A structural comparison of random sequential adsorption and ... - LPTMC

A structural comparison of random sequential configurations of spherocylinders

adsorption

and equilibrium

S. M. Ricci and J. Talbot School of Chemical Engineering, Purdue University West Lafayette, Indiana 47907

G. Tarjus and P. Viot Laboratoire de Physique Thkorique des Liquides, ” Universik Pierre et Marie Curie, 4, place Jussieu 75252 Paris Cedex OS, France

(Received 16 June 1994; accepted 11 August 1994) The structure of two-dimensional configurations of spherocylinders (discorectangles) generated by random sequential adsorption (RSA) is analyzed via the circular harmonic expansion of the pair distribution function and compared to that of equilibrium fluids at the same density. The structural differences are minimal for short particles but become more pronounced as the aspect ratio of the particles increases. An analysis of the correlations between particles which adsorb at high coverage with their nearest neighbors in saturated RSA configurations, reveals that the most probable relative orientation for particles with aspect ratio a-2 is perpendicular. This observation helps to explain the maximum in saturation coverage as a function of particle elongation near the aspect ratio a=2.

I. INTRODUCTION Random sequential adsorption (RSA) has become popular in recent years as a model for the irreversible, monolayer adsorption of large molecules (e.g., proteins, bacteria, and latexes) from solution to solid surfaces.‘-3 In RSA, hard-core objects are added sequentially in random positions to a surface (or volume) subject to the constraint of no overlap. Once placed, the position of an object remains fixed. Eventually, the surface saturates and it is impossible to add another particle-a condition commonly known as the jamming limit. Thus, RSA is a simple nonequilibrium model which incorporates the exclusion or blocking effect of adsorbed particles and represents the limit of complete irreversibility where particles bind so strongly to the surface that they neither diffuse on the surface nor desorb from it. While there is as yet no analytical solution for the kinetics of RSA over the entire coverage range in dimensions greater than one, extensive 2D studies using both spherica13-9 and nonspherical”-I6 particles have produced exact results in the limits of low and high coverage. Various approximate equations have also been developed which interpolate between the low- and high-coverage limits and reproduce simulation data with good accuracy over the entire coverage range.8*‘6*‘7 In addition, simulation studies have revealed that, as the aspect ratio of particles increases, the saturation coverage passes through a maximum near aspect ratio (u=2 and then decreases algebraically to zero in the limit of infinite elongation.“*‘3~15 The structure of surface phases generated in RSA has been investigated for spherical particles and it has been shown that, at intermediate and high coverages, RSA configurations are different from their equilibrium counterparts5s6 In particular, one may show analytically that the pair distribution function for surfaces saturated with disks diverges logarithmically at contact. No such study has as yet ‘knit& 9164

de Recherche Associ6e au CNRS (URA 765). J. Chem. Phys. 101 (iO), 15 November 1994

been reported for RSA surfaces composed of unaligned nonspherical bodies. Equilibrium fluids composed of hard nonspherical bodies have been shown to exhibit a rich phase behavior. Despite the fact that particle-particle interactions are purely repulsive, both computer simulations’* and various theories” indicate that 3D fluids of hard convex bodies may form a variety of liquid crystalline phases. Two-dimensional simulation studies of ellipses by Cuesta and Frenke12’revealed a continuous, isotropic-to-nematic (I-N) transition for (r=6 at a coverage 0=0.59. In addition, a first-order I-N transition was observed for a=4 at 8---0.76, but not for cr=2. Various theories, such as the hypernetted chain (HNC) and PercusYevick (PY) integral equations,2’*22 as well as density functional theories= have also been applied to 2D fluids. In the study by Ferreira et al.,22 it was found that HNC predicts the occurrence of an orientational freezing transition for n=4 and a=6 while PY does not. The degree of surface diffusion present in irreversible adsorption may have profound consequences for the structure and properties of monolayer films. Certainly, if a nematic phase were to form in a system where particles diffuse rapidly on the surface, one would expect that the properties of such a film to be vastly different from one where the particles remain immobile after adhesion. This is indeed what we show below. However, for equiIibrium fluids of hard ellipses and the known RSA jamming coverages, nematic phases are not expected to be present within the RSA coverage range for c&6. In these cases, one would like to know the extent of the structural differences between the RSA and equilibrium configurations at the same coverage. In particular, might the structure be used to distinguish between the two adsorption mechanisms? We also investigate the diffusional relaxation of nearly jammed RSA configurations, a situation which could arise if the surface diffusion of the adsorbed molecules is small compared with the adsorption rate. Since there is a

