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A model for the low interfacial tension of the hydrocarbon-water-surfactant system
A Model for the Low Interfacial Tension of the Hyd rocarbon-Water-Surfactant System 1 P. R. ANTONIEWICZ AND R. RODRIGUEZ Department o f Physics, University o f Texas, Austin, Texas 78712 Received April 20, 1977; accepted S e p t e m b e r 27, 1977 A p h e n o m e n o l o g i c a l model of the h y d r o c a r b o n - w a t e r - s u r f a c t a n t s y s t e m is presented. Postulating a surfactant m o l e c u l e - i n t e r f a c e interaction and lateral interactions a m o n g the surfactant molecules which are confined to a lattice on the interface, a free energy for the surface p h a s e is calculated and t h e interfacial tension is found. In particular, the model s e e m s to a c c o u n t for the striking m i n i m a in the interfacial tension o f the s y s t e m as a function o f alkane carbon n u m b e r , which are o b s e r v e d experimentally. Several interaction energy p a r a m e t e r s are obtained by fitting t h e theoretical model to the experimental curves.
Recently there has been great interest in the observed reduction of hydrocarbonwater interfacial tensions to very low values by introducing appropriate surfactants (1- 5). Part of the interest is due to the potential value in tertiary oil recovery. The mechanism itself, however, is not well understood. A group headed by R. S. Schechter and W. H. Wade has made a systematic study (1-3) of the trends of the surface tensions for various homologous hydrocarbon series with several surfactant formulations. For example, they observe a dramatic dip in the interfacial tension as a function of the number of carbon atoms, in the alkane (5) used in the mixture. There is a decrease of several orders of magnitude in the interfacial tension at a particular value of the carbon number (see Fig. 1) (6). A number of hydrocarbon series and surfactants have been investigated by Wade and Schechter. Seve r n empirical rules for the prediction of the interfacial tension minimum have been formulated and can predict, with good accuracy, for a given mix of hydrocarbons, the combination of surfactants which will yield a tension minimum.
The microscopic behavior of such systems is not understood. The theoretical underpinnings for the empirical relations are not known and consequently insight into the behavior of the systems is missing. A number of studies of surface and interfacial tensions have appeared recently (7-11);, however, they deal principally with relatively simple liquids and do not address the surfactant problem directly. In this communication, we present a simple model of a surfactant-interface system, which nevertheless shows many of the features observed experimentally. Extensive experimental work has been done by Cayais et al. (6), in observing the systematics of interfacial tension minima using the spinning drop technique. They have developed the concept of the equivalent alkane carbon number (EACN) which is the parameter varied in the experiments (3). They obtain a minimum in the interfacial tension for a particular EACN. Any hydrocarbon may be assigned an EACN value and a simple averaging rule has been established for hydrocarbon mixtures: (EACN)m~xtu~e = • (EACN),X~,
1 Supported in part by the Office o f N a v a l Research.
[1]
i 320
0021-9797/78/0642-0320502.00/0 Copyright © 1978by AcademicPress, Inc; All fights of reproduction in any form reserved.
Journal of CoUoidand Interface Science, Vol. 64, No. 2, April 1978
H Y D R O C A R B O N - W A T E R - S U R F A C T A N T I N T E R F A C I A L TENSION
where X~ is the mole fraction of the i-th component. For instance, a mixture of undecane (EACN 11) and heptyl benzene (EACN 7) with X~ = ½ has an EACN of 9. Therefore, if a surfactant gives a minimum interfacial tension for nonane (EACN 9), it will also give a low interfacial tension for the above mixture. Mixtures of different surfactants also behave in such an averaged way. A typical graph of the interfacial tension of an a l k a n e - w a t e r - s u r f a c t a n t system plotted as a function of EACN is shown in Fig. 1. There is a dip of two orders of magnitude in the interfacial tension in the vicinity of a particular value of the EACN which in this case corresponds to heptane. It seems evident that the surfactant interface interaction cannot be highly specific but must depend on some averaged properties of the interface. A model describing the behavior of the system must therefore depend on average interaction energies, average chemical potentials and so on.
