Percolation cluster numbers and effective surface tension

Volume 95A, number 1

PHYSICS LETTERS

11 April 1983

PERCOLATION CLUSTER NUMBERS AND EFFECTIVE SURFACE TENSION H. FRANKE 1 and J. KERTESZ 2 Institut ffir Theoretische Physik der Universitiit KOln, Ziilpicher Strasse 77, D-5000 Cologne 41, Fed. Rep. Germany

Received 11 November 1982

We analyse Monte Carlo calculated cluster numbers ns for site percolation up to size s ~ 200. For the square lattice at concentrationp = 0.65 and the simple cubic lattice at p = 0.35 we find the asymptotic behaviour n s ~xs -s/4 exp(_0.44s 1/2) and n s o : s 1 / 9 exp(_0.09s 2/3 + 0 . 8 s 1 / 3 _ 13.1s-1/3), respectively. Applying the surface concept we compute an effective surface tension on the basis of these results. Because of the large negative S - 1 / 3 term, however, the concept does not work for small clusters in three dimensions.

The nucleation rate for gas-to-liquid phase transitions depends on the average number n s of liquid droplets containing s molecules [1 ]. The surface concept assumes on the coexistence curve for large droplets n s cx exp(_/~Fs), where the free energy F s should contain no bulk term, F s ,~ asO b (a s is the surface of a droplet with s molecules, o b is the (bulk) surface tension,

= 1 / k B r ). Percolation and thermal systems can be brought into a close connection by identifying - l n ( 1 - p) with ( p concentration of occupied sites) [2]. Hence, from the expression for percolation cluster numbers of asymptotically large clusters [2], n s o~ s - O e x p ( - A s l - 1 / d ) ,

(1)

it follows A = - T r O r p / 4 ) - l 1 2 o b ln(1 - p )

(2)

and A = - T r ( n p / 6 ) -213 o b ln(1 - p )

(3)

for d = 2 and d = 3 dimensions, respectively, where we I Permanent address: Bach 16, D-5204 Lobmar 1, Fed. Rep. Germany. 2 Permanent address: Research Institute for Technical Physics of the HAS, H-1325, Hungary; present addres: Physikdepartment T30, Technical University Munich, D-8046 Garching, Fed. Rep. Germany. 52

have u s e d a s = 2rrRs, n R 2 p = s a n d a s =4rrR 2, 4 r r R 3 p / 3 = s, respectively, with R s being the radius of the cluster. O is a so-called non-critical exponent, which should be constant for Pc < P ~< 1 [2]. Lubensky and McKane [3] gave arguments that ®(d = 2) = 5/4 and O(d = 3) - 1/9 exactly. By Monte Carlo (MC) techniques we simulated [ 4 6] the surface region of very large clusters above the percolation threshold Pc of the square and simple cubic site problem and calculated the bulk surface tension o b, summarized in fig. 1. Therefore the constant A in (1) is known by (2) and (3). Correction terms in In n s are of the order s l-2/d, s 1 - 3 / d , ... [3]. In order to determine the coefficients of these terms we calculated cluster numbers up to size s ~ 200 by MC method, since exact data are available only up to s = 17 for the square lattice [7] and s = 1 t for the simple cubic lattice [8,9]. It is, however, very difficult and computer time expensive to get reliable data for concentrations far enough above PcWe restricted the simulations therefore to p = 0.65 and p = 0.70 for d = 2 and to p = 0.35 for d = 3.350 runs were carried out on a 1000 × 1000 square lattice, and 600 runs on a 100 X 100 × 100 simple cubic lattice. For the square lattice we replace now ( l ) by

0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland

Volume 95A, number !

11 April 1983

PHYSICS LETTERS

(4)

rts = s - 5 / 4 exp [-(Asl/2 + Bs-1/2 + K ) ] I

I

I

I

and obtain y = Bx + K by setting y = - [ln (nssS/4) +Asl/2],x = s -1/2. F o r p = 0.65 a n d p = 0.70 we have o b = 0 . 0 9 5 , A = 0.438 and o b = 0 . 1 8 0 , A = 0.918, respectively. Fig. 2 shows plots o f y versus x. F o r large e n o u g h clusters (s > 30) the slope vanishes roughly, thus B ~ 0 and K = 5.7, K = 5.2 for p = 0.65, p = 0.70, respectively.

