Structure of spherical interphase at high temperatures

3 March 1995

ELSEVIER

CHEMICAL PHYSICS LETTERS Chemical Physics Letters 234 (1995) 25-28

Structure of spherical interphase at high temperatures Douglas Henderson a, Jacqueline Quintana b, Stefan Sokotowski a,1 a Departamento de Fisica, UniversidadAut6noma Metropolitana/Iztapalapa, Apdo. Postal 55-534, 09340 DF Mexico, Mexico b Instituto de Qulmica, UNAM, Coyoac{m, DF 04510 Mexico, Mexico

Received 14 September1994; in final form 29 December1994

Abstract

We present an application of a version of the density functional theory to describe the interphase of Lennard-Jones fluid in contact with a large sphere. A comparison of theory and Monte Carlo data is shown. The theory is generally good and becomes more satisfactory as the radius of the large sphere is increased.

Density functional theories (DF) have been proved to be successful in treating adsorption at planar interphases and in slitlike and cylindrical pores, when simple fluids are considered [1-6]. However, the problem of non-planar interphases is still far from being completed. By this, we mean problems involving colloidal particles as well as adsorption in cavities [7-12]. One of the important points to consider is how the size of colloidal particles would affect properties like wettability or critical adsorption of the solvent [13-15]. A knowledge of the interphase around the colloidal particle is also important because it determines the solvation forces acting between them [16]. This topic is relevant for many practical problems [17]. For example, colloidal particles are involved in biological and chemical reactions. Also such particles are used to purify some substances. Moreover adsorption on colloidal particles of soils is important from the nutritional point of view of plants.

1Permanent address: Computer Laboratory, Faculty of Chemistry, MCS University, 20031 Lublin, Poland.

Our principal goal in this work is to test a version of a DF theory in the case of the adsorption of a Lennard-Jones fluid on a large spherical particle interacting via repulsive-attractive forces. Among the usual techniques in liquid theory, density functional theories have shown to be reliable to describe simple liquids in the inhomogeneous phase. For this case, the most successful density functional are theories which involve a coarse grained average density. One example of this case is the theory by Evans and Tarazona. In fact, this approach has been shown to be accurate and computationally inexpensive schemes [18,19]. We briefly recall this theory. Let us consider the definition of the grand potential $2=F +

- Ix],

(1)

where IX is the chemical potential, n ( r ) is the inhomogeneous density and v(r) is the fluid-solid interaction potential. We divide the functional F in two parts: the contributions due to repulsive forces, Fa, and the attractive forces, FA, between the molecules. The former is modeled by hard-spheres with suitable

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f drn(r)[v(r)

26

D. Henderson et al. / Chemical Physics Letters 234 (1995) 25-28

diameter d and the latter is treated in the mean-field approximation. To calculate F R a smoothed density function h(r) is introduced in a non-local way

h(r)

=f dr' n(r')w(

I r - r ' l , n(r)),

(2)

where w is a weight function. This function is assumed to be described by a power series expansion w(r, n) = wo(r) + wa(r)n + w2(r)n 2, and the coefficients wo, w 1 and w 2 are given in Refs. [18,19]. The free energy takes then the form

F=

f dr n(r){kT[ln n(r)A 3 -

1] + f ( h ( r ) ) }

+ ½ f drdr' n ( r ) n ( r ' ) u A ( I r - r ' l ),

(3)

where A is the thermal wavelength, u g the attractive part of the interparticle potential and the free energy density of hard-spheres. To calculate f the Carnahan-Starling equation [20] is used. f ( n ) / k T = 1 T / ( 4 - - 3 r / ) / ( 1 - - 1 7 ) 2, where r/=gTrd3n0 is the packing franction and n o is the bulk fluid density. The equilibrium density profile minimizes the grand potential ~ , thus the local density is evaluated from the condition ~g2(n(r))/Sn(r)=O, and as usual, the excess adsorption isotherm F is defined by the equation

ar= f

dr [ n ( r ) - n o ]

(4)

where A is the area. The integration is performed over the entire volume available to particles of the fluid. In this Letter, we consider Lennard-Jones fluid with a cutoff

u(r) = 4~[(trs//r)12 -- ( ~ / / r ) 6 ] ,

for r 4%
u(r) = O,

for r > r e ,

(5) where the cutoff distance re has been assumed to be r e - - 2 . 5 ~ . The division of the potential u(r) we employ here has been used in some DF calculations (cf. Ref. [19]) UA(r) ---- -- e,

for r ~ 21/6O's,

UA(r) = u(r),

for r > 2 1 / 6 % ,

and the reference hard-sphere diameter is ~ .

