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Mass transfer and dynamic liquid—liquid interfacial tension. III. Theory of nonequilibrium pressure coefficient of interfacial tension
Mass Transfer and Dynamic Liquid-Liquid Interfacial Tension III. Theory of Nonequilibrium Pressure Coefficient of Interfacial Tension M. V. OSTROVSKY 1 AND R. J. G O O D Department of Chemical Engineering State University of New York, Buffalo, New York 14260 Received A u g u s t 22, 1983; accepted N o v e m b e r 1, 1984
Using a molecular-kinetic method of derivation, we have found the general equations that describe the pressure coefficient of interfacial tensions (IFT) in nonequilibrium conditions. These equations show the factors which govern the process of change of interfacial tension with pressure in multicomponent nonequilibrium systems. We have explained the nonequilibrium behavior of multicomponent liquid-liquid systems with change in pressure: the equations proposed explain the cases of positive, negative, and zero values of the pressure coefficient of IFT, as well as the influence of agitation on changes in interfacial tension. © 1985AcademicPress,Inc. change in pressure was not an essential part of the observations reported in Ref. (3).) A rapid alteration of pressure in a multiphase system will, in general, put the system into a nonequilibrium distribution of mass. For the purpose of studying nonequilibrium behavior per se, it has the advantage over change of temperature or concentration, o f not being limited by heat conduction or mass transport in the establishment of the initial nonequilibrium state. There are two consequences that are to be expected, from a change in pressure: one may be considered an equilibrium effect, and the other is a function of nonequilibrium behavior.
1. I N T R O D U C T I O N
Until recently, there has been no strong reason to expect prolonged transient time effects in interfacial tension (IFT) measurements. A simple diffusion-controlled transient is a well-known phenomenon (1, 2). Such transients would be expected to pass in times that are not more than a few orders of magnitude larger than the mechanical relaxation time of the system, e.g., in minutes, with low-viscosity liquids. Recently, we have reported (3, 4) a strong, quasi-periodic variation of IFT that persisted for several days, and gave evidence that it would continue for long after the runs were terminated. The systems were dodecane-water with the model surfactant, 8-phenylhexadecane sodium sulfonate [Refs. (3, 5)] and octane-brine with the surfactant, Witco TRS 10-80, and as cosurfactant, n- and isopropyl alcohol (4, 6). These studies were a part of a program of measurements in which pressure was varied, as a fundamental investigation o f effects that would be relevant to enhanced oil recovery by surfactant flooding. (However,
(1) An equilibrium thermodynamic analysis has recently been made (7, 8) of the effect of pressure on the interface between two phases; the principal components of which were, respectively, substances i and j. It was shown that if X21 is the distance between the r l = 0 surface and the r2 = 0 surface, then the pressure coefficient of IFT is related to ~k21 by
i Present address: Petroferm Research, Inc., P.O. Box 1567, Cambridge, Mass. 02238, USA. 140 0021-9797/85 $3.00 Copyright © 1985 by Academic Press, Inc. All fights of reproduction in any form reserved.
Journalof ColloidandInterfaceScience,Vol. 106, No. 1, July 1985
141
LIQUID-LIQUID INTERFACIAL TENSION, III
