Adsorption of fluids on hard wall: Application of inhomogeneous pair correlation function

Indian Journal of Chemistry

Vol. 41 A, November 2002, pp. 2216-2222

Adsorption of fluids on hard wall: Application of inhomogeneous pair correlation function

A Guha '

*

& Ashish Mukheljee²

Deparlment of Chemistry, KalY~lIli University, Kalyani. Nadia,West Bengal 741 235, India IEmail Address: drasitguha@vsnl.net

2Email Address:askm@vsnl.com Receil'ed 19 N(!I'ellliJer 2001: rel'ised 5 August 2002

The local density of non-uniform Iluids is important in IIlderstanding iluid behaviours in contact with the planar walls or slit likl: porco The density profile of the iluid near the hard wall is obtained starling from the direct correlation function defined by Ornstein-Zernike equation with Percus Yevick analytical solution. The bridge function, which is u~eful to obtain the cavity fUllction or radial distribution function, has been calculated. In the case of outside the hard sphere cere, the mean spherical model approximation (MSMA) has been used to calculate the direct correlation function. The adsorption of a model L<.!nnard -Jones Iluid ncar a hard wall has been calculated. The results obtained for the generalized adsorption and agree well with the standard results.

The structure of the i nterphase around a colloidal particle determines the solvation force acting between them. The colloidal particles are involved in several biological and chemical reactions. The adsurption of the fluid on colloidal particle or soils is significant considering the nutritional point of view of the plant.

Presently the properties of the bulk fluids are well understood from the statistical thermodynamics t.

Attard" calculated the full inhomogeneous pair correlation using several closure relations of Ornstein- Zernike (O-Z) convolution integral. The Hamiltonian, which specifies the system need, only depends on positions of pairs of fluid particles via the two radial coordinates and the mutual angle. Thus, image interactions may be included in describing the electric double layer around a micelle, vesicle, macro ion etc. Allard also calculated the inhomogeneous density profiles corresponding to the inhomogeneous pair correlation function in the presence of a fixed particle in a bulk.

The local density of nOll-uniform fluid has been studied^J to understand fluid behaviour in contact with planar walls and in slit-like pores or cylindrical geometry.

Among several methods to study the mieros«Jpic structure and dynamics of liquids, the method of integral equations in the theory of liql1id~ is attractive because of two reasons: a) the po~sihilll)' of ublaining allalytically representable result~ concerning the structure and thermodynamic properties of liquids and

b) the possibility of solving the inverse problem, that of reconstructing the form of intermolecular potential u(r), if g(r) is known, where g(r) is the radial distribution function (p(r)lpo, per)

=

^density

distribution function and

Po

= bulk density). The two most common closed equations, that of Percus-Yevick (PY) and hypernetted chain (HNC) approximation are used here. Before we present these two equations we would like to discuss the Ornstein-Zernike (O-Z) equation which introduces the two important functions namely the direct correlation function (DCF) C(r) and the total correlation function her) and are useful in deriving the PY and HNC equations. The O-Z integral equation⁴ is defined by the relation

h(rt2) = C(rI2) + p

f

^{h(rI3) C(r23) dr).}

where, her) = g(r) - I and C(r), defined by the O-Z equation is the direct correlation Function (DCF).

The contribution of total correlation function between particles I and 2 can be wrirten as - (i) a direct effect of particle I on particle :2 given by direct correlation function, C(rI2), which is short range like potential function, u(r) and (ii) an indirect effect, in

"vhieh particle I is influenced by any third particle 3.

'vvhic\l in turn exerts intluence on particle 2. This Indirect effect is the sum of all contributions from all other (N-2) particle!', canonically ~lveraged over the

Wlh.lk voiume.

The general form of O-Z equation can be written as:

GUHA el 01.: ADSORPTIO OF FLUIDS ON HARD WALL 2217

where hij ⁼ hij (rij) = gij(rij) - I, is the total correlation function; Cij is the direct correlation function for a pair (i and j) of molecules of species i and j, Pk = Nk/V [Nk

= No. of molecules of species k and V = volume]; rij = I ri - rj I, ri is the position of centre of molecule 'i' . .

The DCF is important in theory of liquids due to two reasons: (i) It can be calculated directly from experimental data and the same scattering data can be used to determining both g(r) and C(r); and (ii) the range of C(r) is shorter in comparison to her). Since C(r) is short ranged, it is expected that the theories based on it will be less sensitive to the accuracies of the approxi mation used.

