Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids
Molecular Dynamics simulations of a Lennard-Jones system with different range of attraction show that the attractive forces modify the radial distribution of the particles. For condensed liquids only, the forces within the the first coordination shell (FCS) are important, but for gases and moderate...
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irk-123456789-1534692019-06-15T01:29:53Z Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids Toxvaerd, S. Molecular Dynamics simulations of a Lennard-Jones system with different range of attraction show that the attractive forces modify the radial distribution of the particles. For condensed liquids only, the forces within the the first coordination shell (FCS) are important, but for gases and moderate condensed fluids, even the attractive forces outside the FCS play a role. The changes in the distribution caused by neglecting the attractive forces, lead to a too high pressure. The weak long-range attractions damp the dynamics and the diffusion of the particles in gas-, super critical fluid- and in liquid states. The values of self-diffusion coefficients (SDC) agree qualitatively with a modified Cohen-Turnbull model. The SDC-s along the critical isotherm show no anomaly at the critical point in agreement with experimental data. Моделювання методом молекулярної динамiки системи Лєннарда-Джонса з рiзною областю притягання показують, що сили притягання видозмiюють радiальний розподiл частинок. Сили в межах першої координацiйної сфери (ПКС) є важливими лише для густих рiдин, в той час, як для газiв i помiрно густих плинiв, навiть сили притягання за межами ПКС вiдiграють певну роль. Змiни у розподiлi, спричиненi нехтуванням силами притягання, приводять до занадто високого тиску. Слабке далекосяжне притягання гасить динамiку i дифузiю частинок у станах газу, надкритичного плину та рiдини. Значення коефiцiєнтiв самодифузiї якiсно узгоджуються з модифiкованою моделлю Коена-Турнбулла. Коефiцiєнти самодифузiї вздовж критичної iзотерми не показують нiякої аномалiї у критичнiй точцi, що узгоджується з експериментальними даними. 2015 Article Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids / S. Toxvaerd // Condensed Matter Physics. — 2015. — Т. 18, № 1. — С. 13002:1-11. — Бібліогр.: 37 назв. — англ. 1607-324X PACS: 05.20.Jj, 47.11.Mn, 64.70.pm, 65.20.De, 66.10.cg, 82.20.Wt DOI:10.5488/CMP.18.13002 arXiv:1504.00809 http://dspace.nbuv.gov.ua/handle/123456789/153469 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Molecular Dynamics simulations of a Lennard-Jones system with different range of attraction show that the attractive forces modify the radial distribution of the particles. For condensed liquids only, the forces within the the first coordination shell (FCS) are important, but for gases and moderate condensed fluids, even the attractive forces outside the FCS play a role. The changes in the distribution caused by neglecting the attractive forces, lead to a too high pressure. The weak long-range attractions damp the dynamics and the diffusion of the particles in gas-, super critical fluid- and in liquid states. The values of self-diffusion coefficients (SDC) agree qualitatively with a modified Cohen-Turnbull model. The SDC-s along the critical isotherm show no anomaly at the critical point in agreement with experimental data. |
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Toxvaerd, S. |
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Toxvaerd, S. Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids Condensed Matter Physics |
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Toxvaerd, S. |
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Toxvaerd, S. |
| title |
Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids |
| title_short |
Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids |
| title_full |
Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids |
| title_fullStr |
Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids |
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Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids |
| title_sort |
role of attractive forces in determining the equilibrium structure and dynamics of simple liquids |
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Інститут фізики конденсованих систем НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/153469 |
| citation_txt |
Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids / S. Toxvaerd // Condensed Matter Physics. — 2015. — Т. 18, № 1. — С. 13002:1-11. — Бібліогр.: 37 назв. — англ. |
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Condensed Matter Physics |
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AT toxvaerds roleofattractiveforcesindeterminingtheequilibriumstructureanddynamicsofsimpleliquids |
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2025-07-14T04:59:03Z |
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2025-07-14T04:59:03Z |
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1837597085358096384 |
| fulltext |
Condensed Matter Physics, 2015, Vol. 18, No 1, 13002: 1–11
DOI: 10.5488/CMP.18.13002
http://www.icmp.lviv.ua/journal
Role of attractive forces in determining the
equilibrium structure and dynamics of simple liquids
S. Toxvaerd
DNRF centre “Glass and Time”, IMFUFA, Department of Sciences, Roskilde University,
Postbox 260, DK-4000 Roskilde, Denmark
Received July 31, 2014, in final form October 2, 2014
Molecular Dynamics simulations of a Lennard-Jones system with different range of attraction show that the
attractive forces modify the radial distribution of the particles. For condensed liquids only, the forces within
the the first coordination shell (FCS) are important, but for gases and moderate condensed fluids, even the
attractive forces outside the FCS play a role. The changes in the distribution caused by neglecting the attractive
forces, lead to a too high pressure. The weak long-range attractions damp the dynamics and the diffusion of
the particles in gas-, super critical fluid- and in liquid states. The values of self-diffusion coefficients (SDC) agree
qualitatively with a modified Cohen-Turnbull model. The SDC-s along the critical isotherm show no anomaly at
the critical point in agreement with experimental data.
