attila r. imre and thomas kraska- stability limits in binary fluids mixtures
TRANSCRIPT
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Stability limits in binary fluids mixtures
Attila R. ImreKFKI Atomic Energy Research Institute, H-1525 Budapest, P.O. Box 49, Hungary
Thomas Kraskaa
Department of Physical Chemistry, University Cologne, Luxemburger Strasse 116, D-50939 Kln, Germany
Received 7 September 2004; accepted 17 November 2004; published online 2 February 2005
The stability limits in binary fluid mixtures are investigated on the basis of the global phase diagram
approach employing a model for the attracting hard-sphere fluid. In addition to the diffusion
spinodals the mechanical spinodals are included. As a result one finds topologically different types
of the diffusion spinodals while only one shape exists for the mechanical spinodals which are
present in the region of liquid-vapor equilibria only. The diffusion spinodals represent the
underlying properties of the phase behavior. The types of stable phase behavior therefore resemble
that of the spinodal behavior. The different shapes of the spinodals can be important for
nonequilibrium processes in nature and technology. 2005 American Institute of Physics.
DOI: 10.1063/1.1847651
I. INTRODUCTION
Besides the stable homogeneous liquid and gas phases
fluids can also exist under metastable condition.1
The border-
line between stable and metastable states at positive pressure
is the coexistence or saturation curve. A part of the meta-
stable region of the liquid phase is located at negative
pressures14 Fig. 1. The metastable region is limited by
another curve called spinodal separating the metastable states
from the unstable region in which states cannot exist. As a
limit of the metastable region the spinodal also reaches nega-
tive pressure on the liquid side. Hence, if a liquid is isotro-
pically stretched it is transferred into a metastable state either
at positive or negative pressure. By overheating only the
metastable region at positive pressure can be reached. In ex-periments one can usually not reach the spinodal but a so-
called attainable stability limits, also called homogeneous
nucleation limit.1,5
The presence of a heterogeneous nucleus
such as an impurity lowers the critical work of the formation
of a vapor nucleus and therefore leads to a phase separation
at lower supersaturation. A vapor can be transferred into a
metastable state by undercooling or pressurizing. The meta-
stable region of the vapor phase also has a limiting spinodal
which is entirely located at positive pressures. The vapor and
the liquid spinodal meet continuously at the critical point in
the pand Tprojections. This is the only point in the phase
diagram where the spinodal touches the stable region. Hence,
it is the only point at which the diverging density fluctuations
at the spinodal reach the stable region of the phase diagram.
All other points of the spinodal are screened from the stable
gas- or liquid-phase regions by metastable regions as shown
in Fig. 1. In the pressure-temperature pT projection the two
branches of the spinodal meet in a cusp at the critical point.
Some similarities exist for the behavior close to a critical
point and close to a spinodal such as the diverging compress-
ibility and isobaric heat capacity. However, other properties
such as the exponents of the powers laws differ when ap-proaching the critical point and the spinodal.
5
For pure substances two topologically different liquid-
vapor spinodal types are discussed which are marked as type
A and B in Fig. 1a.4
The difference between types A and B
is the existence of a minimum in type B. Since the experi-
mental determination of the spinodals is difficult and only
possible by extrapolation,6
the spinodals of real materials are
not well known. The three best known systems are water and
the two isotopes of the helium. The spinodals of the helium 3
and helium 4 are type B and type A, respectively, while for
water it is still under discussion whether it is type A or
B.1,4,7,8
In binary liquid mixtures not only liquid-vapor LV butalso liquid-liquid LL phase transition can take place.
Therefore, it is also possible to reach a metastable liquid
state by crossing the LL coexistence curve. The limit of such
metastable liquid is a LL spinodal. When reaching the LL
spinodal the system immediately splits into two liquid phases
with different compositions. Although experimentally the
spinodal cannot be reached or jumped over, due to the slow
kinetics of phase separation, in some systems one can ob-
serve the so-called spinodal decomposition. Such jump can
be done by changing the temperature in a very fast manner or
changing the pressure which can be faster and easier for LL
phase separation than for LV phase separations.
