Monte Carlo simulation of phase equilibria in Ising fluids and their mixtures
The mean field theory for the pure Ising fluid was recently extended to binary mixtures of an Ising and a van der Waals fluid. Depending on the relative interaction strengths, their three dimensional phase diagrams show lines of tricritical consolute and plait points, lines of critical end points...
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Інститут фізики конденсованих систем НАН України
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nasplib_isofts_kiev_ua-123456789-1207662025-06-03T16:30:13Z Monte Carlo simulation of phase equilibria in Ising fluids and their mixtures Моделювання фазових рівноваг в ізинґівських плинах та їх сумішах методом Монте-Карло Fenz, W. Folk, R. The mean field theory for the pure Ising fluid was recently extended to binary mixtures of an Ising and a van der Waals fluid. Depending on the relative interaction strengths, their three dimensional phase diagrams show lines of tricritical consolute and plait points, lines of critical end points and magnetic consolute point lines. Our current efforts are to compare these mean field results with different Monte Carlo simulation techniques, investigating both first order (liquid-vapor and demixing) and second order (paramagneticferromagnetic) phase transitions. We show the resulting ρ, T phase diagrams of the pure Ising fluid for different magnetic interaction strengths R and constant pressure cross-sections of the x, T, p phase diagrams of Ising mixtures for different relative interaction strengths. The methods we have used include Gibbs Ensemble MC, Multihistogram Reweighting, Hyper-parallel Tempering, the cumulant intersection method and the newly developed Density of States MC technique. Область застосування теорії середнього поля для чистого ізинґівського плину нещодавно поширилась на бінарні суміші ізинґівських і ван-дер-вальсівських плинів. Залежно від сил відносної взаємодії, їхня тривимірна фазова діаграма демонструє такі лінії: трикритичних точок розчинення і критичних точок рідина-пара, критичних кінцевих точок та точок розшарування у феромагнітній фазі. У даній роботі ми порівнюємо результати, отримані в наближенні середнього поля, з моделюванням різними методами Монте-Карло, досліджуючи фазові переходи і першого (рідина-пара або розшарування), і другого (парамагнетик-феромагнетик) роду. Нами побудовано результуючі фазові діаграми ρ, T для чистого ізинґівського плину при різних силах магнітної взаємодії R та перерізи фазових діаграм x, T, p при сталому тиску для ізинґівських сумішей при різних силах відносної взаємодії. Серед використаних нами методів є МК для ансамблю Гіббса, мультигістограмний перерозподіл, гіперпаралельне згладжування, метод кумулянтних перерізів і недавно створений метод МК для густини станів. This work was supported by the Fonds zur F¨orderung der wissenschaftlichen Forschung under Project No. 15247-TPH. We would like to thank I.Mryglod and I.Omelyan for helpful discussions. Updated version of a talk given at the 15th Symposium on Thermophysical Properties, June 22–27, 2002. 2003 Article Monte Carlo simulation of phase equilibria in Ising fluids and their mixtures / W. Fenz, R. Folk // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 675-686. — Бібліогр.: 33 назв. — англ. 1607-324X PACS: 05.70.Fh, 82.60.Lf, 64.60.-i, 64.60.Kw, 02.70.Uu DOI:10.5488/CMP.6.4.675 https://nasplib.isofts.kiev.ua/handle/123456789/120766 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The mean field theory for the pure Ising fluid was recently extended to binary
mixtures of an Ising and a van der Waals fluid. Depending on the relative
interaction strengths, their three dimensional phase diagrams show lines of
tricritical consolute and plait points, lines of critical end points and magnetic
consolute point lines. Our current efforts are to compare these mean field
results with different Monte Carlo simulation techniques, investigating both
first order (liquid-vapor and demixing) and second order (paramagneticferromagnetic)
phase transitions. We show the resulting ρ, T phase diagrams
of the pure Ising fluid for different magnetic interaction strengths
R and constant pressure cross-sections of the x, T, p phase diagrams
of Ising mixtures for different relative interaction strengths. The methods
we have used include Gibbs Ensemble MC, Multihistogram Reweighting,
Hyper-parallel Tempering, the cumulant intersection method and the newly