Key words: simple liquids, equation of state, perturbation theory, self-diffusion
PACS: 05.20.Jj, 47.11.Mn, 64.70.pm, 65.20.De, 66.10.cg, 82.20.Wt
1. Introduction
Ever since van der Waals (vdW) [1] in 1873 formulated his theory “the continuity of the gaseous and
liquid states”, the understanding of simple fluids has been that repulsive and attractive forces give rise to
separate modifications of the ideal-gas equation of state and to separate contributions to the free energy.
The harsh, repulsive forces reduce the available volume and determine the structure of the fluid and
thus its configurational entropy; the weaker, longer-range attractive forces give rise to an energy effect
that reduces pressure and energy compared to those of an ideal gas at the same temperature and density.
With respect to the effect of the longer-range attractive forces, van der Waals imagined that he could
(quoting from Ref. [2]) “. . . define an element of volume in a liquid which is small compared . . . ” with
“. . . the range of the intermolecular force, but large enough for it to contain sufficient molecules for us
to assume that there is within it a uniform distribution of molecules of number density, ρ = n/V ”. The
pressure p(n/V ,T ) is in accordance with van der Waals
(
p +
n2a
V 2
)
(V −nb) = nRT. (1.1)
He assumed that the main effect of the attractive forces originates from the molecules within this local
sphere and estimated this sphere to have a range of 3–6 Å [2]. For condensed liquids, the sphere can be
identified as the first coordination shell [3].
It is well known that the vdW equation of state reproduces the qualitative behavior of the fluid state
[4, 5]. Van der Waals’ idea about the role of the repulsive and attractive forces lies behind Zwanzig’s high-
temperature expansion [6], in which the contribution to the free energy from the long-range forces is
expressed in powers of the inverse temperature, the reference high-temperature system being a system
with infinitely strong, purely repulsive forces. The success of this perturbation theory (PT) was demon-
strated by Longuet-Higgens and Widom [7], Barker and Henderson [8], and soon after by Weeks, Chan-
dler, and Andersen (WCA) in their seminal paper “Role of repulsive forces in determining the equilibrium
structure of simple liquids” [9].
© S. Toxvaerd, 2015 13002-1
http://dx.doi.org/10.5488/CMP.18.13002
http://www.icmp.lviv.ua/journal
S. Toxvaerd
In PT, a fundamental problem is how to separate the strong repulsive attractions from the weaker
long-range attractions. One possible separation was proposed by Barker and Henderson, who marked
the separation at the distance where the potential is zero. WCA demonstrated, however, that by choosing
instead the separation at the potential minimum, one obtains a much better agreement between the par-
ticle distributions of the system and the reference system. Doing so, the forces are separated into purely
repulsive and purely attractive forces, and one may say that the original idea of van der Waals is here
captured in its purest form. Numerous refinements of perturbation theory have appeared since then.
An excellent summary of perturbation approaches until 1976 can be found in Barker and Henderson’s
classic review [10].
Whereas the vdW equation is used for simple systems in gas-, liquid- and coexisting liquid-gas states,
the PT theories are normally only used for condensed liquid states. In WCA [9], only fluid densities
ρ Ê 0.65 were included in the investigation, corresponding to the densities more than twice the criti-
cal density. Here, we shall determine the effect of the attractions in a simple Lennard-Jones system on the
structure (section 3) and dynamics (section 4), as well as on gas-, liquid- and super-critical fluid states.
