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Evaluation of boundary effects in monomer-dimer mixtures by the method of lattice constants I. The plane-square lattice
Physica
50 (1970) 348-355
o North-Holland
EVALUATION
Publishing
OF BOUNDARY
IN MONOMER-DIMER BY
THE
METHOD
EFFECTS
MIXTURES
OF LATTICE
I. THE PLANE-SQUARE A. BELLEMANS Universite’
Co.
CONSTANTS
LATTICE
and S. FUKS
Libre de Bruxelles,
Belgique
Received 8 June 1970
synopsis It is shown how edge effects in monomer-dimer mixtures on the square lattice can be obtained by means of the strong lattice constants of a particular graph. Series expansions of the edge tension and adsorption are given up to the sixth power in the dimer density. The subsequent analysis shows that these series are still too short to account for the actual value of the edge tension of a pure dimer assembly derived by Fisher and Ferdinand.
1. Introduction. avzd description of the method. The recent and remarkable advances in the three-dimensional Ising problem (and lattice statistics in general) are largely due to the increasing number of available coefficients in appropriate series expansions at low and high temperatures. In this respect the works of Dombl), Sykes et al. 2), and Essam3) on the enumeration of the characteristic lattice constants of a given lattice are of fundamental importance. We intend here to discuss edge effects in monomerdimer mixtures on a square lattice by a method which relies essentially on the knowledge of the lattice constants of a particular lattice graph which will be defined in the next section. We shall first describe the method formally and therefore we need to recall a few definitions used in references 2 and 3. A linear graph G is an abstract concept represented by a set of vertices V and the associated pairings of these vertices represented by edges. TWO graphs G and G’ are isomorplzic when there is a one-one correspondence
IZIP
Fig. 1. Example of isomorphic graphs.
348
BOUNDARY
between
EFFECTS
the vertex
sets
IN MONOMER-DIMER
V and
V’ such that
MIXTURES.
corresponding
I
349
vertices
are
joined by edges in G only if they are joined by edges in G’ and conversely (see fig. 1). A graph H is a subgraph of G when its vertex set V(H) is contained in V(G) and all the edges of H are edges of G. A section graph G(A) of G is defined as the subgraph of G whose vertex set is A and whose edges are all those edges of G which connect two vertices in A. Any section graph (subgraph) G’ of G which is isomorphic with a graph g represents a strong (weak) embedding of g in G. The stwng (weak) lattice constant of a graph g on a graph G is defined as the number of section graphs (subgraphs) of G isomorphic with g. Consider now a lattice of which each vertex can be occupied by one molecule at most, subject to the condition that some definite pairs of vertices cannot be simultaneously occupied. Linking together each such pair of vertices by an edge we obtain a connected lattice graph G which is characteristic of the system. Let us call z the activity of the molecules. The grand partition function of the system EG(.z) can then be obtained by enumerating the number of ways of distributing molecules on G so that no edge has its two ends occupied at the same time. Applying the technique of cluster expansion we find In &(.z)
= F Bz(G) zr,
(1)
where Br(G) is the contribution of the so-called reducible clusters of I molecules on G. Consider next a connected graph g isomorphic with certain section graphs of G and, following a notation introduced by Domb, let us denote its strong lattice constant as [g; G]. We may rewrite (1) as ln
BG(z)
=
c
B
[g ;
Gl '@'&),
(2)
where wg is the total contribution of all clusters involving an arbitrary number of molecules distributed on the graph g, so that each vertex of g is occupied by one molecule at least. As relation (2) applies to any graph G, the weights ZQ(Z) can be derived by recurrence, starting from the simplest connected graphs (see table I). This operation is greatly simplified on account of a theorem proved by Essams). Defining Pg(z) = In Z&z), we have zQ(z) = I: P,(z)(c
l)V=-w, [c; g]F,
(3)
where vc and vg are, respectively, the number of vertices of the graphs c and g, and [c; g]F, called the strong full perimeter lattice constant of c in g, is the number of section graphs of g isomorphic with c, with the restriction
A. BELLEMANS
350
AND
TABLE
List graph
of all r,
and wg of connected
G of fig. 2b, with
between
i
graphs
6 vertices
S. FUKS
I
g isomorphic
and less. (For
We and r,
with
simplicity
for vg I
section
graphs
we give
4 only)
Eg = exp r,
gr 1
1
2
fz
191
=
r1 z -
1 + 22
w2
=
r2 - 2rl
3
1 + 32 + 22
w3
=
r3 - 2r2 + rl
4
1 + 32
w4
=
r4 - 3r2 + 3rl
5
1 + 42 + 3~2
w5
=
$z3 +
Qz3 -
$z4 +
@5 -
&z6 +
--22+2z-$z4+6~5-?+6+ 23 -
424 +
1225 -
.