We now present the aforesaid two integral equations, which _are close in nature. They are kn;wn as PY equation)·6 and hypernetted chain (HNC) equation7-~.IO. These equations utilise the concept of C(r), the direct correlation function and relate it to the radial distribution function of the systcm. The approximate forms of C(r) used in obtaining HNC and PY equations are as follows:

~

PY : C(r) = g(r) [1- exp (u(r)1 Kil T}:I H C: C(r) = g(r) - I - Ing(r) - {u(r)/KIlT}

Substituting these forms in the O-Z equation, one gets the PY and HNC integral equations as -

PY: g(rd exp {u(rdIKBT} = 1+ P

J

^{Ig(r~.\) - II g(ru)}

II-exp {u(rl,)IKI1T}] dr,

HNC: In g(rl~) + {u(rdlKBT} = P

f

[g(r2,) - I] [g(rl,) - In g(r1.1) - I - (u(ru)IKI3/l l dr,

The PY equation has acquired importance since it can be solved with a hard sphcre potential and one can obtain an analytical solution for C(r). Thus Wertheim

I l'h' I II I' .

anc Ie e . - have obt~lIned an exact analytical solution for the C(r) in the case of hard sphere potential.

I n the hard core region i.e. r<a,

- C(r) =

a..

+

B

^{(ria) + Y (ria)' ... (I} a) where,

a..

= {( I + 211/1( I - 11)4)

Y = 0.511a..

11

=

^{(1/6) rrpa3}

=

packing fraction and, C(r) = 0 I' >

a

where,

a

is the hard sphere diameter

For the range, r >

a

here the well known mean spherical model approximation (MSMA) has been applied. The mean spherical model approximation for the pair distribution function with impenetrable hard core was first formulated by Lebowitz and Percusl3 The importance of MSMA lies in the fact that it can be solved analytically for a number of pair potentials.

The pair potential is assumed to consist of a hard core.

U(r) =

a..

^{for r} <

a

and an attractive tail, which may not be spherically symmetric, e.g. the dipole- dipole interaction. The MSMA can be expressed by two equations, namely

g(r) = 0 for r <

a

C(r) = -{ u(r)1 KI3 T} = - Bu(r) for r>

a .. .

(I b)

I n fact the PY equation for the hard sphere system is a special case of MSMA, where u(r) = 0 for r> a.

To calculate C(r) and g(r). the O-Z equation must be supplemented by a closure relating C(r) and g(r).

For that reason O-Z relation can be turned into an integral equation by means of some approximation between hij and Cij. Two of the 1110st common closure or approx i mations are -

a) PY approximation: h - C = Y - I, where, y = y(r) = exp [Bu(r)] g(r)

b) HNC approximation: h - C

=

^Iny

Two of the above equations are approximations to the exact solution -

y(r) = h - C = In y - B ... (2) This is the starting point of a reliable closure. The above equation is a definition of bridge function, B(r).

INDIAN J CHEM, SEC A, NOVEMBER 2002

Thus the development of a closure comes down to the approx i Illation of B(r).

The purpose of this paper is to present the results of adsorption of Lennard-Jones fluid near a hard wall and the density profile of the hard sphere fluid in the presence of a wall, which is both smooth and hard. The wall is located at Z=O and the corresponding external potential is given bis,

u(z) = C/. , z< al2

u(z)

=

0 , z> a/2

where 'a' is the hard sphere diameter. For the sake of si mpl i fication planar symmetry and densi ty variation only in the direction perpendicular to the wall are considered i.e. the fluid structure parallel to the wall neglected. li.e. Per) = P(z» I.

The paper in divided into two sections : (I) Determination the radial distribution function of L-Z fluids near a hard wall and (2) calculation of adsorption of the same. In first section, direct correlation !"unction, C(r) inside and outside the hard core region is calculated from the knowledge of Bridge function, B(r) which is defined Eg. (2) for a hard sphere ina hard sphere sol vent and then usi ng hydrostatic hypernetted chain approximation (HHNC) which was formulated by Zhou and Stell¹⁶ for the system of L-Z fluids ncar a hard wall to obtain g(z).

In the second section, density functional theory is applied to obtain the adsorption or L-J fluid near a hard wall interacting via L-Z type potential.

Mclhod

I. Co/cil/olioll ofg(-;.)

To obtain g(r) andlor y(r) from Eg. (2) for a hard sphere in a hard sphere solvent an approximation is needed to the bridge function. There are no explicit expressions for B(r), other than those, which come from approximations. However, explicit expressions can be obtained at low densities, where the leading terms, of order p2 and p3, can be written explicitly. These low order coefficients are difficult to calculate and the convergence of the series may not be rapid.

As a result it IS common to use simple approximations. The Verlet approximation seems useful for calculating B(r) for hard sphere systems.