developed Density of States MC technique. |
| format |
Article |
| author |
Fenz, W. Folk, R. |
| spellingShingle |
Fenz, W. Folk, R. Monte Carlo simulation of phase equilibria in Ising fluids and their mixtures Condensed Matter Physics |
| author_facet |
Fenz, W. Folk, R. |
| author_sort |
Fenz, W. |
| title |
Monte Carlo simulation of phase equilibria in Ising fluids and their mixtures |
| title_short |
Monte Carlo simulation of phase equilibria in Ising fluids and their mixtures |
| title_full |
Monte Carlo simulation of phase equilibria in Ising fluids and their mixtures |
| title_fullStr |
Monte Carlo simulation of phase equilibria in Ising fluids and their mixtures |
| title_full_unstemmed |
Monte Carlo simulation of phase equilibria in Ising fluids and their mixtures |
| title_sort |
monte carlo simulation of phase equilibria in ising fluids and their mixtures |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120766 |
| citation_txt |
Monte Carlo simulation of phase
equilibria in Ising fluids and their
mixtures / W. Fenz, R. Folk // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 675-686. — Бібліогр.: 33 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT fenzw montecarlosimulationofphaseequilibriainisingfluidsandtheirmixtures AT folkr montecarlosimulationofphaseequilibriainisingfluidsandtheirmixtures AT fenzw modelûvannâfazovihrívnovagvízingívsʹkihplinahtaíhsumíšahmetodommontekarlo AT folkr modelûvannâfazovihrívnovagvízingívsʹkihplinahtaíhsumíšahmetodommontekarlo |
| first_indexed |
2025-11-30T16:57:14Z |
| last_indexed |
2025-11-30T16:57:14Z |
| _version_ |
1850235248261988352 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 4(36), pp. 675–686
Monte Carlo simulation of phase
equilibria in Ising fluids and their
mixtures
W.Fenz, R.Folk
Institute for Theoretical Physics, Johannes-Kepler-University Linz,
Altenbergerstr. 69, A-4040 Linz, Austria
Received September 23, 2003
The mean field theory for the pure Ising fluid was recently extended to bina-
ry mixtures of an Ising and a van der Waals fluid. Depending on the relative
interaction strengths, their three dimensional phase diagrams show lines of
tricritical consolute and plait points, lines of critical end points and magnetic
consolute point lines. Our current efforts are to compare these mean field
results with different Monte Carlo simulation techniques, investigating both
first order (liquid-vapor and demixing) and second order (paramagnetic-
ferromagnetic) phase transitions. We show the resulting ρ, T phase dia-
grams of the pure Ising fluid for different magnetic interaction strengths
R and constant pressure cross-sections of the x, T, p phase diagrams
of Ising mixtures for different relative interaction strengths. The methods
we have used include Gibbs Ensemble MC, Multihistogram Reweighting,
Hyper-parallel Tempering, the cumulant intersection method and the newly
developed Density of States MC technique.
Key words: magnetic fluid, binary mixture, Monte Carlo simulation, phase
diagram, tricritical point
PACS: 05.70.Fh, 82.60.Lf, 64.60.-i, 64.60.Kw, 02.70.Uu
1. Introduction
The investigation of continuum fluid models with coupled translational and spin
degrees of freedom is of current theoretical interest [1–3]. The importance of such
models lies in their display of a rich variety of transitions between solid, liquid, and
gas, as well as magnetically ordered and disordered phases, which may occur in
real systems. The liquid-gas and magnetic para-ferro phase transitions in spin fluids
were previously studied using the mean field theory [4–6], the method of integral
equations [7–11], and Monte Carlo (MC) simulation techniques [2,7,10,12–15].
In [3] we presented the mean field theory for a binary mixture of an Ising fluid and
a van der Waals fluid and studied the topology of the resulting three-dimensional
c© W.Fenz, R.Folk 675
W.Fenz, R.Folk
phase diagrams. Our interest in this system is founded, on the one hand, on its
extremely rich phase behavior, including, for example, lines of tricritical points and
consolute point lines in the ferromagnetic phase, and, on the other hand, in its
connection to ternary mixtures which is due to the fact that the Ising fluid can be
mapped onto a symmetric binary mixture [16]. Moreover, the Ising system exhibits
features that are also found in other more realistic models, like, for example, the
restricted primitive model (RPM) of ionic fluids [17].
