6876
J. Phys. Chem. 1994,98, 68766884
Thermal Expansion and Stability Limits of Generalized van der Waals Fluids Anantbu S. Cbakravarthi,t Pablo G. Debenedetti,’lt Srikantb Sastry,ht and Sang-Do Yeof** Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544-5263, and Center for Polymer Studies, Department of Physics, Boston University, Boston, Massachusetts 02215 Received: January 25, 1994; In Final Form: April 27, 19948
Generalized van der Waals equations of state can exhibit density anomalies, as characterized by a negative thermal expansion coefficient in some region of the phase diagram, when either the hard core volume, b, or the mean field attractive parameter, a, are made temperature-dependent. If 6 decreases as the temperature increases, negative thermal expansion occurs a t all temperatures. At a given temperature, the liquid becomes anomalous when the pressure is increased; hence the P-T projection of the locus of density maxima has positive slope. The locus of density maxima extends to zero temperature when b is finite a t T = 0, and it terminates a t a minimum temperature if b diverges at T = 0. If, on the other hand, u increases with temperature while the core volume remains constant, generalized van der Waals fluids exhibit density anomalies that disappear at high pressure. Hence, in this case the locus of density maxima has a negative slope in the P-Tplane. The liquid spinodal is re-entrant both when a increases as the temperature increases and when b increases as the temperature decreases, becoming infinite at T = 0.
Introduction The purpose of this paper is to investigate the thermodynamics of generalized van der Waals fluids with temperature-dependent attractive and repulsive parameters. This class of model fluids can expand when cooled at constant pressure, and we focus here on the relationship between expansion upon cooling, equilibrium phase behavior, and the limits of stability of the liquid phase. The resulting behavior includes some features (such as a retracing liquid spinodal) that are of interest in the study of more complex systems, for example water. Such behavior had not been hitherto obtained with simple and general equations of state. The transparent physical meaning of the van der Waals attractive and repulsive parameters, combined with the use of thermodynamic consistency arguments, allows us to draw conclusions on fluids that contract when heated that are not always limited to simple, spherically-symmetricsystems, to which the van der Waals model in principle applies. If a liquid contracts when heated at constant pressure, its thermal expansion coefficient ap= -(a In p/aT)pis negative, and the liquid is said to exhibit density anomalies. Stable and supercooled water’ and heavy water,* supercooled silica,” and most dilute water solutions show this type of behavior over some region of their phase diagram. If the density anomaly disappears at high temperature, the density will first increase, then reach a maximum and eventually decrease upon isobaricheating (density maxima). If the density anomaly disappears at low temperature, the density will first decrease, then reach a minimum, and eventually increase upon isobaric heating (density minima). Water, heavy water, aqueous solutions, and silica exhibit density maxima. In this paper we concentrate mostly on density maxima. There are two known mechanisms that can cause density anomalies: directional bonding and core-softening.4 In water, heavy water, and silica, directional bonding stabilizes open, lowdensity networks. Water forms a network in which each oxygen atom is hydrogen-bonded to four other oxygen atoms through an 0-H-0 bridge.5 The oxygen atoms occupy the vertices of a To whom correspondence should be. addressed. Princeton University. t Boston University. Current address: Physical Sciences Laboratory, Division of Computer Research andTechnology,National Institutesof Health, Beth&. MD 20892. 1 Current address: Department of Chemical Engineering, Kyungpk National University, Taegu, 702-701, Korea. @Abstractpublished in Advance ACS Abstracts, June 15, 1994. t
tetrahedron in the resulting three-dimensional arrangement. At low enough temperatures, this arrangement is regular and is called hexagonal ice (Ih). Upon melting, the spatial regularity of the network is destroyed, but local tetrahedrality persists. In silica, each Si atom is bonded to four other Si atoms through a Si-oSi bridge to form a three-dimensional, tetrahedrally-coordinated network.6 Thermal disruption gradually causes open networks to collapse into denser and more disordered arrangements; macroscopically,this translates into a negative thermal expansion coefficient. Core-softening, the second mechanism capable of causing negative thermal expansion, is mainly7 of theoretical interest.&l5 As shown by Debenedetti et aL4 for systems interacting via spherically-symmetric and pairwise additive potentials, if the repulsive force between two atoms decreases as they approach each other, density anomalies can occur. In order for the repulsive force not to increase monotonically at low distances, there must be an inflection point in the repulsive core of the pair potential. Density anomalies are closely related to the stability of supercooled liquids. Ever since Speedy and Angel116 postulated that liquid water can become thermodynamically unstable when supercooled, the general question of the stability of supercooled liquids has attracted considerable attention. One interpretation of the properties of supercooled water, due to Speedy,’7J8 postulates the existence of a continuous spinodal curve bounding the superheated, stretched (negative pressure), and supercooled states of water. Power-law extrapolations of the isobaric specific heat and the isothermal compressibility of supercooled water are not inconsistent with Speedy’s conjecture. In this picture, the P- T projection of water’s density maxima intersects the spinodal at two points: the liquid’s point of maximum tensile strength and the low-temperature, high-pressure termination of the liquid spinodal. The former intersection causes the sign of the slope of the P-T projection of the spinodal to change (retracing or reentrant spinodal). A lattice model with water-like bonding characteristics has recently been shown to yield a re-entrant liquid spinodal when solved in the mean-field approximation.19 Recently, Poole et al.” have proposed a different picture. In an extensive study of ST2 water**by molecular dynamics, these authors found no evidence of a retracing spinodal. Instead, they proposed that a novel, metastable critical point is responsible for water’s anomalous behavior. This critical point is associated with the first-order transition between two amorphous forms of ice. Poole