The Verlet approximation with C/. constant is identical to the PY approximation (i.e. B(r) = Iny(r) -y(r) +1) at low densities and gives an incorrect fourth virial

coefficient. This is not too serious for hard spheres since the PY fourth virial coefficient can have substantial errors for potentials with attractive regions. The Verlet approximation may well be useful for B(r), the bridge function for large hard spheres in a small hard spheres fluid especially if 'c/.' is not kept constant and also for bridge function for a pai r of large hard spheres in a fluid of small hard spheresl

-1.

An approximation due to Verletl5 is used here which was obtained from ref. 16.

B(r) = _ y2(r) 2[1

+

ay(r)J

... (3)

where,

a

= (17/120 11) + 0.5150 - 0.2210 11

From Eg. (2), the definition of bridge function, y(r) = her) - C(r) = In y(r) - B(r)

Let us consider here a single solute hard sphere in a solvent of hard spheres. Let R be the ratio of the diameter of the solute hard sphere to that of the solvent hard sphere. If R=I, the correlation functions of the pair are those of the sol vent and if R> I, the correlation functions of the pair are those of solvent- solute pair. Outside the hard sphere cores, r>=(R+I/2), the cavity function, y(r) is equal to g(r) = p(r)/p. Here r is distance of separatiorl of the centres of the two spheres.

Once y(r) is known BCr) can be calculated from Eg. (2).

y(r) can be obtained from the expression of C(r), C(r) = - 1- y(r) for r::; (R+ 1)/2

and, CCr) = y(r) - 1- y(r) for r2 (R+I)/2 ... (4)

Here, we are using the value of e(l') obtain from W ertn, . elm ane I 'rhO Ie e I ' s II . -I" exact ana yI lica . I so I ' ullon for C(r) stated in Eg. (I a) in the range r ::;

a

and for r>

a using MSM approximation.

From the work of Zhou and Stell¹⁷ it is known that HHNC approximation is better than the PY ancl H IC approximation for both the homogeneous and inhomogeneous L-Z fluid systems. Both the PY approximation and HNC approximation, which are reasonabl y accurate for harcl spheres near a hard wal L

GUHA el 01.: ADSORPTION OF FLUIDS ON HARD WALL 2219

both fail when applied to L-Z fluids, especially near transitions. Fur a system of fluids (here we consider the Lennard- Jones fluid with a cut off relll = 2.Sa) ncar a planar wall in the HHNC approximation^'7 we have:

h(z) - C(z) = ^pll

f

^Cil (I r-r'l) h(z') dr'

=

^pl3

f

^{h13 (I r-r'l}) C(z') dr'

Ing(z) = _~V"I(Z) + h(z) - C(z) - B(z) ... (S) where, a superscript B denotes the bulk homogeneous properties, C(z) is direct correlation function. h(z)

=

g(z)-I

=

¹ p(z)/p^I31 - I with density profile p(z) and bul k density pll. V'''I(Z) is the interaction potential bet ween a

n

u id particle and the wall and p"'(z) = pll (h"'(z) + I) with h"'(z) =

f

^{w (I r'-rl}) h(z') dr'.

Ih"'(z) = weighted pair correlation function.

w(r) = weight function which satisfies the normalized condition.

f

^{w (r) dr}

=

^I]

To make the above equation simple the HNC approximation is applied. where

B(z)

=

⁰

So Eq. (5) becomes.

Ing(z)

= -

^~V"'I(Z) + h(z) -C(z) or. Ing(z) = _~VC'I(Z) + Iny(z)

I from. Eq. ([3). h(z) - C(z)

=

In y(z) - B(z) or. h(z) - C(z) = Iny(z)l

... (6)

To obtain radial distribution function from Eq. (6) one has to calculate the Cavity function, y(r). Outside the hard sphere cores. r>

a

the cavity function is equal to g(l'). the radial distribution function. III that region y(r) is calculated from c!assical equation,

y(r)=expl Bu(r)Jg(r) and using PY equation it becomes.

expl~u(r)]C(r) y( 1')

rl-expl~u(r)}1

In t he case of inside the hard sphere cores (i .e. r<a) y(r) is calculated from Eqs (2-4) and using the value

of C(r). The graph of y(r) against ria is given in Fig.

0).

For a Lennard - Jones fluid, the pair potential is:

~VCXl(Z) = ~VeXl(z) = 4 l(alz)'2 - (aid' ]/1"* ... (7)

where, T" = reduced temperature =

"13

TIE (E and

a

are parameters in units of energy and distance respectively).

