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Exact solution of restricted cell model for hard parallel rectangles in a “brick-wall” configuration
Volume 47A, number 5
PHYSICS LETTERS
EXACT SOLUTION
OF RESTRICTED
RECTANGLES
22 April 1974
CELL MODEL FOR HARD PARALLEL
IN A "BRICK-WALL"
CONFIGURATION
O.J. HEILMANN, R. MOSS and E.L. PRAESTGAARD Department o f Chemistry, H.C. (Prsted Institute, University o f Copenhagen, DK-2100 Copenhagen G), Denmark Received 8 March 1974 The cell restrictions are defined, and it is shown that the model is essentially equivalent to a model presented in a previous paper. In a recent letter [1 ] a new class o f cell models for liquids was introduced, In a following paper [2] a special version o f this model, a restricted hard square model, was solved rigorously. We consider here a cell model for hard parallel rectangles. The cell boundaries are given by the positions o f the four particles which we have called the topological neighbours. They are the nearest neighbours to the particle in question in the closedpacked configurations; at lower densities they conserve the topology o f the close-packing, but they are not necessarily the physical nearest neighbours to the particle in the cell. In contrast to the model introduced earlier the cell boundaries are drawn parallel to the edges o f the particles and the x- and y-boundaries are slightly different as shown in fig. 1. More precisely the interior of the cell is defined by the conditions:
1
Y
4 X
x > x I +e
x > x 3 +e
x < x 2 +e
x < x 4 +e
Y >Y3
Y >Y4
Y
Y
(1)
where e determines where the x - b o u n d a r y is drawn, -a
Fig. 1. A hard rectangle and its four topological neighbour~ The accessible area for the center is shaded for the unrestricted model and dotted for the restricted model (3a = 5b and e = 0). Let us consider N rectangles arranged in ny topological rows o f n x particles inside a box o f area A = L x X L y . F o r the restricted model the cell condition can be written xl. 1 + a + e < x i <
x'i-a+e,
(2)
where a dash indicates a co-ordinate in an even numbered row, one can solve for the x-dependence independent o f the y-cordinates. This is accomplished 413
Volume 47A, number 5
PHYSICS LETTERS
by a transfer-matrix method identical to the one used in eq. (2) and the solution is given in terms of the free length pr. particle,
Lx -(2n x gx
.
.
.
.
II
1)a
.
.
.
Lx .
X
?1 X
2a
(3)
in the thermodynamic limit. The entropy in the x-direction is
Sx/k = 1 + log(rx/rr ).
(4)
To obtain they-dependence the particles are considered to be arranged in rows along the y-axis, 2n X rows of ½ny particles, since this gives the cell conditions the form t
Yi 1 + 2 b < Y i
t
t
2b,
(5)
and the same transfer-method solution. The free length is L), - (ny - 1 ) 2 b ry = ~n = 2 [ L y / n y - 2b ] , (6) y
22 April 1974
Beyerlein 13] and Beyerlein et al. [4] have developed high density expansions for hard squares packed in "brick-wall" as well as in regular square lattice configurations. They find a distinctly lower entropy for the brick-wall configurations. That we find the same entropy for the two restricted but otherwise equivalent models is reasonable as the following argument shows: In the "brick-wall" model with walls parallel to the edges of the particles it is the edges which bunch into the walls while it is the corners for the other model where the walls are perpendicular to the diagonals; consequently the deviation between the restricted and unrestricted model should be smallest for the brick-wall configuration. The result for the equation o f state
pA _ 1 Nk T 1 -,v/"Ao/A ' where A o = N 2a 2b, and the results for some correlation functions can immediately be taken over from eq. (2).
and the total entropy per particle is
s/k = 2 + 21og( r/~xrxry/rr).
(7)
For the case of hard squares, a = b, the entropy per particle agrees with the expression found in eq. (2), when the density is the same in b o t h directions, Lx/n x = Ly/ny. We observe, however, that the entropy from the y-dependence exceeds the entropy from the x-dependence by log 2 reflecting the different cell restrictions in the two directions for the "brick-wall" model.
414
References [1] O.J. Heil,nann, R. Moss and E.L. Praestgaard, J. Phys. C6 (1973) 403. [2] O.J. Heilmann, R. Moss and E.L. Praestgaard, J. Star. Phys. to be published. [3] A.L. Beyerlein, J. Compt. Phys. 7 (1971) 403. [4] A.L. Beyerlein, et al., M. Buynashi, J. Chem. Phys. 53 (1970) 1532.
