Gas-gas phase separation in binary inverse-12 systems

Physica A 169 (1990) North-Holland

GAS-GAS J.A.

17-28

PHASE

SCHOUTEN,

SEPARATION T.F.

Van der Waals Laboratory,

Received

IN BINARY

INVERSE-12

SYSTEMS

SUN and A. DE KUIJPER

University of Amsterdam,

The Netherlands

11 May 1990

The conditions under which gas-gas phase separation will occur in binary systems having intermolecular pair potentials of the form 4(r) = c(air)” h ave been studied using the van der Waals one-fluid theory. Calculations have been carried out for three different ratios of the diameters of the pure components and a series of values for (T,~, the potential parameter for the unlike interaction. It is found that the system exhibits gas-gas equilibrium for ~~/cr, = 3 assuming additivity of the diameters. For large values of (r,? it also happens for ~~/a, = 1.1 and 1.5. Moreover it is shown that under certain conditions the volume of mixing is negative for high concentrations of the larger component but positive for small concentrations of that component at the same temperature and pressure. A comparison with Monte Carlo data of an equimolar mixture show that these conclusions are correct not only for the one-fluid theory but have a general validity.

1. Introduction The interest in the phase behaviour of multi-component systems is increasing rapidly. In particular the influence of pressure on the properties of mixtures is important for geology and planetary sciences. In order to develop reliable models for the giant planets Jupiter, Saturn and Uranus, experimental and theoretical information on the complex behaviour of mixtures under extreme conditions

of pressure

and temperature

is essential.

The geological

and chemi-

cal reactions which have occurred in the deeper layers of the earth’s crust can be understood better if more knowledge about the important mixtures is available. In recent years, experimental study of the influence of very high pressure on the properties of binary systems has produced a number of very interesting results, such as phase density inversions [l], the development of a metastable two-phase equilibrium from a homogeneous system [2], the enormous shift of a phase line due to the solubility of a gas in a solid [3] and the broadening of the fluid-fluid range in helium-hydrogen at very high pressures [4]. A theoretical description of the phase behaviour of helium-hydrogen up to 10 kbar using 0378-4371/90/$03.50

0

1990 - Elsevier

Science

Publishers

B.V. (North-Holland)

1X

J. A.

perturbation

theory

Schouten

et al.

has been

i Gus-gas separation

given

by Van den Bergh and Schouten survey of the experimental results ref. [7]. Although individual

perturbation systems,

theory

in inverse-12

systems

by Ree [.5]. This theory

has been extended

up to 1 Mbar for the same system [6]. A and the theoretical calculations is given in can

describe

it does not give clearer

the

insight

fluid

phase

into the physical

the occurrence of gas-gas (or fluid-fluid) equilibrium, necessary conditions in terms of the intermolecular

behaviour

of

reasons

for

nor does it provide the potential. It is a well-

known fact that hard sphere mixtures do not show the phenomenon of gas-gas phase separation assuming additivity for the unlike interaction parameter. The Gibbs free energy (G) of a system strives for a minimum. In a mixture of hard spheres there is competition only between the - TS and +pV terms. But since hard additive sphere systems have a negative volume of mixing at all pressures for all ratios of the diameters [8], and the communal entropy of mixing the complete mixing of the system will (- RC, X, In xi) is always positive, always be favoured since it results in a minimum in G. In real systems, however, the attractive part of the potential might favour the demixing process due to a negative contribution to the energy. A soft repulsive potential also causes a change of the interaction energy when the two components are mixed. Probably more important, particularly at high pressures, is the fact that the volume of mixing might be positive as in the case of the system helium-hydrogen [6]. The question then arises whether the presence of the attractive part of a potential is a necessary condition for the existence of fluid-fluid equilibrium. Recently, Kerley [9] performed Monte Carlo calculations equimolar mixtures

on equimolar mixtures with inverse-12 potentials and with inverse-6 potentials. He used the usual Berthelot parameter for dissimilar molecules, combining rule for u,?, the interaction namely w,* = (v, + ~~)/2. He concluded that soft-sphere mixtures have a positive volume of mixing at high pressures. The inverse-6 mixture showed fluid demixing below the freezing point. However, the repulsive part of a real system

is not even

approximately

described

by an inverse-6

potential.

How-

ever, in contrast to the inverse-6 case in the inverse-12 case the homogeneous mixture was stable up to the freezing point of the larger component. Since Kerley had only the equimolar mixture and pure component data at his disposal, he could merely calculate an upper bound to the two transition pressures. Moreover, the use of the Berthelot combining rule restricts the possibility of occurrence of gas-gas phase separation. The main purpose of this paper is to investigate whether demixing in the gas phase may occur in purely repulsive systems, with a potential comparable to the repulsive part of a real system. In particular it will be shown that under certain conditions gas-gas phase separation may occur in inverse-12 systems.