fz”
+
.. .
2z3-9z4+30~5-90~~+
1 + 42 + 222
wg
.. .
r5 - 2r3 + r2 -24
6
of the
the relations
= =
r6 -
+
625 -
r4 -
- 224 +
2526 +
2r3 +
1425 -
1..
3r2
7
1 + 42 + 2zs
w7
=
r7 - 4r3 + 4r2
8
1 + 42
w3
=
rs - 4r4 + 6r2 - 4rl
9
1 + 52 + 6z2 + z3
wg
=
25 -
10
1 + 52 + 5zs
WlO =
2~5 -
18~6 +
...
11
1 + 5z + 5~2 + 23
Wl
=
225 -
19~” +
.. .
12
1 + 52 + 422
w12 =
425 -
4226 +
. ..
13
1 + 52 + 4~2
w13 =
625 -
5926 +
.. .
14
1 + 52 + 322
w14 =
625 -
6626 +
1..
15
1 +6z+
w5
-26
16
1 + 62 + 9,s + 2z3
w16 =
-226
+
...
17
1 + 62 + 9zs + 3z3
WI7 =
-226
+
. ..
18
1 + 62 + 9~2 + 4z3
WE
=
-226
+
...
19
1 + 62 + 8~2
w19 =
-4~6
+
...
20
1 + 62 + 8z2 + 2z3
w20 =
-426
+
.. .
21
1 + 62 + 7z2
w21 =
-
-3324 +
20~5 -
9223 +
- 624 + 4825 -
lOzs+4zs
=
826 +
+
260~~ +
...
. ..
1226 + . . .
rl
-
6726 +
... ...
.
BOUNDARY
EFFECTS
IN MONOMER-DIMER
MIXTURES.
I
351
TABLE I (Continusd) i __ 22 23
Sg = exp r,
gc
Iz
wu
1 + 62 + 722 + 23
~22 =
1+6Z+79+2z3
ze1~~=-626+..
-1226
+
24
m
1 + 62 + 622
w24 =
-1226
+ ...
25
a
1 + 62 + 6z2
W25 =
-18Zs
+ . ..
0
t +62+922+223
tt,26=
-5z’3+...
w2, =
-6626 + . . .
W28
-6626 + . . .
26 27 28
Ia m
1 + 62 + 822+ 1 + 62 + 722
that all the vertices
223
=
of g not in c (if any) must be adjacent
to vertices
of c.
An important point is that, on account of the relation between the weights ~~(2) and the cluster formulation, one finds on expanding We in powers of z z4?&) = AZ”~ + . . . . Hence by retaining all connected graphs g involving v vertices or less, we obtain from (2) a power series in z for ro(z) which is correct up to order zV. Obviously this is equivalent to the consideration of the cluster sums BZ for I < v in (1) ; however, the labour involved in such a calculation would be much larger if all possible connected graphs of I < z, vertices must be taken into account, while in the method described above a large number of these graphs are immediately ruled out by the structure of the lattice. Actually when dealing with an homogeneous graph G, a more powerful method, due to Rushbrooke and Scoins, is to construct a density expansion instead of the activity expansion (1). Then only the section graphs of G isomorphic with multiply connected graphs g containing no cut-complete star need to be counteds-7). (This enormous advantage is, however, partially lost due to an increase of algebraic manipulations.) Yet the method described above is well fitted to deal with inhomogeneous graphs, describing inhomogeneous problems where the adequate independent variable is the activity (and not the local density). It is precisely this kind of problem that we consider in the next section. 2. Edge effects in monomer-dimer mixtures on the square lattice. Consider the finite lattice graph of fig. 2a, constituted by an m x n square lattice
352
A. BELLEMANS
-m
AND
,
.._.