If an adsorbed film of component 3 is present, _ d7 = d'/ =-RTPm(1- Z0;) then )k21 c a n be identified with the effective T T i=1 thickness of the film. A value of 1 A for ~,21 corresponds to 0.01013 dyn/cm per atmo" F2b2 [ C d(PS)~] sphere as the value of 03"/OP. i=1 (2) The masses of the components in the phases, after a change in pressure, will no ; -Rrrm(l - 2 0,) longer be in their equilibrium distribution. T i=1 Diffusional process will start, powered by the " F2b'( d(rS)~l [la] gradients of chemical potential across the x Z k Vls C2 -- C1 ~11 interface. Redistribution of mass will be pari=1 ticularly significant in a system near a critical In the case of spherical drop we have point. d(rS) _ 2 r dr The concentrations of chemical species in + [21 dVl r dr' the interfacial region will vary with time, as the various species diffuse to and from that where region; this transport to and from the interface 3' is interfacial tension, is an essential component of transport of R is the gas constant, species from phase a to phase /3 and vice S is area of interface, versa; see Part II of this series for a theoretical T is temperature, analysis of the effect of that transport on r is the radius of a spherical drop, adsorption and interfacial tension (9). It may be expected also that the changes of IFT 0i = r J P m is the degree of surface coverage owing to the nonequilibrium state may exceed by molecules of transferred component i (i the changes due to equilibrium factor, for = 1,2 . . . n ) , a n d the same change in pressure. Fi and rm are the adsorption under dyWe will report, here, a theoretical investi- namic conditions and in a condensed monogation of the change in IFT that follow a molecular layer, respectively. change of pressure in a multicomponent, We denote the transferring and receiving two-phase system. The strong evidence that phases by the subscripts 1 and 2, respectively; we have observed for nonequilibrium behav- V1 and V2 are the volumes of the two bulk ior (4) indicates that an equilibrium ther- phases, neglecting the volume of the adsorpmodynamic approach (and even a approach tion layer and the volumes of the diffusion using nonequilibrium thermodynamics at the layers (Vls and Vzs). level presently available) would be unfruitful. We make the following definition [Ref. But a molecular-kinetic approach can be very (9)] useful. Such an approach has not been re/32i b2i [3] ported in the literature, to date. We will P m ( a l i -[- a2i ) consider the case of mass transfer with intense agitation, using a kinetic approach. flu b,i [3a]
"]di"
rm(OQi 71- OL2/) '
II. THEORY OF NONEQUILIBRIUM PRESSURE COEFFICIENT OF INTERFACIAL TENSION
We have shown earlier [Ref. (9)] that isothermal mass transfer processes, in a closed multicomponent systems with mixing, are described by the equations
where fli and at are, respectively, the adsorption and desorption constants for component i, from or to, each phase. We will use Eq. [1 ] below retaining all the assumptions made in Ref. (9), including the employment of agitation. Journal of Colloid and Interface Science, Vol. 106, No.
1, July 1985
142
OSTROVSKY
~ Let us multiply both sides of Eq. [1] by the derivative, (dVddP)T. Then we can represent Eqs. [1] and [la] in the form dy
= -RTPm(1 - ~ 0,) T
2r
dr)
r
Tr
(a>)
1 0
= RTPm(1 - ~ Oi)
d3` T
Fvq l X ~-~]TJi : d3"
where ~ is the coefficient of isobaric expansion
Cp and Cv are the heat capacity coefficients at constant pressure and volume, respectively. Using this estimate, Eqs. [4] and [4a] become
i=1
×~F2b2[c
AND GOOD
i=1
[4]
× i=1 ~ L v2s C2 Cl -
=-RTrm(1-
T
~r
i
[5l
= RTFm(i - Z Oi)
d3`
x i=1 ZL-~l ~ G-G
T
2r r
~rr
(dVl~ l
1. The degree of surface coverage by molecules that are being transferred, ~ 0 i. 2. The ratio of adsorption and desorption kinetic constants (b2 or bi values). 3. Intensities of agitation, and the viscosity near the surface since these control Vls and V2s. 4. The radius (or radii) of curvature of the surface. 5. The absolute concentrations, and the ratio of concentrations, of transferable substances in the bulk phases, as well as the values of ri, the adsorption of these substances. The value of derivative dVl/dP in Eq. [4] may be estimated qualitatively using an equilibrium approach. In cases far from any critical point, we may use the equilibrium value of an isothermal compressibility coefficient, in the form
Journalof Colloidand InterfaceScience,Vol. 106, No. 1, July 1985
× ~ L--~-l~ G - C1
r
~rr
i"
[5a]
[4a]
We can see from Eqs. [4] or [4a] that changes of IFT with pressure will depend on