9
A jump into the nonstable region can lead to two differ-
ent structures which are nucleation-and-growth type or spin-
odal type. The nucleation-and-growth type of phase separa-
tion happens in the metastable region and forms islands of
the new phase in the mother phase. At the spinodal the phase
separation is spontaneous and happens immediately within
the whole body of liquid leading to two bicontinuous, spon-
gelike phase.1
Although these structures disappear quickly, it
is possible to follow LV phase separations using very fast
photographic technique.10
In case of LL phase separationsaElectronic mail: [email protected]
THE JOURNAL OF CHEMICAL PHYSICS 122, 064507 2005
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the two different topologies, nucleation-and-growth type, and
spinodal decomposition type can be long lasting, especially
when very high molecular weight components are involved.
There are processes such as membrane production or phase
separation in polymer blends in which the partly phase-
separated liquid is further cooled down until one or both
phases solidify. In this way the initial structure can be frozen,
leading to different mechanical behavior of the solid. Hence,
the morphology and properties of the composite material
strongly depends on the type of the phase transition. There-
fore, the knowledge of the location of the spinodal is crucial
for the understanding and prediction of the properties of new
materials.
Furthermore, there are a lot of important liquid mixtures
including binary ones which can be temporarily in meta-
stable condition at which the physical and chemical proper-
ties can change very fast. Stretched binary liquids at absolute
negative pressure can easily split into two liquid phases
which can cause changes in the viscosity and several other
properties abruptly. Prominent examples for multicompo-
nents fluid are biological fluids such as blood or sap, crude
oil or aqueous solutions in soil. Blood or other body fluids
can be in metastable state locally when they are exposed to
medical ultrasound see Ref. 4 and references therein. Sap is
under moderate negative pressure down to 4 bar in the
xylem,11
crude oil can experience negative pressure during
flow or sudden decompression,12
while aqueous solutions
can be under deep negative pressure 100 MPa in the cap-
illaries of the soil.13,14
Miscibility measurements as well as
other measurements of other properties under metastable
conditions are very difficult.1520
Therefore, methods have
been developed in which properties measured at positive
pressure are extrapolated into the region of negative
pressure.14,2125
For this kind of extrapolation the knowledge
of the spinodals is crucial, because extrapolating beyond the
spinodal would yield artificial results.25
While there are several investigations of the stability
limits in pure substances experimentally and theoretically5,26
there is yet no thorough investigation on the stability limits
in binary fluid mixtures. For such study the systematic be-
havior of the global phase diagram approach is very useful.
The global phase behavior of binary fluid mixtures has beeninvestigated over the last decades in a general fashion
2732as
well as applied to specific problems and phenomena in phase
behavior.3336
In such investigations usually only the stable
parts of the phase diagram are considered. In some cases
nonstable parts are included such as metastable and unstable
parts of a critical curve.31,36,37
Although there are studies about the LV spinodal of pure
liquids and some more about the LL spinodal of binary liq-
uids, the relative position and topology of LL and LV spin-
odals in binary mixtures has been rarely studied.1,38,39
In this
work we study systematically the spinodals exemplary for
attracting hard-sphere fluid binary mixtures.
II. METHOD
The conditions for the stability of phases can be obtained
from the second law of thermodynamics saying that a closed
system is stable if the entropy is at its maximum value. From
the second-order variation of the entropy with respect to the
volume, the condition for the mechanical stability limit, the
mechanical spinodal, can be obtained:1,38,40
pV
T
pV= 0 . 1
A phase is mechanically stable for pV0 and mechanically
unstable for pV0. This is the case for pure substances and,
in principle, also for mixtures. Integration of Eq. 1 gives
2A/V2TA2V= 0. The mechanical spinodal is therefore
given on the liquid side by the minima and on the gas side by
the maxima of an isothermal van der Waals loop Fig. 1b.
As the maximum and the minimum approach each other with
increasing temperature and eventually coincide, the spinodal
goes through the critical point Fig. 1b.