J.A.

Schouten et al. I Gas-gas

separation in inverse-12 systems

19

The van der Waals one-fluid theory is used for the description of the mixture. The possibility for phase separation was calculated for different ratios of the potential parameters of the pure components and different values of 5, where 5 is defined by V,* = 5(a, + 412

.

(1)

As Kerley pointed out [9], for the inverse-12 system with a ratio of 3 for the interaction parameters, the one-fluid theory is in good agreement with Monte Carlo calculations at low densities but the quantitiative agreement is much poorer at high densities. It will be shown that this has no influence on the main conclusions of this paper.

2. The model Hoover et al. [lo, 111 performed computer simulations using the Monte Carlo method to determine accurate thermodynamic properties for an inverse12 potential of the form c$(F-)=

E(g2.

(2)

This potential is of special interest because it is part of the Lennard-Jones 12-6 potential which provides a realistic description of the thermodynamic properties of simple systems at pressures of a few kilobar and which has been used in numerous computer simulation experiments. An enormous advantage of the inverse-n system above a Mie potential is the fact that 4(r) of eq. (2) is a homogeneous function of r. As shown in ref. [lo] such a potential leads to an important reduction of the calculations since the excess thermodynamic properties are functions of p4.zlkT only, where p is a reduced density defined as p = Na3/fi V. In this case, the equation of state can be determined by simulation of a single isotherm or a single isochore. Moreover, once the results are known for a certain value of U, they are consequently known for any value of U. If, therefore, the mixture can be described as a one-fluid system with an effective value for u, the equation of state of the mixture is also available. The van der Waals one-fluid model gives 3

u, = x2& + 241 - x)ai,

+ (1 - x)*4,

where (+rl and u2* are the potential parameters

,

(3) of the pure components,

(+12the

20

J. A. Schouren

parameter

for the unlike

composition The

fluid

termined

et al.

interaction

separation

in invrrse-12

systems

and (T, for the homogeneous

x (X is the mole fraction

of component

1).

phase

homogeneous

mixture

compressibility

factor

properties

in the following of p’e/kT

that the following

exact

Z = nEl3NkT

of the way.

given as a function

The

by Monte

relation

exists

Carlo between

simulation.

mixture

can

now

with be de-

(2 = pV/NkT)

It can be shown

2 and the internal

is [lo]

energy

+ 1 - nl2.

for an inverse-n potential. The Gibbs free energy relations

I tias-gas

E: (4)

G can be obtained

from

standard

thermodynamic

with the result

G/NkT=-

J r;,

[(Z-

I)/V]dV-InV+

Z+xlnx+(l

-,r)In(l

-n)+C. (5)

V, is a large volume at which the entropy of mixing can be expressed by the logarithmic terms in eq. (5) and C is an integration constant depending on 7‘ and V,,. The Gibbs free energy of mixing at constant pressure and temperature. AG,

is defined

as

AG = G, ~ XC; - (1~

x)G;

.

(6)

where G’; and GI’ are the Gibbs free energies of the pure components at the pressure and temperature under consideration and G, that of the homogeneous mixture with composition x. AG can be easily calculated as a function of x from eqs. (5) and (6). A double

tangent

construction

tion of the two co-existing phases. Other important quantities are the volume AV=V,-XV:‘-(I-# and the energy AE= However

will then provide

the composi-

of mixing (7)

of mixing

E, -xE:‘-(l-x)E;.

(8)

from eq. (4) follows

AE = 3A( pV) In

(9)

J. A. Schouten et al. I Gas-gas separation in inverse-12 systems

21

As expected, in the case of the inverse-n systems, both the energy and volume influence AG and, therefore, the onset of a phase separation. According to eq. (9), the contributions of the energy and p AV terms to AG have the same sign, and the ratio is fixed. It is, therefore, sufficient to study only the behaviour of AV.

Besides the fluid-fluid equilibrium, the fluid-solid equilibrium will also be calculated in order to complete the isotherms and to determine the position of the triple point fluid-fluid-solid. The melting line of the inverse-12 system can be written as [lo] p,,,g3k

=

16~‘2 (k7’&)‘.*”

,

(10)

where p, and T, are the melting pressure and temperature, respectively. Using eq. (3) for reference vX of the mixture, eq. (10) provides the pressure at which a fluid mixture of a certain composition starts to solidify, given the temperature. (It is not the melting pressure of a solid mixture!)