THEORY
In order to explain the observed changes in the interfacial tension and in particular the striking minimum for a particular hydrocarbon (or EACN) we have used the following model of the physical situation. Even though separation cannot in principle be precise, we assume three distinct phases; an aqueous phase with surfactant molecules present, a hydrocarbon phase, and an interface upon which surfactant molecules may be adsorbed. The waterhydrocarbon interface has an interfacial tension y = y0 in the absence of surfactant molecules. We do not address ourselves to the calculation of 3'0 but only the changes in the interfacial tension due to surfactant adsorption. Upon introduction of the surfactant, the aqueous phase acts as a reservoir for the surfactant molecules which occur both as single molecules and as members of micelles. These molecules are
321
described by a chemical potential /x. The surfactant molecules are free to adsorb on the hydrocarbon-water interface. The surfactant molecules have a hydrophobic hydrocarbon tail and a hydrophilic polar head aligning themselves therefore with the head in the aqueous phase and the tail in the hydrocarbon. There is an energy of intercation e for the adsorption, and the surfactant molecules adsorbed on the interface interact among themselves through the intermediary of the interface. To facilitate the calculation, we have assumed a lattice model for the surface phase. A surfactant molecule may be adsorbed on a lattice site or the site may be empty. Surfactant molecules adsorbed on adjacent lattice sites interact with an energy w and there are Z nearest neighbors. One could consider more sophisticated lattice models, however, we feel it is probably not justified in this situation. In addition, since surfactant molecules have long hydrocarbon chains, we include a parameter representing the change in the internal free energy of the surfactant molecule. Considering the above model, we calculate the change in the free energy of the interface from which we can then find the change in the interracial tension. The problem then is analogous to the adsorption isotherm derived initially by Fowler and Guggenheim (12) and elaborated more recently by Honig (13). Fowler and Guggenheim as contrasted to Langmuir (14) include lateral interactions between neighboring molecules in the analysis of the adsorption isotherm. We use the technique as presented by Honig to calculate the change in the free energy of the interface. The Fowler-Guggenheim approximation considers two entities in the model: sites and bonds. The sites are individual lattice sites which can be either occupied or empty; bonds are the interactions between two neighboring sites. The occupational probability of a bond between two empty sites is /30, between one empty and one Journal of Colloid and Interface Science, VoI. 64, No. 2, April 1978
322
ANTONIEWlCZ AND RODRIGUEZ
filled site is 2/31, and between filled sites is /32- The occupational probability of a filled site is oq and an empty site is ot0. The total energy of an adsorbing interface is: NZ
E = ~ -
2
N(Z
(/30Egg 4- 2/31EAB 4- /32EBB) -
1)(ozoEA + OQEB) 4- N~I/~, [2]
where N is the total number of sites, Z is the number of nearest neighbors, EAA is the energy of two nearest neighbor empty sites (it is equal to twice the surface energy per
site of the bare interface), EAB is the energy of one filled and one empty nearest neighbor sites, EBB is the energy of two filled nearest neighbor sites, including w, the interaction energy between nearest neighbors. Also, EA is the energy of an empty site and EB is the energy of a filled site. Finally, /~s is the change in the internal energy of an adsorbed molecule. It is important to keep track of the number of bonds and sites since their total is larger than the number of sites in the first place (13). The entropy due to the presence of the surfactant molecules is:
(NZ/2)!
S = kin (/3oNZ/2)
!{(/31NZ/2)!}2(/32NZ/2)! -
kln
where Ss is the change of the internal entropy of the adsorbing molecule. This term will principally arise from the restriction on the number of configurations which the tail of the surfactant molecule may have. The various occupational probabilities are not independent quantities but have the following restrictions: t~0 + ~1 = 1,/30 + 2/31 +/32 = l, which are statements of the conservation of probability; and, al =/31 +/32 is a consistency relation requiring that the number of filled sites be the same whether obtained by counting filled sites or counting appropriate bonds between pairs of sites. This analysis leaves two independent variables which have to be obtained by minimizing the free energy with respect to the n u m b e r of filled nearest neighbor pairs, /32, and by finding the chemical potential of the adsorbed surfactant molecules and equating it to the chemical potential of the surfactant molecules in the aqueous phase. The Helmholtz free energy for the surface phase is F = E - T S and we have to apply the following two thermodynamic equilibrium conditions mentioned above: OF tZaq.eo.~
=
tZ~.a.ce
--
-
-
ONfill~d
,
[4]
Journal of Colloid and Interface Science, V o l . 64, N o . 2, A p r i l 1978
{(Z-
{(Z-
1)N}!
1)Nao}!{(Z- 1)Na~}!
+ NoqSs,
[3]
and -
OF -
-
0.
[5]
ONpairs We thus have five equations in five unknowns So, ~1, /30, /31, /32 a system which is solved numerically. The interfacial tension, y, is defined as the change in the
i 0 -j.