In binary fluid mixture phase separation usually leads to
two phases with different mole fraction with exception of
azeotropy. The second-order variation of the entropy with
respect to the mole fraction in a closed system gives the
thermodynamic condition of the diffusion spinodal inmixtures:1,38,40
2Gmx2
p,T
G2x = 0 . 2
In analogy to the extrema of the van der Waals loop for the
mechanical spinodal the diffusion spinodal is given by the
extrema of the chemical potentials of the two compounds as
function of the mole fraction. At the diffusion spinodal the
mole fraction exhibits diverging fluctuations. In pure sub-
stances only LV equilibria exist. In mixtures there is the pos-
sibility either for LL or LV phase separations. Although ther-
modynamically there is no difference between these two
FIG. 1. Schematic sketch of the vapor-liquid phase diagram of a pure sub-
stance in a pT projection and b pprojection. a Dotted, vapor pressure
curves; dashed, different types of spinodals; and filled circle, liquid-vapor
critical point. b Solid, coexistence curve; long dashed, mechanical spin-
odal; and short dashed, isotherm.
064507-2 A. R. Imre and T. Kraska J. Chem. Phys. 122, 064507 2005
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cases, i.e., Eq. 2 always holds, experimentally the differ-
ence can be crucial. While in LL phase transitions the vol-
ume of the new phases is in the same order of magnitude as
that of the original supersaturated phase, in case of LV tran-
sition the volume of the vapor phase can be orders of mag-
nitudes larger.
While Eq. 1 can be easily applied to an equation of
state, Eq. 2 has to be rewritten in terms of the Helmholtz
energy A by Jacobi transformation:
G2x = A2x AVx
2
A2V= 0 . 3
The abbreviations of the derivatives in general are given41
by
n+kA/Vn xkTAnVkx. All other natural variables of the
functions are kept constant at differentiation. The Helmholtz
energy can be calculated from the equation of state by inte-
gration as for the residual part:
Ares = presdV. 4After adding the ideal mixing term one can compute the
mechanical and the diffusion spinodal for a given equation of
state. The calculations here are performed for the attracting
hard-sphere fluid modeled by the Carnahan-Starling-van der
Waals equation of state:42
p =RT
Vm1 + y + y2 y3
1 y3 a
Vm2
. 5
Here a is the attraction parameter and y = b/ 4Vm the packing
fraction with the covolume parameter b. The choice of this
equation is twofold: first it is a simple and good description
the attracting hard-sphere model fluid, second the global
phase behavior of this equation is very wellinvestigated.
32,43,44The latter point makes it possible to lo-
cate easily the different types of stable phase behavior in
terms of the global parameters , , and . These global
parameters are reduced differences of the equation of state
parameters a and b,
=b22 b11
b22 + b11, 6
=
a22
b222
a11
b112
a22
b222
+ a11
b112
, 7
=
a22
b222
2a12
b11b22+
a11
b112
a22
b222
+a11
b112
. 8
The parameters of the mixture are calculated from the corre-
sponding parameters of the pure substances aii, bii and the
cross-interaction parameters a12, b12 by the one-fluid mixing
theory with quadratic mixing rule
a =i
j
xixjaij , 9
b =i
j
xixjbij . 10
III. RESULTS
The diffusion and the mechanical spinodals are calcu-
lated by the computer algebra program MAPLE Ref. 45 with
a code developed for this investigation. Although the me-
chanical spinodal is hidden behind the diffusion spinodal inmixtures, its location in phase space can be important be-
cause it influences the magnitude of the density fluctuation in
the system at a metastable state point. A third type of spin-
odal, the isentropic spinodal, is not considered here. In an
recent investigation of pure substances26
the isentropic spin-
odal has been located far below the mechanical spinodal and
therefore its influence on the metastable region is expected to
be negligible.
A. Type I phase behavior
In order to analyze the spinodal behavior we start with
the simplest possible binary system, the type I system in theclassification of van Konynenburg and Scott.