0,8

0.6

Fig. 2 gives also y values calculated from exact cluster numbers (s ~< 17). As can be seen a discontinuity appears b e t w e e n exact and MC data, which indicates a certain inaccuracy o f the MC values. It m a y be caused by finite size effects and statistical errors. Since the j u m p is particularly clear for p = 0.70, the value K = 5.2 seems less reliable. In the case o f the simple cubic lattice (1) is replaced by

0./,

0.2

0 0

0.2

0.4 T/To 0.6

0.8

ns = s 1/9 e x p [ - ( A s 2/3 +Bs 1/3 +Cs -1/3 + K ) ] .

Fig. 1. Longitudinal bulk surface tension o 2 (surface paxallel to a lattice vector) and diagonal bulk surface tension ad (surface parallel to the sum of the lattice vectors) for the square lattice, plotted versus "temperature" TIT c = ln(1 - pc)/ln(1 p). Obviously the difference between a~ and Od is rather small for the concentrations (arrows) we are interested in. Taking o b = (o~ + Od)/2 an isotropic mean cluster shape can be supposed. For the simple cubic lattice we know [6] only the longitudinal bulk surface tension, denoted by a3.

y

]"

r

(5)

Defining y = - [In (nss-l/9) + As 2/3 + Bs 1/3 ], x = s-I~ 3

onegetsy =(3: +K. N o w i t is a b =0.02• andA = 0 . 0 8 . In fig. 3 y is p l o t t e d versus x for several parameters B. F o r large enough clusters (s > 20) the plots exhibit curvature e x c e p t for B = 0.8. Thus this choice for B seems to be the best fit. The plot at this value has the slope C = - 1 3 . 1 and the c o n s t a n t K = 12.6.

'1

[

I

6 I~= 0.70 •-°

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Fig. 2. y = -[in(ns s-5/4) + As ]/2 ] versus x s - 1 / 2 plots for the square lattice. The dots up to s = 17 correspond to exact cluster numbers. The strong fluctuations for large s indicate overrepresentation of such large clusters in our MC statistics. =

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Volume 95A, number 1

PHYSICS LETTERS I

Y

11

I

I

I

April 1983

I

12 o. • o• °o• -@~•

10

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l 100

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] versus x = s -1/3 plots for several parameters B for the simple cubic lattice. Dots up to correspond to exact cluster numbers. Testing for curvature, only dots belonging to s > 20 should be taken into account.

Fig. 3. y = - [ l n ( n s s l / 9 ) + A s 2/3 + Bs 1/3

= 11

To check these results, in fig. 4 we define y = -[ln(nsS-1/9) + A s 2/3 + Cs-ll3], x = s 1/3, h e n c e y = Bx + K, and we arrive at the same values again. A discontinuity between exact (s < 11) and MC data is visible in figs. 3 and 4 too. Nevertheless, our results confirm that correction terms cannot be neglected for dimensions d > 2 [3]. The validity of the asymptotic formulae (4) and (5) can also be extended to finite s, if an effective expo1

Y

I

nent Oeff(s ) is introduced: ns = s - ° e r r e x p [ - ( A s 1/2 + g ) l ns

=

(d = 2 ) ,

(d = 3 ) .

I

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•C. .C~/(

12 0

54

4. y = - [ l n ( n s

• • • •

:.:-:-""i:."'...'"

i.

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Fig.

° "'.o.•.

.~o-,~,. . . . . . . . . . .

I 1

s l / 9 ) + A s 2/3 + Cs -1/3

I 11

I

I

I

20

50

100

] versusx = s 1/3

(7)

Figs. 5 and 6 show O e f f , computed from (6) and (7). It is remarkable that (gel f has reached the asymptotic value ® already for s ~ 30.

....~J

•

(6)

$ - 0 eff exp [-(As2/3 + Bs 1/3 + Cs-1/3 + K )

18

16

s

I1 s

200

plots at several parameters C for the simple cubic lattice.