2. Perturbation theory for simple liquids
In physical chemistry, the van derWaals force (or van derWaals’ interaction), is the sum of the attrac-
tive or repulsive forces between molecules (or between parts of the same molecule) other than those due
to covalent bonds or the electrostatic interaction of ions with one another or with neutral molecules or
charged molecules [11]. The term includes: force between two permanent dipoles (Keesom force), force
between a permanent dipole and a corresponding induced dipole (Debye force) and force between in-
stantaneously induced charge distributions (London dispersion force). The attractive London dispersion
forces are the weakest forces. When considering “simple liquids”, one normally refers to a system of par-
ticles with van der Waals’ interactions, and the perturbation theories are often formulated for systems
of spherically symmetrical particles and with only the first induced dipole-dipole term in the expansion
of the attractive London dispersion force [8, 9]. The potential energy for such a system of N spherical
symmetrical particles in the volume V (density ρ = N /V ) and with interactions, given by a pair potential
u(ri j ) between particle No. i and j separated the distance ri j , is as follows:
U =
N
∑
i< j
u(ri j ), (2.1)
and the equation of state is obtained by the virial equation for the pressure
p = ρT +pw (ρ,T ) = ρT +ρw(ρ,T ), (2.2)
where the contribution to the pressure from the intermolecular forces, pw (ρ,T ) is given by the value of
the virial, w(ρ,T )
w(ρ,T ) =−
2π
3
ρ
∞
∫
0
g (r )
du(r )
dr
r 3dr (2.3)
by integration over the radial distribution g (r ) of the virial between particles at the distance r .
The strength of the force f (r )=−du(r )
dr
can be separated into short range force f (r ) for pair distances
r É r̃c and the long range force for r > r̃c. The separation between short- and long range forces, given by
r̃c is in the PT taken either to be where the potential u(r ) is zero [8] or where the force f (r ) is zero [9].
In PT, the mean distribution of the particles, g̃ (r ) is obtained by taking only forces for r É r̃c into
account in the dynamics, and treating the virial contribution to the pressure for distances r > r̃c as a
mean field contribution, i.e., by assuming that
g̃ (r ) ≈ g (r ). (2.4)
The pressure is approximately obtained in the PT as follows:
p(ρ,T ) ≈ p̃(ρ,T )+∆p(ρ,T ), (2.5)
13002-2
Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids
where
p̃(ρ,T ) = ρT −
2π
3
ρ2
r̃c
∫
0
g̃ (r )
du(r )
dr
r 3dr (2.6)
is the pressure for a system with only short range forces for r É r̃c and the value of the last term
∆p(ρ,T ) =−
2π
3
ρ2
∞
∫
r̃c
g̃ (r )
du(r )
dr
r 3dr (2.7)
is obtained as a mean field term and using the the distribution g̃ (r ) for a system with only short range
interactions.
The pressure (and free energy) can be derived from the exact “λ-expansion” [12]. The thermodynamic
connection between any two systems with pair potential functions, u(r ) and ũ(r ) can be obtained from
the pressure of a system with the potential
u(r,λ) = ũ(r )+λ [u(r )− ũ(r )], (2.8)
where u(r,λ) is equal to a reference potential ũ(r ) for λ= 0 and equal to u(r ) for λ= 1. The equilibrium
pressures for systems with the pair potential u(r,λ) and with values of λ in the interval [0,1] establish
a reversible path between the two systems, and the total differences in pressure, mean energy and free
energy are then determined by numerical integration over different values of λ.
The second step in the PT is to relate p̃(ρ,T ) with the pressure p0(ρ0) for a system of hard sphereswith
a diameter r0 and corresponding density ρ0, i.e., p̃(ρ,T ) ≈ p0(ρ0,T ). In this article we shall investigate
only the validity of the first assumption in the PT, i.e., to what extent the structure of a simple liquid is
given only by the forces f (r ) for r É r̃c.
2.1. Separation of pair interactions into short- and long range interactions
The effect of the attractive forces on the structure and dynamics is obtained for a system of particles
with the standard Lennard-Jones (LJ) pair potential
uLJ(r ) = 4ε
[
(σ/r )12 − (σ/r )6
]
, (2.9)
and the equation of state and the diffusion are determined using Molecular Dynamics (MD) simulations.