3. Results and discussion The easiest way to calculate G from eq. (5) is to express Z in an analytical form. We performed all the resulting calculations for E = kT. The compressibility factor of the inverse-12 potential can then be expressed as a fourth order series expansion in p within the accuracy of the Monte Carlo calculations of Hoover et al. [lo]. Our first goal was to calculate gas-gas equilibrium for a system comparable to helium-hydrogen since it is known that this system shows such a phase separation. Moreover, it is a well known fact that the parameters for the interaction between unlike molecules play an essential role in the phase behaviour of mixtures. In the case of helium-hydrogen, these parameters were determined from experimental data at very high pressures [6] using the exponential-6 potential. The combining rule for the volume parameter obtained from the experiment can be considered as a reasonable guide for the case that only a repulsive potential is used. Some experimental isotherms are shown in fig. 1. It is shown that, apart from the occurrence of fluid-fluid equilibrium, typical characteristics are the nearly symmetrical behaviour of the coexistence lines around the critical point at a mole fraction of helium of about 0.6, that the critical composition is approximately constant at high pressures, that the region of fluid-fluid equilibrium increases with increasing temperature and pressure. The composition of the hydrogen-rich fluid at the triple point is nearly constant at high pressures. In order to facilitate a qualitative comparison, we took o1 = 2.5 x lo-” m,

J. A. Schouten

22

et al.

I Gas-gas

separation

in mverse-12

systems

P t

6.5 GPO 6.0

He-H2

F+F 12

. a

51'5

A

S,+F

q

melting point H2

-.-

-

S,+F,+F2 three phase equil.

2.5 0

20

LO

60

80

100

mole % He 4 Fig. 1. Experimental hydrogen

isotherms

[4c] for gas-gas

and fluid-wlid

equilibrium

in the system helium-

at high pressure.

al = 2.89 x 10~ “’ m and pIz = 2.76 X 10 “’ m. These values are comparable with the values obtained for the volume parameter using the exponential-6 potential in the case of helium-hydrogen [6]. In fig. 2, the results are presented for three temperatures, 150, 225 and 300 K. We have taken ilk = T = 1.50 K. The results for the other two temperatures were calculated using the scaling properties of the inverse-12 system. It should be pointed out that the value of g,? is larger than the arithmetic mean value. For smaller values of v,~ the fluid-fluid range becomes smaller, disappearing completely under the solidification surface for v,? < 2.74. Comparison of figs. 1 and 2 shows a striking similarity in the general shape of the experimental and calculated isotherms.

J.A.

Schouten et al. I Gas-gas separation in inverse-12 systems

23

GPa

/

3.5

3

2.5

-x I

I 0.2

I

I 0.L

I

I

I

0.6

Fig. 2. Calculated isotherms for gas-gas and fluid-solid equilibrium with o, = 2.5 X 10-l” m, a, = 2.89 x lo-‘” m and flIz = 2.76 X lo-”

I 0.8

I

1

at high pressures m.

in the system

All the characteristic features mentioned above for the experimental isotherms are also present in the calculated curves. Of course, we do not expect quantitative agreement between theory and experiment. The results show that demixing in the fluid phase can be obtained in purely repulsive systems with repulsions comparable to real systems. There are, however, some interesting qualitative differences too. Due to the scaling properties, the critical line, three phase line, melting curve and the lines of constant composition all have the

J. A. Schouten et al. I Gas-gas

24

separation in inverse-12 systems

system. In contrast, all these shape p - T’.” in the case of the inverse-12 curves are much steeper for the system helium-hydrogen [12]. Of course, the results greatly depend upon the ratio of the (T’S of the pure components

and

performed range

upon

a number

from

the

value

of u,~ for given

of calculations

(a, + 0;)/2

to

a,.

for a,/~, We

(T’S, Therefore,

we have

= 1.1. 1.5 and 3, and c,: in the

always

take

(T, = 2.5 x 1OY”’ m

and

ilk = T = 150 K; the pressures are given in GPa. Table I gives the results for rr,/a, = 1.1. The second column of table I shows the critical pressure, the third the critical composition, the fourth column gives the solidification pressure of the homogeneous mixture with the critical composition, and the last column gives the effective one-fluid parameter for a mixture It should be noted that, since the fluid-fuid with the critical composition. equilibrium curves are very flat (see fig. 2), it is difficult to determine the critical composition accurately. On the other hand, the results are not very sensitive to a change in x,. As long as p, < p,, , the system exhibits fluid-fluid equilibrium. From table I, we can conclude that this will occur for values of (T,~ > 2.67. For smaller values the fluid-fluid equilibrium disappears under the solidification surface. Thus for c~/c, = 1.1, the phenomenon is not observed is about 0.5 for the arithmetic mean value (T,~ = 2.625. The critical composition and not very dependent

on the value

of c,?.