T=dp_dv ,
i=1
i=1
× \ dP IT-]i"
,=-p
f
EOi) i=1
F2bt
-
In conditions near a critical point (an important case for tertiary oil recovery processes) the value of derivative (dV1/dP)r in Eqs. [4] and [4a] may be also estimated qualitatively, using scaling theory [Ref. (10)]. The isothermal compressibility of a singlecomponent system in the neighborhood of its critical point is at the form
r/=
--g
T
where Tc is the critical temperature, the value of T, where 3' --' 0. g is a constant very close to ~ for all substances [Ref. (10)]. The authors (10) showed that the same exponent is correct in multicomponent surfactant-oil-brine systems. Using this semiempirical approach, we convert Eqs. [5] and [5a] to the form d3"
= RTrm(1 T
_
~ Oi) i=1
× ~ [2V, LV2s b2L Cz - C1 i=1
2P r
drr
x I T - Tel-g]
[6] l
LIQUID-LIQUID
(;)T
INTERFACIAL TENSION,
(;)
= R T r m ( l - Z 03 i=1
[-2V1
2r
143
III
T
as T ~ To, or as 3' ---' 0. This conclusion is based on the thermodynamic analysis given in Ref. (11): the author in (11) shows that derivative aT/dr ¢ 0 in the extreme case × I T - T~l-g]i, [6a] when interfacial tension on the plane interface tends to zero. where L and g are the empirical constants of Experiment (see Table I) corroborates this the scaling theory. conclusion: there is an increase in (d3"/dP)x Equations [6] and [6a] show that the value values with decrease in 3", in the cases of low of the derivative, d3"/dP, depends on the interfacial tension. The highest value of (d3"/ displacement from the critical point. The dP)r occurs in the system with lowest 3", as absolute value of the derivative must increase was predicted by our theory. with decrease in the difference, T - To. We can see from our experimental data The variation of interfacial tension near that in systems without surfactants, IFT is critical point can be represented by equation nearly independent of pressure; see the binary [Refs. (11-13)] systems in Table I. Adding surfactants leads to an increase in (d3"/dP)T, even if the values ",[ = Q(Tc- T) ~, [71 of the IFT are the same; compare the third where Q and v are constants. The authors of and sixth lines in the table. Refs. (11, 12) show that, close to the critical Examination of Eqs. [6] or [6a] and [9] point, and [9a] shows also that the sign of the 3 derivative (d3"/dP)T may be either positive or [8] negative regardless of the interracial tension. 2 This sign depends on the system and condiUsing Eqs. [7] and [8], we can transform tions of experiment: the values of C~, C2, F, Eqs. [6] and [6a] r for all the components in the system. Experiments corroborate this conclusion: d3" = - R T r m ( 1 Oi) there are cases for the derivative (d3"/dP)x T i=1 with a negative sign in liquid-gas systems, where × ~ [_ g2s b2LO2g/3
× i=l ~ [ Vls blL C2 - C1
r
-~r
i=1
x ( C 2 - C1
2r r
dr]
Ji
[91 [experimental data in Refs. (14-23)]. References (3, 4, 7-9), and our experimental data shown in Table I, report that in liquid-liquid systems the derivative (d3"/dP)T has a positive sign at the condition
°
d'g
= -RZIam( T
1 __ £ Oi) i=l
× n~ [2Vl blLQ2g/3
i=l I. Vls × (C2 - C1
2F
r
d P] _2g/3"]
dr]
[9a]
-]i"
Examination of Eqs. [6] or [6a] and [9] or [9a] shows that
F
Equations [9] and [9a] contain volumes Vls or V2s that are functions of mixing intensity (24). They show that the value of the Journal of Colloid and InterfaceScience, VoL 106, No. 1, July 1985
OSTROVSKY AND GOOD
144
TABLE I Measurement of Interfaciat Tension as a Function of Pressure in Two-Phase Systemsa Number of oom °0ots in system
Heterogeneous system
Surfactant and its 0o°=tratioa (wt%)
~'p=~,
T
1
aT
dyn (atm -~)
None Isolubanol-water n-Butanol-water Isopentanol-water
2 2 2
0 0 0
2.03 1.75 4.94
Dodecane-watersurfactant b
3 3 3
Texas No. 1 0.0002 0.0010 0.0050
n-Octane-brinesurfactantn-propyl alcohol
5 5 5
TRS 10-80 2.1 3.3 5.1
19.0 15.0 4.7 0.550 0.135 0.015
0 0 0
0 0 0
0.0007 0.0190 0.0320
0.00004 0.00130 0.00670
0.0003 0.0030 c 0.0036
0.002 q.005 0.024
a Methods of experiment were described in Ref. (4). b Data of Ref. (3). c 50 to 100 atm.