In MD, the interactions for distances rc ≫ r̃c are neglected. There are two standard ways to neglect these
(very) long-range interactions. The usual way is to truncate and shift the potential, whereby the force,
fSP(r ), is as follows:
fSP(r ) =
{
fLJ(r ) if r < rc ,
0 if r > rc .
(2.10)
This is referred to as a SP cut-off because it corresponds to shifting the potential below the cut-off and
putting it to zero above which it ensures a continuity of the potential at rc and avoids an infinite force
herein. Another and more accurate way is to truncate and shift the forces (SF for shifted forces) [13] so
that the force goes continuously to zero at rc, which is obtained by subtracting a constant term:
fSF(r ) =
{
fLJ(r )− fLJ(rc) if r < rc ,
0 if r > rc .
(2.11)
This corresponds to the following modification of the potential [13]:
uSF(r ) =
{
uSF(r )= uLJ(r )− (r − rc)u′
LJ
(rc)−uLJ(rc) if r < rc ,
0 if r > rc .
(2.12)
The advance of the SFmethods is that one can use any value of r̃c or rc without introducing discontinuities
into the forces. Here, we shall use SF with rc = 4.5σ as a good approximation for r∞. The maximum
attraction of LJ forces at r = (26/7)1/6σ = 1.244σ is f (max. att.) = −2.3964ǫ/σ and the attraction at rc =
4.5σ is f =−0.000642ǫ/σ. The role of the attractions is obtained for the attractions in the interval [r̃c,rc].
The longer range attractions for r Ê 4.5σ contribute only marginally to the structure and dynamics except
in very diluted gases (see later).
13002-3
S. Toxvaerd
Figure 1. (Color online) The phase diagram T (ρ) for a Lennard-Jones system. Red (solid) lines: coex-
isting liquid-gas [14] with critical point for rc = 2.5 (red circle) [15] and (red square) [16] for rc = ∞;
blue (dashed) lines coexisting solid-liquid lines [20] with triple point (ρtr(l ),Ttr) and (ρtr(s),Ttr) [22] [23];
green (dotted) line: Zeno line (virial= 0) and light blue (dash-dotted) line and points where excess energy
and the virial (still) are strongly correlated R = 0.9 [24].
3. The effect of attractions on structure and pressure
3.1. The phase diagram of a LJ system
The attractive forces can cause a condensation of particles in a gas state into liquid for sufficient low
temperatures. The phase diagram T (ρ) for a LJ system is shown in figure 11. The coexisting lines and
points are all obtained by various MD simulations. There are several points to notice. The bimodals for
coexisting gas and liquid [red (solid) lines] are sensitive to the value rc for the truncation of the force
field. The lines in the figure are for rc = 3σ [14] connected with red short dashes to the critical point
(ρc,Tc), obtained by Smit [15] for rc = 2.5σ. Also shown in the figure (with a square) is the critical point
(ρc,Tc) = (0.316,1.326) for rc = ∞ obtained by Caillol [16]. The long range attractions behind a cut-off
at rc = 2.5σ significantly change the location of the critical point [16], as well as the surface tension and
interfacial structure [17]. And for systems where the attractions are only short-ranged, there is no longer
a critical point and a liquid condensation [18, 19].
The blue (dashed) lines in the figure [20] are the lines for the coexisting liquid and solid determined by
extrapolations to rc = r∞. The location of the bimodals for coexisting liquid and solid (fcc) is less sensitive
to the value of rc [21], but the location of the triple point is also sensitive to the range of the attraction
and is shifted from Tc ≈ 0.69 for rc = r∞ (blue square) [22] to Tc = 0.618 for rc = 2.5 (blue circle) [23]. The
light blue (dash-dotted) line and the points are the borderline where the excess potential energy and the
virial still are strongly correlated [24, 25]. The forces within the first coordination shell are dominant and
ensure a strong correlation between the potential energy and the virial for state points to the right of this
line [3].