27,28In such sys-
tems the critical points of the pure substances are connected
by a continuous binary liquid-gas critical curve. Besides this
critical curve there are no further critical curves. Such system
can be obtained with the parameter set =0.2, =0.1, and
=0.0 as shown in Fig. 2. One can see in Fig. 2b that in the
limits of the pure substances the mechanical and the diffu-
sion spinodal approach each other. In case of equal covol-
umes =0.0 the cusps of the mechanical spinodal in the pT
projection form a straight line connecting the critical points
of the pure substances Fig. 2a, dotted line. In the px pro-
jection shown in Fig. 2b it appears as a continuous curve.
FIG. 2. a Type I phase behavior in a pressure-temperature projection for
= 0, =0.2, and =0.1. Solid curve, binary vapor-liquid critical curve;
long-dashed curves, diffusion spinodal isopleths; short-dashed curves, me-
chanical spinodal isopleths; dotted line, pseudocritical curve; and dot-
dashed, pure substance vapor pressure curves. b The same phase diagram
in pressure-mole fraction diagram. The numbers mark different T/Tc1values.
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Along this curve also A3V=0 holds, which corresponds to the
pure substance critical condition. Therefore it is occasionally
called pseudocritical curve. It is a straight line in the pTprojection because the critical conditions for pure substances
A2V=A3V=0 give Tc =0.3773 a/Rb and pc =0.070669 a/b2
for the CarnahanStarlingvan der Waals equation. Here Tcand pc both depend linearly on the attraction parameter ax
which is the only mole fraction dependence in case of equal
sized molecules. Therefore the pseudocritical curve is a lin-
ear interpolation of the pure critical points in the pT projec-
tion. It should be emphasized that the pseudocritical curve is
by no means a real critical curve. The binary critical curve is
rather located on the surface to the diffusion spinodal surface
in pTx space.
In Fig. 2 one can see that for a jump from the stable gas
phase into the nonstable region one first reaches the diffusion
spinodal. The mechanical spinodal is located behind the dif-
fusion spinodal. It follows that the vapor-liquid phase transi-
tion in binary mixtures is dominated by concentration fluc-
tuations and the system is decomposed already before at the
diffusion spinodal. The same can be observed for a jump
from the liquid phase into the two-phase region. This is in
agreement with inspection of Eq. 3 which shows that first
G2x and than A2V vanishes.46
For a mechanically stable phase
A2V is positive and because of the square the complete sec-
ond term of Eq. 3 AVx2 /A2V is positive. It follows that for
vanishing G2x the first of term Eq. 3 A2x has to be positive
as well. At the mechanical spinodal A2V vanishes and henceG2x diverges to minus infinity. Therefore, vanishing G2x re-
quires positive A2V and hence first the diffusion spinodal and
then the mechanical spinodal is reached. In Fig. 3 a sche-
matic sketch of an isothermal vapor-liquid equilibrium in-
cluding the spinodals is shown. The chosen temperature is
between the critical temperatures of the pure substances. The
diffusion spinodal long dashed touches the binodal solid
at the binary critical point. The mechanical spinodal short
dashed is in this case present only in the lower part of the
phase equilibrium. Half of the diffusion spinodal belongs to
the liquid phase LV spinodal; from the bottom-left corner to
the critical point; the other half belongs to the vapor phase
VL spinodal. The liquid part can be approached only com-
ing for the liquid side and vice versa as indicated by the
arrows in Fig. 3. When jumping into the two-phase region by
pressure increase from the vapor phase or pressure decrease
from the liquid phase one eventually reaches the diffusion
spinodal. Figure 3 also shows that a spinodal can be located
in the one-phase region in such diagram, for example, the
gas-phase spinodal in the liquid phase region. This may ap-
pear unusual but makes sense because the spinodals belongto the mother phase on the other side of the corresponding
branch of the coexistence curve. The limit of both, the dif-
fusion and the mechanical spinodal for vanishing mole frac-
tion is the pressure of mechanical spinodal of the pure sub-
stance at given temperature. This is shown in the left
diagram in Fig. 3. The fact that the spinodals can be located
outside the vapor-liquid coexistence region is inseparably re-
lated to this limiting behavior.
