On the thermodynamic properties of the generalized Gaussian core model
We present results of a systematic investigation of the properties of the generalized Gaussian core model of index n. The potential of this system interpolates via the index n between the potential of the Gaussian core model and the penetrable sphere system, thereby varying the steepness of the...
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irk-123456789-1194882017-06-08T03:06:21Z On the thermodynamic properties of the generalized Gaussian core model Mladek, B.M. Fernaud, M.J. Kahl, G. Neumann, M. We present results of a systematic investigation of the properties of the generalized Gaussian core model of index n. The potential of this system interpolates via the index n between the potential of the Gaussian core model and the penetrable sphere system, thereby varying the steepness of the repulsion. We have used both conventional and self-consistent liquid state theories to calculate the structural and thermodynamic properties of the system; reference data are provided by computer simulations. The results indicate that the concept of self-consistency becomes indispensable to guarantee excellent agreement with simulation data; in particular, structural consistency (in our approach taken into account via the zero separation theorem) is obviously a very important requirement. Simulation results for the dimensionless equation of state, βP/%, indicate that for an indexvalue of 4 a clustering transition, possibly into a structurally ordered phase might set in as the system is compressed. Ми представляємо результати системного дослідження властивостей узагальненої моделі гаусового кора з індексом n. Потенціал такої системи з допомогою індексу n, що визначає крутизну притягання, дозволяє інтерполювати взаємодії від моделі гаусового кора до системи проникних сфер. Для розрахунку структурних і термодинамічних властивостей нами використовувалися як традиційні, так і самоузгоджені версії теорії рідкого стану, а система відліку бралася з комп’ютерного експерименту. Результати показують, що для отримання доброго узгодження з даними моделювання концепція самоузгодження стає обов’язковою; зокрема структурне узгодження, яке у нашому підході проводиться через теорему нульового розділення, є надзвичайно важливою вимогою. Результати моделювання для обезрозміреного рівняння стану, βP/%, вказують на те, що для моделі з індексом 4 може виникати структурно впорядкована фаза (перехід кластерування) при стисканні системи. 2005 Article On the thermodynamic properties of the generalized Gaussian core model / B.M. Mladek, M.J. Fernaud, G. Kahl, M. Neumann // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 135–148. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 61.20.Gy, 61.20.Ja, 64.10.+h DOI:10.5488/CMP.8.1.135 http://dspace.nbuv.gov.ua/handle/123456789/119488 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
We present results of a systematic investigation of the properties of the
generalized Gaussian core model of index n. The potential of this system
interpolates via the index n between the potential of the Gaussian core
model and the penetrable sphere system, thereby varying the steepness
of the repulsion. We have used both conventional and self-consistent liquid
state theories to calculate the structural and thermodynamic properties of
the system; reference data are provided by computer simulations. The results
indicate that the concept of self-consistency becomes indispensable
to guarantee excellent agreement with simulation data; in particular, structural
consistency (in our approach taken into account via the zero separation
theorem) is obviously a very important requirement. Simulation results
for the dimensionless equation of state, βP/%, indicate that for an indexvalue
of 4 a clustering transition, possibly into a structurally ordered phase
might set in as the system is compressed. |
| format |
Article |
| author |
Mladek, B.M. Fernaud, M.J. Kahl, G. Neumann, M. |
| spellingShingle |
Mladek, B.M. Fernaud, M.J. Kahl, G. Neumann, M. On the thermodynamic properties of the generalized Gaussian core model Condensed Matter Physics |
| author_facet |
Mladek, B.M. Fernaud, M.J. Kahl, G. Neumann, M. |
| author_sort |
Mladek, B.M. |
| title |
On the thermodynamic properties of the generalized Gaussian core model |
| title_short |
On the thermodynamic properties of the generalized Gaussian core model |
| title_full |
On the thermodynamic properties of the generalized Gaussian core model |
| title_fullStr |
On the thermodynamic properties of the generalized Gaussian core model |
| title_full_unstemmed |
On the thermodynamic properties of the generalized Gaussian core model |
| title_sort |
on the thermodynamic properties of the generalized gaussian core model |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119488 |
| citation_txt |