3.2. The zeno line
Green full triangles in figure 1 are points on the “zeno line” [26] where the pressure equals the pres-
sure of an ideal gas. The points are fitted with a second order polynomial (green line) and the zeno line
is weakly curved. All points at the zeno line are obtained for SF with rc = 4.5σ. The point (0,TB) with the
Boyle temperature TB = 3.2310 is obtained from the expression for the second virial coefficient. The Boyle
temperature for a LJ system (rc =∞) is TB = 3.418, and the value of the Boyle temperature for rc = 4.5σ is
TB = 3.2310. It shows that long-range attractive forces behind rc = 4.5σ play a role in (very) diluted gases.
1Units: length in unit of σ; time in units of σ
p
ǫ/m; energy in units of ǫ/kB.
13002-4
Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids
The virial, w and the contribution ρw to the pressure is zero for state points at the zeno line. The con-
tribution pw = ρw to the pressure from the forces can be divided into contributions from the repulsive
and the attractive forces
pw (+) = ρw(+) =−
2π
3
ρ2
rWCA
∫
0
g (r )
du(r )
dr
r 3dr > 0, (3.1)
pw (−) = ρw(−) =−
2π
3
ρ2
∞
∫
rWCA
g (r )
du(r )
dr
r 3dr < 0, (3.2)
where rWCA = 21/6 for a LJ system. Hence, the zeno line is a borderline between the dominance of the
repulsive and the attractive forces on the pressure. For state points above the line, the effect on the
pressure from the virial of the repulsive forces is dominant. The structure of the fluid, given by the radial
distribution function ensures a positive value of the the virial, i.e. pw (+) > −pw (−), and visa verse for
state points below the line.
The central assumption in PT is that the structure is given by the repulsive forces, and thus it is natural
to investigate whether it is the case for state points along the zeno line, where the contributions to the
virial from the repulsive and the attractive forces cancel. The LJ potential with only the first term in the
induced multipole interactions is the potential for the weakest London forces, and all real (simple) van
der Waals liquids have stronger attractions than a LJ system.
3.3. The structure of the fluid at the zeno line
The impact of the attractive forces on the structure of the fluid can be determined by obtaining the
structure of the fluid with and without the attraction included in the dynamics. The attractive force
changes the distributions of particles. Figure 2 and figure 3 show the radial distribution functions for
the condensed liquid state (ρ,T ) = (0.85,0.7453) and for the diluted gas state (ρ,T ) = (0.20,2.6680), re-
spectively. Both state points are on the zeno line, and the effect of the attractive forces on the structure
is bigger for the points below this line. The black (solid) line in figure 2 is the distribution g (r ) for a LJ
fluid, and the red (dashed) line is the distribution g̃ (r ) when only forces within the FCS are included in
the dynamics. The green (dotted) line is g̃ (r ) for r̃c = 21/6, where only the repulsive forces are included
in the dynamics. The inset shows the corresponding differences g̃ (r )−g (r ). For this condensed fluid, the
attractive forces behind the FCS have only a marginal effect on the distribution, but the attractive forces
within the FCS change the structure of the fluid significantly [3].
Figure 2. (Color online) Radial distribution function, g (r ), for ρ = 0.85 and T = 0.7453 (at the zeno line).
Black (solid): g(r); red (dashed): g̃ (r ) for r̃c = 1.54 (first coordination shell) and green (dotted): r̃c = 21/6
(WCA). The inset show the difference, g̃ (r )−g (r ).
13002-5
S. Toxvaerd
Figure 3. (Color online) Radial distribution function, g (r ), for ρ = 0.20 and T = 2.668 (at the zeno line).
Black (solid): g(r); red (dashed): g̃ (r ) for r̃c = 1.75 (first coordination shell) and green (dotted): r̃c = 21/6
(WCA). Also shown is the distribution for r̃c = 2.5 in blue (dash-dotted) and r̃c = 3.5 in magenta (dash-
double-dotted). The inset show the differences, g̃ (r )−g (r ).
Figure 3 shows the corresponding distributions at the diluted state with ρ = 0.2. Also included [with
blue (dash-dotted)] is the distribution when the effect of the forces from the second coordination shell
is included in the dynamics, and the magenta (dash-double-dotted) curve is for r̃c = 3.5. The two figures
demonstrate that nowhere on the zeno line there is a structure of the fluid given only by the repulsive
forces. The insets show that the main differences in the distributions appear for distances within the
FCS, and that the differences are of the same order for the two states. The distributions with blue (dash-
dotted) and magenta (dash-double-dotted) in figure 3 show that whereas at condensed states it is the
forces within the FCS which change the structure compared to the structure of a repulsive system, the
longer range attractions behind the FCS also have an effect on the structure of a more diluted fluid.