0022-3654/94/2098-6816~04.50~0 0 1994 American Chemical Society
Generalized van der Waals Fluids et a1.20observed that the P-T projection of the density maxima locus, which is negatively sloped at positive pressures, becomes infinite1y-sloped, then positively-sloped with decreasingpressure, and does not intersect the spinodal. The absence of a retracing spinodalis a major difference between Speedy’s and Poole et a1.b interpretation of the properties of supercooled water. An important question that we address in this work is whether Speedy’s continuous spinodal or Poole et al.’s infinitely-sloped locus of density maxima can be obtained from simple equations of state and, if so, over what parameter ranges. D’Antonio and DebenedettiZZand Debenedettiand D’Antonio23 investigated the question of the stability of supercooled liquids using thermodynamic consistency arguments. They showed that Speedy’s picture can apply to any liquid that expands when cooled isobarically. They did not consider the possibility that the locus of density maxima could become infinitely sloped, and thus their analysis did not address the behavior observed by Poole et al.20 The thermodynamic constraints on the ways in which loci of density extrema and of stability limits can intersect provide a powerful analytical tool for testing and discriminating theories of metastable water. In spite of the interest and importance of density anomalies, there is no simple equationof state that can model negative thermal expansion. Existing equations of state that can show density anomalies are specific to water2’26 and are obtained by fitting the relevant Helmholtz energy derivative to volumetric, thermal, speed of sound, heat capacity, and throttlingdata. Suchequations aredue to Haar, Gallagher, and KelP (HGK), Saul and WagnerZS (SW), and Hi1126(H). The number of fitting coefficients is very large (48 in the HGK equation,24 58 in the SW equationZs),and the resulting equations, though highly accurate and useful for the representation of data, lack generality. Some aspects of the Helmholtz energy functions in the HGK, SW, or H equations have theoretical significance(for example,the use of a rearranged Ursell-Mayer virial expansion form for the reference pressure function in the HGKequation). In general,though, the functional form of the density and temperature-dependentterms is empirical, and the fitted parameters in these multiterm equations have no physical significance. In contrast, our purpose is to investigate density anomaliesusing simple, substance-independentequations of state. For our analysis we select three members of the family of generalized van der Waals fluids: the van der Waals, Guggenheim?’ and CarnahanStarling-van der Waal~2&3~ fluids. Their equations of state have two parameters with a clear physical meaning: the hard-core volume and the mean-field attraction. We introduce density anomalies by making the hard-core and mean-field attraction temperature-dependent.32 In what follows, we begin by definingthe family of generalized van der Waals fluids, and we derive expressions for the isothermal density dependence of the entropy in this class of fluids. From these expressions we derive necessary conditionsfor the existence of density anomalies and for the locus of density anomalies to have either positive or negative slope when projected onto the P-T plane. These necessary conditions involve explicitly the temperature dependence of the van der Waals attractive and repulsive parameters. The bulk of the paper is devoted to an analyticaland numericalanalysis of the phase boundaries, stability limits, and density anomalies in van der Waals fluids with temperature-dependent parameters. An important result of this work is to show that water-type density maxima, in which the fluid becomes normal upon compression and hence has a negatively-sloped P-T projection of the locus of density anomalies, can be obtained by making the mean-field attraction an increasingfunctionof temperature, while the excluded volume remains independent of temperature. Density maxima in which the fluid becomes normal upon lowering the pressure and hence has a positively-sloped locus of density maxima in the P-T projection can be obtained by making the
The Journal of Physical Chemistry, Vol. 98, No. 27, 1994 6877 excluded volume a decreasing function of temperature. For the Guggenheim and CarnahanStarling-van der Waals fluids, this can also give rise to a low-temperature region of density minima. Several interesting featurescan result from making thegeneralized van der Waals parameters temperature-dependent, includingthe asymptotic vanishing of the difference between coexistingvapor and liquid densities at low temperature. Some of these features may suggest the development of microscopic liquid models with interesting new properties.
Generalized van der Waals Fluids The family of fluids of interest here is described by the generalized van der Waals partition function, Q(N,V,q33
where 8 = 1/kT, k is Boltzmann’s constant, Vr is the free volume, N is the number of molecules, A is the Helmholtz energy, p is thenumberdensity,p = N/V, Visthevolume,A = (h2/2?rmkT)1/2 is the deBroglie wavelength, m is the mass of a molecule, h is Planck’s constant, and a is the attractive constant in the equation of state (Le., @ / N = -pa is the mean potential energy per molecule). Equation 1 is based on two approximations: each molecule experiences a uniform background potential energy that does not depend on the details of microscopic configurations;the structure of the fluid is determined by hard-core repulsions that can be described by the free volume available to the center of mass of a molecule as it moves in the fluid. The equation of state follows from eq 1,
where uf and u are the free volume and volume per molecule, respectively. The van der Waals, Guggenheim, and CarnahanStarling-van der Waals equations follow from eq 2 upon using the free volume expressions
of = 0 - b e p = -_UkT - b u2 a (van der Waals)
kT P = -( 1 u
(3)
- a (Guggenheim) (4) V2
(CarnahanStarling-van der Waals) ( 5 ) where r ) ( = b/4u) is the volume fraction occupied by N spheres of diameter (3b/2~)1/3in a volume V, and bo, a characteristic volume, can be any positive number. The entropy also follows from eq 1:
S N,V
=-+pa‘+ 2 k
a In uf ) + l n 2 ( 6 ) (alnT p
~3
where a’= da/dT. Differentiating with respect to density gives
6878 The Journal of Physical Chemistry, Vol. 98, No. 27, 1994
Chakravarthi et al.
