square lattice of width M, with first- and second-neighbor interactions. ... sumed to be repulsive, and allowing second-neighbor interactions to be attractive or ...
PHYSICAL REVIEW E
VOLUME 55, NUMBER 3
MARCH 1997
Monomer adsorption on a square lattice with first- and second-neighbor interactions Alain J. Phares and Francis J. Wunderlich Department of Physics, Mendel Hall, Villanova University, Villanova, Pennsylvania 19085-1699 ~Received 2 October 1996! We obtain the low-temperature phases and phase transitions of monomer adsorption on a semi-infinite square lattice of width M , with first- and second-neighbor interactions. With first-neighbor interactions assumed to be repulsive, and allowing second-neighbor interactions to be attractive or repulsive, six sets of surface adsorption phases have been identified. Most of the numerical results conducted up to M 512 are found to fit exact closed-form expressions in M , thus allowing exact analytic extrapolations to the infinite twodimensional case ~M 5`!. @S1063-651X~97!03103-6# PACS number~s!: 02.50.2r, 05.50.1q, 05.70.2a, 64.60.Cn
I. INTRODUCTION
A year ago, we reported a number of crystallization patterns ~phases! of monomers with nearest- ~first-! neighbor interaction on a multilayered semi-infinite square lattice @1#. This article goes a step further, analyzing the more realistic model of monomer surface adsorption including nearest- and next-nearest- ~second-! neighbor interactions. Lattice models have been used for a very long time; a brief summary of lattice calculations done by others can be found in Ref. @1#, and the notation used here is very similar to that of Ref. @1#. The surface is a semi-infinite M 3N square lattice ~N→`! in the presence of a gas containing one molecular species and the adsorbed molecules occupy one site. For this reason, we refer to them as monomers. The system is at thermal equilibrium and the chemical potential energy m of a monomer depends on the external gas pressure. The interaction energies of an adsorbed monomer are V 0 with the lattice, V with any first-neighbor monomer at a distance a, and W with any second-neighbor monomer at a distance a &. The activities associated with these three interactions are
F G
F G
m 1V 0 , k BT
x5exp
V , k BT
y5exp
F G
z5exp
W , k BT
~1!
where k B is Boltzmann’s constant and T the absolute temperature. Here the transfer matrix T 1M for a lattice of width M is of rank D(M )52 M . It is recursively constructed as in Ref. @1#, and we find T 1M 5
T 2M 5
S
T 1M 21 T 2M 21 P 1M 5
P 2M 5
S
T 1M 21 zT 2M 21
S
S
T 1M 21 T 2M 21
D
x P 1M 21 , xy P 2M 21
xz P 1M 21 xyz P 2M 21
D
T 1M 21 zT 2M 21
xy P 1M 21 , xy 2 z P 2M 21
xyz P 1M 21 xy 2 z 2 P 2M 21
1063-651X/97/55~3!/2403~6!/$10.00
The numbers of monomers, first-neighbor monomermonomer interactions, and second-neighbor monomermonomer interactions per site are u0 , u, and b, respectively. In the limit ~N→`!, these quantities are related to the largest eigenvalue R of T 1M according to
u 05
S5
D
u5
y ]R , MR ]y
b5
z ]R , M R ]z
1 lnR2 u 0 lnx2 u lny2 b lnz, M
~3!
where S is the entropy per site divided by k B . For finite length N, all the eigenvalues contribute to the expressions of u0 , u, b, and S in a manner discussed in Ref. @2#. For given monomer-monomer interactions V and W and with the temperature of the system set to be below a certain value as dictated by the relation V ~ 2V ! ,210⇒T, , k BT 10k B
~4!
adsorption patterns are observed to occur sequentially with increasing external chemical potential m. The Cray C90 supercomputer of the Pittsburgh Supercomputing Center was used with EISPACK for the numerical computations. In one dimension ~M 51!, next-nearest-neighbor monomers are at a distance 2a and their interaction is neglected, leaving only first-neighbor interactions. This case has an exact analytic solution and has been fully discussed in Ref. @1#. The case M 52 is the only other case for which an exact analytic solution can be derived including first- and secondneighbor interactions, since the eigenvalues of T 1M are the solutions of the secular equation
$ R2x ~ y2z ! %$ R 3 2R 2 @ 11x ~ y1z ! 1x 2 y 3 z 2 #
with T 10 5T 20 51;
D
x ]R , M R ]x
1Rx @~ y1z22 !~ 11x 2 y 3 z 2 ! 1xy ~ y 2 z 2 21 !#
~2!