2. Ca/clI/atioll of adsorptioll

(n

According to Lane and Spurling'lJ, the adsorption

r

is defined as the surface excess of gas particles. with respect to a Gibbs dividing surface²⁰ coincident ",ith the plane containing the centers of the carbon atoms nearest the gas phase. dividing by the area, A of the two interfaces in the periodic box. The gas density is very low in all these calculations so thaI the number of particles N_gas• in a gas phase of the same volume as the particle box, is very small compared with the ensemble average particle number <N>: hence adsorption is given with sufficient accuracy by.

r =

<N>/A

The average <N> is directly related to the probability P(NJ of finding Nj particles in the box by the expression:

<N> = L P(NJ Ni

i=1

The probabilities, P(N^{j ),} were evaluated for each Monte Carlo (MC) run and they were proved to be very useful in assessing the reliability of the Markov chain generated during the run.

In experimental gas adsorption studies. the solid is generally in the form of a fine powder the surface area of which cannot be determined accurately and therefore the absolute adsorption is subject to a similar uncertainty. In an attempt to avoid this problem, the adsorption is often reportee! in terms

or

the coverage

e

^{defi ned by.}

e = ArJ(Ar^lll)

where Ar^lll is the adsorption when the solid has been covered completely by a single layer of closed-packed particles. The problem is not completely removed by

2220

INDIAN J CHEM, SEC A, NOVEMBER 2002

this device as the experimental value of Arm is dependent upon the use of a model of gas adsorption.

The MC calcul::nion gives r directly but there is some arbitrariness in defining rm. According to Rowley el a/.21

r is the number density per unit area of triangularly close-packed spheres having a diameter 2^1/6 CJ (Kr, Kr).

To calculate adsorption (1) of simple fluids on colloidal particles, the version of density functional theory developed by Evans and Tarazona has been used. 1t is proved to be one of the most accurate versions and it keeps the computational expense within reasonable limits2~.2:l.

Densitv jilnCliono/ theory

To know the theory briefly let us start from the definition of the grand potential Q

Q = F +

f

^{dr per) rver) - 11],}

where vCr) is the fluid- solid interaction potential, and

.u

^is the chemical potential. The Helmholtz free energy F is divided into two parts. The first part F_R,

representing the contribution due to repulsive forces, is calculated by invoking the smoothed local density PICr). and is defined as.

PI(r) = fdr' per') W rl 1'- 1" I, p(r)l,

where W is a 'vVeigh! function. According to Evans - Tarazona approach the weight function is assumed to be a power series expansion

w(r, p) = wo(r) + wl(r) P + w2(r) p2,

where the coefficients Wo. Wl.W2 are available in literature²².2

'. The attractive force contribution, however. is treated in the mean-field approximation.

Thus, the free energy takes the form, F = Jd(r) p(r){ KT [lnp(r)A' - IJ + ffpl(r)]}

+ ^1/2 J drdr' per) per') U;\ (I r-r' I)

In this equation, A is the thermal wavelength, u,\

represents the attractive part of the interparticle potential and the free energy density of hard spheres f

S · . '-l

is calculated from the Carnahan- tarllng equatlon- f(p)/kl = Y (4-]y)/(I-y)~ ,

where, y= (1/6) npCJ.\ = packing fraction and CJ is the hard sphere diameter.

The equilibrium density profile m1l11mlZeS the grand potential Q; thus the local density is evaluated from the condi tion :

8Q[p(r)] = 0 8p(r) ,

and the excess adsorption isotherm r is defined by, Ar =

L

dr [per) - Po],

where A is the surface area and Po is the density of a bulk fluid at a given temperature and chemical potential. The integration is performed ver the entire volume V available to the particles of the fluid.

or,

u

rCJ" = pll CJ3

f

^{h(z) dz}

o

Results and Discussion

... (8)

The method of calculation for adsorption of Lennard-Jones fluid near a hard wall has been presented above and to obtain it one has to know the full density profile of hard sphere near a large hard sphere.

There are so many methods to obtain density profi les for a hard sphere fluid near a large hard sphere. For example Attard" has obtained density profile using integral equation techniques, Degreve and Hendersonl-l have obtained density profile using the Monte Carlo method. But here we started from the direct correlation function, C(r), which was introduced by Ornstein and Zernike in their integral equation and which was used in obtaining two important closed integral equation - HNC and PY. From Eq. (I). the solution of Wertheim and Thiele in the case of hard sphere potential, the C(r) for r :::; CJ can be easily obtained. But in the case of r>CJ, definite value of C(r) could not be obtained. In this case C(r) = O. For that region (outside the hardcore region) the mean spherical model approximation, where,

C(r) = - {u(r)/KRTJ = -Bu(r) [B = (1IKI3T)] where r > cr is used.