On the thermodynamic properties of the generalized Gaussian core model / B.M. Mladek, M.J. Fernaud, G. Kahl, M. Neumann // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 135–148. — Бібліогр.: 23 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT mladekbm onthethermodynamicpropertiesofthegeneralizedgaussiancoremodel AT fernaudmj onthethermodynamicpropertiesofthegeneralizedgaussiancoremodel AT kahlg onthethermodynamicpropertiesofthegeneralizedgaussiancoremodel AT neumannm onthethermodynamicpropertiesofthegeneralizedgaussiancoremodel |
| first_indexed |
2025-07-08T15:57:44Z |
| last_indexed |
2025-07-08T15:57:44Z |
| _version_ |
1837094945255587840 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 1(41), pp. 135–148
On the thermodynamic properties of
the generalized Gaussian core model
B.M.Mladek 1,2 , M.J.Fernaud 1 , G.Kahl 1 , M.Neumann 2
1 Center for Computational Materials Science
and Institut für Theoretische Physik,
Technische Universität Wien,
Wiedner Hauptstraße 8–10, A–1040 Wien, Austria
2 Institut für Experimentalphysik,
Universität Wien,
Strudlhofgasse 4, A–1090 Wien, Austria
Received November 24, 2004
We present results of a systematic investigation of the properties of the
generalized Gaussian core model of index n. The potential of this system
interpolates via the index n between the potential of the Gaussian core
model and the penetrable sphere system, thereby varying the steepness
of the repulsion. We have used both conventional and self-consistent liquid
state theories to calculate the structural and thermodynamic properties of
the system; reference data are provided by computer simulations. The re-
sults indicate that the concept of self-consistency becomes indispensable
to guarantee excellent agreement with simulation data; in particular, struc-
tural consistency (in our approach taken into account via the zero separa-
tion theorem) is obviously a very important requirement. Simulation results
for the dimensionless equation of state, βP/%, indicate that for an index-
value of 4 a clustering transition, possibly into a structurally ordered phase
might set in as the system is compressed.
Key words: soft matter, integral equations, computer simulations,
clustering transition, Gaussian core model
PACS: 61.20.Gy, 61.20.Ja, 64.10.+h
1. Introduction
Soft matter physics has become a rapidly developing, and challenging field for
both experimentalists and theoreticians. For several reasons, particular effort has
been made in recent years to investigate this topic. From the applied viewpoint, soft
matter – bean ubiquitous in our daily life – plays a key role in many technological
applications and processes. From the academic point of view, it offers a broad range
of possibilities where experiment and theory can cooperate in a very constructive and
c© B.M.Mladek, M.J.Fernaud, G.Kahl, M.Neumann 135
B.M.Mladek et al.
complementary manner. Typical examples for soft matter systems are suspensions
of mesoscopic particles (their size ranging from 1 µm to 1 nm) immersed in a solvent
formed by particles of atomic size. The fact that they are considerably larger than
particles in atomic systems makes experimental investigations much easier, using
modern tools such as video microscopy or optical tweezers (for a recent overview see
[1]). Further, by suitable modifications of the solvent and/or by systematic synthesis
of the aggregates (see, for example, recent progress for dendrimers [2]), the physical
properties of the mesoscopic particles can be tailored at will, which brings a direct
comparison between theory and experiment within reach.
Since typical soft matter particles are themselves complex aggregates built up
from several thousands of atoms or molecules, it is impossible to use potentials in
theoretical investigations that take explicitly into account all the degrees of free-
dom of the constituent particles. Coarse graining methods have turned out to be
a powerful tool to cope with this problem [3]; using suitable averaging procedures,
these approaches drastically reduce the huge number of degrees of freedom and one
finally arrives at effective (pair) interactions Φ(r): they depend typically on the di-
stance between the centers of mass of the aggregates and thus represent potentials
between artificial, “effective” particles. For a few soft matter systems such effective
interactions have been proposed or derived in literature, sometimes even in closed,
analytical forms: neutral and charged star polymers and microgels [4,5] are a few
examples (for an overview we refer to [3]).
In contrast to atomic systems where the interatomic potentials are harshly repul-
sive at short distances, the effective potentials of soft matter systems show complete-
ly new features: as a consequence of a complex internal structure of the mesoscopic
aggregates these particles are allowed to overlap, to mutually penetrate, or to even
interweave when being compressed, which is expressed by the fact that frequent-
ly the effective potentials only weakly diverge or even remain finite at the origin.