derivative (d'y/dP)Td e p e n d s on the i n t e n s i t y o f agitation in the system. T h e higher the intensity o f agitation, t h e larger the value o f the ratio V1/V2~. In o r d e r to e x p l a i n this effect, we m a y t r a n s f o r m Eq. [6] to the f o r m
m i x i n g intensity, b e c a u s e it d e p e n d s o n the V1/V2sratio. 2 Thus, t h e equations, which we w o r k e d o u t above, describe the e x p e r i m e n t a n d give a p r e d i c t i o n o f influence o f agitation o n I F T in n o n e q u i l i b r i u m conditions. CONCLUSIONS U s i n g a m o l e c u l a r - k i n e t i c m e t h o d o f derivation, we have f o u n d t h e general e q u a t i o n s t h a t describe the pressure coefficient o f interfacial tensions in n o n e q u i l i b r i u m c o n d i t i o n s . These e q u a t i o n s show the factors w h i c h govern the process o f change o f IF1" with pressure in m u l t i c o m p o n e n t n o n e q u i l i b r i u m systems. These are
where //
a = 2Rrrm(1
- ~ 03 i=1
i=1
x IT- T¢i-gl/
[t01
T h e values o f ~2 a n d Ap d o n o t d e p e n d on m i x i n g intensity, b u t the ratio V1/V2~does. C o n s e q u e n t l y , in the case P = constant, a n d with c o n s t a n c y o f all o t h e r p a r a m e t e r s in ~2, the difference A~ (Eq. [10]) d e p e n d s on Journal of Colloid and Interface Science, Vol. 106, No. 1, July 1985
a. Interracial tension. b. A d s o r p t i o n a n d fractional surface coverage. 2 This conclusion is a qualitative one. The quantitative analysis in this direction is limited because parameters Vis and V2~are unknown. We have no basis for quantifying these volumes in our experiment.
LIQUID-LIQUID INTERFACIAL TENSION, IlI
c. d. e. f.
Drop dimensions. Concentration of transferred substances. Intensity of agitation. Influence of the other components.
We have explained the nonequilibrium behavior of multicomponent liquid-liquid systems with change in pressure: the equations proposed, above, explain the cases of positive, negative, and zero values of the pressure coefficient of IFT, as well as the influence of agitation on changes in 1FT. ACKNOWLEDGMENT This research was supported by the U.S. Department of Energy, Grant DE-ASI9-80BC10326, and by the National Science Foundation, Grant CPE-82-17971. REFERENCES 1. Ostrovsky, M. V., Theor. Found Chem. Eng. (USSR) 11, 431 (1977). 2. (a) Ostrovsky, M. V., Frumin, G. T., Kremnev, L. J., and Abramzon, A. A., J. Appl. Chem. USSR, 40, 1319, 1328 (1967); (b) Ostrovsky, M. V., and Ostrovsky, R. M., J. Colloid Interface Sci. 93, 392 (1983). 3. Good, R. J., and Sun, C. J., J. Colloid Interface Sci. 91,341 (1983). 4. Good, R. J., and Ostrovsky, M. V., J. Colloid Interface Sci. 106, 122 (1985). 5. Wade, W. H., Morgan, J. C., Schechter, R. S., Jacobsen, J. K., and Salager, J. L., SPE 6844 (1977).
145
6. Wickert, B. L., Willhite, G. P., Green, D. W., and Black, S. L., "Tertiary Oil Recovery Project." University of Kansas 1980. 7. Good, R. J., Z Colloid Interface Sci. 85, 128 (1982). 8. Good, R. J., Buff, F. P., in "The Modern Theory of Capillarity" (A. I. Rusanov and F. C. Goodrich, Eds.). Academic Verlag, Berlin, 1981. 9. Ostrovsky, M. V., Good, R. J., J. Colloid Interface Sci. 106, 131 (1985). 10. Fleming, P., and Vinatiery, J., J. ColloM Interface Sci. 81, 319 (1981). 11. Rusanov, A. I., "Phase Equilibrium and Interfacial Phenomena," pp. 70-95. Pub. Khmia, Lenigrad, USSR, (1967). [in Russian] 12. Cahn, J. W., and Milliard, J. E., J. Chem. Phys. 28, 258 (1958). 13. Rusanov, A. I., Vesm. LGU N. 4, 84 (1958). [in Russian] 14. Kundt, A., Ann. Phys. Chem. 12, 538 (1881). 15. Richards, T. W., and Carver, E. K., J. Amer. Chem. Soc. 827, (1921). 16. Pollara, L. Z., J. Phys. Chem. 46, 1163 (1942). 17. Hough, E. W., Wood, B. B., and Rzasa, M. J., J. Phys. Chem. 56, 996 (1952). 18. Cutting, C. L., and Jones, D. C., J. Chem. Soc., 4067 (1955). 19. Slowinski, E. J., Gates, E. E., and Waring, C. E., J. Phys. Chem. 61, 808 (1957). 20. Gielessen, J., and Schmatz, W. Z., Z. Phys. Chem. N. F. 27, 157 (1961). 21. Masterton, W. L., Bianchi, J., and Slowinski, E. J., J. Phys. Chem. 67, 615 (1963). 22. Rice, O. K., J. Chem. Phys. 15, 333 (1947). 23. Eriksson, J. C., Acta Chem. Scand 16, 2199 (1962). 24. Levich, V. G., "Physicochemical Hydrodynamics," Phizmatgiz, Moscow, 1959. [in Russian]
Journal of Colloid and Interface Science, Vol. 106, No. 1, July 1985
Using a molecular-kinetic method of derivation, we have found the general equations that describe the pressure coefficient of interfacial tensions (IFT) in nonequilibrium conditions. These equations show the factors which govern the process of change of interfacial tension with pressure in multicomponent nonequilibrium systems. We have explained the nonequilibrium behavior of multicomponent liquid-liquid systems with change in pressure: the equations proposed explain the cases of positive, negative, and zero values of the pressure coefficient of IFT, as well as the influence of agitation on changes in interfacial tension. © 1985AcademicPress,Inc. change in pressure was not an essential part of the observations reported in Ref. (3).) A rapid alteration of pressure in a multiphase system will, in general, put the system into a nonequilibrium distribution of mass. For the purpose of studying nonequilibrium behavior per se, it has the advantage over change of temperature or concentration, o f not being limited by heat conduction or mass transport in the establishment of the initial nonequilibrium state. There are two consequences that are to be expected, from a change in pressure: one may be considered an equilibrium effect, and the other is a function of nonequilibrium behavior.