The outline of this paper is as follows: In section 2 we summarize the mean
field theory of Ising mixtures from [3]. Section 3 deals with MC simulations for the
pure fluid system, the applied methods and the results, and section 4 contains the
corresponding treatment for binary mixtures.
2. Mean Field theory
We consider the general case of a binary mixture in the molar volume V consisting
of a van der Waals fluid (fluid 1) and an Ising fluid (fluid 2). The mole fraction of
the second component shall be denoted as x and its magnetization per particle as
m. The case of a pure Ising fluid follows for x = 1. We describe our system with a
van der Waals-like equation of state,
p (T, V, x, m) =
RT
V − b
−
a (x, m)
V 2
, (1)
where the attraction parameter a is defined according to the quadratic mixing rule
as
a (x, m) = a11 (1 − x)2 + 2a12x (1 − x) +
(
a22 +
1
2
amm2
)
x2 , (2)
and the size parameter b is assumed constant. In equation (2), a11 and a22 denote the
nonmagnetic interactions between particles of the same kind, a12 the nonmagnetic
interaction between unlike particles and am the magnetic interaction in the Ising
fluid. For the magnetization, the equation of state at zero magnetic field reads
m = tanh
(
amxm
V RT
)
. (3)
The corresponding molar Helmholtz free energy of the system described by (1) and
(3) with respect to the reference state of an ideal unmixed gas with molar volume
V 0 = b is given by
Arel(T, V, x, m) = A(T, V, x, m) − A0(T, V 0)
= xAs (T, m) + RT [(1 − x) ln (1 − x) + x ln x] − RT ln
(
V − b
b
)
−
a (x, m)
V
,
where
As(T, m) = RT
(
1 − m
2
ln
1 − m
2
+
1 + m
2
ln
1 + m
2
)
(4)
676
MC simulation of Ising fluids
is the entropy part of the free energy of the Ising fluid component. The mixture can
be described by the three reduced parameters ζ, Λ and Rm,
ζ =
a22 − a11
a11 + a22
, Λ =
a11 − 2a12 + a22
a11 + a22
, Rm =
1
2
am
a22
, (5)
and we introduce dimensionless variables scaled by the critical parameters of com-
ponent 2,
Tr =
27
8
bRT
a22
=
27
4
Rm
bRT
am
, pr =
27b2p
a22
, Vr =
V
b
. (6)
As was shown in [4], the phase diagram of the pure van der Waals Ising fluid can
have three different topologies:
• type I: For Rm > 0.304 the second order magnetic transition line Tλ ends
in a tricritical point below which a first order phase transition in the liquid
appears. This is the scenario also valid in the “ideal” case when → ∞ [18].
• type II: For 0.211 < Rm < 0.304 the second order magnetic transition line
Tλ ends in a tricritical point below which a first order phase transition in the
liquid appears which ends in a triple point on the gas liquid first order phase
transition. This first order line ends in a critical point. Thus, a tricritical and
a critical point coexist.
• type III: For Rm < 0.211 the second order magnetic transition line Tλ ends
in a critical end point on the gas liquid first order phase transition. This first
order line ends in a critical point.
In the mixture, the conditions for the coexistence of two phases α and β at a
temperature T0 and pressure p0 are given by
p (xα, Vα, mα) = p (xβ, Vβ, mβ) = p0 , (7)
µ1 (xα, Vα, mα) = µ1 (xβ, Vβ, mβ) , (8)
µ2 (xα, Vα, mα) = µ2 (xβ, Vβ, mβ) , (9)
mα = m (xα, Vα) , mβ = m (xβ, Vβ) . (10)
By solving these equations numerically, one can find the first order phase transition
surfaces in x, T, p-space.