-2 to v z o
I 0 -z"
10`3.
t.u z
EACN
FIG. 1. Sample graph taken by Cayais, Schechter, and Wade (6). The surfactant was petronate at 0.2% concentration with NaCl at 1% concentration and T = 27°C. The abscissa is a simple alkane scan.
HYDROCARBON-WATER-SURFACTANT INTERFACIAL TENSION
323
To be able to plot the interfacial tension as a function of E A C N , we assume that changing the h y d r o c a r b o n in the substrate changes the interaction linearly between adsorbed molecules and likewise between the adsorbed molecules and the substrate, that is: w = w0 + Wl x ( E A C N ) ,
[8]
e = e0 + el x (EACN).
[9]
and
C3 5
7
9
11
13
15
17
EnCN
FIG. 2. Theoretical fit to experimental data (+) of Wade et al. The surfactant was sodium 4[1-pentylnonyl]benzene sulfonate, which we will denote as Cs-C-Cs. The surfactant concentration was 0.7% with 2% isopropyl alcohol and 0.32% NaC1 as solutes. The temperature was 27°C and the area we chose here was 40 A~. free energy as a function of the total surface area,
OF Y = O(AN)
1 A
OF ON
[63
where A is the area per lattice site and N is the n u m b e r of lattice sites. Hence:
The calculated change in w and e over the span of alkanes is relatively small making the use of higher order terms unimportant. At this point, we have five energy parameters (w0, w~, Fs - eo, et, /x) which we choose in a way to minimize the error between the experimental points and the calculated curves. A simplex minimization procedure is used in order to obtain the best fit. Figures 2 and 3 show the fits to the data for 0.07% surfactant solution in water in the presence of 0.32% sodium chloride versus the equivalent alkane carbon n u m b e r of the hydrocarbon. The parameters for the fit in the Figs. 2 and 3 are given in Table I, along with the calculated parameters for different values of the area per molecule, A.
3/ = [(BoEAA -4- 2fl,EAB + fl~EnB)Z/2 r cz_
+ (a0EA + a~En)'(1 - Z) + a~Ps + kT{(/30 In/30 + 2fll In/31 c3
+ r2 In flz)Z/2 + (a0 In ao + a~ In a0(1 - Z)}]/A.
[7]
Further simplifications and details of the numerical solution are indicated in the Appendix. COMPARISON WITH EXPERIMENT The parameters in the equations above were chosen so that the surface tension, computed from Eq. [A3], fit the experimental data. The choice of parameters for a given set of experimental data appears unique and the detailed shape of the c o m p u t e d curves is sensitive to their values.
c3
aJ
+
uJ
co i -
10
-
T
~
~3
i
i
12
13
~
-
-
i
14 ~s ERCN
i
i
1~
17
FIG. 3. Same as Fig. 2 except the surfactant was sodium 4[1-hexylnonyl]benzenesulfonate, abbreviated as Cs-C-Cs. Journal of Colloid and Interface Science,
Vol.
64, No.
2, April
1978
324
ANTONIEWICZ AND RODRIGUEZ TABLEI
Energy Parameters Obtained by Fitting the Interfacial Tension Eq. [A3] to Experimental Data for Two Kinds of Surfactant Molecules" Area A s 30
40
50
60
- 15.36 -12.67 0.0836 -0.984 0.0328
- 19.56 -16.83 0.0826 -0.925 0.0336
C~-C-C8 /x ffs - eo E1 w0 wl
- 10.22 -7.51 0.0556 -0.922 0.0234
- 12.98 -10.30 0.0669 -0.904 0.0272
CrC-C8 /~ Fs - eo el w0 wl
-10.30 -7.70 0.0520 -0.874 0.0219
-11.89 -9.67 0.0631 -0.717 0.0242
-15.36 -12.39 0.0921 - 1.016 0.0360
-17.33 -13.41 0.1475 - 1.364 0.0549
a All values are in units of kT (dimensionless). We chose the close packed value of Z = 6 for this calculation.