1. I N T R O D U C T I O N
Until recently, there has been no strong reason to expect prolonged transient time effects in interfacial tension (IFT) measurements. A simple diffusion-controlled transient is a well-known phenomenon (1, 2). Such transients would be expected to pass in times that are not more than a few orders of magnitude larger than the mechanical relaxation time of the system, e.g., in minutes, with low-viscosity liquids. Recently, we have reported (3, 4) a strong, quasi-periodic variation of IFT that persisted for several days, and gave evidence that it would continue for long after the runs were terminated. The systems were dodecane-water with the model surfactant, 8-phenylhexadecane sodium sulfonate [Refs. (3, 5)] and octane-brine with the surfactant, Witco TRS 10-80, and as cosurfactant, n- and isopropyl alcohol (4, 6). These studies were a part of a program of measurements in which pressure was varied, as a fundamental investigation o f effects that would be relevant to enhanced oil recovery by surfactant flooding. (However,
(1) An equilibrium thermodynamic analysis has recently been made (7, 8) of the effect of pressure on the interface between two phases; the principal components of which were, respectively, substances i and j. It was shown that if X21 is the distance between the r l = 0 surface and the r2 = 0 surface, then the pressure coefficient of IFT is related to ~k21 by
i Present address: Petroferm Research, Inc., P.O. Box 1567, Cambridge, Mass. 02238, USA. 140 0021-9797/85 $3.00 Copyright © 1985 by Academic Press, Inc. All fights of reproduction in any form reserved.
Journalof ColloidandInterfaceScience,Vol. 106, No. 1, July 1985
141
LIQUID-LIQUID INTERFACIAL TENSION, III
If an adsorbed film of component 3 is present, _ d7 = d'/ =-RTPm(1- Z0;) then )k21 c a n be identified with the effective T T i=1 thickness of the film. A value of 1 A for ~,21 corresponds to 0.01013 dyn/cm per atmo" F2b2 [ C d(PS)~] sphere as the value of 03"/OP. i=1 (2) The masses of the components in the phases, after a change in pressure, will no ; -Rrrm(l - 2 0,) longer be in their equilibrium distribution. T i=1 Diffusional process will start, powered by the " F2b'( d(rS)~l [la] gradients of chemical potential across the x Z k Vls C2 -- C1 ~11 interface. Redistribution of mass will be pari=1 ticularly significant in a system near a critical In the case of spherical drop we have point. d(rS) _ 2 r dr The concentrations of chemical species in + [21 dVl r dr' the interfacial region will vary with time, as the various species diffuse to and from that where region; this transport to and from the interface 3' is interfacial tension, is an essential component of transport of R is the gas constant, species from phase a to phase /3 and vice S is area of interface, versa; see Part II of this series for a theoretical T is temperature, analysis of the effect of that transport on r is the radius of a spherical drop, adsorption and interfacial tension (9). It may be expected also that the changes of IFT 0i = r J P m is the degree of surface coverage owing to the nonequilibrium state may exceed by molecules of transferred component i (i the changes due to equilibrium factor, for = 1,2 . . . n ) , a n d the same change in pressure. Fi and rm are the adsorption under dyWe will report, here, a theoretical investi- namic conditions and in a condensed monogation of the change in IFT that follow a molecular layer, respectively. change of pressure in a multicomponent, We denote the transferring and receiving two-phase system. The strong evidence that phases by the subscripts 1 and 2, respectively; we have observed for nonequilibrium behav- V1 and V2 are the volumes of the two bulk ior (4) indicates that an equilibrium ther- phases, neglecting the volume of the adsorpmodynamic approach (and even a approach tion layer and the volumes of the diffusion using nonequilibrium thermodynamics at the layers (Vls and Vzs). level presently available) would be unfruitful. We make the following definition [Ref. But a molecular-kinetic approach can be very (9)] useful. Such an approach has not been re/32i b2i [3] ported in the literature, to date. We will P m ( a l i -[- a2i ) consider the case of mass transfer with intense agitation, using a kinetic approach. flu b,i [3a]