Lines of second order phase transitions (plait point and consolute point lines)
are calculated from the equations
(
∂2Gm
∂x2
)
T,p,m
= 0,
(
∂3Gm
∂x3
)
T,p,m
= 0, (11)
where Gm = Am−∂Am/∂V is the Gibbs free energy. In the case of a tricritical line the
first of these conditions taken in the limit of m → 0 yields an analytic expression
677
W.Fenz, R.Folk
0
0.2
0.4
0.6
0.8
1
x0.2
0.4
0.6
0.8
1
Tr
0
0.5
1
1.5
2
pr
0
0.2
0.4
0.6
0.8
1
x
0
0.5
1
1.5
2
pr
0
0.2
0.4
0.6
0.8
1
x
0.5 1 1.5
Tr
0
2
4
6
8
pr 0
0.2
0.4
0.6
0.8
1
x
0
2
4
6
8
pr
(a) (b)
Figure 1. x, Tr, pr-diagram of a mixture with ζ = 0.5, Λ = −0.05 and Rm = 0.2
(a) and ζ = 0.5, Λ = −0.25 and Rm = 0.5 (b). Thick lines, liquid-vapor curves of
the pure substances; thin lines, isobaric curves on the first order surface; dotted
lines, tricritical lines; dashed lines, critical lines.
for the tricritical temperature as a function of the mole fraction x. It turns out
that depending on the three mixture parameters, the tricritical line has either the
character of a consolute line and goes to infinite pressure, or it resembles a plait
point line and eventually becomes unstable in a tricritical endpoint. In the three
dimensional parameter space, the boundary between these two types of behavior
is described by the equation Λ = −Rm(1 + ζ)/4. Figures 1 (a) and (b) show the
complete three-dimensional phase diagrams of systems corresponding to the two
topologies.
3. Monte Carlo simulations I – Pure Ising fluid
To investigate the phase diagram of a pure magnetic fluid we have modelled
the pair potential between two particles i and j as a sum of a nonmagnetic and a
magnetic interaction part. For the first contribution we used a conventional Lennard-
Jones potential, while for the second we used a Yukawa potential multiplied by the
two Ising spins connected with the particles:
uij (r) =
{
uLJ (r) + sisjuY (r) , r < rc ,
0 , r > rc
(12)
with
uLJ (r) = 4ε
[
(
σ
r
)12
−
(
σ
r
)6
]
, (13)
678
MC simulation of Ising fluids
uY (r) = −Rε
e−(r−σ)/σ
r/σ
. (14)
Here, σ is the size of the particles, r = |~ri − ~rj| is the inter-particle distance and si
is the spin of particle i, which is either +1 or −1. The potential was truncated at
a cutoff-distance rc equal to half the box length. This truncation was accounted for
by adding a tail correction
utail
i = 2π
[
ρ
∫
∞
rc
uLJ (r) r2dr + simρ
∫
∞
rc
uY (r) r2dr
]
(15)
to the potential energy of each particle. The parameter R was used to vary the
relative strength of the magnetic interaction. Simulations were performed for systems
with R = 0 (pure Lennard-Jones fluid), R = 0.05, R = 0.075, R = 0.1, R = 0.115
and R = 0.1333.
The phase diagrams of these fluids were investigated via three different simulation
methods: Gibbs Ensemble MC, Grand Canonical simulations with Multihistogram
reweighting and Density of States MC.
In the GEMC [19,20] simulations, the total number of particles N was usually
500 or 1000. The simulations were performed in cycles consisting of N trial particle
displacements, N spin flip attempts, one volume change attempt and a number
of particle swap attempts which was chosen to yield about 1–3% acceptance and
depended strongly on the temperature. Typically, the number of the performed MC
cycles was about 5 × 105.
Additionally, simulations in the Grand Canonical Ensemble have been performed.
These were conducted with a system size of L = 8σ or L = 10σ. Typical runs con-
sisted of 2 × 109 MC steps, where one step was either an attempted deletion or
insertion of a random particle. Particle displacements and spin flip attempts were
not implemented explicitly. The resulting histograms were reweighted using the al-
gorithm from [21,22]. In order to speed up the simulation process, the simulation
runs at different state points for the same system were performed in parallel and con-
figurations of two such runs could be interchanged according to the Hyper-Parallel
Tempering scheme described in [23].
Since at low temperatures and high densities of the liquid phase, both the GEMC
and the GC simulations suffer from a poor acceptance rate of particle insertion at-
tempts, we also applied another Monte Carlo method which was recently introduced
for lattice [24,25] and off-lattice systems [26,27] and which is especially well suited for
low temperatures, namely the Density of States (DOS) method. This approach uses
the so-called uniform ensemble in which all configurations are equally probable. The
considered MC moves consist of random particle displacements, particle insertions
and deletions and in our case also spin flip attempts. The discretization of energy
values was made with a bin size equal to the reduced unit of energy, ε. For each
system and particle number, the energy range corresponding to the temperature in-
terval we were interested in was determined beforehand from short simulation runs
in the canonical ensemble at the highest and the lowest considered temperature.