Table II shows the result for cr,/rr, = 1.5. Gas-gas phase separation occurs if LT,~> 3.20. which is again larger than the arithmetic mean value. In this case the critical composition is sensitive to the value of (T,? and becomes about 0.75

Table I Critical properties for (T,/w = I.1 and various values of q?. component I and 2 are respectively 3.0 and 2.25 GPa. cl = 2.5 (r,?

(10

I” m)

2.75 2.72 2.6’) 2.68 2.67

Table II As in table

I for az/u,

The x IO

melting pressures m. T = 150 K.

p,,,(Gpa)

P, (GPa) 0.6’)

0.49

2.40

I .(I5

0.5I

2.35

1.7 2.2 2.Y

0.51 0.5 I 0.51

2.44,

2.w 2.67 2.66

2.51

2.65

2.52

2.05

= 1.5, melting

of pure

I”

pressure

of component

2 is 0.89 GPa.

(r,? (1W’“m)

P, (GW

I‘

P,, (GW

~7~~ ( 10 “’ m)

3.75 3.30 3.24 3.20

0.06 0.44 0.92 2. I

0.56 0.62 0.65 0.70

1.23 1.53 1.65 1.80

3.36 3.13 3.05 2.91

J.A.

Schouten et al. I Gas-gas separation in inverse-12 systems

25

for or2 = 3.20. It shifts to a larger mole fraction of the smaller component in agreement with the results for real systems. In table III the results are presented for the case uJur = 3. Gas-gas phase separation occurs if urz > 4.8, so, in this case, the phenomenon also occurs for the arithmetic mean value with a relatively low value for the critical pressure. However, the critical increases rapidly if u,* is decreased. The critical composition is very sensitive to the value of ur2 and is even about 0.95 for (a, + 02)/2. It should b e noted that Kerley [9] did not find any phase a;2 = separation from his computer simulations for a,/~, = 3 and the arithmetic mean for urz. This will be explained below. The results obtained so far were acquired using the one-fluid approximation. As Kerley [9] pointed out, the one-fluid theory is good at low densites but much poorer at high densities. For an equimolar mixture the error in the compressibility factor may be as high as 30% near the solidification point. Kerley performed computer simulations on a system with u, = 1, a, = 1/3 and u,~ = (u, + ~~2)/2 = 0.667. From fig. 1 of ref. [9], it is clear than the effective value a, calculated from the one-fluid theory is too low. Combining the results of Hoover et al. [lo] for the pure r-l2 system and those of Kerley for the equimolar mixture, we have calculated the value for a, needed to get the correct compressibility factor for the equimolar mixture, if this mixture could be described as one fluid. The second column of table IV gives the comTable III As in table ciz (lo-”

I for ~~/a, = 3, melting m)

7.5 6 5.0 4.9 4.825

Table IV Effective value NU3

pressure

of component

2 is 0.11 GPa.

P, (GW

XC

P, (GP4

a+ (lo-‘”

0.005 0.024 0.32 0.76 2.3

0.62 0.78 0.95 0.96 0.97

0.18 0.45 1.73 1.94 2.17

6.43 4.70 3.00 2.89 2.79

(T
mixture

calculated

from

simulation

results,

m)

elk = T.

VV?

pl, NkT

fix

ULZ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.65

1 1.448 2.121 3.101 4.557 6.641 9.460 11.357

0.749 0.755 0.761 0.764 0.766 0.769 0.769

0.686 0.700 0.713 0.719 0.724 0.731 0.731

26

pressibility

J. A. Schouten

factor

et al.

for a series of values

we take

ilk = T, it provides

reduced

density.

agreement independent

I Gas-gas

The

between

by a single effective

in inverse-12

of the scaled

the compressibility

third

column

Hoover

et al.

of the density,

separation

presents and

Kerley.

value

as a function for u,,

It is shown

so that the equimolar

value for cr. It is, therefore,

p(e/kT)’ “‘. If

parameter

factor the

swtems

mixture

that

cannot

not possible

of the

which

gives

q, is not

be described

to modify

eq. (3)