Invoking the thermodynamic identity [(as/aN)T = (BP/dT),; s = S / w results in 11
T
where upand KT are the thermal expansion coefficient and the isothermal compressibility;we see that the sign of the isothermal density derivativeof the entropy is oppositeto that of the thermal expansion coefficient for a stable or metastable fluid. Thus, when a stable or metastable fluid exhibits density anomalies (up< 0), it becomes more disordered when compressed isothermally. The first term on the right-hand side of eq 7 is always negative. Hence, density anomalies can only occur in van der Waals fluids if the second and/or third terms on the right-hand side of eq 7 are positive. Substituting the respective free volume expressions gives the isothermal density derivatives of the entropy as
0.260 88T,b’,(qZ - 2q - 2)
-
equation of state with a temperature-independent repulsive core (b, = 1 in our notation) and a van der Waals-type mean-field attraction that is a monotonically increasing function of temperature (a’, > 0 in our notation). 3. Ceneraliwtioasand Exact Results. The above-discussed conditionsfor a’and b‘are necessary but not sufficient: examples of density minima with b’< 0 will be given below; also, as discussed above for the simple van der Waals form, a negatively-sloped density maxima locus requires a’, > 32/27, not just al, > 0. The type of density anomaly of interest here is density maxima. This corresponds to pressure minima along isochores in the P-T plane. Therefore, a fluid that becomes anomalous upon compression has a positively-sloped locus of density maxima on the P-Tplane, and a fluid that, like water, becomes anomalous upon decompression has a negatively-sloped locus of density maxima in the P-T plane. This behavior is summarized in Figure 1. We refer to water-like behavior with a negatively-sloped density maxima locus as low-pressure anomaliesand to a positively-sloped density maxima locus as high-pressure anomalies. We now derive two exact results for the van der Waals form, with a‘ = 0. In this case,
+
(*)a: (11) (1 -VI4 kTC where eqs 9-1 1 apply to the van der Waals, Guggenheim, and CarnahanStarling-van der Waals fluids, respectively: pr= PIPc; 0, p c / p ; br = b(Tr)/bc; Or a(Tr)/ac; Tr = T/Tc; b‘, = dbr/dTr; a’, = da,/dTr; and where subscript c denotes the critical point. 1. a’, = O$’, < 0. Consider first the case a’, = 0. From eqs 9-1 1, we must have b’, < 0 for density anomalies. However, as the density is increased isothermally, the second term on the right-hand side diverges faster than the first one, so generalized van der Waals fluids with temperature-independent a and b’, < 0 become anomalous (ap< 0) upon compression. In contrast, water, heavy water, and silica become normal (ap> 0) at high enough pressures. This behavior of generalized van der Waals fluids is unchanged when a’, # 0 (as long as a’, is finite), since the divergenceinthesecond termon the right-hand sidedominates the behavior of the entropy derivative at high density. 2. a’r > O;b‘, = 0. If, however, b‘, = 0 and a‘, > 0, the entropy derivative is large and negative (up> 0) at low density (v, m) and at very high density (u, b,/3 or q l), but there exists a temperature-dependent range of densities where the fluid is anomalous (ap< 0). For the simple van der Waals fluid with b: = 0 and a’, > 0, density anomalies exist, provided a’, > 32/27, over the density range
-
-
From this, using (aP/aT), = 0, we obtain the expression for v along the locus of density extrema, u, =
where
$:- $)
Interestingly, when the water-like lattice model of Sastry et al.19is solved in the mean-field approximation after orientationdependent interactions have been averaged, there results an
6, - Trb:
3
Differentiating P, once again and substituting into eq 15, we obtain, for the curvature of isochores at the locus of density extrema,
Thus, for density maxima, we need b”, > 0. Using eq 15, we obtain the equation of the locus of density extrema, in P-T projection,
8 27 p, = --b’, (b, - Trb’,)’ From this, we obtain the slope of the density extrema locus as (%)pa=
6’ =
T
Figure 1. Schematicrepresentationof water-like,low-pressureanomalies (left), and of high-pressure anomalies (right) in fluids with a negative thermal expansion coefficient.
[&-(br-Trb’J3 54Tr
]b’:
(18)
We have, in general, the condition for stability,
Substituting the expression for v, at the locus of density extrema (eq 15) gives the stability condition along this locus
Generalized van der Waals Fluids
The Journal of Physical Chemistry, Vol. 98, No.27, 1994 6079
Using eq 20 in eq 18, we see that in the stable region (dP/dT), > 0 (since b” > 0 for density maxima). Thus, low-pressure anomalies [(dP/dT),, < 01 are unstable. In summary, for the van der Waals equation with b’, < 0 and a, = 1, (i) for density maxima, one must have b’: > 0, and (ii) low-pressure anomalies, if they exist, are always unstable. If a’, # 0, or if the Guggenheim or Carnahan-Starling-van der Waals forms are used, a similar analysis leads to expressions that must be solved numerically, and such general conclusions as reached above cannot be derived. However, the conclusion that b’, < 0 leads to high-pressure anomalies is, as was already shown, valid for the three types of van der Waals fluids.
\
-1.2
/
0
I
I
1
0.2
0.4
0.6
1
0.8
T,
Figure 2. Pressure-temperature projection of the binodal (-), spinodals (- -), and locus of density maxima (- * -) for the infinitely softened van der Waals fluid. a = l / 2 ; a ’ = 0. 1.2
High-Pressure Anomalies To illustrate high-pressure anomalies, we consider two types of b,( T I )functionality,
b,=
/
‘--s’
,
I
I
/,
I
\,
I
i
I
1 I
\
-0.6
(21)
-1.2
I I
‘, .
\
/
.
I
, 1
I
i
I’ /’
-/‘
-1 0
I
1
0.4
0.8
I
I
1.2
1.6
2
p,
Note that br(1) = 1 in both cases. The physical basis for eq 21 follows from defining a temperature-dependentdistance of closest approach,34 d( T )
mT)1= 2kT
Figure 3. Pressure-density projection of the binodal (-), spinodals (- -), and locus of density maxima (- -) for the infinitely softened van der Waals fluid. a = l/2; a’ = 0.