679
W.Fenz, R.Folk
0.0 0.2 0.4 0.6 0.8
0.8
1.0
1.2
1.4
1.6
R=0.1333
R=0.115
R=0.1
R=0.075
R=0.05
R=0
T*
ρ∗
(a)
0.2 0.4 0.6 0.8
Ρ*
0.6
0.8
1
1.2
1.4
1.6
1.8
Tr
CCC M
M
C
M
(b)
Figure 2. (a) Results of GEMC simulations (diamonds), multihistogram reweight-
ing (squares) and DOS MC simulations (lines) for Ising fluids with different inter-
action ratios R. The paramagnetic-ferromagnetic phase transition points (circles)
were found via the Binder crossing technique, the dashed lines are linear extra-
polations. T ∗ = kBT/ε, ρ∗ = ρσ3. (b) Mean field phase diagrams of pure Ising
fluids with different interaction ratios Rm = 0, 0.1333, 0.2, 0.26, 0.35, 0.5. M, tri-
critical point; C, critical point; solid lines, first order phase transitions; dashed
lines, second order (magnetic) phase transitions. The line of all tricritical points
for Rm > 0.211 is also shown.
680
MC simulation of Ising fluids
The paramagnetic-ferromagnetic phase transition line (Curie line) was investi-
gated using the Binder crossing technique [28]. During simulations in the canonical
ensemble, the two-dimensional energy-magnetization histograms were obtained and
extrapolated to other temperatures via the multi-histogram reweighting technique.
Thus, the fourth-order cumulant ratio of the magnetization distribution P (m) could
be calculated as a function of temperature. This was done for three different system
sizes, N = 150, N = 300 and N = 500, and the intersection point of the resulting
curves was determined as an estimate for the critical temperature of the infinite
system. We found that the slope of the Curie line is in fact proportional to the
parameter R, as predicted by mean field theory. From the systems with R = 0.05
and R = 0.075, where we calculated two points of the Curie line, we obtained a
proportionality constant c = 27.1. For the other parameters we assumed the Curie
line to be given by Tr = cRρ∗ and determined one transition point in order to check
if it really falls onto this line.
The phase diagrams we found in the simulations (figure 2 (a)) correspond well
to the ones predicted by mean field theory (figure 2 (b)). For small values of R,
0.05 and 0.075, we find a critical end point where the Curie line intersects the phase
coexistence curve. Above the critical end point temperature, these phase diagrams
are almost indistinguishable from the case of the pure Lennard-Jones fluid (R = 0)
which is also included in figure 2 (a). For larger values of R, e. g. R = 0.133, the shape
of the coexistence curve changes and there is no critical point any more. Instead, a
tricritical point seems to occur where the Curie line meets the binodal. For R values
that lie in between these two regimes, mean field theory predicts the existence of
both a critical and a tricritical point. For the parameter we considered, R = 0.1,
our simulations in fact showed a triple point at T ∗ = 1.32 with para-para as well as
para-ferro transitions directly above the triple point temperature. However, a direct
comparison between phase diagrams with a certain value of R in MC simulations
and Rm in mean field theory is not possible since there is no direct correspondence
between the two ratios. We demonstrate in figures 2 (a) and (b) that the same
topologies are found in the global phase diagram.
At H = 0 the Ising fluid may be mapped onto a symmetric binary mixture and
comparison with other methods is possible. Our results show the same scenario as
MC simulations in an Ising liquid with both interactions, the non-magnetic and the
magnetic one, of square well [16] or Lennard-Jones type [29]. It turned out that
the hierarchical reference theory (HRT) [30] did not agree with the scenario found
in MC simulations. In the language of the magnetic fluid, for small values of R no
critical end point [29], but a phase diagram of type II was found [30].
The complete phase diagram of a magnetic fluid includes the magnetic field H.