(or the one-fluid theory) to get agreement with computer simulations. We then used eq. (3) to calculate the value for cIz which provides the correct value for o-~ (fourth column). It is shown that this value is always larger than the arithmetic mean. For the equimolar mixture, this result suggests that, if one applies the one-fluid theory, the unlike interaction parameter should be taken larger than the arithmetic mean, at least for systems with CT~/(T,= 3. It also implies that the conclusions we have drawn in the preceding section for the one-fluid approximation should be even more valid for the computer simulations and, thus, for real experiments on inverse-12 systems. The behaviour of the volume of mixing AV given by eq. (7) is quite interesting. In fig, 3 the relative volume change at 1 GPa is given for (T, = 2.5, a? = 3.75 and u,?_ is 3.75, 3.20 and 3.125. As expected, for c,? = 3.75 the relative volume change is very large and reaches a maximum for x about 0.7.

t

-X

Fig. 3. Relative volume of mixing AV* = AV/[xV:’ + (1 - x)Vq] at 1 GPa (T, = 2.5 x lo-‘” m, a, = 3.75 x lO_‘” m and various values for CT,?.

for the system

with

J.A.

Schouten et al. I Gas-gas

separation in inverse-12 systems

27

For a,, = 3.2 the volume of mixing is slightly positive but this is not enough to produce a phase separation. For (+r2= 3.125 the volume of mixing is negative over the entire concentration region. The effect of pressure is relatively small, amounting to about 15% in the range from 0.2 to 2 GPa. There is no change in sign in any of these cases. The behaviour is more complicated for the ratio uz2/u1= 3. Fig. 4 shows the results for U, = 2.5, a, = 7.5 and (or* = 5.3 and three pressures (0.008, 0.004 and 0.2 GPa). At a pressure of 0.008 GPa, AV is negative over the entire composition range. At 0.04 GPa, AV is negative for x < 0.65 and positive for x > 0.65. At 0.2 GPa, AV is negative over a larger composition range and the positive maximum is less pronounced. As the pressure is increased further, this tendency continues but AV is always positive for the highest x values. In fact, the situation is essentially the same for u112= (or + ~~2)/2 = 5.0, but the composition range where AV is positive is smaller. This figure also explains why Kerley [lo] did not find phase separation in his simulations of an equimolar mixture. Fig. 4 shows that for x = 0.5 the volume of mixing is negative at all pressures. However, as can be seen from table III, the critical composition for the system under consideration is about 0.95 and the phase separation will therefore only occur for the higher values of x. As expected, fig. 4 shows a positive AV in that region. In conclusion, we can say that a reasonable purely repulsive potential is sufficient to obtain fluid-fluid equilibrium. The simple inverse-12 potential can

Fig. 4. Relative volume of mixing AV* = AV/[xV~ + (1 - x)VfJ at various pressures for the system with u1 = 2.5 x lo-” m, q = 7.5 x lo-” m and cIz = 5.3 x lo-” m. -. -. - 0.008 GPa; 0.04 GPa: --0.2 GPa.

J. A. Schouten

28

produce

a complicated

et al.

cubic

1 Gu.s-gusseparation dependence

in inverse-12

of the

volume

systems

of mixing

on the

composition.

References [l] J.A. Schouten and L.C. van den Bergh. Fluid Phase Equilibria 32 (19X6) 1. [2] L.C. van den Bergh. J.A. Schouten and N.J. Trappeniers, Physica A 132 (1985) 549. [3] W.L. Vos and J.A. Schouten, Phys. Rev. Lett. 64 (lY90) X9X. [4a] W.B. Streett. Astrophys. J. 1X6 (1973) 1107. [4b] J.A. Schouten. L.C. van den Bergh and N.J. Trappcniers, Chem. Phys. Lett. II4 (IYXS) 52. [4c] L.C. van den Bergh, J.A. Schouten and N.J. Trappenicrs, Physica A 141 (lYX7) 524. [4d] P. Loubeyre, R. Le Toullec and J.P. Pinceaux, Phys. Rev. B 36 (1987) 3723. 151F.H. Ree. J. Phys. Chem.. X7 (1983) 2846. [61 L.C. van den Bergh and J.A. Schouten. J. Chem. Phys. XY (198X) 2336. [71 J.A. Schouten, Phys. Rep. 172 (19X9) 33. [Xl K.S. Shing and K.E. Gubbins. in: Molecular Based Study of Fluids. J.M. Haile and G.A. Mansoori. cds. (Am. Chem. Sot.. Washington, DC, 1983) p. 73. PI G.I. Kerley, J. Chem. Phys. 91 (1989) 1204. J.A. Barker and B.C. Brown. J. 1w W.G. Hoover, M. Ross. K.W. Johnson, D. Henderson. Chem. Phys. 52 (1970) 493 I. [Ill W.G. Hoover. S.G. Gray and K.W. Johnson. J. Chem. Phys. 55 (1971) 1228. IO (198’)) I. 1121J.A. Schouten, Int. J. Thcrmophys.