S. FUKS
-m-
x
>
a Fig. 2. (a) m x n
b
square-lattice
graph
with
periodic
boundary
conditions
in
the
x direction. (b) Corresponding
decorated
lattice
graph.
periodic in the x direction only, where each pair of first-neighbouring sites is joined by an edge. Our problem is to evaluate the grand partition of a mixture of monomers (each occupying one site) and dimers (each occupying one edge) on this lattice. Actually the monomers may be treated as holes for the present purpose. All positions available to the dimers are exactly matched by the 2mn - m vertices of the decorated lattice graph of fig. 2b, which is such that a pair of joined vertices cannot be occupied simultaneously. This is consequently the graph G characteristic of the problem; it is obviously inhomogeneous on account of the boundary edges parallel to the x direction. In the limit of m and n large one has In E&z)
= 2mna(z) -
2mb(z) + . . . .
(4)
with the correspondence b(z) = y/kT.
~(2) = p/Q,
(5)
Here p and y respectively denote the pressure and edge free energy) when taking the distance between the square lattice equal to one. The values of [g; combined with the weights wg of table I, allow the expansions of a(z) and b(z) up to 26. One finds a(z) = z b(z) z
The density
$22 +
+{z -
58 -52
i!$" +
3 _&+4 L+"
$ -
163324 -4
+
17pz5
construction
47:4Q6 _
+
of series
... .
"$f.X,S
+
...}.
of dimers on the graph G, in the limit m, n + co, is given by
p = .zd‘zz1d.z
which, after inversion,
(P I
t)>
leads to
z = p + 7~2 + 40~3 + 206~4 + in agreement
4g50ti25_
the edge tension (or neighbouring sites of G] listed in table II,
with results obtained
1000~5 +
4678~6 +
by other methodsa).
....
Substituting
in the
BOUNDARY
EFFECTS
IN MONOMER-DIMER TABLE
MIXTURES.
I
353
II
Strong lattice constants of the graph G of fig. 2b for all connected graphs g with 6 vertices and less k; Gl
g
. -
g
k; Gl
2mn -
m
390mn -
990m
6mn -
m
268mn -
676m
25m
520mn -
1276m
A
18mn -
A
4mn -
6m
84mn -
202m
iv
50mn -
90m
38mn -
9am
36mn -
64m
184mn -
436m
N tI
b(z) series we finally
8mn -
16m
2m
16mn -
30m
142mn -
307m
50mn -
llam
92mn -
202m
12mn -
28m
1OOmn-
208m
4mn -
am
18mn -
37m
2mn -
3m
8mn -
12m
24mn -
44m
12mn -
26m
28mn -
72m
mn -
m
mn -
obtain
y/,$T = ${p + Qp2 N
&{-ln(l
-
Q,,3 -
p) -
yp4
2p3 -
Note further that the adsorption dynamic relationship r = _
d(ylkT) d In z
--z---,
_
"2p5 -
5~4 -
yp6
6~5 -
of dimers
r
_
__.}
4~6 follows
. ..>.
(6)
from the thermo-
db dz
whereof we find I’ = - ${p -
6p2 +
6p3 +
4p4 +
12~5 + 2p6 + . . .}.