B. Type II phase behavior
After having analyzed a type I liquid-vapor system the
next step is to turn to a system with a liquid-liquid immisci-
bility. Type II is such system. In Fig. 4 different projections
of a type II system as calculated with Eq. 5 for the global
parameters = 0, =0.396, and =0.005 are plotted. In Fig.
4a the pT projection of the binary critical curve is shown.
One can see a type I VL critical curve and in addition a LL
critical curve at low temperatures. At high pressure the LL
critical curve goes to infinite pressure within a fluid model,
however, it is terminated by solidification in real systems. In
some cases this LL branch of the critical curve behaves mo-
notonously in the pT projection, in other cases it can exhibit
a temperature minimum.36
At low pressure the LL-critical
curve goes trough a minimum to a cusp in the pT projection.This cusp is the stability limit of the critical curve G4x = 0
and hence the remaining branch of the LL-critical curve go-
ing to negative pressure is an unstable critical curve. While a
normal phase becomes unstable at G2x =0 a critical phase
becomes unstable at G4x = 0. Somewhere, typically below
and close to the vapor pressure curve of the more volatile
substance a critical endpoint separates the stable and the
metastable branch of the LL-critical curve. In the px projec-
tions in Figs. 4a and 4c as well as in the pTx diagram in
Fig. 4e one can recognize that the cusp of the LL-critical
curve in the pT projection is actually a continuous curve.
Figure 4e also shows that in case of type II the diffusion
spinodals have an additional branch going to high pressure.Together with the spinodal branches known from type I the
diffusion spinodal isopleths form a &-like shape in the con-
stant mole fraction section. Further analysis shows that the
high pressure spinodal is also present in type I systems but
located at negative absolute temperature which is of course
outside the physical meaningful range in these systems.
When varying the molecular parameters the high pressure
branch of the spinodals can move to positive temperature
which results in the appearance of the LL-critical curve. This
appearance corresponds to the hypothetical transition from
type I to type II which is called zero-Kelvin transition.47
At
this transition the system has a LL critical curve at zero
FIG. 3. Schematic vapor-liquid phase diagram with the coexistence curve
solid, the mechanical spinodal short dashed, and the diffusion spinodallong dashed. The arrows symbolize a jump into the two-phase region from
the vapor or the liquid phase. In all cases first the diffusion spinodal is
reached. The pressure-temperature diagram on the left side corresponds to
one of the pure substances with x =0. Here the vapor pressure curve solid
curve of the pure substance and the mechanical spinodals short dashed are
plotted.
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Kelvin, an approach which neglects of course the solidifica-
tion. In the isothermal sections at high temperature such as
T/Tc1 =2.0 the diffusion spinodals behave as in type I sys-
tems shown in Fig. 4a. With decreasing temperature the
spinodals in the isothermal section undergo a topological
transition.
At T/Tc1 =1.03 close to the critical temperature of the
more volatile substance Fig. 4b, the looplike topology of
the LV diffusion spinodals has changed to a curve with two
maxima and one minimum. These extrema of the isothermal
diffusion spinodal are points on the critical curves: the maxi-
mum at lower pressure is a critical point on the metastable
branch of the LL-critical curve, the maximum at higher pres-
sure is a stable VL critical point, and the minimum is a point
on the unstable branch of the LL-critical curve. These three
points are marked by symbols in Fig. 4b and by a dot-
dashed line in Fig. 4d. At slightly higher temperature below
T/Tc1 = 1.07 the maximum and minimum corresponding to
the unstable and metastable LL-critical points meet and the
diffusion spinodal exhibits a saddle point. This saddle point
corresponds to the cusp of the LL-critical curve in the pTprojection with G4x =0 connecting the metastable and un-
stable branch of the critical curve. In Fig. 4c spinodals at
lower temperature are shown. The spinodals are similar as
for T/Tc1 = 1 Fig. 4a but exhibit a pronounced maximum
at high pressure corresponding to a stable LL-critical point.
For T/Tc1 =0.5 the extrema are marked by symbols in Fig.