3.4. Pressure along the zeno line
According to figure 2 and figure 3, the radial distribution g̃ (r ) differs from g (r ) if the attractive forces
are neglected in the dynamics. The integral effect on the pressure is given by
p̃w (+) = ρw̃(+) =−
2π
3
ρ2
rWCA
∫
0
g̃ (r )
du(r )
dr
r 3dr, (3.3)
p̃w (−)= ρw̃(−) =−
2π
3
ρ2
∞
∫
rWCA
g̃ (r )
du(r )
dr
r 3dr. (3.4)
The two contributions and their sum are shown in figure 4. With black (solid) is pw , pw (+) and pw (−)
for a LJ fluid, with red (dashed) p̃w , p̃w (+) and p̃w (−) for g̃ (r ) with forces only from the FCS. The green
(dotted) curves are p̃w , p̃w (+) and p̃w (−) for a system with only repulsive LJ forces. Distributions are
changed by neglecting the attractive forces in the dynamics, but the integral effect of the attractions on
the pressure, p̃w (−) is close to zero. This is, however, not the case for the effect on the pressure p̃w (+)
from changes in the distribution for distances where the forces are repulsive. Figure 4 reveals that the
pressure obtained by PT fails, not at diluted supercritical states, but for condensed states ρ > 0.6, and it
fails because the structure of the fluid for distances where the forces are repulsive is changed due to the
effect of the attractive forces! Figure 2 shows [green (dotted) lines] that the radial distribution function
g̃ (r ) for r < 21/6 is “parallel shifted” toward shorter distances by neglecting the attraction in the dynamics,
and it is this effect which leads to the deviation in the virial and pressure p̃w (+)−pw (+) for a condensed
fluid.
13002-6
Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids
Figure 4. (Color online) Contribution to the pressure from the virial, pw = ρw , along the zeno line. The
contributions are divided into the contribution from the repulsive forces, ρw(+) and the attractive forces,
ρw(−). With black (solid) is the two contributions and their sum (which is equal to zero), and with red
(dashed) are the corresponding contributions obtained from g̃ (r ) by neglecting the attractive forces be-
hind the first coordination shell in the dynamics. Green (dotted) are obtained from g̃ (r ) for a system of
repulsive LJ particles.
The apparent agreement between the radial distribution function of a LJ system and the correspond-
ing distribution for a system with only repulsive forces for a liquid is striking (figure 2); but there are
systematic differences at short distances and these differences lead to a too high pressure. The standard
perturbation theory (PT) (Barker-Henderson [8] and Weeks-Chandler-Anderson [9]) is not quantitatively
correct. The central assumption in PT is that the effect of the attraction can be obtained from a fluid with
only repulsive forces. It is, however, possible to modify the PT by replacing the reference system with a
reference system with the forces within the first coordination shell [3]. Doing so, the pressure obtained
by PT is not only qualitatively and to a high extent also quantitatively correct for diluted gas and super
critical states as well as for condensed liquid states.
The standard perturbation theory is the “zero” term in an exact λ expansion [12], and it is, of course,
possible to obtain the pressure from the pressure p̃(ρ,T ) for the short range potential by determining the
distribution for the potential u(r,λ) for sufficient values of λ [27, 28].
4. The role of attractions on self-diffusion
The attractive forces change the dynamics of a system. The effect of the attractive forces on the self-
diffusion coefficient (SDC) is determined for the critical isotherm T = Tc and the triple point fluid isochore
ρtr for a LJ system. The values of the SDC for the critical LJ isotherm Tc(LJ) = 1.326 are shown in figure 5.
The values are obtained from the mean displacements of the N = 2000 particles and for 1.6×106 time
steps with δt = 0.0025. For the system with only short-range attractions [green (dotted) line], the temper-
ature is well above its critical temperature, and the system with no attractions [blue (dashed) line] has no
critical temperature. The SDC for all three systems decline exponentially with the density in the interval
ρ ∈ [0.2,0.8] (inset). For densities ρ > 0.5, there are no significant differences between the coefficients
for the LJ system [red (solid) line] and the coefficients for the system with only forces within the FCS in
the dynamics [green (dotted) line] [13], but for smaller densities also the longer ranged forces play a role
and damp the diffusion. The diffusion in a LJ system is systematically smaller than the diffusion for a
system with only repulsive forces [red (solid) and blue (dashed) lines], and the differences increase with
decreasing density.