We see first that the net interaction is attractive between two surfactant molecules. For increasing EACN, E, the binding energy to the surface, increases (i.e., the surfactant molecules are more tightly bound) and - w , the attractive interaction energy, decreases (weaker interaction). Another observation which emerges from the calculation is that the change in coverage for the whole span of EACN is small, on the order of 1%. We have included in the calculation the tension of the bare h y d r o c a r b o n - w a t e r interface which varies from 50 dyn/cm for a p e n t a n e water interface (EACN = 5) to 54 dyn/cm for a h e x a d e c a n e - w a t e r interface (15). The numerical results indicate that the changes in the adsorption energy of the surfactant molecules, ~, and their lateral interactions, w, are linear functions of the EACN of the constituent alkanes. (This correlates with the empirically derived equivalent EACN of mixtures of alkanes.) We get the same result if we take a linear combination of the interaction parameters of the constituents of the mixture. The Journal of Colloid and Interface Science, Vol. 64, No. 2, April 1978
assumption is that interaction energies are an average over a finite volume of the interface. Thus the interaction between surfactant molecules is through the intermediary of the average interface. We believe, therefore, that we have a simple yet consistent model of the anomaly in the interfacial tension of a hydrocarbon with water in the presence of a surfactant. The parameters which we obtain from fitting to experimental data are physically reasonable. Presently we are working to expand this simple model, in particular to represent the surface phase as a liquid. We will then fit this model to the large amount of experimental data which is now becoming available. From systematic data on the energy parameters, we hope to elaborate upon the theory of interfacial adsorption and interactions. APPENDIX
In order to do the numerical work, we rewrite the pertinent equations in the following way. First of all one can write: EAA =
2EA;
EA 8 = E A + EB; EB = E A --
EB~ = 2EB + w;
e; EA = 70A. [A1]
Consequently one has the following set of equations to solve: ao + % = l, /3o + 23~ + 132 = 1, oq =/31 +/32,
fiofl~ = fix 2 e x p [ - w / k T ], • l Z a o z - 1 --__ fl0ZOgl z - 1
. . . . . . × exp[(iz + e - i:s)/kT].
[A2]
The interfacial tension 3' is evaluated finally as ')/ ~" 3"0 ÷
{(I~'S - - E ) a i
÷
Zwfl2/2
+ kT[(fioln[3o + 2fillnfll + f121n[32)Z/2 + (% In ~ + oq In al)(1 - Z)]}/A. [A3]
HYDROCARBON-WATER-SURFACTANT INTERFACIAL TENSION
In the above equations Z is the number of nearest neighbors; Z = 4 for a square lattice and Z = 6 for a hexagonal lattice. The temperature, T, is 300°K. The area per lattice site has to be chosen, compatible with the physical area covered by a surfacrant molecule. In this calculation we have used area A = 30, 40, 50, and 60 .~2 in order to see the area dependence of the parameters. We have also used the data of Legrand and Gaines (15) to find 3/0 as a function of E A C N , namely, Y0 = 48.57 + (0.2912) × ( E A C N ) (dyn/cm). There are essentially three parameters which are obtained from fitting the calculated interfacial tension to the experimental curves. These are/x, the chemical potential in the aqueous phase, w, the interaction energy b e t w e e n neighboring surfactant molecules, and Fs - ~, the change in the energy of the system w h e n an isolated surfactant molecule is adsorbed on the interface. ACKNOWLEDGMENT We would like to thank Professor W. H. Wade and his group for their encouragement and for supplying us with the experimental data.
325
and Wade, W. H., Paper SPE 5813, Presented: SPE Symposium on Improved Oil Recovery, March 22-24, 1976. 2. Morgan, J. C., Schechter, R. S., and Wade, W. H., in "Improved Oil Recovery by Surfactant and Polymer Flooding" (D. O. Shah and R. S. ScheCter, Eds.), pp. 101-118. Academic Press, N. Y., 1977. 3. Cayais, J. L., Schechter, R. S., and Wade, W. H., ACS Symp. Ser. No. 8, 234 (1975). 4. Melrose, J. C., and Brandner, C. F., J. Canad. Petrol. Technol. 54 (Oct.-Dec. 1974). 5. Taber, J. J., Soc. Petrol. Eng. J. 9, 3 (1969). 6. Cayais, J. L., Schechter, R. S., and Wade, W. H., ACS Syrup. Set. No. 8, 235 (1975). 7. Davis, H. T., J. Chem. Phys. 62, 3412 (1975). 8. Bongiorno, V., and Davis, H. T. Phys. Rev. A 12, 2213 (1975). 9. Yang, A. J. M., Fleming, P. D., III, and Gibbs, J. H., J. Chem. Phys. 64, 3732 (1976). 10. Percus, J. K., J. Statistical Phys. 15, 423 (1976). 11. Ebner, C., Saam, W. F., and Strond, D., Phys. Rev. A 14, 2264 (1976). 12. Fowler, R. H., and Guggenheim, E. A., "Statistical Thermodynamics," Chap. 12. Cambridge Univ. Press, New York, 1939. 13. Honig, J. M., in "The Solid-Gas Interface" (E. Alison Flood, Ed.), Vol. 1, pp. 371-396. Dekker, New York, 1967.