4c and as well as by a dot-dashed line in Fig. 4 d.
C. Type V phase behavior
Type V phase behavior consists of a LV-critical curve
which is interrupted by a LLV three-phase curve as shown
schematically in Fig. 5d. The calculations for = 0, =0.5,and =0.2 shown in Fig. 5a indicate that the spinodals at
high temperature are similar to the ones of type I. At T/Tc1=1 the diffusion spinodal has a topologically similar shape as
at higher temperature but the maximum is more pronounced.
Therefore, the mechanical spinodal is at much lower pressure
than the diffusion spinodal compared to the type I phase
diagram described above. This distance between mechanical
and diffusion spinodal, which is also present for the type II
phase diagram, appears to be typical for LL equilibria. The
appearance of such LL type diffusion spinodal corresponds
to the LL type critical curve of type V at T/Tc11. The
maximum is maintained with decreasing temperature. Fur-
thermore, an additional minimum at high mole fraction be-comes visible, for example, at T/Tc1 =0.8 in Fig. 5c. These
two extrema coincide in a saddle point at negative pressure
corresponding to a cusp in the critical curve G4x = 0 in the
pT projection at which the metastable and unstable branches
meet. In Fig. 5d the situation corresponding to T/Tc1 =0.8
is shown schematically. In Fig. 5b the region close to the
critical point of the more volatile substance is enlarged. At
T/Tc1 =1.01 one can recognize one minimum corresponding
to the unstable branch of the critical curve and a maximum
corresponding to the very short LV-critical branch connected
to the critical point of the more volatile substance Fig. 5d.
At T/Tc1 =1.03 these two extrema have vanished and hence
the cusp of the critical curve is located between T/Tc1=1.01 marked by a dot-dashed line in Fig. 5d and
T/Tc1 =1.03.
D. Type III phase behavior
In type III systems the LV-critical curve starting at the
critical point of the less volatile substance is continuously
connected to the LL-critical curve going to high pressure.
The critical curve is monotonously going to high pressure,
i.e., there is no pressure minimum or maximum.32
Figure
6a shows the spinodals of such system = 0, =0.45, and
=0.1 which are at high temperature as T/Tc1 =2.0 also
FIG. 4. Type II spinodal isotherms calculated for =0 , =0.396, and
=0.005. a High temperature isotherms; b enlargement of the region near
the critical point of the more volatile substance; c low temperature iso-
therms; and d pT projection. Legend as for Fig. 3. The numbers markdifferent T/Tc1 values. The dot-dashed lines at T/Tc1 =0.5 and 1.03 in d
correspond to the diffusion spinodal isotherms plotted as bold dashed curves
in b and c. e Three dimensiona1 plot: bold solid curve, critical curve;
light solid curves, diffusion spinodal isopleths; and dashed curves, mechani-
cal spinodal isopleths.
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similar to those of type I. With decreasing temperature, but
still above the critical temperature of the less volatile sub-
stance, the diffusion spinodal forms a pronounced maximum
at high pressure corresponding to a LL-critical point. The
continuous appearance of the LL type spinodal represents the
continuous transition from LV to LL phase equilibria. At
T/Tc1 = 1 the spinodals reach the critical point of the less
volatile substance in a cusp as for the other types of phase
behavior mentioned above. This diffusion spinodal is not
continuously connected to the less volatile substance. It
rather diverges to infinite pressure coming from both pure
substances. Figure 6b shows how the spinodals behave at
high pressure. At T/Tc1 =1.28 the spinodals go to high pres-
sure as for T/Tc1 =1, while at T/Tc1 =1.29 they consist of
two parts, one at high pressure with a minimum and one
below with a maximum. Since these extrema correspond toLL-critical points the LL-critical curve exhibits a critical
temperature minimum between T/Tc1 =1.28 and T/Tc1=1.29. The topology of this transition resembles that of the
binodals at a critical temperature minimum.48
E. Type IIIm phase behavior
Type IIIm phase behavior is similar to that of type III but
the critical curve exhibits a pressure minimum and a maxi-
mum in Fig. 7b. In Fig. 7a some diffusion spinodal iso-
therms are shown for a type IIIm phase diagram near the
region of the critical point of the more volatile substance
= 0, =0.4, and =0.05. At T/Tc1 =1.05 and T/Tc1 =1.1one can recognize two maxima and one minimum which are
marked by symbols for T/Tc1 =1.1. The maximum at low
pressure and the minimum has vanished for T/Tc1 =1.2
FIG. 5. Type V phase behavior calculated for =0 , =0.5, and
=0.2. aHigh temperature diffusion and mechanical spinodal isotherms. b Enlarge-
ment of the region near the critical point of the more volatile substance. The
bold dashed curve calculated for T/Tc1 =1.01 shows the existence of three
extrema of the diffusion spinodal isotherm together with a maximum athigher pressure not shown in this scale. c Low temperature diffusion and
mechanical spinodal isotherms. The minimum of the diffusion spinodal iso-
therms close to x =1 corresponds to the unstable branch of the critical curve.