(23)
where 4 is the pair potential. Since at closest approach the force between two atoms is repulsive, T
k T = 2t[u/d(T)]”
(24)
where t is a characteristic energy (for example, the well depth of the Lennard-Jones fluid), and u, a characteristic size (for example, the separation at which the pair potential vanishes). In eq 24 we have neglected the attractive term, since at closest approach it is negligiblewith respect to the repulsivecontribution. Thus, with Ts = k T / t and b = d3, we obtain35
b(T) 0: (T*)-3/”
(25)
Equation 25, with n = 12, is the high-temperature limit of the commonly used e x p r e s s i ~ n ~ ~ . ~ ~
b
0:
[l
+
It is always possible to define,via eq 23, a temperature-dependent distance of closest approach. The type of behavior described below arises when we use use this distance as the definition of hard-core radius in the generalized van der Waals equations. as T, We refer to eq 21 as infinite core-softening (b, 0) and to eq 22 as finite core-softening. The significance of the parameter y in eq 22 follows from writing
- -
Thus, the larger thevalueof y,thesmaller the overall temperature sensitivity of the core. The infinitely core-softened van der Waals, Guggenheim, and Carnahan-Starling-van der Waals fluids display qualitatively identical behavior. Figure 2 shows the binodal, spinodals, and locus of density maxima for the infinitely softened van der Waals fluid with a, = 1 (eqs 3 and 21). The superheated liquid spinodal
0.4r
o
.
6
.
;
’
/
0.2
/
/
0
u
0
0.4
.-- , 0.8
1.2
1.6
2
p,
Figure 4. Temperature-density projection of the binodal (-), spinodals (- -), and locus of density maxima (- -) for the infinitely softened van der Waals fluid. a = l/2; a’= 0.
-
is re-entrant. The fluid’s maximum tensile strength corresponds to the point of tangency between the spinodal and the positivelysloped locus of density maxima. The corresponding behavior is shown in Figures 3 and 4 in pressure-density and temperature density projections, respectively. Note the density extremum along the liquid branch of the binodal and the vanishing of the density difference between the saturated liquid and vapor at low tempertures. This follows from the fact that the core diverges at T = 0 and permissible densities tend to vanish as T 0. Since the density along the liquid spinodal has an extremum (Figures 3 and 4), isochores behave in one of three ways. Let pax and pspmrxdenote, respectively, the density at the fluid’s tensilestrength maximum (minimum pressure along the spinodal in Figure 3) and the maximum density along the spinodal. For p > pspmx,isochores do not touch a spinodal. Instead, upon isochoric cooling, the pressure decreases until the locus of density maxima is encountered, whereupon apbecomes negative and the pressure increases upon cooling thereafter. Isochorea p, = 2 (Figure 5a; Guggenheim fluid; eqs 4 and 21) and pr = 1.7 (Figure 5b; van der Waals fluid; eqs 3 and 21) show this typeof behavior. Thus, it is possible to cool isochoricallyan infinitelycore-softened generalizedvan der Waals fluid without ever reaching a spinodal. For densities such that p,> p > p-, there are two limits of stability (Th, ph, p ; T I ,PI,p ) . The fluid is metastable for T > Tb and T < T I and unstable for T I < T < Tb. Isochore pr = 1.1 (Figure 5b, where pI,IpIuu= 1.266) shows this type of behavior.
-
Chakravarthi et al.
6880 The Journal of Physical Chemistry, Vol. 98, No.27, 1994 2 1 ,
I
I\
0.2
0.4
I /
I
I
0.8
1
1.2
1
-1
-
I
-7
0
0.6
0
/
0.2
I
I
I
0.6
0.8
1
I
0.4
Tr 1
Figure 6. Effect of a on the liquid spinodal of the infinitely softened Guggenheim fluid; u’= 0. The dotted line is the locus of tensile strength
.s 1
0.5
0.3
I
I
I
I
0.2 -
-0.5 -1
-1.5
I
0
I
1
I
I
I
0.2
0.4
0.6
0.8
1
1.2
? a
1
Figure 7. Effect of a on the temperature at the tensile strength maximum for the CarnahanStarling-van der Waals fluid. u’ = 0.
-0.4
P
3 -0.5
\.
\
\
-0.6
0
I
0.02
\
I
’. \.
\
0.04
0.06
0.08
0.1
Tr
Figure 5. (a, Top) Pressuretemperature projection of the binodal (-), spinodals (- -), isochores, and locus of density maxima (- -) for the infinitelysoftenedGugsenheimfluid.a = l/*;u’=0. Thep, = 2 isochore never becomes unstable. (b, Middle) Pressuretemperature projection of the binodal (-), spinodals (- -), isochores,and locusof density maxima -) for the infinitely softened van der Waals fluid. a = l/z; u t = 0. The p, = 1.7 isochorenever becomes unstable,and the p, = 1.266 isochore has only one point of tangency with the liquid spinodal. (c, Bottom) Low-temperature portionof thep, = 0.5 isochoreof theinfinitelysoftened van der Waals fluid. a = */2; u’= 0. The minimum pressure along the unstable isochore occurs along the unstable continuationof the locus of density maxima (- -); the isochore becomes metastable after touching the spinodal (- -). (-a
Note the double tangency between this isochoreand the spinodal, which follows from the fact that, in a P-Tprojection, the spinodal is an envelope of i s o c h ~ r e s .The ~ ~ minimum pressure along any isochoreoccurs along the locus of density maxima. For densities such that p < pthere are also two limits of stability, but the minimum pressure occurs along the unstable low-temperature branch of the locus of density maxima (Figure 5c; eqs 3 and 21). Note that the limit of lowest attainable pressures for any given density (Figure 3) is the locus of density maxima (if p > p-) and the liquid spinodal (if p < pmX). Figure 6 shows the effect of a on the liquid spinodal of the Guggenheim fluid with a’ = 0 (eqs 4 and 21). The spinodal retraces for any a > 0. The maximum tensile strength decreases with increasing a;however, the temperature corresponding to the tensile strength maximum shows a maximum with respect to a. This is highlighted by the dotted line in Figure 6, which is the
-10
I
0
/
I
I
I
1
I
0.2
0.4
0.6
0.8
1
T,
Hgure 8. Effect of a on the stable portion of the locus of density maxima for the infinitely softened van der Waals fluid. u‘ = 0.