These particular features lead, in turn, to unexpected, surprising effects both in their
structural properties as well as in their phase behaviour: for example, the structure
factor of star polymers shows an anomalous behaviour upon compression, which is
reflected in a decrease of the height of the main peak for densities larger than the
overlap density [6]; in the phase diagram of these polymers and charged microgels,
new and completely unexpected features were encountered such as re-entrant mel-
ting processes or clustering transitions [7,8]. In addition, well-established criteria
that indicate the onset of a freezing transition in atomic systems (such as freezi-
ng criteria due to Hansen & Verlet or Lindeman [9]) break down completely and
therefore loose their significance in soft systems.
These new and particular features have also to be taken into account from the
conceptual point of view when calculating the properties of these systems: liquid
state theories that were originally designed for systems with harshly repulsive po-
tentials and have proved to be reliable there have to be thoroughly reconsidered;
a straightforward extension to soft systems is not justified. Examples that justify
this scepticism are, for instance, closure relations to the Ornstein-Zernike relation
including the bridge function, B(r): in those relations this function plays the role of
136
Thermodynamics of the generalized Gaussian core model
an additional, effective potential. In systems with harshly repulsive interactions the
particular shape of B(r) was irrelevant at short distances, where the diverging po-
tential dominates; this contributed to the justification of the universality hypothesis
[10] which states that B(r) can be approximated reasonably well by the bridge func-
tion of a suitably chosen hard sphere reference system. In soft systems, however, the
situation is completely different: the particular shape of the bridge functions (which
is now of the same order of magnitude as the potential at short distances) is now
crucial and the universality hypothesis can no longer be maintained.
In several recent studies the properties of systems with soft potentials have been
investigated using different liquid state theories [11,12], but no systematic investiga-
tions are available where the degree of softness of Φ(r) can be varied in a continuous
way. In the present contribution we report on the structural and thermodynamic
properties of a particular model system of soft matter, where the functional form
of the interaction permits such a variation, the so-called generalized Gaussian core
model with index n (GGCM-n):
ΦGGCM−n(r) = εe−(r/σ)n
. (1)
ε is an energy parameter and σ represents a length scale. The potential interpolates
smoothly, via the index n, between two model systems that have been widely used
in literature: the Gaussian core model (GCM) is recovered for n = 2 while for
n → ∞, we obtain the penetrable sphere model (PSM). Further one can show
[13], that the Fourier transform of the potential, Φ̃(q), is a positive, monotonously
decaying function for n 6 2, while for n > 2, Φ̃(q) can also attain negative values.
These two scenarios correspond exactly to the classification introduced by Likos et
al. [14] to distinguish between two classes of systems, the Q+ and the Q± systems;
upon compression these systems show a completely different behaviour: while Q+
systems show re-entrant melting, a clustering transition is predicted for the Q±
class. Although a detailed investigation of these phenomena would clearly exceed
the limits of the present contribution we shall briefly come back to this feature in
section 3.
Restricting ourselves to the liquid phase, we intend to fill the gap between the
two limiting cases, the GCM and the PSM – which have already been studied in
detail [11,12] – by considering the thermodynamic properties of GGCM-n for two
different intermediate values of n, i.e. for n = 4 and n = 10. We use different liquid
state theories and put emphasis on a critical test and revision of these frameworks
and on a thorough discussion of the results, for the reasons mentioned above. To
assess our results and to test the reliability of our numerical approaches we have
performed computer simulations.
The paper is organized as follows: in the subsequent, theoretical section we shall
present the model and briefly outline the liquid state concepts we have used to
calculate the properties of the system. These will be discussed in section 3, starting
with the structural properties and focusing then on the thermodynamics, including
a brief outlook on further implications on the phase behaviour. The paper is closed
with a summary of our findings.
137
B.M.Mladek et al.
2. Theory
2.1. The system
The functional form of the GGCM-n potential has been presented in (1); system
parameters are the temperature T [where β = (kBT )−1] and the number density
%, or, equivalently, the following reduced quantities, ε∗ = βε and %∗ = %σ3. As
mentioned above, for n → ∞ we recover the potential of PSM, being given by
ΦPSM(r) =
{
ε, r 6 σ,
0, r > σ.
(2)
In figure 1 we compare the GGCM-n potentials for those n-values we consider in
the present contribution along with the PSM potential.