"]di"
rm(OQi 71- OL2/) '
II. THEORY OF NONEQUILIBRIUM PRESSURE COEFFICIENT OF INTERFACIAL TENSION
We have shown earlier [Ref. (9)] that isothermal mass transfer processes, in a closed multicomponent systems with mixing, are described by the equations
where fli and at are, respectively, the adsorption and desorption constants for component i, from or to, each phase. We will use Eq. [1 ] below retaining all the assumptions made in Ref. (9), including the employment of agitation. Journal of Colloid and Interface Science, Vol. 106, No.
1, July 1985
142
OSTROVSKY
~ Let us multiply both sides of Eq. [1] by the derivative, (dVddP)T. Then we can represent Eqs. [1] and [la] in the form dy
= -RTPm(1 - ~ 0,) T
2r
dr)
r
Tr
(a>)
1 0
= RTPm(1 - ~ Oi)
d3` T
Fvq l X ~-~]TJi : d3"
where ~ is the coefficient of isobaric expansion
Cp and Cv are the heat capacity coefficients at constant pressure and volume, respectively. Using this estimate, Eqs. [4] and [4a] become
i=1
×~F2b2[c
AND GOOD
i=1
[4]
× i=1 ~ L v2s C2 Cl -
=-RTrm(1-
T
~r
i
[5l
= RTFm(i - Z Oi)
d3`
x i=1 ZL-~l ~ G-G
T
2r r
~rr
(dVl~ l
1. The degree of surface coverage by molecules that are being transferred, ~ 0 i. 2. The ratio of adsorption and desorption kinetic constants (b2 or bi values). 3. Intensities of agitation, and the viscosity near the surface since these control Vls and V2s. 4. The radius (or radii) of curvature of the surface. 5. The absolute concentrations, and the ratio of concentrations, of transferable substances in the bulk phases, as well as the values of ri, the adsorption of these substances. The value of derivative dVl/dP in Eq. [4] may be estimated qualitatively using an equilibrium approach. In cases far from any critical point, we may use the equilibrium value of an isothermal compressibility coefficient, in the form
Journalof Colloidand InterfaceScience,Vol. 106, No. 1, July 1985
× ~ L--~-l~ G - C1
r
~rr
i"
[5a]
[4a]
We can see from Eqs. [4] or [4a] that changes of IFT with pressure will depend on
T=dp_dv ,
i=1
i=1
× \ dP IT-]i"
,=-p
f
EOi) i=1
F2bt
-
In conditions near a critical point (an important case for tertiary oil recovery processes) the value of derivative (dV1/dP)r in Eqs. [4] and [4a] may be also estimated qualitatively, using scaling theory [Ref. (10)]. The isothermal compressibility of a singlecomponent system in the neighborhood of its critical point is at the form
r/=
--g
T
where Tc is the critical temperature, the value of T, where 3' --' 0. g is a constant very close to ~ for all substances [Ref. (10)]. The authors (10) showed that the same exponent is correct in multicomponent surfactant-oil-brine systems. Using this semiempirical approach, we convert Eqs. [5] and [5a] to the form d3"
= RTrm(1 T
_
~ Oi) i=1
× ~ [2V, LV2s b2L Cz - C1 i=1
2P r
drr
x I T - Tel-g]
[6] l
LIQUID-LIQUID
(;)T
INTERFACIAL TENSION,
(;)
= R T r m ( l - Z 03 i=1
[-2V1
2r
143
III
T
as T ~ To, or as 3' ---' 0. This conclusion is based on the thermodynamic analysis given in Ref. (11): the author in (11) shows that derivative aT/dr ¢ 0 in the extreme case × I T - T~l-g]i, [6a] when interfacial tension on the plane interface tends to zero. where L and g are the empirical constants of Experiment (see Table I) corroborates this the scaling theory. conclusion: there is an increase in (d3"/dP)x Equations [6] and [6a] show that the value values with decrease in 3", in the cases of low of the derivative, d3"/dP, depends on the interfacial tension. The highest value of (d3"/ displacement from the critical point. The dP)r occurs in the system with lowest 3", as absolute value of the derivative must increase was predicted by our theory. with decrease in the difference, T - To. We can see from our experimental