(7)
A. BELLEMANS
354
0
_lp_I_ 0.05 0.1
0.15
0.2
AND
S. FUKS
0.25
-L
Fig. 3. Edge
tension
y/kT vs. p from Fisher
Fig. 4. Adsorption
of dimers
means
(6); the full dot
at p = & is the exact
value
of
and Ferdinand.
r vs. p from
(7) by straightforward
of the Pad4 approximants
P(3,
evaluation
and by
2) and P(2, 3).
The edge tension y/kT and the adsorption r are plotted as functions of the density in figs. 3 and 4, respectively. Both curves have been obtained by straightforward evaluation of the finite polynomials (6) and (7) of degree six in p; the more sophisticated method of Pad& approximants was also used but it gives no detectable difference except for r very near to close-packing. At first sight the behaviour of r is roughly the expected one. For dilute systems of dimers r must be negative as dimers near to the edge lose their freedom of orientation to some extent. However, for sufficiently large p values, r must increase and ultimately become equal to zero for the pure dimer case. This is precisely what appears in fig. 4. However, this apparently sound behaviour of r must be regarded as fortuitous for the following reason. From fig. 3 the value of y/kT at close-packing is very nearly 0.115; actually edge effects for the pure dimer case have been analyzed exactly by Fisherg) and Ferdinandlo) and it follows from their works that the correct value of y/kT is a ln( 1 + J2)
-
G/~Tc= 0.074563..
.
(8)
(G = l-2 - 3-s + 5-Z - 7-s + . ..). H ence our estimated value of y/k?’ for pure dimers is about 50% too high and both curves of figs. 3 and 4 must become appreciably wrong as p approaches close-packing. In order to allow y/kT to reach finally the exact value (8) at close-packing, the higher coefficients of the series (6) should become strongly negative. Similarly the higher coefficients of the series (7) for r should become strongly positive. It could even be possible that at very high concentration
BOUNDARY
of dimers,
EFFECTS
the few remaining
IN MONOMER-DIMER
monomers
would be preferentially
within the bulk of the system,
giving rise to a positive
in t,his range, i.e. an S-shaped
curve for Yll)
y/KT should then pass through
a maximum
MIXTURES.
I
355
located
adsorption
of dimers
*. The corresponding
curve for
for same high value of p.
3. Conclusions. The preceding discussion does not preclude the usefulness of the proposed method for calculating edge tensions but merely shows that the series (6) and (7) are far too short to account correctly for the actual behaviour of concentrated systems of dimers. Up to the power p6 considered here the required graphs were counted by hand, but this procedure becomes prohibitively long at higher orders. It seems that a few more terms (say, e.g., four) in series (6) and (7) could be obtained by counting the relevant graphs with the aid of an electronic computer. This problem is presently under consideration as well as the calculation of the surface tension of monomer-dimer mixtures on the simple cubic lattice.
REFERENCES 1) Domb, C., Phil. Mag. Suppl. 9 (1960) 149. 2) Sykes, M. F., Essam, J. W., Heap, B. R. and Hiley, B. J., J. math. Phys. 7 (1966) 1557. 3) Essam, J. W., J. math. Phys. 8 (1967) 741. 4) See also O.Ore, Theory of Graphs, American Mathematical Society Colloquium Publications, Vol. XXXVIII, American Mathematical Society (Providence, Rhode Island, 1942). 5) Rushbrooke, G. S. and Scoins, H. J., Proc. Roy. Sot. A 230 (1955) 74. 6) Temperley, H. N. V., Proc. Phys. Sot. 80 (1962) 813; 86 (1965) 185. 7) Bellemans, A. and Nigam, R. K., J. them. Phys. 46 (1967) 2922. 8) Gaunt, D. S., Phys. Rev. 179 (1969) 174. 9) Fisher, M. E., Phys. Rev. 124 (1961) 1664. 10) Ferdinand, A. E., J. math. Phys. 8 (1967) 2332. 11) Fisher, M. E. and Stephenson, J., Phys. Rev. 132 (1963) 1411.
$ The correlation between a pair of monomers embedded in a dimer lattice is discussed in this paper but only in the limiting case where the edges recede to infinity.