(6)

To calculate the attractive interaction between the large sphere and each of the small spheres in the fluid, we assume that the surface of the large sphere is 'divided' into attractive elements smeared on the surface. Therefore we perform an integral over the whole set of attractive elements given by [21]

= fs( 4 ~ ' s [ ( t r ' s / ' ~ ( r ) ) 1 2 - ( t r t j ~ ( r ) ) 6 ] }

dr,

(7) where ej~ and o-1~ are the potential parameters, R is the distance from the centre of the large sphere. ~ ( r ) is the distance from the point (0, 0, R) to a point on the surface of the large sphere of the radius t~1. In the case of a planar surfaces (i.e. when oq ~ oo) the last equation leads to the well-known Lennard-Jones (10, 4) function [21]. To test the accuracy of the DF theory results we performed some Monte Carlo simulations considering a macrosphere in a Lennard-Jones solvent for a range of orb diameters. We use periodic boundary conditions and the minimum image convention for the solvent molecules. We put the large sphere at the centre of the simulational cell. Because we are interested in the zero limit concentration of the solute, we do not consider any images of the large sphere. Each of Xj, YI and Z 1 dimensions of the simulational cell was at least three times larger than the large sphere diameter. The simulational scheme is similar to that described in Refs. [22,23]. In all cases we performed at least several millions of Monte Carlo steps to reach the equilibrium of the system and also to get averages. In Figs. la and lb we show examples of a comparison between DF and MC profiles, for two sizes of the large particle. These profiles are typical and they show layering connected with subsequent solvation shells. The comparison shows that the agreement is better if the size of the colloidal particle increases. This behavior is expected because this theory is known to work reasonably well when the size of the colloidal particle goes to infinity (planar wall). However, the theory becomes less satisfactory when the size of the molecule producing the external field decreases. As usual the height of the first peak depends on the state conditions, and the spacing in the layering is a larger in DF than in MC.

27

D. Henderson et al. / Chemical Physics Letters 234 (1995) 25-28

In Fig. 2a, we show a sequence of DF local densities evaluated at supercritical temperature T * = k T / e s = 1.295 at subsequently increasing bulk densities. For n o = n 0 ~ 3 > 0.4 a second layer is developed. This temperature is quite close to the bulk critical temperature for a system in which To*= 1.2944 and consequently at densities close to critical density (n o = 0.212) we expect a structureless and long ranged interface. See the profile at n o = 0.2 where the asymptotical decay to bulk values is very slow. Fig. 2b shows density profiles evaluated at constant density n * = 0.2 and temperatures approaching the bulk critical temperature from above. At the highest temperature (lowest curve) the density profile goes to the bulk density in few molecular diameters while as the temperature decreases the approaching is slower. In Fig. 3 we display adsorption isochores evaluated for o"b = 10. At densities far from the critical

T

T

~

~

6

7

8

1

t

n*(r)

05M 1

~

T

9

r*

(b)

0.8 0.6

n*(r)

0.4

6

°

I

!

i

0.2

i

0 5 " - - - 6~

4

0._.

3

' (b) J

2

0 ......II ..... 1'2 13

8

9

r

Fig. 2. (a) Density profiles for T* = 1.295, 8 i s / k T = 2, o"I / ~ = 10, and densities subsequent (from the bottom) bulk densities n~ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8. (b) Density profiles for n~ = 0.2, e l s / k T = 2, o.l = 10, and at different (from the bottom) temperatures T * = 1.35, 1.30, 1.297, 1.295 and 1.29458.

2

n*(r)

7

14

I'5 r*

Fig. 1. A comparison of Monte Carlo data (points) and density functional profiles (line). (a) is for o"1 / ~ = 10, T * = 0.833 and n~ = n 0 ~ 3 = 0 . 8 0 2 , oo~s/kT=2. (b) is for o-l/~s = 20, T* = 0.833 and n~ = 0.761, e l s / k T = 2.

density the adsorption is linearly dependent on temperature while for T * = To* the dependency changes abruptly. This behavior is the so-called supercritical adsorption. It is well known that the supercritical fluids offer many interesting characteristics both from the fundamental and applied point of view. Supercritical fluids have unusual properties that allow one, for example, to learn about solute-solvent interaction. It is known that they posses an enhanced local solvent density [24] which produces unusual solvent effects. This is particularly interesting for many applications [25,26]. Therefore the problem of supercritical adsorption on colloidal particles will be the subject of future work. In conclusion, the comparison shows that DF provides a quite successful way to describe the structure of fluids around colloidal particles. It should be

28

D. Henderson et al. / Chemical Physics Letters 234 (1995) 2 5 - 2 8 i

(a)

40 0.2

F* 20

of their computing facilities. They also thank Dr. Orest Pizio, Instituto de Qulmica, UNAM for valuable discussions.

References o

0 1.294 0.4

1.296

<>

<>

1.298 T* 1.300

i

, o.1

(b)

0.2

A

F*

0.3

0 A

-0.2 0.4 -0.4

1.294

/

1.296

|

1.298

I

T*

1.300

Fig. 3. Adsorption isochores / ' * = Fo-s2 evaluated for cq/o-~ = 10, e l s / k T = 2, and for bulk densities n~ = 0.1, 0.2, 0.3 and 0.4, marked in the figure. The vertical line in part (a) denotes the critical temperature for the system.

said that there is another route to describe this problem, by integral equations, namely using the inhomogeneous Ornstein-Zernike equation [27]. However, this approach seems to require much more numerical effort, i.e. more memory allocation and CPU time. Therefore, because DF is more efficient than other approaches it is advisable to study systems which involve long time computations. For example, investigations of wetting and pre-wetting transitions, adsorption at supercritical temperatures, very near the critical point, and crossover between them. The authors are grateful to CONACYT of M6xico (Grant No. 4186-E9405 and el Fondo para Cfitedras Patrimoniales de Exelencia) for their financial support and to the Departamento de Matematicas y Mecanica, IIMAAS, UNAM for granting us the use

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