locus of tensile strength maxima, and is shown explicitly in Figure 7 (CamahanStarling-van der Waals fluid;eqs 5 and 21). Figure 8 shows the effect of a on the stable part of the locus of density maxima for the van der Waals fluid (eqs 3 and 21). The locus is always positively-sloped; the low-temperature, low-pressure endis thepointwherethelocusofdensitymaximabecomes tangent to the liquid spinodal and is thereforGQ3 the fluid‘s tensile strength maximum and the point where the liquid spinodal retraces. The behavior shown in Figures 2-8 is unchanged in its essentials if 4,= b,. The exception is the temperature at the tensile strength maximum, which increases monotonicallywith a (Figure 9; eqs 4 and 21) as opposed to showing an extremum (compare with Figure 7). The behavior of the finitely-softened Guggenheim and Carnahanatarling-van der Waals fluids is qualitatively similar, but some aspects are quite different from those of the simple van der Waals fluid. The liquid spinodal is monotonic in P-Tprojection for the three fluids: there is no spinodal retracing. However, the van der Waals fluid exhibits positively-sloped density maxima, whereas the Guggenheimand CarnahanStarling-van der Waals fluids show both negatively-sloped density minima and positivelysloped density maxima. Figure 10 shows the effect of finite
The Journal of Physical Chemistry, Vol. 98, No. 27, 1994 6881
Generalized van der Waals Fluids
1 1 2
0.8 0.7
0.6 0.5 0.4
8
0.3
[
0.2
5
0
0.1
-so I
a
Figure9. Effect of a on the temperature at the tensile strength maximum for the infinitely softened Guggenheim fluid. or = b,.
0
I
I
I
I
0.5
1
1.5
2
T,
Figure 13. Pressuretemperature projection of the binodal (-), liquid spinodal(--),isochores,and lociofdensityextremaofthefinitelysoftened CarnahanStarling-van der Waals fluid. y = l/2; a' = 0. 1.2
1
1
I
1
-s
0.8
t,' II
0.6
I
0.4 -10
0
0.4
0.2
0.6
1
0.8
1.2
1.4
0.2
T, Figure 10. Pressuretemperature projectionof the binodal (-), spinodals (- -), locus of density maxima (- -), and isochores of the finitely softened van der Waals fluid. y = l/2; a' = 0.
0
-
pr
Figure 14. Temperaturdensity projection of the binodal (-), spinodals (- -), and loci of density extrema of the finitely softened CarnahanStarling-van der Waals fluid. y = '/2; (I'= 0.
7
T
'0.4 o
,
6
y
,
\
f
0
),/I
\\ \,,;
1
0.2
40
20
p,
\
0
0.5
2
1.5
I
30
1
I
t
I Pmin
,'
, .(
pm.,
PC
Figure 11. Tcmpcraturdensity projection of the binodal (-), spinodals (- -), and locus of density maxima (- -) of the finitely softened van der Waals fluid. y = 1/2; a' = 0.
-
I
-100 -120 -80
I
I
I
I
I
4 0
0.2
0.4
0.6
0.8
1
T,
Figure 12. Effect of y on the liquid spinodal of the finitely softened CarnahanStarling-van der Waals fluid. a' = 0.
softening on the van der Waals fluid (eqs 3 and 22). The locus of density maxima is now stable and positively-sloped down to T = 0, and there is no spinodal retracing. At high enough densities (e,g., pr = 2 in Figure 10) the isochores do not touch the spinodal, and the van der Waals fluid can be cooled isochorically without losing stability. This is also shown in Figure 11 (eqs 3 and 22). The effect of y on the liquid spinodal is shown in Figure 12 for the CarnahanStarling-van der Waals fluid (eqs 5 and 22;
\
-10
I
I \
I
1
2
3
4
PI Figure 15. Pressurdensity projection of the binodal (-), spinodals (- -), isotherms, and locus of density extrema of the finitely softened Guggenheim fluid. y = l/2; a' = 0.
qualitatively similar behavior applies to the van der Waals and Guggenheim fluids). In contrast to the simple van der Waals form, both the Guggenheimand CarnahanStarling-van der Waals fluids, when finitely-softened,exhibit both density minima and density maxima. Figure 13 is the P-T projection of the CarnahanStarling-van der Waals fluid, with y = l / 2 and a' = 0 (eqs 5 and 22). The high-temperature, positively-sloped locus of density maxima merges with the low-temperature, negatively-sloped locus of density minima; the merging point is the point of minimum pressure and density along the loci. Above a minimum density, isochores exhibit P-T minima (density maxima) and maxima (density minima), The corresponding T-p behavior is shown in Figure 14. The merging of the loci of density anomalies projects as a cusp in the P-p plane; this is shown in Figure 15 for the Guggenheim fluid (eqs 4 and 22). Note the crossingof isotherms (density anomalies), as well as the tangency between isotherms and loci of density extrema (Tr = 0.5 tangent to pmin;Tr = 1 and 1.5 tangent to pmx). The latter feature follows from the fact that
Chakravarthi et al.