0 0.5 1 1.5 2 2.5
r/σ
0
0.2
0.4
0.6
0.8
1
Φ
(r
)/
ε
Figure 1. Potential Φ(r) of the GGCM-n as a function of r/σ [cf. (1)] for different
values of n: full line – n = 2 (GCM), broken line – n = 4, dotted line – n = 10,
and dot-dashed line – n → ∞ (PSM).
2.2. Liquid state theories
We have used conventional as well as self-consistent liquid state theories to cal-
culate the structural and thermodynamic properties of the system [9,15]. The con-
ventional closure relations to the Ornstein-Zernike equation we have used are the
hypernetted-chain (HNC) and the Percus-Yevick (PY) relations, which are given by
cHNC(r) = −βΦ(r) + h(r) − log [h(r) + 1] , (3)
cPY(r) =
[
1 − eβΦ(r)
]
g(r) (4)
138
Thermodynamics of the generalized Gaussian core model
and the mean spherical approximation (MSA), where the closure for a system with
a soft potential reads
cMSA(r) = −βΦ(r). (5)
Here, c(r) and h(r) are the direct and the total correlation functions, and g(r) is the
pair distribution function (PDF). In general a numerical solution of the Ornstein-
Zernike equation with one of the closure relations is required to calculate these
functions. Once these functions are known, one can calculate thermodynamic pro-
perties via different thermodynamic routes. Note, however, that as a consequence
of the simplifying assumptions made in their derivation, the above closure relations
are only approximations of exact statistical mechanics relations; hence, the resulting
correlation functions are only approximative.
The thermodynamic routes we have used to calculate the equation of state are
the following [9],
%kBTκT = 1 + %
∫
[g(r) − 1] dr and κT =
1
%
(
∂P
∂%
)
T
, (6)
U ex
N
= 2π%
∫
g(r) Φ(r) r2 dr and %
∂2
∂%2
(
%
U ex
N
)
=
∂2
∂β∂%
(%P ex) , (7)
βP vir
%
= 1 −
2π
3
%
∫
r2
[
r
dβΦ(r)
dr
g(r)
]
dr, (8)
where κT is the isothermal compressibility, U ex and P ex are the excess (over ideal gas)
internal energy and pressure, and P vir is the virial pressure. In the order presented
above, they are called compressibility, energy, and virial routes.
If g(r) were exact, then the different routes would lead to the same thermody-
namic properties. However, since we obtain for g(r) only approximate results, the
different routes might lead to different values for a given thermodynamic proper-
ty, a fact that in literature is called thermodynamic inconsistency. Advanced liquid
state theories try to remove this deficiency by introducing more complex closure
relations that enforce thermodynamic consistency between at least two different
thermodynamic routes. The self-consistent liquid state theories we have used are
the Zero-Separation integral-equation approach (ZSEP) [16] and the Self-Consistent
Ornstein-Zernike approximation (SCOZA) ([17] and references therein). In the clo-
sure relations of these theories parameters or functions are introduced, which are
chosen to enforce certain consistency relations.
The ZSEP approach is based on the following closure relation,
g(r) = exp[−βΦ(r) + h(r) − c(r) + BZSEP(r)], (9)
where for the bridge function BZSEP(r) an approximate parameterization proposed
by Verlet [18] was chosen:
BZSEP(r) = −
As(r)2
2
[
1 −
BCs(r)
1 + Bs(r)
]
, (10)
139
B.M.Mladek et al.
s(r) = h(r) − c(r) is the indirect correlation function. The parameters A, B, and C
are adjusted to enforce three different self-consistency conditions: two thermodynam-
ic relations and one structural requirement. Firstly, we require consistency between
the virial and the compressibility routes, secondly, we fulfill the Gibbs-Duhem rela-
tion, and thirdly, we enforce the so-called zero separation theorem (for details see
[12]). The ZSEP approach offers not only information about the structural and the
thermodynamic properties. The bridge function – which plays a considerably more
important role in soft matter than in the systems with harshly repulsive systems –
can also be analyzed. We shall compare it with the data extracted from computer
simulations.