1x 3 y ~ yz21 ! „~ yz11 !~ y1z ! 24yz…% 50,
~5!
where one root is immediately identified as x(y2z). The first-neighbor interaction is repulsive ~V,0! and, consequently, the second-neighbor interaction W must be algebraically greater whether it is repulsive or attractive (V,W).
with P 10 5 P 20 51. 55
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© 1997 The American Physical Society
ALAIN J. PHARES AND FRANCIS J. WUNDERLICH
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TABLE I. Characteristics of the phase adsorptions on square lattices of finite width M >3. The quantity @M /2# refers to the integer part of M /2 and the question mark stands for the lack of a closed-form expression for the entropy. S Phases p0 p1 p2 ~M p3 ~M p4 ~M p5 p6 p7 ~M p8 p9 ~M p 10 ~M p 11 ~M p 12 ~M
u0
u
b
M odd
M even
0 M 2 @ M /2# 2M (M 12)/4M
0 0
0 0
0 0
0 ~1/2M !ln@~M 12!/2#
0
1/M
?
(M 12)/4M
(M 22)/2M 2
2/M 2
~1/2M !ln~M /2!
(M 12)/4M
1/2M
0
~1/2M !ln~M /2!
1 2
0 3/M
even! even! even! (M 12)/2M 1 2
(M 21)/M (M 21)/M 2u1b5(M 21)/M
0 0
0 0 ?
0
0
even! M 2 @ M /2# M (M 12)/2M
3M 24 @ M /2# 21 2M (M 13)/2M
0 2/M
0
even! 2u1b5(M 15)/M
(M 12)/2M
?
even! 3 4
(M 21)/M
(M 21)/M
?
3 4
(2M 25)/(2M 24)
M 2 24M 16 M (M 22)
?
2M 2 @ M /2# 2M (3M 12)/4M
3M 22 @ M /2# 21 2M (M 11)/M
2M 22 @ M /2# 22 M 1
0
1
(2M 21)/M
(2M 22)/M
0
even! even.4!
p 13 p 14 ~M even! p 15
Thus z.y and the root x(y2z) is negative and cannot be the largest root of Eq. ~4!. Indeed, according to the FrobeniusPerron theorem, the eigenvalue of the largest modulus of the family of matrices of which the transfer matrices belong must be real and positive. The energy per site must be continuous across a phase boundary. Thus, with Du0 , Du, and Db being the corresponding changes of u0 , u, and b across a given boundary, no change in the energy per site requires ~ m 1V 0 ! D u 0 1VD u 1WD b 50.
~6!
This equation has been verified to hold in all cases. Section II provides most of the lattice configurations corresponding to the possible phases encountered for M .2. Which phases and phase transitions are observed should depend on the first- and second-neighbor interaction energies. These questions will be answered in Sec. III. Section IV discusses the limit as M →` and Sec. V is the summary and conclusion. II. LATTICE CONFIGURATIONS OF THE OBSERVED PHASES
With the physical constraints V,0 and V,W, numerical calculations were carried out up to and including M 512.
~1/2M !ln@~M 22!/2# ~1/2M !ln~M /2! 0
The characteristics of all the phases p encountered for any width M of the lattice greater than 2 are found in Table I. The phases are ordered by increasing values of the coverage u0 of the lattice, and distinct phases with the same coverage are generally ordered by increasing the number of first neighbors per site. In most cases, the characteristics u0 , u, b, and S of the phases for a lattice of width M were observed to fit exactly closed-form expressions as reported in Table I. The quantity @M /2# refers to the integer part of M /2 and the question marks stands for the lack of a closed-form expression for the entropy. For phases p 7 and p 10 , we obtain an exact closed-form fit for ~2u1b!, but not for the separate values of u and b. With these two exceptions, the knowledge in a given phase of u0 , u, and b analytically in terms of M should, in principle, allow the construction of all the corresponding lattice configurations. Samples of these configurations are provided in Figs. 1 and 2. They exclude phases p 7 and p 10 for the reason mentioned above and the trivial cases of the empty ~p 0! and the completely filled ~p 15! lattice. In these figures, a diagram represents one possible configuration associated with a given phase whose characteristics u0 , u, and b have been determined numerically. A diagram only shows the section of the lattice whose configuration repeats throughout the lattice. We were able to enumerate all distinct
MONOMER ADSORPTION ON A SQUARE LATTICE WITH . . .