data The variation of interfacial tension near that in systems without surfactants, IFT is critical point can be represented by equation nearly independent of pressure; see the binary [Refs. (11-13)] systems in Table I. Adding surfactants leads to an increase in (d3"/dP)T, even if the values ",[ = Q(Tc- T) ~, [71 of the IFT are the same; compare the third where Q and v are constants. The authors of and sixth lines in the table. Refs. (11, 12) show that, close to the critical Examination of Eqs. [6] or [6a] and [9] point, and [9a] shows also that the sign of the 3 derivative (d3"/dP)T may be either positive or [8] negative regardless of the interracial tension. 2 This sign depends on the system and condiUsing Eqs. [7] and [8], we can transform tions of experiment: the values of C~, C2, F, Eqs. [6] and [6a] r for all the components in the system. Experiments corroborate this conclusion: d3" = - R T r m ( 1 Oi) there are cases for the derivative (d3"/dP)x T i=1 with a negative sign in liquid-gas systems, where × ~ [_ g2s b2LO2g/3
× i=l ~ [ Vls blL C2 - C1
r
-~r
i=1
x ( C 2 - C1
2r r
dr]
Ji
[91 [experimental data in Refs. (14-23)]. References (3, 4, 7-9), and our experimental data shown in Table I, report that in liquid-liquid systems the derivative (d3"/dP)T has a positive sign at the condition
°
d'g
= -RZIam( T
1 __ £ Oi) i=l
× n~ [2Vl blLQ2g/3
i=l I. Vls × (C2 - C1
2F
r
d P] _2g/3"]
dr]
[9a]
-]i"
Examination of Eqs. [6] or [6a] and [9] or [9a] shows that
F
Equations [9] and [9a] contain volumes Vls or V2s that are functions of mixing intensity (24). They show that the value of the Journal of Colloid and InterfaceScience, VoL 106, No. 1, July 1985
OSTROVSKY AND GOOD
144
TABLE I Measurement of Interfaciat Tension as a Function of Pressure in Two-Phase Systemsa Number of oom °0ots in system
Heterogeneous system
Surfactant and its 0o°=tratioa (wt%)
~'p=~,
T
1
aT
dyn (atm -~)
None Isolubanol-water n-Butanol-water Isopentanol-water
2 2 2
0 0 0
2.03 1.75 4.94
Dodecane-watersurfactant b
3 3 3
Texas No. 1 0.0002 0.0010 0.0050
n-Octane-brinesurfactantn-propyl alcohol
5 5 5
TRS 10-80 2.1 3.3 5.1
19.0 15.0 4.7 0.550 0.135 0.015
0 0 0
0 0 0
0.0007 0.0190 0.0320
0.00004 0.00130 0.00670
0.0003 0.0030 c 0.0036
0.002 q.005 0.024
a Methods of experiment were described in Ref. (4). b Data of Ref. (3). c 50 to 100 atm.
derivative (d'y/dP)Td e p e n d s on the i n t e n s i t y o f agitation in the system. T h e higher the intensity o f agitation, t h e larger the value o f the ratio V1/V2~. In o r d e r to e x p l a i n this effect, we m a y t r a n s f o r m Eq. [6] to the f o r m
m i x i n g intensity, b e c a u s e it d e p e n d s o n the V1/V2sratio. 2 Thus, t h e equations, which we w o r k e d o u t above, describe the e x p e r i m e n t a n d give a p r e d i c t i o n o f influence o f agitation o n I F T in n o n e q u i l i b r i u m conditions. CONCLUSIONS U s i n g a m o l e c u l a r - k i n e t i c m e t h o d o f derivation, we have f o u n d t h e general e q u a t i o n s t h a t describe the pressure coefficient o f interfacial tensions in n o n e q u i l i b r i u m c o n d i t i o n s . These e q u a t i o n s show the factors w h i c h govern the process o f change o f IF1" with pressure in m u l t i c o m p o n e n t n o n e q u i l i b r i u m systems. These are
where //
a = 2Rrrm(1
- ~ 03 i=1
i=1
x IT- T¢i-gl/
[t01
T h e values o f ~2 a n d Ap d o n o t d e p e n d on m i x i n g intensity, b u t the ratio V1/V2~does. C o n s e q u e n t l y , in the case P = constant, a n d with c o n s t a n c y o f all o t h e r p a r a m e t e r s in ~2, the difference A~ (Eq. [10]) d e p e n d s on Journal of Colloid and Interface Science, Vol. 106, No. 1, July 1985
a. Interracial tension. b. A d s o r p t i o n a n d fractional surface coverage. 2 This conclusion is a qualitative one. The quantitative analysis in this direction is limited because parameters Vis and V2~are unknown. We have no basis for quantifying these volumes in our experiment.