Recently the magnetic field dependence of the critical point location in an ideal
Ising fluid has been considered by MC simulations [31] supporting the existence of
a tricritical point in the Ising fluid at H = 0. The H-dependence at small fields is in
agreement with the H2/5 law for the wings from MF theory. In the limit H → ∞,
a monotonic decrease of the wing line to the critical point of the Yukawa fluid has
been found, being also in agreement with mean field theory. Moreover, it turned out
681
W.Fenz, R.Folk
that the MF phase diagram topology of type II splits into two parts depending on
whether the critical point or the tricritical point is connected to the critical point at
saturation (H → ∞). This leads to the appearance of a van Laar point in the global
phase diagram and at this point to a phase diagram containing an unsymmetrical
tricritical point at finite magnetic field [32]. This is in agreement with the scenario
found in mean field theory for symmetrical mixtures [33] and the HRT [30].
4. Monte Carlo simulations II – Mixture with an Ising fluid
For binary mixtures, we used the same interaction potential as for the pure fluid,
uij (r) =
{
uLJ (r) + sisjuY (r) , r < rc ,
0 , r > rc .
(16)
However, the spin interaction part only enters if both particles belong to the mag-
netic component. Also, the Lennard-Jones interaction strength ε now depends on
the species a and b of the two particles,
uLJ (r) = 4εab
[
(
σ
r
)12
−
(
σ
r
)6
]
(17)
and the Yukawa potential is accordingly given by
uY (r) = −Rε22
e−(r−σ)/σ
r/σ
. (18)
We can then define parameters ζ and Λ in analogy to (5) with ε11, ε12 and
ε22. Again, we have added a tail correction term to the energy to compensate the
truncation of the potential.
In our simulations we looked at the two systems with parameter values we ex-
pected to lead to similar phase diagram topologies as the two types in our mean
field theory, namely ζ = 0.5, Λ = −0.05, R = 0.05 and ζ = 0.5, Λ = −0.25,
R = 0.1333. These values of R were chosen as representatives of type I and III Ising
fluids according to the results of the simulations for the pure Ising fluid.
The phase diagrams of these systems were investigated with GEMC simulations
(N = 1000 or 500) at constant pressure in the case of first order phase transitions, as
well as simulations in the constant NpT ensemble (N = 150, 300 and 500) and the
cumulant crossing technique for the second order para-ferro phase transitions. More-
over, the first order transition was also studied within an isobaric Semigrand Canon-
ical ensemble (N = 300) at constant N, ∆µ, p, T using Multihistogram reweighting.
The MC moves performed in this ensemble consist of particle displacements, spin
flips, particle identity changes and volume rearrangements. Thus, both liquid-vapor
and demixing transitions could be studied. Figures 3 and 4 show the results for the
first system at p∗ = pσ3/ε22 = 0.08 and p∗ = 0.15, figure 5 for the second one at
p∗ = 0.24.
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MC simulation of Ising fluids
0.0
0.2
0.4
0.6
0.8
1.0
0.4
0.6
0.8
1.0
1.2
T
*
x
0.2 0.4 0.6 0.8 1
x
0.3
0.4
0.5
0.6
0.7
0.8
Tr
CEP
(a) (b)
Figure 3. (a) Results of GEMC simulations (squares) at p∗ = 0.08 for a mixture
with ζ = 0.5, Λ = −0.05, R = 0.05. The light points correspond to paramagnetic
phases, whereas the dark ones correspond to a paramagnetic and a ferromagnetic
phase in coexistence. The para-ferro phase transition points (circles) were found
via the Binder crossing technique. Figure (b) shows the corresponding mean field
diagram (ζ = 0.5, Λ = −0.05, Rm = 0.15) at pr = 0.3. CP, critical point; CEP,
critical end point; dashed line, Curie line.
0.0
0.2
0.4
0.6
0.8
1.0
0.4
0.6
0.8
1.0
1.2
T
*
x
0.2 0.4 0.6 0.8 1
x
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Tr
TCP
TCP
CP
(a) (b)
Figure 4. (a) Results of GEMC simulations (squares) and SGC simulations (small
squares) at p∗ = 0.15 for a mixture with ζ = 0.5, Λ = −0.05, R = 0.05. The light
points correspond to paramagnetic phases, whereas the dark ones correspond to
a paramagnetic and a ferromagnetic phase in coexistence. The para-ferro phase
transition points (circles) were found via the Binder crossing technique. Figure
(b) shows the corresponding mean field diagram (ζ = 0.5, Λ = −0.05, Rm = 0.15)
at pr = 0.6. CP, critical point; TCP, tricritical point; dashed line, Curie line.