Additionally, we report the preliminary results obtained via the SCOZA. In a
straightforward generalization of the original SCOZA idea [17] the following closure
to the Ornstein-Zernike relation, which can be viewed as a generalization of the
MSA closure relation (5), is used:
c(r) = ξ(%, β)βΦ(r). (11)
The function ξ(%, β) is chosen to enforce thermodynamic self-consistency between the
compressibility and the virial routes. In conventional applications of parameterized
integral-equations (such as the Rogers-Young scheme [19]), thermodynamic self-
consistency was enforced only for a local state, i.e. ξ was then a simple, state-
independent mixing parameter; in SCOZA, however, ξ = ξ(%, β) is explicitly state-
dependent, which leads to a partial differential equation for ξ(%, β) that in general
has to be solved numerically. Details will be presented in a later publication [20].
To assess our data we have performed standard Monte-Carlo simulations in a
canonical ensemble [21]. Typically, around 1000 particles were used, simulations ex-
tended over 200000 sweeps and observables were averages over 20000 configurations.
The usual tail corrections were applied when calculating the pressure and the energy.
3. Results
We have calculated structural and thermodynamic properties of the GGCM-
n with the liquid state concepts outlined above and computer simulations, which
provide reference data. For our investigations we have chosen three different n-values,
n = 2 (i.e. the GCM), n = 4, and n = 10, and two different energy parameters for
the potential, i.e., ε∗ = 0.1 and ε∗ = 2, for each of the three choices of n.
3.1. Structural properties
In figure 2 we display g(r) for n = 4, ε∗ = 2, and %∗ = 2. We observe that
PY fails distinctly, and that MSA still shows some discrepancies with respect to
the simulation data, in particular at short distances; HNC thus remains the only
conventional liquid state theory that provides good overall agreement. Regarding the
two self-consistent concepts, ZSEP and SCOZA, we observe that SCOZA slightly
improves the MSA data, in particular at short distances, which can be attributed
140
Thermodynamics of the generalized Gaussian core model
0 1 2 3 4
r / σ
0
0.5
1
1.5
g(
r) 0 0.1 0.2
1.5
1.75
Figure 2. Pair distribution function g(r) of a GGCM-n for n = 4 at ε∗ = 2 and
%∗ = 2. The different curves correspond to the HNC (solid line) and PY (dotted
line) approximations and to the MSA (dashed line), as well as to the two self-
consistent approaches considered: SCOZA (dot-dashed line) and ZSEP (solid line
with crosses). The dots are the MC simulation results. In the inset a magnified
view for short distances is depicted.
0 1 2 3
r / σ
0
0.5
1
1.5
g(
r) 0 0.4 0.8
0.4
0.6
0.8
Figure 3. Pair distribution function g(r) of a GGCM-n for n = 10 at ε∗ = 2 and
%∗ = 0.5. The labels are the same as those used in figure 2. SCOZA results are
not available for this system.
141
B.M.Mladek et al.
to the thermodynamic self-consistency requirement; ZSEP shows perfect agreement
with the simulation data. In figure 3, where we depict the results for the g(r) of
a GGCM-10 system at ε∗ = 2 and %∗ = 0.5, ZSEP and Monte Carlo data agree
perfectly, HNC again gives reasonable results, PY fails and MSA even goes negative.
This unphysical behaviour of the MSA was already observed for the GCM [11]
and there is evidence [20] that the SCOZA-PDF can also attain negative values
for the GGCM-n in dependence on the system parameters. We conclude that, for
soft systems, self-consistency requirements (both thermodynamic and structural)
are effective concepts to provide accurate agreement with computer simulations.
However, for the increasing values of n, the requirement of thermodynamic self-
consistency alone might not be a sufficient requirement to provide physical data
for g(r). The requirement of the structural consistency becomes then particularly
relevant (note the good agreement of ZSEP-data with simulation results for the
PSM, i.e. as n → ∞ [12]).
In figure 4 we display the bridge function B(r) of the GGCM-4 system with ε∗ = 2
and %∗ = 2, i.e. for a system which is considerably softer than the PSM studied
in [12]. We compare B(r) as predicted from the ZSEP concept and as extracted
from computer simulation data (for details of the extraction procedure see [12] and
references therein). The agreement is rather on a qualitative level; B(r) is quite
small, a fact which indicates that HNC [where B(r) ≡ 0] shows already a large
degree of self-consistency in the sense of the ZSEP concept. At present, investigations
are carried out to perform systematic studies of the bridge function in soft matter
systems [22], since these functions show a behaviour which is distinctly different
from the bridge functions in systems with harshly repulsive potentials.
