Range effect on percolation threshold and structural properties for short-range
attractive spheres
Jiachen Wei,1 Limei Xu,2, 3, a) and Fan Song1, b)
1)
State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics,
Chinese Academy of Sciences, No. 15 Beisihuanxi Road, Beijing 100190,
China
2)
International Center for Quantum Materials and School of Physics,
Peking University, No. 5 Yiheyuan Road, Beijing 100871,
China
3)
Collaborative Innovation Center of Quantum Matter, Beijing,
China
(Dated: 4 January 2015)
a)
To whom correspondence should be addressed. E-mail: [email protected]
b)
To whom correspondence should be addressed. E-mail: [email protected]
1
l
0
1 0
1 0
-2
1 0
-3
1 0
-4
1 0
-5
1 .0
1 .0
1 .0
1 .0
B
1
1
5
5
*
2
0 .9
-0 .7
0 .9
-0 .7
n
0
(
s
)
1 0
-1
1
1 0
1 0 0
1 0 0 0
s
FIG. 1. The distribution of contact cluster size, n0 (s), along the contact percolation boundary for
λ = 1.05 and λ = 1.01. See the legend for the meaning of the symbols.
I.
NUMBER OF CONFIGURATIONS USED IN SIMULATION
Number of configurations used in simulation are listed in Table I:
TABLE I. Number of analyzed configurations (in multiples of 105 )
λ N = 864 N = 2916 N = 6912
1.01
4
2
1
1.03
6
3
1.5
1.05
8
4
2
II.
SUPPLEMENTAL FIGURES
At the same B2∗ but different λ, the cluster size for systems along contact percolation
boundary also share the same distribution form, see Fig. 1. It shows that, at the same B2∗ ,
the same slope is obtained for two different well-width. The value of the Fisher exponent τ
near percolation is within the range 2.08 ∼ 2.17.
The distribution of bond cluster size n∗0 (s) along the bond percolation boundary is provided in Fig. 2. It can be seen that the distribution is almost the same as n0 (s) along contact
percolation boundary. We obtain the universal Fisher exponent τ near both contact and
bond percolation boundary.
Fig. 3 presents Π(z) along the isoline hzi = 2. The functional form of Π(z) are also
2
0
1 0
-1
1 0
-2
1 0
-3
1 0
-4
1 0
-5
s
(
0
2
n
*
*
0 .7
0 .5
0 .2
0 .0
-0 .2
-0 .5
-0 .7
)
1 0
B
1
1 0
1 0 0
s
1 0 0 0
FIG. 2. The distribution of bond cluster size, n∗0 (s), along the bond percolation boundary. All the
results are obtained for λ = 1.05. See the legend for the meaning of the symbols.
l
0 .3
B
1 .0
1 .0
1 .0
1 .0
1 .0
1 .0
1 .0
1 .0
P (z )
0 .2
5
5
5
5
1
1
1
1
*
2
-0 .5
0 .0
0 .5
0 .9
-0 .5
0 .0
0 .5
0 .9
0 .1
0 .0
0
1
2
z
3
4
5
6
FIG. 3. The distribution of contact coordination number, Π(z), along hzi = 2 isoline. The solid
and the dashed lines represent Π(z) for λ = 1.05 and λ = 1.01, respectively. See the legend for the
meaning of the symbols.
asymmetric, regardless of B2∗ . At the same B2∗ , Π(z) for different λ are less dispersed,
compared with Π(z) along contact percolation boundary.
Fig. 4 provides the maximum contact cluster size, smax , for systems with λ = 1.05, at
different B2∗ . At contact PT, the critical maximum contact cluster size, scmax ∼ 2500. The
larger the value of B2∗ , the smaller the values of scmax , highlighted by downward-sloping
dashed lines in Fig. 4.
3
B
3
*
2
1 0
2
1 0
1
1 0
0
-0 .7
-0 .5
-0 .2
0 .0
0 .2
0 .5
0 .7
0 .9
s
m a x
1 0
0 .1
f
0 .2
0 .3
0 .4
0 .5
FIG. 4. The maximum contact cluster size, smax , as a function of packing fraction, φ, at different
B2∗ . Error bars provide the standard deviation after averaging at least 105 configurations. All the
results are obtained for λ = 1.05. See the legend for the meaning of the symbols. All lines are
guides to the eyes.
III.
EQUIVALENCY OF INTERACTION
We note that the effective force1–3 used by Babu et. al. is equivalent to our square-well
interaction, for reasons as follows.
The contribution of the interaction energy U0 is taken into account by introducing
the bond probability P for bond percolation. Our primary study shows that percolation boundaries of different λ would collapse to that for Baxter sticky model only when
P = 1 − exp (−U0 / kB T ), the same expression employed by Babu et. al.
Let εb be the number of active bonds with equivalent energy Eb , εi the number of inactive
bonds with equivalent energy Ei , and ε = εb + εi the total number of free contacts (pairs of
particles in range to interact), then
εb ∝ e−βEb , εi ∝ e−βEi ,
(1)
where β = 1/kB T . The energy difference ∆E = Eb − Ei , so the bond probability P can be
written as
P =
1
εb
=
.
ε
1 + eβ∆E
(2)
For square-well interaction, all neighbors in range are in interaction −U0 , while for on-andoff bonding interaction, neighbors in range are in interaction ∆E with probability P . The
4
formation of εb randomly distributed bonds over ε contacts leads to a decrease in the free
energy equal to U0 per contact. When the system is in equilibrium, with ε contacts we have:
ε (−U0 ) = εb ∆H − T ∆S,
(3)
where ∆H = εb ∆E, and the change in entropy is determined by the number of ways εb
bonds can distributed over contacts:
∆S = kB lnΩ = −εkB [P lnP + (1 − P )ln(1 − P )].
(4)
Combination of Eq. (1) to Eq. (4) yields
P = 1 − e−βU0 .
(5)
Therefore, the bond-making/breaking effective force1–3 is equivalent to square-well potential.
REFERENCES
1
S. Babu, J. C. Gimel, and T. Nicolai, The Journal of Chemical Physics 125, 184512 (2006).
2
S. Babu, J. C. Gimel, and T. Nicolai, The Journal of Chemical Physics 127, 054503 (2007).
3
S. Babu, J. C. Gimel, and T. Nicolai, The European Physical Journal E 27, 297 (2008).
5