6882 The Journal of Physical Chemistry, Vol. 98, No. 27, 1994
the locus of density extrema is an envelope of isotherms in the P-p plane. To see this, we write dP =
(g)p d T + (E) dp aP
T
and therefore, along a locus of density extrema,
As y increases (core sensitivity to T decreases; see eq 27), both the pressure and the density at which the loci of density extrema merge increase. This means that the range of pressures and densitiesover which the fluidcan exhibit density anomaliesshrinks or, equivalently, that more extreme conditions of pressure and density must be imposed to make the fluid anomalous. As noted by Trebble and Bishnoi,37the temperature dependence of b can give rise to unphysical predictions for the heat capacity. To check this we write the identity
where c: is the ideal-gas heat capacity at constant pressure. Substituting the van der Waals, Guggenheim, and CarnahanStarling-van der Waals equations results in expressions for the respective heat capacity deviations that diverge at close packing to --OD for the van der Waals fluid and to for the Guggenheim and CarnahanStarling-van der Waals fluids (of course, this divergence at close packing is different from the expected divergence of cpalong the spinodal). For the Guggenheim and CarnahanStarling-van der Waals fluids, the density at close packing is given by
-
where qc = bc/(4vc),with qc = 0.126 59 (Guggenheim) and qc = 0.130 44 (Carnahan-Starling-van der Waals). Equation 31 is equivalent to writing q 1. For the van der Waals fluid, the close packing condition is pb = 1 or, in reduced units,
-
3 Pr = -
‘
-2.5
0.2
0. 6
0.4
01
I
0
\
i 1.0
1.5
+ w tanh({T,) + w tanh({)
2.0
2.5
3.0
Figure 17. Temperature-density projection of the binodal (-), spinodals (- -), and locus of density maxima (- -) of the van der Waals fluid with 6’= 0, w = 40, = 2.
-
-1.5
1 0
1
1
‘.I*
0.5
1.5
1.0
’
.
2.0
2.5
3.0
p,
Low-Pressure Anomalies Here, we consider the van der Waals fluid with b, = 1, a, # 0, and a’, > 0. Except where otherwise noted, the essential features of the behavior to be discussed below do not depend on the specific a(“) functionality. For our examples we use the following function for ar,3*
1
I
I
0.5
P,
-2. 5
Thus, the heat capacity of the van der Waals fluid with temperature-dependent b becomes large and negative at high densities (unphysical thermal behavior since stability is not violated). As an example, the divergence for the infinitelysoftened van der Waals fluid at T, = 0.5 ( a = O S ) occurs at p r = 2.12; for finite softening at T, = 0.5 (y = OS), the divergence occurs at p r = 2.353. The density at which this negative divergence occurs increases with temperature, since b, decreases with temperature.
1
I
1.0
Figure 16. Prwure-temperature projection of the binodal (-), spinodals (- -), and locus of density maxima (- -) of the van der Waals fluid with 6’ = 0, w = 40, J; = 2.
br
a, =
0.8
T,
(33)
where the parameter o is a measure of the overall range of a,
Figure 18. hcasure-density projection of the binodal (-), spinodals (- -), and locus of density maxima (- -) of the van der Waals fluid with 6’ = 0, w = 40, { = 2.
-
values spanned as T, changes from 0 to
-,
or(-) (34) ar(0) and { is proportional to the steepness of the temperature dependence of a, in the low-temperature limit, -=l+w
W
(35) = 1 o tf;nh({) Figure 16 shows the P-Tprojection of the binodal, spinodals, and locus of density maxima for the van der Waals fluid with a, as per eq 33. There is a Speedy-like retracing liquid spinodal and a negatively-sloped locusof density maxima. The region of density anomalies is entirely metastable; in P-T projection it is bounded by the retracing spinodal on the low-temperature side and the locus of density maxima on the high-temperature side. T-p and P-p projections are shown in Figures 17 and 18. The density increases monotonically along both the liquid spinodal and the
+
The Journal of Physical Chemistry, Vol. 98, No. 27, 1994 6883
Generalized van der Waals Fluids 2
1
-1
-2 0
0. 25
0. M
0. IS
I. 00
Tr
Figure 19. Pressuretemperatureprojection of the binodal (-), spinodals
(- -), and locus of density maxima (- e -) of the van der Waals fluid with b' = 0, w = 40, { = 2. There are no density anomalies along the pr = 1.5 isochore; at pr = 2 and 2.2 the fluid becomes unstable with a negative
thermal expansion coefficient.
liquid branch of the binodal as the temperature is lowered. This is in contrast with the high-pressure anomalies case, where the binodal and the spinodal can exhibit a density maximum (see Figures 3,4, and 11). For reduced densities between 1.935 and 2.465, isochoric cooling of the stable fluid leads initially to a pressure decrease ( a p> 0); the minimum pressure (maximum tension) is reached along the locus of density maxima (Figure 18). If the metastable liquid is cooled further at constant density, the tension decreases (pressure increases) until stability is lost at the spinodal. The same behavior is shown in Figure 19, with isochores included for clarity. Since low-pressure anomalies require u: > 32/27 and u; decreases monotonically as T, increases, the maximum reduced temperature at which density anomalies occur is
In Figures 16-19 we use w = 40 and f = 2, which gives T,,,, = 0.382. However, this represents an unstable condition: the maximum temperature of stable density maxima is the tensile strength maximum, which, for the given choices of w and S; occurs at pr = 1.935, Pr = -2.061, and T, = 0.347. The low-temperature intersection of the liquid spinodal and the locus of density maxima is quite interesting. The density maxima locus of a van der Waals fluid with temperaturedependent u and u: > 32/27, regardless of the a( 7') functionality, satisfies
-
This function increases monotonically with u: and approaches P, = 3 for u: -. The locus density maxima approaches a limiting reduced density ( p r = 2.465 for w = 40 and l = 2) where it becomes tangent to the retracing spinodal (in P-T projection), causing the slope of the spinodal to change sign, as required by thermodynamic consistency.23 Thus, for any finite u;(O), there is no stable density maxima locus below its high-density, lowtemperature encounter with the liquid spinodal. Figure 20 shows the low-temperature tangency between the liquid spinodal and the locus of density maxima at T, = 0.0069 and pr = 2.465. Note that the pr = 2.6 isochore becomes tangent to the spinodal with a positive slope, indicating ap> 0 in the metastable region. The same behavior can be inferred from Figure 17, where it takes the form of a high-density, low-temperature intersection between
Figure 20. Pressuretemperature projection showing low-temperature
tangencybetween theliquid spinodal(- -) and the locusofdensitymaxima (- -). The density at tangency is pr = 2.4649. At a density of pr = 2.6 the thermal expansion coefficient is always positive, and the isochore becomes tangent to the spinodal with positive slope. Unstable and metastable portions of the pr = 2.2 isochore are shown: the former is the low-temperaturecontinuationof the isochore beyond its tangency to the spinodal; the latter is the high-temperature extension.
