Surface density profile

However, the surface coverage is the same for both copolymers when weakly adsorbed to the surface. Surface density profiles were also compared. Finally, scaling relationships for triblock copolymer adsorption under weak adsorption conditions were derived (Haliloglu et al. 1997). In a related paper (Nguyen-Misra et al. 1996), adsorption and bridging of triblock copolymers in an athermal solvent and confined between two parallel flat surfaces were studied, and the dynamic response of the system to sinusoidal and step shear was examined. 

Chambers J. E. and Cassen P. (2002) Planetary accretion in the inner solar system dependence on nebula surface density profile and giant planet eccentricities. Lunar Planet. Sci., abstract 1049. 

Figure 1. Cumulative p.d.f. (see insets) GRB - SNIb/c, GRB - SNIa, GRB - XB, GRB -two DM models. Smooth solid black curve is a GRB surface density profile with localization errors taken into account.

Equations (54) and (55) are applicable both for an ordinary cluster and for a cluster with a bubble. To characterize the density profile for the cluster with a bubble, we choose the helium atom density function in the form of a void at r < Rb — t /I, a rising profile toward a constant density with increasing r beyond the void boundary at r > Rb — t jl, and an onset of the cluster exterior decreasing density profile for r > / — tijl. Here Rb is the bubble radius, R is the cluster radius, t is an effective thickness parameter for the density profile of the bubble wall, and 2 is the thickness of the cluster surface density profile. The explicit form of the helium density profile was taken as... 

The parameter b in Eq. (56b) specifies spatial samration taking b =[Rb — h/2]. For sufficiently large clusters the density in the interior of the cluster [Eq. (56b)] converges to the bulk value hq. The parameter c in Eq. (56c) is taken as (c) = 2no/n). Equation (56b) was advanced on the basis of previous work on nonuniform He near a hard wall [247]. Equation (56c) represents the surface density profile of the cluster with a bubble in the form of the gudermannian function [178-180]. 

The interior contribution (r) to the potential [Eq. (60a)] is given by the superposition of electron-atom pseudopotentials exerted on the electron by the helium atoms within the surface density profile of the bubble walls and by the electronic polarization potential V,(r) induced within the region of the bubble, which is represented in terms of a cluster image potential... 

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. 

Fig. 21 Density profiles p z) and running integrals Njo (z) of the ion densities for cations (full lines) and anions (dashed lines) at three different surface charge densities in units of pCcm as indicated. Left NaCi solutions right CsF solutions. The top graphs show, for reference, the corresponding oxygen density profile near the uncharged surface. Density profiles are normalized to the bulk densities corresponding to 2.2 molal solutions in each case.

Figure 2.17. The hyperbolic tangent surface density profile.

Surface Density Profile in Liquid Metals A Semi-empirical Treatment of the Screening Effect, in Proc. 6th Int. Conf. on Liquid and Amorphous Metals, Z. Phys. C 1S6S, 445-450. 

These predictions from mean field lattice models are consistent with density functional calculations of the crystal surface 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).]... 

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. 

Fig. XI-7. Volume fraction profile of 280,000-molecular-weight poly(ethylene oxide) adsorbed onto deuterated polystyrene latex at a surface density of 1.21 mg/m and suspended in D2O, from Ref. 70.

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. 

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... 

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 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... 

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. 

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. 

Fig. 19 shows an example of the orientational density profile evaluated for the system 10. Most of the particles of the first adlayer assume two limiting configurations parallel to the surface and vertical to the surface. The second adlayer exhibits also well pronounced orientational ordering ... 

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