REFERENCES
14. Langmuir, I., J. Amer. Chem. Soc. 38, 2267 (1916); 40, 1361 (1918).
1. Cash, R. L., Cayais, J. L., Fournier, G., Jacobson, J. K., Schares, T., Schechter, R. S.,
15. LeGrand, D. G., and Gaines, G. L., Jr., J. Colloid Interface Sci. 42, 181 (1973).
Journal of Colloid and Interface Science, Vol. 64, No. 2, April 1978
Recently there has been great interest in the observed reduction of hydrocarbonwater interfacial tensions to very low values by introducing appropriate surfactants (1- 5). Part of the interest is due to the potential value in tertiary oil recovery. The mechanism itself, however, is not well understood. A group headed by R. S. Schechter and W. H. Wade has made a systematic study (1-3) of the trends of the surface tensions for various homologous hydrocarbon series with several surfactant formulations. For example, they observe a dramatic dip in the interfacial tension as a function of the number of carbon atoms, in the alkane (5) used in the mixture. There is a decrease of several orders of magnitude in the interfacial tension at a particular value of the carbon number (see Fig. 1) (6). A number of hydrocarbon series and surfactants have been investigated by Wade and Schechter. Seve r n empirical rules for the prediction of the interfacial tension minimum have been formulated and can predict, with good accuracy, for a given mix of hydrocarbons, the combination of surfactants which will yield a tension minimum.
The microscopic behavior of such systems is not understood. The theoretical underpinnings for the empirical relations are not known and consequently insight into the behavior of the systems is missing. A number of studies of surface and interfacial tensions have appeared recently (7-11);, however, they deal principally with relatively simple liquids and do not address the surfactant problem directly. In this communication, we present a simple model of a surfactant-interface system, which nevertheless shows many of the features observed experimentally. Extensive experimental work has been done by Cayais et al. (6), in observing the systematics of interfacial tension minima using the spinning drop technique. They have developed the concept of the equivalent alkane carbon number (EACN) which is the parameter varied in the experiments (3). They obtain a minimum in the interfacial tension for a particular EACN. Any hydrocarbon may be assigned an EACN value and a simple averaging rule has been established for hydrocarbon mixtures: (EACN)m~xtu~e = • (EACN),X~,
1 Supported in part by the Office o f N a v a l Research.
[1]
i 320
0021-9797/78/0642-0320502.00/0 Copyright © 1978by AcademicPress, Inc; All fights of reproduction in any form reserved.
Journal of CoUoidand Interface Science, Vol. 64, No. 2, April 1978
H Y D R O C A R B O N - W A T E R - S U R F A C T A N T I N T E R F A C I A L TENSION
where X~ is the mole fraction of the i-th component. For instance, a mixture of undecane (EACN 11) and heptyl benzene (EACN 7) with X~ = ½ has an EACN of 9. Therefore, if a surfactant gives a minimum interfacial tension for nonane (EACN 9), it will also give a low interfacial tension for the above mixture. Mixtures of different surfactants also behave in such an averaged way. A typical graph of the interfacial tension of an a l k a n e - w a t e r - s u r f a c t a n t system plotted as a function of EACN is shown in Fig. 1. There is a dip of two orders of magnitude in the interfacial tension in the vicinity of a particular value of the EACN which in this case corresponds to heptane. It seems evident that the surfactant interface interaction cannot be highly specific but must depend on some averaged properties of the interface. A model describing the behavior of the system must therefore depend on average interaction energies, average chemical potentials and so on.