50 (1970) 348-355
o North-Holland
EVALUATION
Publishing
OF BOUNDARY
IN MONOMER-DIMER BY
THE
METHOD
EFFECTS
MIXTURES
OF LATTICE
I. THE PLANE-SQUARE A. BELLEMANS Universite’
Co.
CONSTANTS
LATTICE
and S. FUKS
Libre de Bruxelles,
Belgique
Received 8 June 1970
synopsis It is shown how edge effects in monomer-dimer mixtures on the square lattice can be obtained by means of the strong lattice constants of a particular graph. Series expansions of the edge tension and adsorption are given up to the sixth power in the dimer density. The subsequent analysis shows that these series are still too short to account for the actual value of the edge tension of a pure dimer assembly derived by Fisher and Ferdinand.
1. Introduction. avzd description of the method. The recent and remarkable advances in the three-dimensional Ising problem (and lattice statistics in general) are largely due to the increasing number of available coefficients in appropriate series expansions at low and high temperatures. In this respect the works of Dombl), Sykes et al. 2), and Essam3) on the enumeration of the characteristic lattice constants of a given lattice are of fundamental importance. We intend here to discuss edge effects in monomerdimer mixtures on a square lattice by a method which relies essentially on the knowledge of the lattice constants of a particular lattice graph which will be defined in the next section. We shall first describe the method formally and therefore we need to recall a few definitions used in references 2 and 3. A linear graph G is an abstract concept represented by a set of vertices V and the associated pairings of these vertices represented by edges. TWO graphs G and G’ are isomorplzic when there is a one-one correspondence
IZIP
Fig. 1. Example of isomorphic graphs.
348
BOUNDARY
between
EFFECTS
the vertex
sets
IN MONOMER-DIMER
V and
V’ such that
MIXTURES.
corresponding
I
349
vertices
are
joined by edges in G only if they are joined by edges in G’ and conversely (see fig. 1). A graph H is a subgraph of G when its vertex set V(H) is contained in V(G) and all the edges of H are edges of G. A section graph G(A) of G is defined as the subgraph of G whose vertex set is A and whose edges are all those edges of G which connect two vertices in A. Any section graph (subgraph) G’ of G which is isomorphic with a graph g represents a strong (weak) embedding of g in G. The stwng (weak) lattice constant of a graph g on a graph G is defined as the number of section graphs (subgraphs) of G isomorphic with g. Consider now a lattice of which each vertex can be occupied by one molecule at most, subject to the condition that some definite pairs of vertices cannot be simultaneously occupied. Linking together each such pair of vertices by an edge we obtain a connected lattice graph G which is characteristic of the system. Let us call z the activity of the molecules. The grand partition function of the system EG(.z) can then be obtained by enumerating the number of ways of distributing molecules on G so that no edge has its two ends occupied at the same time. Applying the technique of cluster expansion we find In &(.z)
= F Bz(G) zr,
(1)
where Br(G) is the contribution of the so-called reducible clusters of I molecules on G. Consider next a connected graph g isomorphic with certain section graphs of G and, following a notation introduced by Domb, let us denote its strong lattice constant as [g; G]. We may rewrite (1) as ln
BG(z)
=
c
B
[g ;
Gl '@'&),
(2)
where wg is the total contribution of all clusters involving an arbitrary number of molecules distributed on the graph g, so that each vertex of g is occupied by one molecule at least. As relation (2) applies to any graph G, the weights ZQ(Z) can be derived by recurrence, starting from the simplest connected graphs (see table I). This operation is greatly simplified on account of a theorem proved by Essams). Defining Pg(z) = In Z&z), we have zQ(z) = I: P,(z)(c
l)V=-w, [c; g]F,
(3)
where vc and vg are, respectively, the number of vertices of the graphs c and g, and [c; g]F, called the strong full perimeter lattice constant of c in g, is the number of section graphs of g isomorphic with c, with the restriction
A. BELLEMANS
350
AND
TABLE
List graph
of all r,
and wg of connected
G of fig. 2b, with
between
i
graphs
6 vertices
S. FUKS
I
g isomorphic
and less. (For
We and r,
with
simplicity
for vg I
section
graphs
we give
4 only)
Eg = exp r,
gr 1
1
2
fz
191
=
r1 z -
1 + 22
w2
=
r2 - 2rl
3
1 + 32 + 22
w3
=
r3 - 2r2 + rl
4
1 + 32
w4
=
r4 - 3r2 + 3rl
5
1 + 42 + 3~2
w5
=
$z3 +
Qz3 -
$z4 +
@5 -
&z6 +
--22+2z-$z4+6~5-?+6+ 23 -
424 +
1225 -
.
