Density profile

Density profiles are the central quantity of interest in computer simulation studies of interfacial systems. They describe the correlation between atom positions in the liquid and the interface or surface . Density profiles play a similarly important role in the characterization of interfaces as the radial distribution functions do in bulk liquids. In integral equation theories this analogy becomes apparent when formalisms that have been established for liquid mixtures are employed. Results for interfacial properties are obtained in the simultaneous limit of infinitesimally small particle concentration and infinite radius for one species, the wall particle (e.g., Ref. 125-129). Of course, this limit can only be taken for a smooth surface that does not contain any lateral structure. Among others, this is one reason why, up to now, integral equation theories have not been able to move successfully towards realistic models of the double layer. 

For real corrugated surfaces, in principle the full single particle density p x,y, z), where x,y,z) denote the Cartesian coordinates, has to be considered. The onedimensional normalized density function p z) can be regarded as the lateral average of the full density function p z) according to 

When (42) is used in the derivation of the polymer density profile, the profile appears independent of the degree of polymerization N of the arms. The degree of polymerization N only determines the cut-off distance for the profile via the normalization condition, (2). This implies that the local conformations of the arms at any distance r / are independent of N. In particular, the elastic tension in the chains at any distance r R does not depend on the overall degree of polymerization of the chains. 

Alternatively, implementing (40) and (41) leads to a quite different picture for the star structure. Here, the elastic tension in the arms is determined by the local monomer-monomer repulsion only at the edge of the corona, r = I . At r 7 the arms are stretched more strongly, due to an excess pulling force exerted by the terminal parts of the arms. Therefore, the polymer density profile Cp r,N,R) and the chemical potential A(A,7 ) depend explicitly on N (or the star size R) [123]. 

Closed analytical expressions for the polymer density profiles Cp(r) can be obtained only in certain limiting cases (asymptotic regimes), when the free energy density can be presented as a power law function of the polymer concentration, /int Cp(/ ) Cp(r). The density profiles have the simplest form when they are presented in reduced variables, r/R and Cp(r)/Cp(7 ). 

A simplified quasi-planar approach predicts a power law decay of the density profile for any value of 7 and at any distance r  

A more accurate theory predicts a different functional form for the density profile that depends on the value of 7. The polymer concentration, Cp(r), can be approximated by a power law function only in the central region of the corona. 

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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]. 

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).]... 

Fig. Ill-13. (a) Plots of molecular density versus distance normal to the interface a is molecular diameter. Upper plot a dielectric liquid. Lower plot as calculated for liquid mercury. (From Ref. 122.) (b) Equilibrium density profiles for atoms A and B in a rare-gas-like mixmre for which o,bb/ o,aa = 0.4 and q,ab is given by Eq. III-56. Atoms A and B have the same a (of Eq. m-46) and the same molecular weight of SO g/mol the solution mole fraction is jcb = 0.047. Note the strong adsorption of B at the interface. [Reprinted with permission from D. J. Lee, M. M. Telo de Gama, and K. E. Gubbins, J. Phys. Chem., 89, 1514 (1985) (Ref. 88). Copyright 1985, American Chemical Society.]...

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. 

At finite positive and negative charge densities on the electrode, the counterion density profiles often exhibit significantly higher maxima, i.e. there is an overshoot, and the derived potential actually shows oscillations itself close to the electrode surface at concentrations above about 1 M. 

This covariance ftmetion vanishes as t-5 approaches because the initial density profile has a finite integral, that creates a vanishing density when it spreads out over the infinite volume. 

The scattering from an isolated sphere may be calculated from equation (B1.9.3 2). This derivation assumes that the sphere is iinifonn, with its density profile p(r) = Pq if r < rg and p(r) = 0 if r > Tq (surrounded by a non-scattering material). With this assumption, equation (BE9.32) becomes... 

Figure Bl.9.12. The schematic diagram of the relationships between the one-dimensional electron density profile, p(r), correlation fiinction y (r) and interface distribution fiinction gj(r).

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]. 

Fig. 7. Axial density profiles in the (—) bubbling, (------) turbulent, and (----) fast and ( ) riser circulating fluidization regimes. Typical gas velocities for...

The bed level is not weU defined in a circulating fluidized bed, and bed density usually declines with height. Axial density profiles for different CFB operating regimes show that the vessel does not necessarily contain clearly defined bed and freeboard regimes. The sohds may occupy only between 5 and 20% of the total bed volume. 

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... 

Fig. 11. Laboratory OSB board density profile as a function of board thickness when using 10 s and 50 s press closing times. Note the much higher peaks of surface density for the 10 s case, and the more even density profile for the slower press closing time.

Overall board density will strongly affect core layer plasticization and density profile (Fig. 12), as at the highest overall board density a steep density gradient appears between the surface and core layers of the board. This is due to the greater difficulty encountered by the steam to penetrate and plasticize it. At lower density, the greater mat permeability enables a faster steam throughflow of the board, comparable to a steam injection process. The final result is similar as the overall board density is closer in value to both core and surface densities. 

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]. 

B. Gotzelmann, S. Dietrich. Density profiles and pair correlation functions of hard spheres in narrow slits. Phys Rev E 55 2993-3005, 1997. 

R. Evans, J. R. Henderson, D. C. Hoyle, A. O. Parry, Z. A. Sabeur. Asymptotic decay of liquid structure oscillatory liquid-vapour density profiles and the Fisher-Widom line. Mol Phys 50 755-775, 1993. 

FIG. 6 Density profile of a hard sphere fiuid near a hard wall. The bulk density is pd = 0.6. The curve gives the results of the lOZ equation with the PY closure and the circles give the simulation results. The results obtained using the HAB equation together with the MV closure are very close to the solid curve whereas the results obtained from the HAB equation with the HNC and PY closures are too large and too small, respectively. 

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... 

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