THEORY
In order to explain the observed changes in the interfacial tension and in particular the striking minimum for a particular hydrocarbon (or EACN) we have used the following model of the physical situation. Even though separation cannot in principle be precise, we assume three distinct phases; an aqueous phase with surfactant molecules present, a hydrocarbon phase, and an interface upon which surfactant molecules may be adsorbed. The waterhydrocarbon interface has an interfacial tension y = y0 in the absence of surfactant molecules. We do not address ourselves to the calculation of 3'0 but only the changes in the interfacial tension due to surfactant adsorption. Upon introduction of the surfactant, the aqueous phase acts as a reservoir for the surfactant molecules which occur both as single molecules and as members of micelles. These molecules are
321
described by a chemical potential /x. The surfactant molecules are free to adsorb on the hydrocarbon-water interface. The surfactant molecules have a hydrophobic hydrocarbon tail and a hydrophilic polar head aligning themselves therefore with the head in the aqueous phase and the tail in the hydrocarbon. There is an energy of intercation e for the adsorption, and the surfactant molecules adsorbed on the interface interact among themselves through the intermediary of the interface. To facilitate the calculation, we have assumed a lattice model for the surface phase. A surfactant molecule may be adsorbed on a lattice site or the site may be empty. Surfactant molecules adsorbed on adjacent lattice sites interact with an energy w and there are Z nearest neighbors. One could consider more sophisticated lattice models, however, we feel it is probably not justified in this situation. In addition, since surfactant molecules have long hydrocarbon chains, we include a parameter representing the change in the internal free energy of the surfactant molecule. Considering the above model, we calculate the change in the free energy of the interface from which we can then find the change in the interracial tension. The problem then is analogous to the adsorption isotherm derived initially by Fowler and Guggenheim (12) and elaborated more recently by Honig (13). Fowler and Guggenheim as contrasted to Langmuir (14) include lateral interactions between neighboring molecules in the analysis of the adsorption isotherm. We use the technique as presented by Honig to calculate the change in the free energy of the interface. The Fowler-Guggenheim approximation considers two entities in the model: sites and bonds. The sites are individual lattice sites which can be either occupied or empty; bonds are the interactions between two neighboring sites. The occupational probability of a bond between two empty sites is /30, between one empty and one Journal of Colloid and Interface Science, VoI. 64, No. 2, April 1978
322
ANTONIEWlCZ AND RODRIGUEZ
filled site is 2/31, and between filled sites is /32- The occupational probability of a filled site is oq and an empty site is ot0. The total energy of an adsorbing interface is: NZ
E = ~ -
2
N(Z
(/30Egg 4- 2/31EAB 4- /32EBB) -
1)(ozoEA + OQEB) 4- N~I/~, [2]
where N is the total number of sites, Z is the number of nearest neighbors, EAA is the energy of two nearest neighbor empty sites (it is equal to twice the surface energy per
site of the bare interface), EAB is the energy of one filled and one empty nearest neighbor sites, EBB is the energy of two filled nearest neighbor sites, including w, the interaction energy between nearest neighbors. Also, EA is the energy of an empty site and EB is the energy of a filled site. Finally, /~s is the change in the internal energy of an adsorbed molecule. It is important to keep track of the number of bonds and sites since their total is larger than the number of sites in the first place (13). The entropy due to the presence of the surfactant molecules is:
(NZ/2)!
S = kin (/3oNZ/2)
!{(/31NZ/2)!}2(/32NZ/2)! -
kln
where Ss is the change of the internal entropy of the adsorbing molecule. This term will principally arise from the restriction on the number of configurations which the tail of the surfactant molecule may have. The various occupational probabilities are not independent quantities but have the following restrictions: t~0 + ~1 = 1,/30 + 2/31 +/32 = l, which are statements of the conservation of probability; and, al =/31 +/32 is a consistency relation requiring that the number of filled sites be the same whether obtained by counting filled sites or counting appropriate bonds between pairs of sites. This analysis leaves two independent variables which have to be obtained by minimizing the free energy with respect to the n u m b e r of filled nearest neighbor pairs, /32, and by finding the chemical potential of the adsorbed surfactant molecules and equating it to the chemical potential of the surfactant molecules in the aqueous phase. The Helmholtz free energy for the surface phase is F = E - T S and we have to apply the following two thermodynamic equilibrium conditions mentioned above: OF tZaq.eo.~
=
tZ~.a.ce
--
-
-
ONfill~d
,
[4]
Journal of Colloid and Interface Science, V o l . 64, N o . 2, A p r i l 1978
{(Z-
{(Z-
1)N}!
1)Nao}!{(Z- 1)Na~}!
+ NoqSs,
[3]
and -
OF -
-
0.
[5]
ONpairs We thus have five equations in five unknowns So, ~1, /30, /31, /32 a system which is solved numerically. The interfacial tension, y, is defined as the change in the
i 0 -j.