-
the spinodal and the locus of density maxima. In order for the low-temperature encounter between the spinodal and the density maxima locus of a van der Waals fluid to occur only at T = 0, it is necessary to have an u, function with infinite slope at the origin u:(O) = =, The temperature dependence of u does not introduce heat capacity anomalies.
Conclusion Negative thermal expansion in fluids can be described and modeled with simple equations of state. Generalized van der Waals fluids show density anomalies when either the hard-core diameter or the mean-field attraction is made temperaturedependent. High-pressure anomalies result from allowing the hard core to decrease with temperature, and low-pressure anomalies result from allowing the attractive parameter to increase with temperature. Some of the resulting phenomena, such as a Speedy-likeretracing liquid spinodal, have considerable theoretical interest in studying supercooled water and, more generally, in understanding the thermodynamicstability of liquids that expand when The potential usefulness of these results stems not only from the simplicityof the equations of state, but also on their semitheoretical basis, and on the clear physical meaning of the u and b parameters. Thus, our results may provide insight into the formulationof more fundamental, microscopically-based models for fluids with density anomalies. If the core decreases with increasing temperature, the van der Waals, Guggenheim, and CarnahanStarling-van der Waals fluids can contract when heated isobarically. These density anomalies occur at all temperatures. At a given temperature, the fluid becomes anomalouswhen the pressureis increased (highpressure anomalies). Hence the locus of density maxima projects on the P-T plane with positive slope. The liquid spinodal of infinitely softened fluids is retracing. For this class of fluid, the density does not increase monotonically with distance from the critical point (T, - 7') along either the liquid spinodal or the liquid branch of the binodal: instead, there are density maxima along both curves. For the finitely softened van der Waals fluid, the liquid spinodal is nonretracing,the liquid branch of the binodal exhibits a density extremum, and a positively-sloped locus of density maxima exists at all temperatures. For the finitely softened Guggenheim and CarnahanStarling-van der Waals fluids, both the liquid spinodal and the liquid branch of the binodal are monotonic in density, and negative thermal expansion occurs at all temperatures above a threshold density. However, these anomalies are density minima at low temperatures and density
6884 The Journal of Physical Chemistry, Vol. 98, No. 27, 1994
TABLE 1: Effect of the Temperature F~nctio~lities of P and b on Density Anomalies in Generalized van der Waals Fluids’ Camahancase vdW Guggenheim Starling-vdW b’< 0;b(0) # 0 ~ a’= ; 0 P-(+) b’< 0; b(0) = m; a’= 0 pmu(+) b’= 0; a’> 0
PmuH
~min(-),
pmu(+) Pmu(-)
P-(+)
~min(-),
P-(+)
P-(+) PmU(-)
‘The symbols p- and p- denote density maxima and minima, respectively. The symbols (+) and (-) denote the sign of the slope of the locus of density extrema in the P-T projection. maxima a t high temperatures. The locus of density anomalies is continuous: in P-T projection it is negatively-sloped a t low temperatures (density minima) and positively-sloped at high temperatures (density maxima). Thus, generalizedvan der Waals fluids with temperature-dependent cores do not have water-like density maxima: they become anomalous upon compression (not upon expansion, as is the case with water), and their locus of density maxima is positively-sloped in P-T projection (it is negatively-sloped in the case of water). Generalized van der Waals fluids whose attractive parameter, a, increases with temperature while the core, b, remains temperature-independent exhibit density maxima that disappear upon compression. Their locus of density maxima is negativelysloped in P-Tprojection. Identical features (mean-field attraction increasing with temperature; temperature-independent core) are present in the mean-field solution of a water-like lattice model19 that also predicts low-pressure anomalies. The van der Waals fluid with a‘ > 0 and constant b has a retracing, Speedy-like liquid spinodal. Density anomalies are metastable, occurring only under tension. For finite a( T ) slope in the T = 0 limit, the lowest temperature for which ap< 0 is finiteand theliquid spinodal and the locus of density maxima meet twice. The hightemperature encounter occurs a t the fluid’s tensile strength maximum; below the low-temperature encounter (lowest temperature for which ap< 0) the P-Tprojection of the liquid spinodal is once again positively-sloped. The various types of behavior are summarized in Table 1. An interesting question raised by this study is the possibility of modeling stable water-like density anomalies (that is to say, negatively-sloped P-T projection of locus of density maxima extending to positive pressures). The equation of state derived from the mean-field solution of a water-like lattice model19 has a van der Waals attractive term with a’> 0 and a Bragg-Williams logarithmic repulsive term and yields water-like anomalies a t positive pressure. Thus, the form of the repulsive term appears to be crucial as to the equation’s ability to model water-like anomalies at positive pressures. The numerical investigation of the behavior of the Guggenheim and CarnahanStarling-van der Waals fluids with constant b and a’> 0 and, preferably, the derivation of constraints on the form of the repulsive term necessary todescribe anomalies at positive pressure are interesting open problems.