683
W.Fenz, R.Folk
0.0
0.2
0.4
0.6
0.8
1.0
0.4
0.6
0.8
1.0
1.2
1.4
1.6
T
*
x
0.2 0.4 0.6 0.8 1
x
0.4
0.6
0.8
1
1.2
Tr
TCP
TCP
CP
(a) (b)
Figure 5. (a) Results of GEMC simulations (squares) and SGC simulations (small
squares) at p∗ = 0.24 for a mixture with ζ = 0.5, Λ = −0.25, R = 0.1333. The
para-ferro phase transition points (circles) were found via the Binder crossing
technique. Figure (b) shows the corresponding mean field diagram (ζ = 0.5,
Λ = −0.25, Rm = 0.4) at pr = 2.5; TCP, tricritical point.
It turns out that the observed phase behaviors indeed resemble those of two
mean field systems belonging to either of the two topology classes (at least in the
temperature ranges accessible to our simulations). In the figures, isobaric sections
of the mean field x, T, p-diagrams are presented together with the corresponding
Monte Carlo results in each case. Regarding the values of the ratios R and Rm the
same remark as in the case of pure Ising fluids has to be made. We can only say
that similar topological features are recovered by MC simulations.
At p∗ = 0.08, the first system exhibits a critical end point where the Curie
line intersects the phase coexistence curve, separating first order para-para phase
transitions occurring at temperatures above the critical end point temperature from
first order para gas – ferro liquid transitions below this critical temperature. At
higher pressure, p∗ = 0.15, the Curie line does not connect to the para-para liquid-
vapor two-phase region. Instead, it obviously passes into a para-ferro first order
demixing transition via a tricritical point.
For the second system we found para-ferro phase coexistence in the whole tem-
perature range, in agreement with the mean field prediction. Accordingly, all points
of the Curie line we calculated lie inside the two-phase region.
5. Acknowledgement
This work was supported by the Fonds zur Förderung der wissenschaftlichen
Forschung under Project No. 15247-TPH. We would like to thank I.Mryglod and
I.Omelyan for helpful discussions. Updated version of a talk given at the 15th Sym-
posium on Thermophysical Properties, June 22–27, 2002.
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MC simulation of Ising fluids
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685
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Моделювання фазових рівноваг в ізинґівських
плинах та їх сумішах методом Монте-Карло
В.Фенц, Р.Фольк
Інститут теоретичної фізики, Університет Йогана Кеплера м. Лінц,
Австрія, A-4040 Лінц, вул. Альтенбергерштрассе, 69
Отримано 23 вересня 2003 р.
Область застосування теорії середнього поля для чистого ізинґів-
ського плину нещодавно поширилась на бінарні суміші ізинґівських
і ван-дер-вальсівських плинів. Залежно від сил відносної взаємодії,
їхня тривимірна фазова діаграма демонструє такі лінії: трикритич-
них точок розчинення і критичних точок рідина-пара, критичних кін-
цевих точок та точок розшарування у феромагнітній фазі. У даній ро-
боті ми порівнюємо результати, отримані в наближенні середнього
поля, з моделюванням різними методами Монте-Карло, досліджу-
ючи фазові переходи і першого (рідина-пара або розшарування), і
другого (парамагнетик-феромагнетик) роду. Нами побудовано ре-
зультуючі фазові діаграми ρ, T для чистого ізинґівського плину при
різних силах магнітної взаємодії R та перерізи фазових діаграм
x, T, p при сталому тиску для ізинґівських сумішей при різних силах
відносної взаємодії. Серед використаних нами методів є МК для ан-
самблю Гіббса, мультигістограмний перерозподіл, гіперпаралельне
згладжування, метод кумулянтних перерізів і недавно створений ме-
тод МК для густини станів.
Ключові слова: магнітний плин, моделювання методом
Монте-Карло, фазова діаграма, трикритична точка
PACS: 05.70.Fh, 82.60.Lf, 64.60.-i, 64.60.Kw, 02.70.Uu
686
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