Legend as for Fig. 3. The numbers mark different T/Tc1 values. d Sche-
matic sketch of the pT diagram of this system. The dot-dashed lines mark
the isothermal sections shown in b and c.
FIG. 6. Type III phase behavior calculated for =0 , =0.45, and =0.1. a
Diffusion and mechanical spinodal isotherms in the low pressure region. b
Diffusion spinodal isotherms in the high pressure region. Legend as for Fig.
3. The numbers mark different T/Tc1 values.
FIG. 7. a Type IIIm phase behavior calculated for = 0, =0.4, and
=0.05. The three squares mark the extrema of the diffusion spinodal iso-
therm at T/Tc1 =1.1. Legend as for Fig. 3. The numbers mark different T/Tc1values. b Corresponding pT diagram. Solid curves, critical curves; and
dashed curves, vapor pressure curves of the pure substances. The isotherm at
T/Tc1 =1.1 marks a section shown in a.
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which means that the cusp formed by the unstable critical
curve and the short critical curve connected to the critical
point of the more volatile substance is passed. The maximum
at high pressure corresponds to the stable LL-critical curve.
F. Types VI and VII phase behavior
The high temperature part of the type VII hase behavior
shown in Fig. 8a for = 0, =0.42, and =0.019 re-
sembles that of type V discussed above. In addition type VII
systems have a LL closed loop critical curve. Type VI phase
behavior is similar to type VII but with a continuous LV-
critical curve as in type I. Therefore, it is not discussed here
separately. It has been shown32
and confirmed several
times44,49 that Eq. 5 is able to generate such closed loopphase behavior, however, at very low temperature. Experi-
mentally LL closed loop phase behavior have been found for
water containing systems.50,51
Figure 8b shows the spin-
odals corresponding to a LL equilibria at low temperature
calculated with Eq. 5. One can see a sequence of LL diffu-
sion spinodals forming the closed-loop dome. At these tem-
peratures the mechanical spinodal is at very large negative
pressure not shown here which indicates that this diffusion
spinodal is LL type, being far away from the mechanical
spinodal.
G. Phase behavior in the shield region
The shield region is a small area in the global phase
diagram which exhibits several complex phase diagrams in-
cluding four-phase equilibria. Here, we have investigated the
diffusion spinodals for a system which very close to the sym-
metric systems at =0.0. The chosen parameters are = 0,
= 0.02, and =0.35. The diffusion spinodals and the pT
projection of this system are shown in Fig. 9a. Each re-
duced temperature, for which a diffusion spinodal is calcu-
lated in Fig. 9a is marked by a thin line in Fig. 9b. The
square symbols in Fig. 9a correspond to the same symbols
in Fig. 9b. At T/Tc1 =0.9 we find two spinodals correspond-
ing to LV equilibria ending at the pure substance spinodals at
x =0 and x =1. At higher pressure a spinodal related to a LL
equilibria with a LL-critical point at the minimum exists.