LIQUID-LIQUID INTERFACIAL TENSION, IlI
c. d. e. f.
Drop dimensions. Concentration of transferred substances. Intensity of agitation. Influence of the other components.
We have explained the nonequilibrium behavior of multicomponent liquid-liquid systems with change in pressure: the equations proposed, above, explain the cases of positive, negative, and zero values of the pressure coefficient of IFT, as well as the influence of agitation on changes in 1FT. ACKNOWLEDGMENT This research was supported by the U.S. Department of Energy, Grant DE-ASI9-80BC10326, and by the National Science Foundation, Grant CPE-82-17971. REFERENCES 1. Ostrovsky, M. V., Theor. Found Chem. Eng. (USSR) 11, 431 (1977). 2. (a) Ostrovsky, M. V., Frumin, G. T., Kremnev, L. J., and Abramzon, A. A., J. Appl. Chem. USSR, 40, 1319, 1328 (1967); (b) Ostrovsky, M. V., and Ostrovsky, R. M., J. Colloid Interface Sci. 93, 392 (1983). 3. Good, R. J., and Sun, C. J., J. Colloid Interface Sci. 91,341 (1983). 4. Good, R. J., and Ostrovsky, M. V., J. Colloid Interface Sci. 106, 122 (1985). 5. Wade, W. H., Morgan, J. C., Schechter, R. S., Jacobsen, J. K., and Salager, J. L., SPE 6844 (1977).
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6. Wickert, B. L., Willhite, G. P., Green, D. W., and Black, S. L., "Tertiary Oil Recovery Project." University of Kansas 1980. 7. Good, R. J., Z Colloid Interface Sci. 85, 128 (1982). 8. Good, R. J., Buff, F. P., in "The Modern Theory of Capillarity" (A. I. Rusanov and F. C. Goodrich, Eds.). Academic Verlag, Berlin, 1981. 9. Ostrovsky, M. V., Good, R. J., J. Colloid Interface Sci. 106, 131 (1985). 10. Fleming, P., and Vinatiery, J., J. ColloM Interface Sci. 81, 319 (1981). 11. Rusanov, A. I., "Phase Equilibrium and Interfacial Phenomena," pp. 70-95. Pub. Khmia, Lenigrad, USSR, (1967). [in Russian] 12. Cahn, J. W., and Milliard, J. E., J. Chem. Phys. 28, 258 (1958). 13. Rusanov, A. I., Vesm. LGU N. 4, 84 (1958). [in Russian] 14. Kundt, A., Ann. Phys. Chem. 12, 538 (1881). 15. Richards, T. W., and Carver, E. K., J. Amer. Chem. Soc. 827, (1921). 16. Pollara, L. Z., J. Phys. Chem. 46, 1163 (1942). 17. Hough, E. W., Wood, B. B., and Rzasa, M. J., J. Phys. Chem. 56, 996 (1952). 18. Cutting, C. L., and Jones, D. C., J. Chem. Soc., 4067 (1955). 19. Slowinski, E. J., Gates, E. E., and Waring, C. E., J. Phys. Chem. 61, 808 (1957). 20. Gielessen, J., and Schmatz, W. Z., Z. Phys. Chem. N. F. 27, 157 (1961). 21. Masterton, W. L., Bianchi, J., and Slowinski, E. J., J. Phys. Chem. 67, 615 (1963). 22. Rice, O. K., J. Chem. Phys. 15, 333 (1947). 23. Eriksson, J. C., Acta Chem. Scand 16, 2199 (1962). 24. Levich, V. G., "Physicochemical Hydrodynamics," Phizmatgiz, Moscow, 1959. [in Russian]
Journal of Colloid and Interface Science, Vol. 106, No. 1, July 1985