-2 to v z o
I 0 -z"
10`3.
t.u z
EACN
FIG. 1. Sample graph taken by Cayais, Schechter, and Wade (6). The surfactant was petronate at 0.2% concentration with NaCl at 1% concentration and T = 27°C. The abscissa is a simple alkane scan.
HYDROCARBON-WATER-SURFACTANT INTERFACIAL TENSION
323
To be able to plot the interfacial tension as a function of E A C N , we assume that changing the h y d r o c a r b o n in the substrate changes the interaction linearly between adsorbed molecules and likewise between the adsorbed molecules and the substrate, that is: w = w0 + Wl x ( E A C N ) ,
[8]
e = e0 + el x (EACN).
[9]
and
C3 5
7
9
11
13
15
17
EnCN
FIG. 2. Theoretical fit to experimental data (+) of Wade et al. The surfactant was sodium 4[1-pentylnonyl]benzene sulfonate, which we will denote as Cs-C-Cs. The surfactant concentration was 0.7% with 2% isopropyl alcohol and 0.32% NaC1 as solutes. The temperature was 27°C and the area we chose here was 40 A~. free energy as a function of the total surface area,
OF Y = O(AN)
1 A
OF ON
[63
where A is the area per lattice site and N is the n u m b e r of lattice sites. Hence:
The calculated change in w and e over the span of alkanes is relatively small making the use of higher order terms unimportant. At this point, we have five energy parameters (w0, w~, Fs - eo, et, /x) which we choose in a way to minimize the error between the experimental points and the calculated curves. A simplex minimization procedure is used in order to obtain the best fit. Figures 2 and 3 show the fits to the data for 0.07% surfactant solution in water in the presence of 0.32% sodium chloride versus the equivalent alkane carbon n u m b e r of the hydrocarbon. The parameters for the fit in the Figs. 2 and 3 are given in Table I, along with the calculated parameters for different values of the area per molecule, A.
3/ = [(BoEAA -4- 2fl,EAB + fl~EnB)Z/2 r cz_
+ (a0EA + a~En)'(1 - Z) + a~Ps + kT{(/30 In/30 + 2fll In/31 c3
+ r2 In flz)Z/2 + (a0 In ao + a~ In a0(1 - Z)}]/A.
[7]
Further simplifications and details of the numerical solution are indicated in the Appendix. COMPARISON WITH EXPERIMENT The parameters in the equations above were chosen so that the surface tension, computed from Eq. [A3], fit the experimental data. The choice of parameters for a given set of experimental data appears unique and the detailed shape of the c o m p u t e d curves is sensitive to their values.
c3
aJ
+
uJ
co i -
10
-
T
~
~3
i
i
12
13
~
-
-
i
14 ~s ERCN
i
i
1~
17
FIG. 3. Same as Fig. 2 except the surfactant was sodium 4[1-hexylnonyl]benzenesulfonate, abbreviated as Cs-C-Cs. Journal of Colloid and Interface Science,
Vol.
64, No.
2, April
1978
324
ANTONIEWICZ AND RODRIGUEZ TABLEI
Energy Parameters Obtained by Fitting the Interfacial Tension Eq. [A3] to Experimental Data for Two Kinds of Surfactant Molecules" Area A s 30
40
50
60
- 15.36 -12.67 0.0836 -0.984 0.0328
- 19.56 -16.83 0.0826 -0.925 0.0336
C~-C-C8 /x ffs - eo E1 w0 wl
- 10.22 -7.51 0.0556 -0.922 0.0234
- 12.98 -10.30 0.0669 -0.904 0.0272
CrC-C8 /~ Fs - eo el w0 wl
-10.30 -7.70 0.0520 -0.874 0.0219
-11.89 -9.67 0.0631 -0.717 0.0242
-15.36 -12.39 0.0921 - 1.016 0.0360
-17.33 -13.41 0.1475 - 1.364 0.0549
a All values are in units of kT (dimensionless). We chose the close packed value of Z = 6 for this calculation.