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PHYSICS LETTERS
EXACT SOLUTION
OF RESTRICTED
RECTANGLES
22 April 1974
CELL MODEL FOR HARD PARALLEL
IN A "BRICK-WALL"
CONFIGURATION
O.J. HEILMANN, R. MOSS and E.L. PRAESTGAARD Department o f Chemistry, H.C. (Prsted Institute, University o f Copenhagen, DK-2100 Copenhagen G), Denmark Received 8 March 1974 The cell restrictions are defined, and it is shown that the model is essentially equivalent to a model presented in a previous paper. In a recent letter [1 ] a new class o f cell models for liquids was introduced, In a following paper [2] a special version o f this model, a restricted hard square model, was solved rigorously. We consider here a cell model for hard parallel rectangles. The cell boundaries are given by the positions o f the four particles which we have called the topological neighbours. They are the nearest neighbours to the particle in question in the closedpacked configurations; at lower densities they conserve the topology o f the close-packing, but they are not necessarily the physical nearest neighbours to the particle in the cell. In contrast to the model introduced earlier the cell boundaries are drawn parallel to the edges o f the particles and the x- and y-boundaries are slightly different as shown in fig. 1. More precisely the interior of the cell is defined by the conditions:
1
Y
4 X
x > x I +e
x > x 3 +e
x < x 2 +e
x < x 4 +e
Y >Y3
Y >Y4
Y
Y
(1)
where e determines where the x - b o u n d a r y is drawn, -a
Fig. 1. A hard rectangle and its four topological neighbour~ The accessible area for the center is shaded for the unrestricted model and dotted for the restricted model (3a = 5b and e = 0). Let us consider N rectangles arranged in ny topological rows o f n x particles inside a box o f area A = L x X L y . F o r the restricted model the cell condition can be written xl. 1 + a + e < x i <
x'i-a+e,
(2)
where a dash indicates a co-ordinate in an even numbered row, one can solve for the x-dependence independent o f the y-cordinates. This is accomplished 413
Volume 47A, number 5
PHYSICS LETTERS
by a transfer-matrix method identical to the one used in eq. (2) and the solution is given in terms of the free length pr. particle,
Lx -(2n x gx
.
.
.
.
II
1)a
.
.
.
Lx .
X
?1 X
2a
(3)
in the thermodynamic limit. The entropy in the x-direction is
Sx/k = 1 + log(rx/rr ).
(4)
To obtain they-dependence the particles are considered to be arranged in rows along the y-axis, 2n X rows of ½ny particles, since this gives the cell conditions the form t
Yi 1 + 2 b < Y i
t
t
2b,
(5)
and the same transfer-method solution. The free length is L), - (ny - 1 ) 2 b ry = ~n = 2 [ L y / n y - 2b ] , (6) y
22 April 1974
Beyerlein 13] and Beyerlein et al. [4] have developed high density expansions for hard squares packed in "brick-wall" as well as in regular square lattice configurations. They find a distinctly lower entropy for the brick-wall configurations. That we find the same entropy for the two restricted but otherwise equivalent models is reasonable as the following argument shows: In the "brick-wall" model with walls parallel to the edges of the particles it is the edges which bunch into the walls while it is the corners for the other model where the walls are perpendicular to the diagonals; consequently the deviation between the restricted and unrestricted model should be smallest for the brick-wall configuration. The result for the equation o f state
pA _ 1 Nk T 1 -,v/"Ao/A ' where A o = N 2a 2b, and the results for some correlation functions can immediately be taken over from eq. (2).
and the total entropy per particle is
s/k = 2 + 21og( r/~xrxry/rr).
(7)
For the case of hard squares, a = b, the entropy per particle agrees with the expression found in eq. (2), when the density is the same in b o t h directions, Lx/n x = Ly/ny. We observe, however, that the entropy from the y-dependence exceeds the entropy from the x-dependence by log 2 reflecting the different cell restrictions in the two directions for the "brick-wall" model.
414
References [1] O.J. Heil,nann, R. Moss and E.L. Praestgaard, J. Phys. C6 (1973) 403. [2] O.J. Heilmann, R. Moss and E.L. Praestgaard, J. Star. Phys. to be published. [3] A.L. Beyerlein, J. Compt. Phys. 7 (1971) 403. [4] A.L. Beyerlein, et al., M. Buynashi, J. Chem. Phys. 53 (1970) 1532.
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