0021-9606/94/101(10)/9164/17/$6.00

Q 1994 American Institute of Physics

Ricci et al.: Configurations

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of spherocylinders

r-t2 (see Fig. 1). Given a particle with orientation $t, the probability of finding a second particle in the surface element dr,, at r12 with orientation & is m(rl2,A

7d2)dr12d41d42v

(1)

where p is the system density. The pair distribution function is an important quantity in equilibrium statistical mechanics since thermodynamic properties can be expressed as integrals over g. For nonspherical molecules, however, g (r l 2, 4 t , c$~) is a complicated, threedimensional function which is difficult to calculate numerically. It may be calculated from computer simulation by constructing and normalizing a histogram of particle distances and orientations. However, memory requirements preclude histogram constructions with sufficient resolution for all three variables. A convenient way to overcome this problem is to express g as a series whose coefficients are functions of the inter-particle separation. This is accomplished by expanding a generic configurational property as an infinite series of some suitable orthononnal set. One choice, which has been used extensively in studies of equilibrium fluids,24*25is the spherical harmonics. If &r12,~1,&) is any configurational property, then its circular harmonic expansion is written E(rl2,&,+2)=C

En,m(r12)ein61eimd2, n,m

(2)

where the sums over n and m are from --03 to 00. Note that for particles with an axial plane of symmetry, a( r12, ~6,+~,~2)=~/(r12r451,~2+~)=~3(r12,~lr~2)r so only those coefficients with even n and m are nonzero. The coefficients E:n,m(r12) may be expressed as follows: En,Jr12)=

-$

I)41

/)qb2e-‘“dl

Xe-im42Z(r12,#l,#2). FIG. 1. Illustration of the coordinate system used in the pair distribution function. Also shown are the perpendicular (“T”) and parallel relative orientations, for which the circular harmonic series is summed in Figs. 2 and 3.

high probability that many particles in a jammed RSA configuration are in contact, it is not clear that RSA surfaces are able relax to equilibrium if the constraint of zero surface diffusion is released. Finally, there is still no explanation for the presence of the maximum in e,(m) vs CYX near a=2. A structural analysis may provide some insight as to why the filling process is most efficient for particles with a length-to-width ratio a-2.

If z is a real valued function, then one may show by equating Eq. (2) with its complex conjugate that Z-,,-,= Z,,, . The pair distribution function is used to calculate the ensemble average of B in the shell between r12 and r 12+ dr12 through the following expression: (E)=

SS~!(r12,(61r~2)g(r12,~1,~2)d~1d~2

(4)

SSgtr12941~&)d41d~2

Replacing E with g in Eq. (3) one finds for go,o,

g0,0trl2)

II. THE RADIAL DISTRIBUTION FUNCTION For anisotropic particles, the pair distribution function is given by g(r12, +i, +2)r where y12 is the center-to-center distance between two particles and +1 and CJ&are the orien-

(3)

Letting 8(r,,,+, ranging gives

1 = ;TZ

f

0

wdh

I

o~d4str12~41

.A).

(5)

,+2) = e-in61e-im4z in Eq. (4) and rear-

tationsof the particleswith respectto the interparticlevector J. Chem. Phys., Vol. 101, No. 10, 15 November 1994

Riooi

9166

et

a/.: Configurations

of spherooylinders

1’.

’

*

I’.

‘.

I

*.

.

.-

RS!A a=5 E $ 20 a 22 .(I tn 10.

I\.

0:.

. * I.

I.0

i.2

.

a=lO-

'.S .

* I 1..

I I..

1.4

.i.6

L--z -'*-SW. . t n., f.8

2.0

2.0

2.5

3.0

fll

TlL

40_

40

$

.

$

20

I! 22 h

0 1.4

1.6

1.8

2.0

1.0

1.5

2.0

TlI

rlI

Direct -.----

n

-=12

----

--.*

n

-=20

. __._. .__. _. %=24

FIG. 2. Test of convergence of the circular harmonic series with different numbers of terms by comparison with a direct calculation from simulation for 4,=&=d2 (parallel relative orientation) at surface coverages close to the RSA saturation values. Although g should be finite as r,,+l, the uncertainty resulting from the histogram construction (Ref. 26) is too large here to provide any meaningful result around r12= 1.