0 1 2 3 4
r /σ
-0.1
-0.05
0
B
(r
)
Figure 4. Bridge function B(r) of a GGCM-n for n = 4 at ε∗ = 2 and %∗ = 2.
The solid line corresponds to the ZSEP bridge function, while the circles are the
data extracted from the MC simulations.
142
Thermodynamics of the generalized Gaussian core model
3.2. Thermodynamic properties
In the following we present results for the dimensionless equation of state, βP/%,
a quantity from which one can derive via suitable relations all other thermodynamic
properties of interest. We compare the results obtained via different thermodynamic
routes from the liquid state theories outlined in the preceding section.
0 1 2
1
1.2
1.4
β
P
/ ρ
compressibility route
0 1 2
ρ *
viriral route
0 1 2
energy route
Figure 5. Dimensionless equation of state, βP/%, for a GGCM-10 system as
function of %∗ at ε∗ = 0.1. The results provided by HNC (solid line), PY (dotted
line) and ZSEP (dot-dashed line, only for the compressibility and virial routes)
fall on top of each other and follow accurately the MC simulation data (circles).
The MSA data (dashed line) show small discrepancies between the compressibility
and energy routes.
For ε∗ = 0.1 the situation is rather clear: independent of the thermodynamic
routes, all liquid state theories provide data that are consistent within numerical
accuracy and that agree well with simulation data. However, upon increasing the
n-values the various liquid state theories tend to slightly deviate from the computer
simulations; especially the results obtained via the compressibility and energy routes
exhibit inconsistencies on the order of a few percent. We display three equations of
state for n = 10 in figure 5.
However, as ε∗ increases, differences between the theories start to emerge and
the inconsistencies between the thermodynamic routes become more pronounced.
From our data for ε∗ = 2 we can conclude that the virial route generally provides
the best results: not only the data from different liquid state theories coincide, but
the agreement with computer simulation is very satisfactory as well. Again, as n
increases, the differences between the theoretical approaches become apparent and
are on the order of a few percent. The energy route performs slightly worse, in
particular for large n-values, differences between the various liquid state theories
emerge. In figures 6 and 7 we show the results for the energy equation for n = 4
and for the virial equation for n = 10. In contrast, the data for βP/% calculated via
the compressibility route show a strong dependence on the softness of the potential
143
B.M.Mladek et al.
(i.e. on n) and on the liquid state concept (see figure 8): we note in particular
differences between PY and MSA results and computer simulation data, while the
other concepts show good (HNC) or excellent agreement (ZSEP and SCOZA).
0 2 4 6
ρ ∗
5
10
15
20
β
P/
ρ
Figure 6. Dimensionless equation of state, βP/%, as a function of %∗ at ε∗ = 2.0,
for a GGCM-4 calculated via the energy route. The labels of the curves are the
same as those used in figure 5. Once again HNC, and PY data lie on top of each
other and ZSEP is not depicted.
0 0.2 0.4 0.6 0.8
ρ ∗
1
2
3
4
β
P/
ρ
Figure 7. Dimensionless equation of state, βP/%, as a function of %∗ at ε∗ = 2.0,
for a GGCM-10 calculated via the virial route. The labels of the curves are the
same as those ones used in figure 5. Here HNC and ZSEP lie on top of each other.
Returning to figure 6, where the equation of state, calculated for n = 4 in com-
puter simulations, shows a non-monotonous dependency on the number density for
%∗ ∼ 3.5, we also observe that none of the liquid state theories we have used has
144
Thermodynamics of the generalized Gaussian core model
0 1 2
5
10
15
20
β
P
/ ρ
GGCM-2
0 1 2
ρ *
5
10
15
20
GGCM-4
0 0.5 1
1
2
3
4
5
GGCM-10
Figure 8. Dimensionless equation of state, βP/% as function of %∗, for n = 2, 4
and 10 at ε∗ = 2 obtained via the compressibility route. We show for the three
systems the results provided by HNC, PY, MSA and ZSEP closures, as well as
MC simulation data. SCOZA data are additionally shown for n = 2 and n = 4,
falling on top of HNC and ZSEP in both cases. The labels correspond to the ones
used in the previous figures of this section.
been capable of reproducing – not even in a qualitative way – this particular be-
haviour. The shape of the curve indicates the onset of a phase transition; since the
GGCM-4 belongs to the Q± class [14], the observed effect is obviously a precursor
of a clustering transition. An indication for the onset of such a transition can also
be seen in figure 2, where the main peak of g(r) has shifted to r = 0, suggesting
that clusters with the density higher than the bulk value start to be formed. Prelim-
inary computer simulations have shown that for higher densities clusters are indeed
formed, which appear to arrange themselves on a regular lattice [13]. Further details
on this phenomenon will be reported separately [23].
