tagami, 1971, 1972]. Experimentally Al varies from 2 to 60 run [Kressler et al., 1993 Yukioka and Inoue, 1993, 1994]. Measurements of Al have been recently used to map the miscibility region of PC/SAN blends when varying the AN-content and temperamre [Li et al, 1999]. 

Lattice theory predicts that the density profile across the interface follows an exponential decay function [Helfand andTagami, 1971, 1972] ... 

Figure 9.12. Computed segmental density profile across the interface. The figure defines the interphasial thickness, Al [after Helfand and Tagami, 1971].

Matching the lubrication equation to thermodynamic theory requires some caution, since thermodynamic theory yielding an expression for pL should be applied to the entire system including dense (liquid) and dilute (vapor) phases in equilibrium, whereas only the dense phase may have a suitable aspect ratio. To make the approximation applicable, one has to assume that the interface dividing the dense and the dilute phase is only weakly inclined relative to the substrate and weakly curved, so that its position can be expressed by a function h x, t) with derivatives obeying the above lubrication scahng. Thermodynamic theory, either local or nonlocal, can be used to compute an equilibrium density profile across the interface (in the vertical direction), po z — h x, t)), which is weakly dependent on the horizontal 2D position and time only through its dependence on h, e.g. 

The first information eoming from the application of the method regards the density profile across the interface. Density may be assumed to be constant in bulk liquids at the equilibrium, with local deviations around some solutes (these deviations belong to the family of cybotactic effects, on which something will be said later). At each type of liquid surface there will be some deviations in the density, of extent and nature depending on the system. 

This differential equation allows the calculation of the density profile across the interface, and in light of (75) it implies the following equivalent expressions for the interfacial energy [113],... 

In a series of papers [7,106,107], we have combined our EoS model with the density gradient approximation of inhomogeneous systans [99-105]. In Refs. [7,106,107], we have addressed in three alternative ways the problon of consistency and equivalence of the various methods of calculating the interfacial tension. In the first case [106], we have simulated the number density profile across the interface with the classical hyperbolic tangent expression [92] (Equation 2.138). In the second case [7], this profile was obtained from the free-energy minimization condition [103,105]. 

The values for the consistency parameter p follow the universality shown in Table 2.6 if Equations 2.113 and 2.130 are used for the density profile across the interface. The universality is lost and lower values are obtained [106] if a different ad hoc profile equation is adopted, such as the classical equation [92] ... 

Interest in the multiphase behavior of ILs has grown, as ILs have found applications in extraction and separation. Classic MD studies of the IL-vapor interface of [bmim][PF6] performed by Bhargava and Balasubramanian reported oscillations in the electron density profile across the interface and an ordering of the charges. This study also predicted a hydrophobic surface for [bmimjpFg] consistent with sum frequency... 

The paper is organized as follows. First we recall and discuss SnelVs and FresneFs laws for X-ray optics. We then derive the general relation of the density profile across the surface to specular reflectivity (Fig. 1.1a) and to the Qz-variation in grazing incidence diffraction (Fig. 1.1b). Specular reflectivity is illustrated by two examples. The first is reflection from a bare water surface and the determination of the diffuseness of the air-water interface due to thermally excited capillary waves. In the second example we consider a monomolecular film of an amphiphilic molecule, arachidic acid, floating on water, as the area per molecule is varied by a moveable barrier in a Langmuir trough. ... 

The density profiles across the interface are depicted in Fig. 10(b). At low pressure the coexisting phases differ in the density of polymers. The density of the volatile solvent is almost equal in both phases, and very low. At the center of the interface there is a small excess of solvent. Upon increasing pressure, the density of solvent increases both in the vapor and in the liquid. The density of the polymer in the liquid decreases in turn. The interfacial excess of solvent increases and the profile becomes asymmetric most of the excess is found on the vapor side of the interface. The interfacial excess per unit area can be defined by... 

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

In Fig. III-7 we show a molecular dynamics computation for the density profile and pressure difference P - p across the interface of an argonlike system [66] (see also Refs. 67, 68 and citations therein). Similar calculations have been made of 5 in Eq. III-20 [69, 70]. Monte Carlo calculations of the density profile of the vapor-liquid interface of magnesium how stratification penetrating about three atomic diameters into the liquid [71]. Experimental measurement of the transverse structure of the vapor-liquid interface of mercury and gallium showed structures that were indistinguishable from that of the bulk fluids [72, 73]. 

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).]... 

It was noted in connection with Eq. III-56 that molecular dynamics calculations can be made for a liquid mixture of rare gas-like atoms to obtain surface tension versus composition. The same calculation also gives the variation of density for each species across the interface [88], as illustrated in Fig. Ill-13b. The density profiles allow a calculation, of course, of the surface excess quantities. 

The resolution of infra-red densitometry (IR-D) is on the other hand more in the region of some micrometers even with the use of IR-microscopes. The interface is also viewed from the side (Fig. 4d) and the density profile is obtained mostly between deuterated and protonated polymers. The strength of specific IR-bands is monitored during a scan across the interface to yield a concentration profile of species. While in the initial experiments on polyethylene diffusion the resolution was of the order of 60 pm [69] it has been improved e.g. in polystyrene diffusion experiments [70] to 10 pm by the application of a Fourier transform-IR-microscope. This technique is nicely suited to measure profiles on a micrometer scale as well as interdiffusion coefficients of polymers but it is far from reaching molecular resolution. 

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