We have computed C(r) (or C(z) in case of hard wall, r=z) from Eq. (I) at a fixed P and at a range of pcr) = 0.1 to 1.5 and plotted C(z) against z/CJ at ztlll=2.5CJ (we consider here for Lennard-Jones fluid) in Fig. (I) and

GUHA ela/.: ADSORPTION OF FLUIDS ON HARD WALL 2221

2.000000

0.000000 ~-----

, /

-2.000000 , ,

-4.000000 _, , ^,

E

-6.000000 U

-8.000000

-10.000000 -12.000000 -14.000000

00000 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000 Z

Fig. I- Direct correlation function. C(z) versus zla for T' = 1.5 and p* = 0.6 considering Znll = 2.5a. The dashed (----) curve gives the result of PY2 and solid (-) curve gives our calculated result.

0.500000 .,--- - - -- - - ,

-0.500000

-1.500000

~ -2.500000 N

CO _-3.500000

-4.500000 -5.500000

-6.500000 +---~--~--~--~--~---I

0.0000 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000

z

Fig. 2- Bridge function. B(z) against zla for F = 1.5 and p* = 0.6 considering ZCUI = 2.)a.

also compared it wilh the value obtained from PY2 result. The agreement of our calculated value with the PY2 results is good.

From Eqs (2-4) 8(r) can be calculated for a hard sphere near a large hard sphere. The plot of 8(z) againsl ziG at a fixed P and at a range of pG-' = 0.1 10 1.5 is given ill Fig. (2). Pigure 3 depicts the plot of logarithm of cavity function versus riG.

The plot of g(z) against ziG is given in Fig. (4) and the nature of curve except the height of the first peak is almost similar to that obtained from PY2 result"5

Finally generalized adsorption isotherm, fG² from Eq. (8) is plolted for different P';' = pBG·\ at two different P in Fig. (5).

In this work the density functional theory has been used to calculate the adsorption of fluids on hard wall. The results are the theoretical predictions for

6.0

5.0

4.0 \

8:

_>- ^3.0

E _2.0

1.0 ,

00

-

^:::=

00 0.5 1.0 -"15--- _{2.0 2.5 30}

-1.0

z

Fig. 3 -Logarithm of cavity function. Iny(r) versus ria for F

=

1.5 and p* = 0.6 considering ZCUI = 2.)a. The dashed (----) curve gives the result of PY2 and solid (- ) curve gives our calculated result.

3.000000

2.500000

2.000000

N _1.500000 (;'

1.000000 0.500000 0.000000

0.0000 0.5000 , ' , ' , \

: '

l

^{------ ssenes1enes2J \}

~ ~ .

" ... - - ,

;' '",, ~~---

1.0000 1.5000 2.0000 2.5000 3.0000

Z

Fig. 4- Radial distribution function, g(z) against zla for F = 1.5 and p*=0.6 considering Znll = 2.5a. The dashed (----) curve gives the result of PY2 and solid (- ) curve gives our calculated rcsult.

1.4 , -- - - -

1.2

08

06

0.4

0.2

I - T '=

1 . 5 1

0+-~~-~~-~-~=--=-=--~·T'===~0~.9~

o ^{02 0.4 06 08 1.2 1 ~}

p.

Fig. S- Adsorption isotherm,

ra'

versus p* for two different F.

The dashed (----) curve for T*=0.9 and solid (----) curve for T*=1.5.

INDIAN J ^CHEM. SEC A. NOVEMBER 2002

generalized adsorption of fluids on hard wall. The results can be compared with the adsorption results of L-Z molecules by large hard sphere. The curves are similar ror adsorption isotherm calculated by Ilcnderson ('I al. for flat wall. We observed the jump in the adsorption isotherm at ^pll(J' at about 0.9, this jump is the manifestation of the first order wetting transition. Henderson found the first order prewetting transition at pll = 0.0094 g/ml, which is much lower than the actual liquid state.

Henderson did not calculate the bulk densities close to the coexistence densities. We performed the generalized expression for adsorption and the ^pl1(J"

\'alues have been taken within the liquid range (i.e. A. the packing rraction is taken from 0.2 to 0.7). In the I)revious work27 . ^.\2 it is found that the wellin^o _i::'

tcmper~lture increases when the size or the colloidal particle increases. In our calculation we consider flat, perpendicular wall. which Illay be considered as very large colloidal panicle within small hard sphere fluid

~() the prewelling rounded on curved surfaces docs not (lnse.

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