fz”
+
.. .
2z3-9z4+30~5-90~~+
1 + 42 + 222
wg
.. .
r5 - 2r3 + r2 -24
6
of the
the relations
= =
r6 -
+
625 -
r4 -
- 224 +
2526 +
2r3 +
1425 -
1..
3r2
7
1 + 42 + 2zs
w7
=
r7 - 4r3 + 4r2
8
1 + 42
w3
=
rs - 4r4 + 6r2 - 4rl
9
1 + 52 + 6z2 + z3
wg
=
25 -
10
1 + 52 + 5zs
WlO =
2~5 -
18~6 +
...
11
1 + 5z + 5~2 + 23
Wl
=
225 -
19~” +
.. .
12
1 + 52 + 422
w12 =
425 -
4226 +
. ..
13
1 + 52 + 4~2
w13 =
625 -
5926 +
.. .
14
1 + 52 + 322
w14 =
625 -
6626 +
1..
15
1 +6z+
w5
-26
16
1 + 62 + 9,s + 2z3
w16 =
-226
+
...
17
1 + 62 + 9zs + 3z3
WI7 =
-226
+
. ..
18
1 + 62 + 9~2 + 4z3
WE
=
-226
+
...
19
1 + 62 + 8~2
w19 =
-4~6
+
...
20
1 + 62 + 8z2 + 2z3
w20 =
-426
+
.. .
21
1 + 62 + 7z2
w21 =
-
-3324 +
20~5 -
9223 +
- 624 + 4825 -
lOzs+4zs
=
826 +
+
260~~ +
...
. ..
1226 + . . .
rl
-
6726 +
... ...
.
BOUNDARY
EFFECTS
IN MONOMER-DIMER
MIXTURES.
I
351
TABLE I (Continusd) i __ 22 23
Sg = exp r,
gc
Iz
wu
1 + 62 + 722 + 23
~22 =
1+6Z+79+2z3
ze1~~=-626+..
-1226
+
24
m
1 + 62 + 622
w24 =
-1226
+ ...
25
a
1 + 62 + 6z2
W25 =
-18Zs
+ . ..
0
t +62+922+223
tt,26=
-5z’3+...
w2, =
-6626 + . . .
W28
-6626 + . . .
26 27 28
Ia m
1 + 62 + 822+ 1 + 62 + 722
that all the vertices
223
=
of g not in c (if any) must be adjacent
to vertices
of c.
An important point is that, on account of the relation between the weights ~~(2) and the cluster formulation, one finds on expanding We in powers of z z4?&) = AZ”~ + . . . . Hence by retaining all connected graphs g involving v vertices or less, we obtain from (2) a power series in z for ro(z) which is correct up to order zV. Obviously this is equivalent to the consideration of the cluster sums BZ for I < v in (1) ; however, the labour involved in such a calculation would be much larger if all possible connected graphs of I < z, vertices must be taken into account, while in the method described above a large number of these graphs are immediately ruled out by the structure of the lattice. Actually when dealing with an homogeneous graph G, a more powerful method, due to Rushbrooke and Scoins, is to construct a density expansion instead of the activity expansion (1). Then only the section graphs of G isomorphic with multiply connected graphs g containing no cut-complete star need to be counteds-7). (This enormous advantage is, however, partially lost due to an increase of algebraic manipulations.) Yet the method described above is well fitted to deal with inhomogeneous graphs, describing inhomogeneous problems where the adequate independent variable is the activity (and not the local density). It is precisely this kind of problem that we consider in the next section. 2. Edge effects in monomer-dimer mixtures on the square lattice. Consider the finite lattice graph of fig. 2a, constituted by an m x n square lattice
352
A. BELLEMANS
-m
AND
,
.._.
