Supplementary Tables for “Blowing bubbles in Lennard-Jonesium along the
saturation curve” by Henry S. Ashbaugh
The tables provide here list the simulations that were performed in our studies of cavity
solvation in the Lennard-Jones liquid at point along the saturation curve. Simulation
equation-of-state results and liquid vapor surface tensions are also provided.
1
Table S1. Listing of grand canonical transition matrix Monte Carlo simulations of the
cut-shifted (rc = 2.5) LJ fluid following the algorithm described by Errington1. The range
of temperatures extends from kT/ = 0.65 to close to the critical point. The length of the
cubic simulation box is designated by L = 8. Averages were collected over Nsweeps =
2x109 (one sweep corresponds to either an attempted particle move, particle insertion,
or particle deletion, with each possible move chosen with equal probability) after
equilibration for at least 107 sweeps. The average vapor density, liquid density, and
saturation pressure are listed with the error in the final digit reported in parentheses as
one standard deviation. The critical temperature, pressure, and density determined by
critical scaling are kTc/ = 1.079(1), Pc3/= 0.0959(1), and c3 = 0.321(1)2.
kT/
<vap3>
<liq3>
<P3/>
0.650
0.0044(3)
0.813(2)
0.0034(3)
0.675
0.0058(4)
0.8008(3)
0.0045(4)
0.700
0.0076(4)
0.7877(4)
0.0068(3)
0.725
0.0098(3)
0.7740(6)
0.0075(5)
0.750
0.0126(3)
0.7603(3)
0.0096(6)
0.775
0.0159(2)
0.7459(7)
0.0121(7)
0.800
0.0200(1)
0.7311(6)
0.0150(7)
0.825
0.0247(6)
0.7157(7)
0.0185(8)
0.850
0.03045(1)
0.6995(1)
0.0226(8)
0.875
0.03719(4)
0.6825(1)
0.0273(8)
0.900
0.04520(2)
0.6641(5)
0.0327(8)
0.925
0.05476(4)
0.6444(6)
0.0388(9)
2
0.950
0.0663(2)
0.6232(6)
0.0458(9)
0.975
0.0805(5)
0.5994(3)
0.0537(9)
1.000
0.098(1)
0.572(1)
0.062(1)
1.025
0.124(1)
0.539(2)
0.073(1)
1.050
0.160(2)
0.494(2)
0.084(1)
1.060
0.177(6)
0.472(4)
0.088(1)
3
Table S2. Listing of canonical ensemble Monte Carlo simulations of a slab of liquid in
equilibrium with its vapor at temperatures of kT/ = 0.65 to 1.00 in increments of 0.05.
The initial densities of the vapor and liquid slabs was set to the binodal values
determined from the transition matrix Monte Carlo simulations (Table 1). The bulk
densities were retained during the coarse of the simulations. The x, y, and z box lengths
are Lx, Ly, and Lz, respectively. The liquid-vapor surface tension was averaged over
Ncycles = 107 (one cycle corresponds to one attempted to move for each solvent
molecule), after equilibration for at least 100,000 cycles. The surface tension was
evaluated via the virial formula3. The average surface tension is listed with the error in
the final digit reported in parentheses as one standard deviation.
kT/
N
vap3
liq3
Lx,y/
Lz/
<lv2/>
0.65
1200
0.0044
0.813
10
50
0.690(4)
0.70
1200
0.0076
0.7877
10
50
0.590(4)
0.75
1200
0.0126
0.7603
10
50
0.496(4)
0.80
1200
0.0200
0.7311
10
50
0.406(4)
0.85
1200
0.0305
0.6995
10
50
0.315(2)
0.90
1200
0.0452
0.6641
10
50
0.2361(6)
0.95
1200
0.0663
0.6232
10
50
0.159(1)
1.00
1400
0.0980
0.5718
9
70
0.093(1)
4
Table S3. Listing of isothermal-isobaric ensemble Monte Carlo simulations of N = 1000
LJ particles. Simulations were carried out at kT/ = 0.65 to 1.00 in increments of 0.05
along the saturation curve. Both the equilibrium liquid and gas phase densities were
simulated at each temperature. Cavity insertion probabilities were averaged over Ncycles
= 107 (one cycle corresponds to one attempted to move for each solvent molecule), after
equilibration for at least 100,000 cycles.
kT/
P3/
0.65
0.0034
0.70
0.0068
0.75
0.0096
0.80
0.015
0.85
0.023
0.90
0.033
0.95
0.046
1.00
0.062
5
Table S4. Listing of isothermal-isobaric ensemble Monte Carlo simulations of N LJ
particles and a single hard-sphere of radius R. Simulations were performed at
temperatures from kT/ = 0.65 to 1.00 in increments of 0.05 at the saturation pressure
(Table 3). Averages for the density of the solvent in contact with the cavity were carried
out over Ncycles (One cycle corresponds to one attempted to move for the cavity and
each solvent molecule. Volume moves were attempted every fifth cycle), after
equilibration for at least 100,000 cycles. For some of the larger cavities at temperatures
approaching the critical point additional larger simulations with more solvent particles
were performed to ensure the contact density corresponds to that of a cavity at infinite
dilution. No statistically significant changes were observed.
R/
kT/
N
Ncycles
0.25
0.65 – 1.00
750
106
0.50
0.65 – 1.00
750
106
0.75
0.65 – 1.00
750
106
1.00
0.65 – 1.00
1000
106
1.50
0.65 – 1.00
1000
106
2.00
0.65 – 1.00
1500
106
2.50
0.65 – 1.00
2000
106
3.00
0.65 – 1.00
2000a
106
3.50
0.65 – 1.00
2500b
7.5x105
4.00
0.65 – 1.00
2500c
7.5x105
4.50
0.65 – 1.00
3000d
5x105
5.00
0.65 – 1.00
3500e
5x105
6
kT/ = 0.95 and 1.00 an additional simulation with N = 2500 was performed.
kT/ = 0.95 and 1.00 an additional simulation with N = 3000 was performed.
cFor kT/ = 0.95 and 1.00 an additional simulation with N = 3500 was performed.
dFor kT/ = 0.95 and 1.00 an additional simulation with N = 4000 was performed.
eFor kT/ = 0.95 and 1.00 an additional simulation with N = 4500 was performed.
aFor
bFor
7
References
1
2
3
J. R. Errington, J. Chem. Phys. 118, 9915 (2003).
J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, third ed.
(Butterworth Scientific, London, 1982).
J. S. Rowlinson and B. Widom, Molecular theory of capilarity. (Dover
Publications, Mineola, New York, 2002).
8