Volume 95A, number 1

PHYSICS LETTERS

I

O'eft

i

11 April 1983

I

p=070

03

P=0.65

0.2

Ol

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o

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®eft

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:-:--.-:

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0.5

0

0

I

I

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I

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17

50

100

150

200

S

Fig. 5. Effective exponent @eff and effective surface tension Oeff as function of cluster size s for the square lattice. Note the discontinuity of the ®eft values, which is again stronger for p = 0.70 (crosses) than for p = 0.65 (dots). The broken lines mark the bulk values a b in the ®eft figure. In n u c l e a t i o n t h e o r y o n e maintains the surface conc e p t even for small droplets and d e f i n e s an e f f e c t i v e

l o w this c o n c e p t and obtain an e f f e c t i v e surface tension for finite clusters b y use o f (2), ( 3 ) and (6), ( 7 )

surface t e n s i o n (7ef f through n s = exp(---as(Teff). We fol-

O'eff

I

I

I

f

I

I

I

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0.05

/

/ s-

f

/ / /

-005

I I I

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0.5

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0 11

"---*---.-,..'o ....

.-,-,---,-*-o--.--,--,-----,--

--,--.--

I

I

t

I

50

100

150

200

$

Fig. 6. Effective exponent ®eff and effective surface tension aeff for the simple cubic lattice. The broken straight line marks a b in the aef f figure.

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Volume 95A, number 1

PHYSICS LETTERS

Oeff = o b - (rrp/4s)l/2~effln(s)/rr ln(1 - p)

(a = 2),

(8)

Oeff = Ob -- (~p/6s)2/3 X [Oeffln(s ) + Bs 1/3 + Cs-1/3]/rr ln(1 - p ) (d = 3 ) .

For d = 3 the large negative s -113 term becomes important, since the failure of the surface concept for small clusters arises from it. Finally, the effective exponent IDeff makes clear that the asymptotic behaviour of the cluster numbers already sets in at rather small cluster size.

(9)

In two dimensions the surface concept works for all s. The effective surface tension (fig. 5) has a maximum near s = 10. In three dimensions (fig. 6) the concept breaks down for small clusters s <~ 50, where Oeff becomes unphysical. In the range 50 ~ s ~ 80 the effective surface tension is lower than the bulk value o b and reaches its maximum near s = 500. Similar behaviour was observed for Ising models in two [10] and three [11] dimensions. We note that even with the Tolman correction [ 12] Oeff = Ob/(1-const. s -1/3) we cannot explain the negative Cs -tl3 term in (5). Of course, percolation clusters are only a rough approximation for the droplets occurring in cooperative thermal phenomena like nucleation. But if our results hold also for Ising models, then they would invalidate most existing phenomenological theories of nucleation (see ref. [12] for recent examples) where a small number of parameters is used to fit the surface tension. Agreement between classical nucleation theory and experiment then would have to be regarded as being due to the cancellation of various errors, as it happens e.g. in fig. 6 for our % f f near s = 80. In conclusion, the analysis shows that correction terms to s 1-1M in in n s are negligible only for d = 2.

56

11 April 1983

We are grateful to D. Stauffer for stimulating discussions and for a critical reading of the manuscript. Thanks are due to the SFB 125 Aachen-Ji~lich-K61n for financial support extended to one of us (JK).

References [ 11 K.R. Bauchspiess and D. Stauffer, J. Aerosol Sci. 9 (1978) 567. [2] D. Stauffer, Phys. Rep. 54 (1979) 1; J. Kert6sz, D. Stauffer and A. Coniglio, in: Percolation structures and processes, Ann. Israel Phys. Soc., eds. G. Deutscher, R. Zallen and J. Adler, to be published. [3] T.C. Lubensky and A.J. McKane, J. Phys. A14 (1981) L157. [4] H. Franke, Z. Phys. B40 (1980) 61. [5] H. Franke, Phys. Rev. B25 (1982) 2040. [6] H. Franke, Z. Phys. B45 (1982) 247. [7] M.F. Sykes and M. Glen, J. Phys. A9 (1976) 87. [8] M.F. Sykes, D.S. Gaunt and M. Glen, J. Phys. A9 (1976) 1705. [9] D. Stauffer, Z. Phys. B25 (1976) 391. [10] K. Binder and M.H. Kalos, J. Stat. Phys. 22 (1980) 363. [11] H. Furukawa and K. Binder, Phys. Rev. A26 (1982) 556. [12] D. Stauffer, A. Coniglio and D.W. Heermann, Phys. Rev. Lett. 49 (1 Nov 1982), to be published; D.W. Heermann, J. Stat. Phys., to be published.