The density profile

Three typical profiles are shown in Fig. 6.2. The first is for a film with two surfaces (MD, 1728 molecules) at t = 0 701, and the second and third are for systems in which the liquid is anchored to the bottom of the cell, and which therefore have oscillations in p(z) near the wall at z =0. The second is at t = 0 785 (MD, 1020 molecules) and the third at T = 1 127 (MC, 255 molecules). As the temperature rises, p falls, p rises and the surface becomes thicker. In every case the shape of p(z) is that of 

Three profiles, p(z), for a Lennaid-Jones fluid. Curve 1 is a two-sided film (MD, 1728 molecules) at r = 0 701 curve 2 is for a portion of liquid against a wall (MD, 1020 molecules) at r = 0-785 curve 3 is for a similar configuration but is a MC simulation (255 molecules) at r = 1-127. 

Here 2. is the height of the equimolar dividing surface, and D is a measure of the thickness. Althoi a hyperbolic tangent arises naturally for the penetrable-sphere model (S S.S) and in the van der Waals theory of a system near its gas-liquid critical point (S 9.1) its use here is purely empirical indeed, we shall see in S 7.S that an exponential decay of p(z) at large values of 2 -2. is not correct for a Lennard-Jones potential that runs to r=9c. This equation is, however, a convenient one since it can be fitted to experimental points by inverting it to give 

More interesting is that D increases also with the width of the cell. The repetition of the Cartesian coordinates of molecules in the x- and y-directions means that a simulated surface in a cell of width i is effectively tied to a horizontal square grid of mesh 1 x L Capillary fluctuations of wavelength longer than I are suppressed. As the width of the cell 

Kalos et al. have analysed the results of Rao and Levesque in order to study explicitly the correlations in the x,y-plane. They find the strongest correlations at low values of the wave-vector q, and confirm qualitatively the form of H(q Z], given by Wertheim s expression (4.261). 

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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]. 

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. 

Equilibration of the interface, and the establislnnent of equilibrium between the two phases, may be very slow. Holcomb et al [183] found that the density profile p(z) equilibrated much more quickly than tire profiles of nonnal and transverse pressure, f yy(z) and f jfz), respectively. The surface tension is proportional to the z-integral of Pj z)-Pj z). The bulk liquid in the slab may continue to contribute to this integral, indicatmg lack of equilibrium, for very long times if the initial liquid density is chosen a little too high or too low. A recent example of this kind of study, is the MD simulation of the liquid-vapour surface of water at temperatures between 316 and 573 K by Alejandre et al [184]. 

The press closing time also influences the relative densifications of the surface and core layers of the wood mat during pressing (Figs. 9 and 10). Fig. 11 details the density profile of the particleboard panels prepared at short and longer press closing times [226]. The two cases differ in several aspects. (1) A short... 

The interaction of a simple fluid with a single chemically heterogeneous substrate has also been studied. Koch et al. consider a semiinfinite planar substrate with a sharp junction between weakly and strongly attractive portions and investigate the influence of this junction on the density profile of the fluid in front of the substrate [172-174]. Lenz and Lipowsky, on the other hand, are concerned with formation and morphology of micrometer droplets [175]. 

FIG. 7 Values of the density profile at eontaet for hard spheres in a sht of width H as a funetion of H. The density of the hard sphere fluid that is in equilibrium with the fluid in the slit is pd = 0.6. The solid eurve gives the lOZ equation results obtained using the PY elosure. The broken and dotted eurves give the results of the HAB equation obtained using the HNC and PY elosures, respeetively. The results obtained from the HAB equation with the MV elosure are very similar to the solid eurve. The eireles give the simulation results. 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

In any relation given above, the knowledge of the total or direct pair correlation functions yields an equation for the density profile. The domain of integration in Eqs. (14)-(16) must include all the points where pQ,(r) 0. In the case of a completely impermeable surface, pQ,(r) = 0 inside the wall... 

A set of equations (15)-(17) represents the background of the so-called second-order or pair theory. If these equations are supplemented by an approximate relation between direct and pair correlation functions the problem becomes complete. Its numerical solution provides not only the density profile but also the pair correlation functions for a nonuniform fluid [55-58]. In the majority of previous studies of inhomogeneous simple fluids, the inhomogeneous Percus-Yevick approximation (PY2) has been used. It reads... 

If Bga is set to zero, Eqs. (34) and (32) reduee to the singlet HNCl equation (6). We would like to stress, however, that although two deseribed approaehes may lead to an identieal final equation for the density profile, the approximations involved in their derivation are different. According to the former method of derivation, the HNCl equation is obtained by neglecting the third-order and higher-order direet eorrelation funetions of the bulk fluid, whereas the latter approaeh (Eq. (32)) requires negleeting the wall-a partiele bridge funetions Bg r) to arrive at HNCl. This means that the... 

Let us begin our discussion from the model of Cummings and Stell for heterogeneous dimerization a + P ap described in some detail above. In the case of singlet-level equations, HNCl or PYl, the direct correlation function of the bulk fluid c (r) represents the only input necessary to obtain the density profiles from the HNCl and PYl equations see Eqs. (6) and (7) in Sec. II A. It is worth noting that the transformation of a square-well, short-range attraction, see Eq. (36), into a 6-type associative interaction, see Eq. (39), is unnecessary unless one seeks an analytic solution. The 6-type term must be treated analytically while solving the HNCl... 