There is no observable change in the behavior of the SDC of the LJ system near the critical point. Only
the LJ system exhibits critical fluctuations. The system with only short-range attractions from the FCS
13002-7
S. Toxvaerd
Figure 5. (Color online) Self-diffusion coefficients D(T,ρ) for the critical LJ isotherm T = Tc = 1.326. Red
(solid) line: LJ fluid; green (dotted): for a system with only forces within the FCS included in the dynam-
ics, and blue (dashed) is when only the repulsive forces are included in the dynamics. The inset shows
lnD(T,ρ) and with black (dash-dotted): a exp(−bρ).
has a lower critical temperature, and the system with only repulsive forces has no critical temperature.
Nevertheless, the three systems exhibit a coherent behavior and with no spur of a critical effect, which
should be in only one of the three systems. The critical point marks the location of a thermodynamic
instability and the effect on the SDC is small, if any, and in agreement with experimental data [29–31].
The SDC varies exponentially with increasing density. The inset shows logD for the LJ system [red
(solid)] and the system without attractive forces [blue (dashed)]. The coefficients a and b in the func-
tion ae−bρ are fitted to the values of D for the LJ system, and the function is shown in the inset [black
(dash-dotted) straight line]. The data show that the role of the attraction is given by the value of the pre-
exponential coefficient a, whereas the values of the coefficient b for the two systems are the same within
the accuracy of the MD simulation.
The SDC are usually compared with the coefficients obtained using amodified Cohen-Turnbull theory
[32, 33]. In this theory, the diffusion coefficients are given by
D = D0 exp
(
−α
vc
vf
)
, (4.1)
where D0 is the Chapman-Enskog diffusion for a model pair potential at low density
D0 = DHS
0 /Ω(1,1)(T ). (4.2)
The coefficient DHS
0 is the SDC for a hard sphere system, the temperature dependent collision integral,
Ω
(1,1)(T ), depends on the attractive forces [34], α is an (“overlap”) parameter, vc is “the critical volume”
and vf is the “mean free volume”. The data in figure 5 agree with this model with a = D0, b = αvc and
vf ≈ ρ−1.
The SDC for the triple point (fluid) density ρtr = 0.85 are shown in figure 6. With red (solid) line and
points are D for the LJ system. The triple point temperature is Ttr = 0.69 and the system is supercooled
down below Ttr to T = 0.30 where it crystallizes. The diffusion coefficients for the system with only short-
range attraction (green) are almost equal to the diffusion coefficients for the LJ system, showing that the
long range attractions behind the FCS only have amarginal effect on the diffusion at high temperatures as
well as on the diffusion at super-cooled temperatures. The diffusion coefficients for the system without
attractive forces [blue (dashed) line and points] are, however, systematically higher. The SDC for the
three systems are all given by a simple linear dependence a +bT where b ≈ independent of the range
of the attractive forces, and a’s dependence on the attractive forces to a high accuracy is given by the
short range forces within the FCS. At a deeper super cooling for a system which is more resistant against
13002-8
Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids
Figure 6. (Color online) Self-diffusion coefficients D(T,ρ) for the triple point fluid density ρtr = 0.85. Red
(solid) line and points: LJ fluid; green points: for a system with only forces within the FCS included in
the dynamics, and blue (dashed) line and points are when only the repulsive forces are included in the
dynamics.
crystallization [35, 36], the dynamics and the SDC can still be obtained with only attractive forces within
the FCS included in the dynamics [3], whereas the SDC for a system with only repulsive forces deviates
significantly [3, 37].