4. Conclusions
In this contribution we have investigated the structure and thermodynamic prop-
erties of the GGCM-n; the index n permits a interpolation between the GCM (n = 2)
and the PSM (n → ∞), and thus controls the steepness of the repulsion. For the
present study we have chosen the values 2, 4, and 10 for n. The liquid state methods
we have used are conventional concepts (i.e. the PY, and the HNC approximations,
and the MSA) or self-consistent theories (i.e. the ZSEP approach and SCOZA),
while complementary Monte Carlo simulations provide reference data. Both from
the structural and thermodynamic results we can conclude that the requirement
of self-consistency is a very important issue also in soft systems, which guarantees
an improved agreement with simulation data. Closer investigations indicate in ad-
dition that it is in particular the requirement of structural consistency (which is
145
B.M.Mladek et al.
expressed via the zero-separation theorem) that provides excellent results; SCOZA,
on the other hand, which is based purely on thermodynamic consistency, yields for
certain system parameters negative, i.e. unphysical, pair distribution functions. As
we vary the index n and the energy parameter ε of the potential out of the thermo-
dynamic routes chosen in our study, the virial route shows the best agreement with
simulation data; the energy route performs slightly worse and the compressibility
route often fails. In our computer simulation study of the dimensionless equation of
state, βP/%, we find for the GGCM-4 a non-monotonic behaviour as the density is
increased: such a variation usually indicates the onset of a phase transition. In our
case, we expect a clustering transition: upon compression, the particles start to form
clusters, which themselves are located on a regular lattice. Further investigations of
these preliminary results will be presented elsewhere.
Acknowledgements
The authors are indebted to Dieter Gottwald for numerical help and for criti-
cal comments on the manuscript. This work was supported by the Österreichische
Forschungsfonds under Project Nos. P15758, and P17823: Financial support by the
Hochschuljubiläumsstiftung der Stadt Wien under Proj. No. H–1080/2002 is grate-
fully acknowledged. BMM gratefully acknowledges financial support of the Erwin
Schrödinger International Institute for Mathematical Physics, Vienna, where part
of this work was carried out. MJF thanks the Instituto de Qúımica-F́ısica Rocasolano
(CSIC, Spain) for providing computing resources and the Ministerio de Educación
y Ciencia de España for financial support.
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B.M.Mladek et al.
До термодинамічних властивостей узагальненої
моделі гаусового кора
Б.M.Mладек 1,2 , M.Д.Ферно 1 , Г.Kaль 1 , M.Нойман 2
1 Центр комп’ютерного моделювання матеріалів
та Інститут теоретичної фізики
Віденського технічного університету ,
Відень, Австрія
2 Інститут експериментальної фізики
Віденського університету,
Відень, Австрія
Отримано 24 листопада 2004 р.
Ми представляємо результати системного дослідження властивос-
тей узагальненої моделі гаусового кора з індексом n. Потенці-
ал такої системи з допомогою індексу n, що визначає крутиз-
ну притягання, дозволяє інтерполювати взаємодії від моделі гау-
сового кора до системи проникних сфер. Для розрахунку струк-
турних і термодинамічних властивостей нами використовували-
ся як традиційні, так і самоузгоджені версії теорії рідкого стану,
а система відліку бралася з комп’ютерного експерименту. Ре-
зультати показують, що для отримання доброго узгодження з
даними моделювання концепція самоузгодження стає обов’яз-
ковою; зокрема структурне узгодження, яке у нашому підході
проводиться через теорему нульового розділення, є надзвичайно
важливою вимогою. Результати моделювання для обезрозміреного
рівняння стану, βP/%, вказують на те, що для моделі з індексом 4
може виникати структурно впорядкована фаза (перехід кластеру-
вання) при стисканні системи.
Ключові слова: м’яка речовина, інтегральні рівняння, комп’ютерне
моделювання, перехід кластерування, модель гаусового кора
PACS: 61.20.Gy, 61.20.Ja, 64.10.+h
148
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