Acknowledgment. P.G.D. gratefully acknowledges the financial support of the U.S.Department of Energy, Division of Chemical Sciences, Office of Basic Energy Sciences, through Grant DE-
Chakravarthi et al. FG02-87ER13714. Austen Angell first suggested to P.G.D. the idea of exploring the van der Waals equation with a temperaturedependent core as a model for fluids with density anomalies. It is a pleasure to acknowledge Austen Angell’s influence on our thinking. P.G.D. thanks Stanley Sandler for pointing out the anomalies in the specific heat that can result from making the van der Waals parameters temperature-dependent; and S.S. thanks Francesco Sciortino, H. Eugene Stanley, and Peter Poole for helpful discussions.
References and Notes (1) Speedy, R. J. J. Phys. Chem. 1987, 91, 3354. (2) Kanno, H.; Angell, C. A. J . Chem. Phys. 1980, 73, 1940. (3) Angell, C. A.; Kanno, H. Science 1976, 193, 1121. (4) Debenedetti, P. 0.;Raghavan, V. S.;Borick, S.S. J . Phys. Chem. 1991,95,4540. ( 5 ) Franks, F. Water, Roy. Soc. Chem.: London, 1983. (6) Smith, J. V. Chem. Reu. 1988,88, 149. (7) Liquid metals are an exception. The effectivepair potential of several liquid metals is core-softened. See: March, N. H. Liquid Metals: Concepts and Theory; Cambridge University Press: Cambridge, 1990. (8) Stillinger, F. H.; Weber, T. A. J. Chem. Phys. 1978, 68, 3837. (9) Hemmer, P. C.; Stell, G. Phys. Reu. Lett. 1970, 24, 1284. (10) Stell, G.; Hemmer, P. C. J. Chem. Phys. 1972,56,4274. (1 1) Kincaid, J. M.; Stell, G.; Hall, C. K. J. Chem. Phys. 1976,65,2161. (12) Kincaid, J. M.; Stell, G.; Goldmark, E. J. Chem. Phys. 1976, 65, 2172. (13) Kincaid, J. M.; Stell, G. J . Chem. Phys. 1977, 67, 420. (14) Kincaid, J. M.; Stell, G. Phys. Lett. A 1978, 65, 131. (15) Young, D. A.; Alder, B. J. J. Chem. Phys. 1979, 70,473. (16) Speedy, R. J.; Angell, C. A. J. Chem. Phys. 1976,65,851. (17) Speedy, R. J. J. Phys. Chem. 1982,86,986. (18) Speedy, R. J. J. Phys. Chem. 198186, 3002. (19) Sastry, S.;Sciortino, F.; Stanley, H. E. J. Chem. Phys. 1993, 98, 9863. (20) Poole, P.H.;Sciortino, F.;Essmann,U.;Stanley,H. E. Nature 1992, 360, 324. (21) Stillinger, F. H.; Rahman, A. J . Chem. Phys. 1974,60, 1545. (22) D’Antonio, M. C.; Debenedetti,P. G. J. Chem. Phys. 1987,86,2229. (23) Debenedetti, P. G.; D’Antonio, M. C. AIChE J . 1988, 34, 447. (24) Haar, L.; Gallagher, J. S.;Kell, G. S . In Proceedings of the 9th
Internationul Conference on the Properties of Steam; Straub, J., Scheffler, K., Eds.; Pergamon: Oxford, U.K., 1979. (25) Saul, A,; Wagner, W. J . Phys. Chem. Ref. Data 1989, 18, 1537. (26) Hill, P. G. J. Phys. Chem. Ref. Data 1990, 19, 1233. (27) Guggenheim, E. A. Thermodynamics. An Advanced Treatmentfor Chemists andPhysicfsts,5th ed.: North-Holland Amsterdam, 1967; Chapter 3. (28) Ben-Amotz, D.; Herschbach, D. R. J. Phys. Chem. 1990,94,1038. (29) Ben-Amotz, D.; Willis, K. G. J. Phys. Chem. 1993, 97, 7736. (30) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (31) Johnston, K. P.; Eckert, C. A. AIChE J. 1981,27, 773. (32) In this paper we use the term core-softeningin a more general sense than in ref 4. There, the term means an inflection point in the repulsive core of the pair potential. Here, it means a core whose diameter decreases with temperature. (33) Prausnitz, J. M.; Lichtenthaler,R. N.; de Azevedo, E. G. Molecular Thermodytuamicsof Fluid-Phose Equilibria, 2nd ed.; Prentice Hall: Englewood Cliffs, N J , 1986; Chapter 10. (34) Boltzmann, L. Lectures on Gas Theory; University of California Press: Berkeley, CA, 1964. Speedy, R. J.; Prielmeier, F. X.; Vardag, T.; Lang, E. W.; Ludemann, H.-D. Mol. Phys. 1989,66, 577. (35) It is always pbssible to define, via q 23, a temperature-dependent distance of closest approach. It is only upon adopting this definition as the hard core of the generalized van der Waals fluids that this gives rise to the type of density anomaly that we discuss here. (36) Skripov, V. P. High Temp. 1966, 4, 757. (37) Trebble, M. A.; Bishnoi, P. R. Ffuid Phose Equifib. 1986,29,465. (38) The mean-field solution of the water-like lattice model introduced in ref 19 is another example of low-pressure anomalies caused by a: > 0 and a temperature-independent core. The repulsive term in ref 19 has a BraggWilliams logarithmic form.