5. Conclusion
Liquids are often separated into simple van der Waals liquids with weak intermolecular attractive
forces and into liquids with stronger attractions. The van der Waals theory for simple liquids, where the
repulsive and the weak attractive forces give rise to separate modifications of volume and pressure in the
ideal-gas equation of state, is qualitatively correct and has played a historical role in understanding the
fluid systems. The perturbation theories (PT) for simple liquids, which were formulated late in the twenti-
eth century, showed good agreements with experimental- and computer simulated data, and gave reason
to believe that it is indeed possible to obtain the contribution to the pressure from the weak attractive van
der Waals forces as a mean field contribution from a system with the particle distribution for a system
with only strong repulsive forces. Since then, the development in computer simulations has made it pos-
sible to obtain the equation of state for systems with different strength of attractions with high accuracy,
and to investigate various approximations in the PT theories for fluid systems. The present investigation
shows that, even for a systemwith themost weak attractive London forces, nowhere in the phase space, it
is possible to approximate the radial distribution of particles with the distribution obtained for a system
with only repulsive forces. Even weak attractive induced dipole forces, primarily within the first coor-
dination shell, modify the radial distribution, especially at short intermolecular distances (figure 2), and
with the result that the pressure obtained by the PT is too high (figure 4).
The weak long-range attractions damp the dynamics and the diffusion (figure 5 and figure 6). Since it
is the attractive forces which cause a condensation into a liquid for temperatures below the critical tem-
perature, Tc, one should a priory assume that the attractive forces, which cause long range fluctuations in
densities at Tc would affect the particle diffusion and the value of the self-diffusion constant (SDC) near
the critical point. However, this is not the case, and the result is in accordance with experimental data
for the SDC in the critical region [31].
The SDC along the critical isotherm exponentially varies with the density (figure 5). This is in good
agreement with a modified Cohen-Turnbull model for the SDC [32, 33]. The effect of the attractive forces
on the SDC is a simple damping of the diffusion and with a reduced value of the preexponential constant.
The attractive forces also damp the diffusion in condensed super-cooled fluids. The temperature depen-
13002-9
S. Toxvaerd
dence of the SDC at the triple-point isochore shows a simple linear damping with respect to temperature,
and the damping of the diffusion is caused by the attractive forces within the first coordination shell (fig-
ure 6). For systems which are less prone to crystallization and at a deeper super-cooling, the effect of the
attractive forces is more dominant, but the attractive forces within the FCS still determine the diffusion
and viscosity, as well as the thermodynamics [3].
Acknowledgements
The centre for viscous liquid dynamics “Glass and Time” is sponsored by the DanishNational Research
Foundation (DNRF).
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http://dx.doi.org/10.1126/science.157.3787.375
http://dx.doi.org/10.1063/1.1696618
http://dx.doi.org/10.1063/1.1740409
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http://dx.doi.org/10.1063/1.1674820
http://dx.doi.org/10.1103/RevModPhys.48.587
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Role of attractive forces in determining the equilibrium structure and dynamics of simple liquids
Роль сил притягання у визначеннi рiвноважної структури i
динамiки простих рiдин
С. Токсваерд
Факультет природничих наук, Унiверситет м. Роскiльде, DK-4000 Роскiльде, Данiя
Моделювання методом молекулярної динамiки системи Лєннарда-Джонса з рiзною областю притягання
показують, що сили притягання видозмiюють радiальний розподiл частинок. Сили в межах першої ко-
ординацiйної сфери (ПКС) є важливими лише для густих рiдин, в той час, як для газiв i помiрно густих
плинiв, навiть сили притягання за межами ПКС вiдiграють певну роль. Змiни у розподiлi, спричиненi не-
хтуванням силами притягання, приводять до занадто високого тиску. Слабке далекосяжне притягання
гасить динамiку i дифузiю частинок у станах газу, надкритичного плину та рiдини. Значення коефiцiєнтiв
самодифузiї якiсно узгоджуються з модифiкованою моделлю Коена-Турнбулла. Коефiцiєнти самодифузiї
вздовж критичної iзотерми не показують нiякої аномалiї у критичнiй точцi, що узгоджується з експери-
ментальними даними.
Ключовi слова: простi рiдини, рiвняння стану, теорiя збурень, самодифузiя
13002-11
Introduction
Perturbation theory for simple liquids
Separation of pair interactions into short- and long range interactions
The effect of attractions on structure and pressure
The phase diagram of a LJ system
The zeno line
The structure of the fluid at the zeno line
Pressure along the zeno line
The role of attractions on self-diffusion
Conclusion
|