The behavior of assoeiating fluid near the hard wall was extensively studied in the framework of the theory diseussed above. The model of Cummings and Stell was applied to relatively dense fluids at a high degree of dimerization [33,36]. Fig. 1 presents the density profiles ealeulated within the framework of the eombined PYl/EMSA theory (i.e., the density profiles were evaluated from the PY 1 equation, whereas the bulk direet eorrelation fune-tions follow from the EMSA equation) and HNCl/EMSA approximations [33]. The ealeulations were performed for L = 0.42[Pg.181]

The density profiles are presented for the model of Cummings and Stell [25-27]. The fluid is in eontaet with the (100) plane of the faee-eentered eubie lattiee, the bulk fluid partieles and the solid atoms are assumed of the same size, ai = a2 = infinite dilution (Kq /ksT) [25]. The ealeulations have been done by using PYl... 

The density profiles are shown in Fig. 7(a). Fig. 7(b), however, illustrates the dependenee of the degree of dimerization, x( ) = P i )lon the distance from the wall. It ean be seen that, at a suffieiently low degree of dimerization (s /ksT = 6), the profile exhibits oseillations quite similar to those for a Lennard-Jones fluid and for a hard sphere fluid near a hard wall. For a high degree of dimerization, i.e., for e /ksT = 10 and 11.5, we observe a substantial deerease of the eontaet value of the profile in a wide layer adjacent to a hard wall. In the ease of the highest assoeiation energy,... 

As we have noted in Sec. II, one of the methods leading to the so-called singlet equations for the density profiles, originally initiated for simple fluids in Ref. 24, starts from considering a mixture of fluid particles and another species of hard spheres at density pq and diameter Dq, taking next the limit Pg - 0,Dg- oo. 

The singlet multidensity Ornstein-Zernike approach for the density profile described in this section has also been applied to study the role of association effects in the ionic liquid at an electrified interface [22]. 

We apply the singlet theory for the density profile by using Eqs. (101) and (103) to describe the behavior of associating fluids close to a crystalline surface [120-122], First, we solve the multidensity OZ equation with the Percus-Yevick closure for the bulk partial correlation functions, and next calculate the total correlation function via Eq. (68) and the direct correlation function from Eq. (69). The bulk total direct correlation function is used next as an input to the singlet Percus-Yevick or singlet hypernetted chain equation, (6) or (7), to obtain the density profiles. The same approach can be used to study adsorption on crystalline surfaces as well as in pores with walls of crystalline symmetry. 

The a-s-a and sp-s-sp cuts of the density profiles (Figs. 9(c) and 9(d)) clearly demonstrate that for a highly dimerized fluid the nonassociatively adsorbed dimers have a tendency to orient perpendicularly or slightly tilted... 

The latter relation is the final equation for the density profile, resulting from the modified Meister-Kroll-Groot theory if applied to associating fluids [145]. 

In Fig. 10(b) one can see the density profiles calculated for the system with /kgT = 5 and at a high bulk density, p = 0.9038. The relevant computer simulation data can be found in Fig. 5(c) of Ref. 38. It is evident that the theory of Segura et al, shghtly underestimates the multilayer structure of the film. The results of the modified Meister-Kroll-Groot theory [145] are more consistent with the Monte Carlo data (not shown in our... 

In order to demonstrate that the systems in question exhibit nonzero wetting temperature, we have displayed the results of calculations for one of the systems (with =1 at T = 0.7). Fig. 12 testifies that only a thin (monolayer) film develops even at densities extremely close to the bulk coexistence density (p/,(T — 0.7) — 0.001 664). In Fig. 13(a) we show the density profiles obtained at temperature 0.9 evaluated for = 7. Part (b) of this figure presents the fraction of nonassociated particles, x( )- We... 

In Fig. 15 we show similar results, but for = 10. Part (a) displays some examples of the adsorption isotherms at three temperatures. The highest temperature, T = 1.27, is the critical temperature for this system. At any T > 0.7 the layering transition is not observed, always the condensation in the pore is via an instantaneous filling of the entire pore. Part (b) shows the density profiles at T = 1. The transition from gas to hquid occurs at p/, = 0.004 15. Before the capillary condensation point, only a thin film adjacent to a pore wall is formed. The capillary condensation is now competing with wetting. 

Fig. 17 shows the adsorption isotherms of all (undimerized and dimerized) particles. Except for a very fast increase of adsorption connected with filling of the first adlayer, the adsorption isotherm for the system A3 is quite smooth. The step at p/k T 0.28 corresponds to building up of the multilayer structure. The most significant change in the shape of the adsorption isotherm for the system 10, in comparison with the system A3, is the presence of a jump discontinuity at p/k T = 0.0099. Inspection of the density profiles attributes this jump to the prewetting transition in the... 

Fig. 20 shows the density profiles in the reactive and nonreactive parts of the system. The number density in the reactive part is very high (a one-component density at the center of this part is 0.596, so the number density of two components is twice as high). However, the density in the nonreactive part is much lower and equal to 0.404. The application of the test particle methods is therefore easy. There is a well-established density plateau in the nonreactive part consequently, the determination of the bulk density in this part is straightforward and accurate. 

FIG. 20 Density profiles of particles of one component a ox (3 (solid lines) and the density profile of the center of mass of the dimers (dashed line). The results are for T = 2, = 5, and the total number of particles equal to 3200. The volume of... 

Sec. 4 is concerned with the development of the theory of inhomogeneous partly quenched systems. The theory involves the inhomogeneous, or second-order, replica OZ equations and the Born-Green-Yvon equation for the density profile of adsorbed fluid in disordered media. Some computer simulation results are also given. 

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