S. FUKS
-m-
x
>
a Fig. 2. (a) m x n
b
square-lattice
graph
with
periodic
boundary
conditions
in
the
x direction. (b) Corresponding
decorated
lattice
graph.
periodic in the x direction only, where each pair of first-neighbouring sites is joined by an edge. Our problem is to evaluate the grand partition of a mixture of monomers (each occupying one site) and dimers (each occupying one edge) on this lattice. Actually the monomers may be treated as holes for the present purpose. All positions available to the dimers are exactly matched by the 2mn - m vertices of the decorated lattice graph of fig. 2b, which is such that a pair of joined vertices cannot be occupied simultaneously. This is consequently the graph G characteristic of the problem; it is obviously inhomogeneous on account of the boundary edges parallel to the x direction. In the limit of m and n large one has In E&z)
= 2mna(z) -
2mb(z) + . . . .
(4)
with the correspondence b(z) = y/kT.
~(2) = p/Q,
(5)
Here p and y respectively denote the pressure and edge free energy) when taking the distance between the square lattice equal to one. The values of [g; combined with the weights wg of table I, allow the expansions of a(z) and b(z) up to 26. One finds a(z) = z b(z) z
The density
$22 +
+{z -
58 -52
i!$" +
3 _&+4 L+"
$ -
163324 -4
+
17pz5
construction
47:4Q6 _
+
of series
... .
"$f.X,S
+
...}.
of dimers on the graph G, in the limit m, n + co, is given by
p = .zd‘zz1d.z
which, after inversion,
(P I
t)>
leads to
z = p + 7~2 + 40~3 + 206~4 + in agreement
4g50ti25_
the edge tension (or neighbouring sites of G] listed in table II,
with results obtained
1000~5 +
4678~6 +
by other methodsa).
....
Substituting
in the
BOUNDARY
EFFECTS
IN MONOMER-DIMER TABLE
MIXTURES.
I
353
II
Strong lattice constants of the graph G of fig. 2b for all connected graphs g with 6 vertices and less k; Gl
g
. -
g
k; Gl
2mn -
m
390mn -
990m
6mn -
m
268mn -
676m
25m
520mn -
1276m
A
18mn -
A
4mn -
6m
84mn -
202m
iv
50mn -
90m
38mn -
9am
36mn -
64m
184mn -
436m
N tI
b(z) series we finally
8mn -
16m
2m
16mn -
30m
142mn -
307m
50mn -
llam
92mn -
202m
12mn -
28m
1OOmn-
208m
4mn -
am
18mn -
37m
2mn -
3m
8mn -
12m
24mn -
44m
12mn -
26m
28mn -
72m
mn -
m
mn -
obtain
y/,$T = ${p + Qp2 N
&{-ln(l
-
Q,,3 -
p) -
yp4
2p3 -
Note further that the adsorption dynamic relationship r = _
d(ylkT) d In z
--z---,
_
"2p5 -
5~4 -
yp6
6~5 -
of dimers
r
_
__.}
4~6 follows
. ..>.
(6)
from the thermo-
db dz
whereof we find I’ = - ${p -
6p2 +
6p3 +
4p4 +
12~5 + 2p6 + . . .}.