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FIG. 2. Lattice configurations for phases p 2 , p 3 , p 11 , and p 12 occurring only for lattices of even width. The convention adopted in this figure is the same as in Fig. 1. FIG. 1. Lattice configurations for phases p 1 , p 4 , p 5 , p 6 , p 8 , p 9 , p 13 , and p 14 . A lattice site is represented by a square cell of size a, which is left blank when unoccupied and has a dot in the middle when occupied.
phase configurations and derive the entropy of all but the two phases mentioned above and four others, p 2 , p 3 , p 11 , and p 12 , presented separately in Fig. 2. As a sample of the manner in which we analytically computed the entropy of the adsorbed system in the phases of Fig. 1, we consider phase p 1 . For M odd, the configuration shown is the one for which every other column of the lattice of width M is unoccupied, while the remaining ~M 11!/2 columns have every other site occupied. The remaining configurations having the same values of u0 , u, and b are obtained from this one by vertically shifting by one lattice site the monomers of one or more of these ~M 11!/2 columns. In this manner, we derive that the number C of all the distinct configurations of the p 1 phase is d52(M 11)/2. The value of S follows by dividing the logarithm of C by the number (NM ) of sites ~N is the length of the lattice! and then taking the limit N→`, or
S D
1 S5 lim ln2 ~ M 11 ! /25 lim NM N→` N→`
HS D J
M 11 ln2 50. ~7! 2NM
In the even case, the number of columns having every other site occupied by a monomer is ~M /2!. Thus there is a total of d52M /2 distinct configurations generated by shifting vertically by one site the monomers of one or more of these ~M /2! columns. In addition, any two consecutive lattice rows of each of these d configurations may have their configura-
tion changed without creating additional first or second neighbors. This is done by shifting horizontally one monomer at a time by one cell, an operation that could not be done for M odd. For any two consecutive rows, the number of such restricted horizontal shifts is ~M /2!, leading to a total of g5~M 12!/2 distinct configurations. Since there are ~N/2! sections of two consecutive rows, there are gN/2 distinct configurations in each of the previously identified d configurations. Thus the final number C of all possible configurations in phase p 1 with M even is C5 d g N/2.
~8!
The value of S follows as
S D S D
S5 lim
N→`
5
1 lnC5 lim NM N→`
1 lng . 2M
HS D S D J 1 1 lnd 1 lng NM 2M
~9!
Equations ~7! and ~8! are easily generalized for other phases, and the corresponding values of g and d are indicated in Fig. 1, accordingly. The theoretical values of S obtained in this manner are listed in Table I and have been numerically verified to hold within the accuracy of the Cray C90 supercomputer. For the phases of Fig. 2, a number of configurations can easily be constructed from those presented in this figure. We could not find the total number C that reproduces the observed numerical values. However, we list in Table I the closed-form expression of the entropy for p 3 ,
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ALAIN J. PHARES AND FRANCIS J. WUNDERLICH
~1/2M !ln~M /2!, which fits exactly the numerical results for even M up to and including M 512. The configurations provided for phases p 3 and p 12 require some explanation. Phase p 3 exists for lattices of even width M . The configuration shown exhibits highlighted sections of the lattice that are 434, 636, etc., up to M 3M . This indicates that the section of the lattice that repeats throughout the lattice is the highlighted 434 for the lattice of width M 54, 636 for the lattice of width 6, and so on. Phase p 12 appears only for lattices of even width M >6. Figure 2 exhibits one possible configuration for each of M 56 and 8, which is then easily generalized for higher even values of M .
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Set (b). This set is observed only for M odd and corresponds to 2 21 ,a,2f (M ). The phases encountered sequentially are p 0 , p 1 , p 5 , p 8 , p 6 , p 13, and p 15 and, as predicted by Eq. ~6!, the transitions occur exactly at
m 1V 0 50;
p 0 →p 1 ,
m 1V 0 524W;
p 1 →p 5 , p 5 →p 8 ,
m 1V 0 52V ~ M 11 ! 12W ~ M 21 ! ;
p 8 →p 6 ,
m 1V 0 51V ~ M 25 ! 22W ~ M 21 ! ; m 1V 0 524V;
p 6 →p 13 ,
III. ENERGY REGIONS, SETS OF PHASES, AND PHASE TRANSITIONS
p 13→p 15 ,
First- and second-neighbor adsorbate-adsorbate interaction energies V and W depend on the molecular properties of the monomers and the lattice spacing a. We were able to identify numerically six interaction energy regions. In each region, only a certain number or set of the phases are observed. In a given set, a phase transition occurs at a certain external gas pressure, or chemical potential m, as required by Eq. ~6!. For convenience, we introduce the quantities
m 1V 0 524V24W.
Set (b8). This set is observed only for M even and corresponds to a52 21 . The phases encountered sequentially are p 0 , p 1 , p 3 , p 7 , p 10 , p 12 , p 14 , and p 15 and, as predicted by Eq. ~6!, the transitions occur exactly at
m 1V 0 50;
p 0 →p 1 , p 1 →p 3 ,
m 1V 0 52V;
~10!
p 3 →p 7 ,
m 1V 0 522V;
where a is negative for the repulsive second-neighbor interaction and positive otherwise, with a.21. The six sets of phases are ordered by increasing values of a. Set (a). This set corresponds to 21,a