With decreasing temperature these three spinodals intersect
in two steps. First the LV spinodal at low mole fraction in-
tersects with the LL spinodal as one can see in Fig. 9 a for
T/Tc1 = 0.85. For lower temperatures the VL spinodal at high
mole fractions joins the remaining part of the LV spinodal at
low mole fraction. These two branches do not form a cusp at
T/Tc1 =0.85 as Fig. 9a suggests. Actually both have a con-
tinuous maximum being LV-critical points. At T/Tc1 =0.8 the
system has already passed the second transition. One diffu-
sion spinodal connects the pure substance spinodals continu-
ously while the diffusion spinodals starting at the pure sub-
stance spinodal at low pressure go to very high pressure at
about x =0.18 and x =0.82. The remaining island of the dif-
fusion spinodal between x =0.4 and 0.6 around p/pc1 =0.5 is
related to partly unstable critical curves G4x0 as one can
see in Fig. 9b.
IV. CONCLUSIONS
The spinodals represent the underlying property of a
phase diagram. They form planes in pTx space which are
envelops of the critical curves. There are a limited number ofspinodal shapes such as the LV-type spinodal, the continuous
LL spinodal of types II, V, or III or the diverging spinodal of
type III at low temperature. The different types of phase
behavior as known from the classification of van Konynen-
burg and Scott further differentiate similar types of spinodal
topologies. As expected, the mechanical spinodal shape
marked as type B in pure substances Fig. 1 has not been
found with the attracting hard-sphere model fluid. In case of
a LV equilibrium a mechanical spinodal is located near the
diffusion spinodal. In case of a LL equilibrium the mechani-
cal spinodal is typically far away from the diffusion spinodal.
For type II or III phase behavior the reason for the large
FIG. 8. Type VII phase behavior calculated for = 0, =0.42, and
=0.019. a High temperature diffusion spinodal isotherms. The numbers
mark different T/Tc1 values. b Low temperature diffusion spinodal iso-
therms. Legend as for Fig. 3.
FIG. 9. a Diffusion spinodals for a phase diagram in the shield region
calculated for =0 , =0.02, and =0.35. b Corresponding pT diagram.Symbols, binary critical points forming the critical curves; and dashed
curves, vapor pressure curves of the pure substances. The isotherms at
T/Tc1 =0.8, 0.85, and 0.9 mark the sections shown in a.
064507-7 Stability limits in binary mixtures J. Chem. Phys. 122, 064507 2005
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distance to the mechanical spinodal can be deducted from the
pressure-volume sections. A mechanical spinodal requires a
van der Waals loop in the pressure-density diagram which is
not present at high pressure in these phase diagram types. At
low pressure the diffusion spinodal exhibits a pronounced
maximum which causes the distance to the mechanical spin-
odal.
The unusual shape of the diffusion spinodal of type II or
type IIIm systems in the vicinity of the critical temperatureof the more volatile substance is of interest for processes
including a supercritical solvent such as carbon dioxide. Car-
bon dioxide-solute systems are often type II, IV, or III and
process conditions a usually slightly above the critical tem-
perature of carbon dioxide. The spinodal isotherm at T/Tc1=1.03 in Fig. 4b showing a type II system has already two
maxima and one minimum. However, only one maximum is
related to a stable LV-critical point. In Fig. 5b showing a
type V system at T/Tc1 = 1.01 the second maximum has
emerged to high pressure and is here a stable LL-critical
point. Such type of topology can be found in mixtures of
carbon dioxide with certain organic solvents which exhibit a
LL immiscibility on top of the LV equilibria. In case of thegas antisolvent crystallization such LL decomposition leads
to undesired large agglomerated paracetamol particles rather
than fine ones as discussed recently.52
The location of the
diffusion spinodal is expected to affect the precipitation pro-
cess in such cases.
ACKNOWLEDGMENTS
This work was partially supported by the Hungarian Re-
search Fund OTKA under Contract No. T 043042, by a
bilateral program of the German Science Foundation DFG
and the Hungarian Academy of Science HAS, Grant No.
436 UNG 113/150/n-1. A.R.I. was also supported by theBolyai Research Fellowship.
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