(7)
A. BELLEMANS
354
0
_lp_I_ 0.05 0.1
0.15
0.2
AND
S. FUKS
0.25
-L
Fig. 3. Edge
tension
y/kT vs. p from Fisher
Fig. 4. Adsorption
of dimers
means
(6); the full dot
at p = & is the exact
value
of
and Ferdinand.
r vs. p from
(7) by straightforward
of the Pad4 approximants
P(3,
evaluation
and by
2) and P(2, 3).
The edge tension y/kT and the adsorption r are plotted as functions of the density in figs. 3 and 4, respectively. Both curves have been obtained by straightforward evaluation of the finite polynomials (6) and (7) of degree six in p; the more sophisticated method of Pad& approximants was also used but it gives no detectable difference except for r very near to close-packing. At first sight the behaviour of r is roughly the expected one. For dilute systems of dimers r must be negative as dimers near to the edge lose their freedom of orientation to some extent. However, for sufficiently large p values, r must increase and ultimately become equal to zero for the pure dimer case. This is precisely what appears in fig. 4. However, this apparently sound behaviour of r must be regarded as fortuitous for the following reason. From fig. 3 the value of y/kT at close-packing is very nearly 0.115; actually edge effects for the pure dimer case have been analyzed exactly by Fisherg) and Ferdinandlo) and it follows from their works that the correct value of y/kT is a ln( 1 + J2)
-
G/~Tc= 0.074563..
.
(8)
(G = l-2 - 3-s + 5-Z - 7-s + . ..). H ence our estimated value of y/k?’ for pure dimers is about 50% too high and both curves of figs. 3 and 4 must become appreciably wrong as p approaches close-packing. In order to allow y/kT to reach finally the exact value (8) at close-packing, the higher coefficients of the series (6) should become strongly negative. Similarly the higher coefficients of the series (7) for r should become strongly positive. It could even be possible that at very high concentration
BOUNDARY
of dimers,
EFFECTS
the few remaining
IN MONOMER-DIMER
monomers
would be preferentially
within the bulk of the system,
giving rise to a positive
in t,his range, i.e. an S-shaped
curve for Yll)
y/KT should then pass through
a maximum
MIXTURES.
I
355
located
adsorption
of dimers
*. The corresponding
curve for
for same high value of p.
3. Conclusions. The preceding discussion does not preclude the usefulness of the proposed method for calculating edge tensions but merely shows that the series (6) and (7) are far too short to account correctly for the actual behaviour of concentrated systems of dimers. Up to the power p6 considered here the required graphs were counted by hand, but this procedure becomes prohibitively long at higher orders. It seems that a few more terms (say, e.g., four) in series (6) and (7) could be obtained by counting the relevant graphs with the aid of an electronic computer. This problem is presently under consideration as well as the calculation of the surface tension of monomer-dimer mixtures on the simple cubic lattice.
REFERENCES 1) Domb, C., Phil. Mag. Suppl. 9 (1960) 149. 2) Sykes, M. F., Essam, J. W., Heap, B. R. and Hiley, B. J., J. math. Phys. 7 (1966) 1557. 3) Essam, J. W., J. math. Phys. 8 (1967) 741. 4) See also O.Ore, Theory of Graphs, American Mathematical Society Colloquium Publications, Vol. XXXVIII, American Mathematical Society (Providence, Rhode Island, 1942). 5) Rushbrooke, G. S. and Scoins, H. J., Proc. Roy. Sot. A 230 (1955) 74. 6) Temperley, H. N. V., Proc. Phys. Sot. 80 (1962) 813; 86 (1965) 185. 7) Bellemans, A. and Nigam, R. K., J. them. Phys. 46 (1967) 2922. 8) Gaunt, D. S., Phys. Rev. 179 (1969) 174. 9) Fisher, M. E., Phys. Rev. 124 (1961) 1664. 10) Ferdinand, A. E., J. math. Phys. 8 (1967) 2332. 11) Fisher, M. E. and Stephenson, J., Phys. Rev. 132 (1963) 1411.
$ The correlation between a pair of monomers embedded in a dimer lattice is discussed in this paper but only in the limiting case where the edges recede to infinity.