Since g is real valued, the imaginary part of Eq. (6) can be ignored and gn,Jrlz) may be rewritten as

g,,,(~lz)=go,o(~l2)(cos a41cm 42 -sin a4l sin =go,o(P.l2)(cos(n~l+m~2)).

xcos(d,+w52),

(8)

rncp2)

(7)

It is now possible to express the pair distribution function as an infinite harmonic series. In accordance with Eq. (2),

where we have used the fact that all of the particles are identical so that g,,, = g,,, . The coefficients possess the following properties:

J. Chem. Phys., Vol. 101, No. 10, 15 November 1994

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of spherocylinders

9167

a=10

6.5

6.0

7.0

6.6

7.5

rlI

L

a=10

F;‘

$

!! k. 03

,:

-

-e-w---

00 3.0

3.6

4.0

4.5

5.0

6.5

6.6

6.0

rlI

----_

J

7.0

7.5

rlz

Direct

-----.--

-=I2

._.._..__... n-=24

n-=20

FIG. 3. Same as Fig. 2 using- the perpendicular relative orientation (i.e., I$, =?r/2 and &=O). The convergence is poor for l+,,bn,mtr12>~b0,00.

In the simulations to be discussed here, 500 nearly saturated RSA configurations of spherocylinders (more precisely discorectangles) of various aspect ratios cy were analyzed. The configurations were generated using the methods outlined in Ref. 15. The aspect ratio is defined as a=alb, where a is the half-length of a particle and b is the half-width. The particle width was fixed at unity in the simulations so that b = l/2 and cu=2a. The initial configurations of the equilibrium simulations consisted of a number of particles equal to the average number of particles deposited in the RSA simulations, each centered on a square lattice and aligned with one of the box sides. The configuration was then allowed to equilibrate using the Metropolis Monte Carlo method with combined translational and rotational displacements. Melting

was monitored by calculating the average y component of the particle orientation vector and equilibrium was assumed when the average leveled off to a relatively constant value (which was zero for all aspect ratios except ~310). Once equilibrated, 200 000 cycles of particle displacements were performed (a cycle is an attempted move for every particle) and histogram data was taken every 200 cycles. In all of the simulations, the side of the simulation surface was set to ten times the particle length in order to minimize finite size effects.

iii. CONVERGENCE OF THE CIRCULAR HARMONIC EXPANSION The circular harmonic expansion is known to converge slowly at small r12 and large a.26 More specifically, the resummations of the circular harmonic expansion fall smoothly from a maximum value to zero as r12 approaches the contact

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Ricci et a/.: Configurations

‘~~“I*‘*~,~=~~,~~~-

lo-

9169

of spherocylinders

:

10.

I~~~‘I~‘~~i~“~,-~~’ ~~i~“~,~‘~’ :

a=l.25 8-

i

RSA EQU

a=2 a-

- -

2-

Cd Cd

0 ....i....,....,....l....l ....i....,....,....I.... -0.5

0.0

0.5

1.0

1.5

2.0

a=5

RSA EQU

15 -

I G

- -

3

3

.

$

10

-1

0

1

2

3

R,!+A EQU - -

f

4

\

20

-1

0

SIX

I

2

3

4

S fX

FIG. 5. Isotropic component, g(s,,), of the surface-surface distribution function for RSA and equilibrium at the RSA saturation coverage for various aspect ratios. The RSA curves display a logarithmic divergence as sIz-+l.

value, while a direct calculation leads to a finite value at contact. Increasing the number of terms in the series has the effect of moving the maximum toward the contact value. In order to test the convergence of the series, the pair distribution function was calculated directly for the parallel (& = ?r/2 and &= 7r/2) and perpendicular (& = 7r/2 and &= 0) relative orientations (see Fig. 1). In these calculations, histograms of center-center distances were constructed for both relative orientations. Orientation “bins” with widths of 10” and centered on each of the above angles were also defined. Whenever either of the relative orientations occurred at a given center-center distance (as defined by the bins), the corresponding histogram was incremented. Before proceeding with the calculations of the coefficients of the harmonic expansion, test runs were performed in which the series was calculated to a given number of terms. A given number of terms in the harmonic series was

then summedfor the above two relative orientationsand

compared to the direct calculation. Figures 2 and 3 show the results for RSA and equilibrium configurations of aspect ratios cu=5 and CY=10. In these plots, the series calculated with nmax, lmmaxj= 12,20, and 24 are shown along with the results for the direct calculation. As seen in Fig. 2, the convergence improves as the number of terms in the series increases. The series with it,,, =24 (361 terms) compares well with the direct calculation. For perpendicular relative orientations the convergence of the series is poor in the region IQ-i2