We see first that the net interaction is attractive between two surfactant molecules. For increasing EACN, E, the binding energy to the surface, increases (i.e., the surfactant molecules are more tightly bound) and - w , the attractive interaction energy, decreases (weaker interaction). Another observation which emerges from the calculation is that the change in coverage for the whole span of EACN is small, on the order of 1%. We have included in the calculation the tension of the bare h y d r o c a r b o n - w a t e r interface which varies from 50 dyn/cm for a p e n t a n e water interface (EACN = 5) to 54 dyn/cm for a h e x a d e c a n e - w a t e r interface (15). The numerical results indicate that the changes in the adsorption energy of the surfactant molecules, ~, and their lateral interactions, w, are linear functions of the EACN of the constituent alkanes. (This correlates with the empirically derived equivalent EACN of mixtures of alkanes.) We get the same result if we take a linear combination of the interaction parameters of the constituents of the mixture. The Journal of Colloid and Interface Science, Vol. 64, No. 2, April 1978
assumption is that interaction energies are an average over a finite volume of the interface. Thus the interaction between surfactant molecules is through the intermediary of the average interface. We believe, therefore, that we have a simple yet consistent model of the anomaly in the interfacial tension of a hydrocarbon with water in the presence of a surfactant. The parameters which we obtain from fitting to experimental data are physically reasonable. Presently we are working to expand this simple model, in particular to represent the surface phase as a liquid. We will then fit this model to the large amount of experimental data which is now becoming available. From systematic data on the energy parameters, we hope to elaborate upon the theory of interfacial adsorption and interactions. APPENDIX
In order to do the numerical work, we rewrite the pertinent equations in the following way. First of all one can write: EAA =
2EA;
EA 8 = E A + EB; EB = E A --
EB~ = 2EB + w;
e; EA = 70A. [A1]
Consequently one has the following set of equations to solve: ao + % = l, /3o + 23~ + 132 = 1, oq =/31 +/32,
fiofl~ = fix 2 e x p [ - w / k T ], • l Z a o z - 1 --__ fl0ZOgl z - 1
. . . . . . × exp[(iz + e - i:s)/kT].
[A2]
The interfacial tension 3' is evaluated finally as ')/ ~" 3"0 ÷
{(I~'S - - E ) a i
÷
Zwfl2/2
+ kT[(fioln[3o + 2fillnfll + f121n[32)Z/2 + (% In ~ + oq In al)(1 - Z)]}/A. [A3]
HYDROCARBON-WATER-SURFACTANT INTERFACIAL TENSION
In the above equations Z is the number of nearest neighbors; Z = 4 for a square lattice and Z = 6 for a hexagonal lattice. The temperature, T, is 300°K. The area per lattice site has to be chosen, compatible with the physical area covered by a surfacrant molecule. In this calculation we have used area A = 30, 40, 50, and 60 .~2 in order to see the area dependence of the parameters. We have also used the data of Legrand and Gaines (15) to find 3/0 as a function of E A C N , namely, Y0 = 48.57 + (0.2912) × ( E A C N ) (dyn/cm). There are essentially three parameters which are obtained from fitting the calculated interfacial tension to the experimental curves. These are/x, the chemical potential in the aqueous phase, w, the interaction energy b e t w e e n neighboring surfactant molecules, and Fs - ~, the change in the energy of the system w h e n an isolated surfactant molecule is adsorbed on the interface. ACKNOWLEDGMENT We would like to thank Professor W. H. Wade and his group for their encouragement and for supplying us with the experimental data.
325
and Wade, W. H., Paper SPE 5813, Presented: SPE Symposium on Improved Oil Recovery, March 22-24, 1976. 2. Morgan, J. C., Schechter, R. S., and Wade, W. H., in "Improved Oil Recovery by Surfactant and Polymer Flooding" (D. O. Shah and R. S. ScheCter, Eds.), pp. 101-118. Academic Press, N. Y., 1977. 3. Cayais, J. L., Schechter, R. S., and Wade, W. H., ACS Symp. Ser. No. 8, 234 (1975). 4. Melrose, J. C., and Brandner, C. F., J. Canad. Petrol. Technol. 54 (Oct.-Dec. 1974). 5. Taber, J. J., Soc. Petrol. Eng. J. 9, 3 (1969). 6. Cayais, J. L., Schechter, R. S., and Wade, W. H., ACS Syrup. Set. No. 8, 235 (1975). 7. Davis, H. T., J. Chem. Phys. 62, 3412 (1975). 8. Bongiorno, V., and Davis, H. T. Phys. Rev. A 12, 2213 (1975). 9. Yang, A. J. M., Fleming, P. D., III, and Gibbs, J. H., J. Chem. Phys. 64, 3732 (1976). 10. Percus, J. K., J. Statistical Phys. 15, 423 (1976). 11. Ebner, C., Saam, W. F., and Strond, D., Phys. Rev. A 14, 2264 (1976). 12. Fowler, R. H., and Guggenheim, E. A., "Statistical Thermodynamics," Chap. 12. Cambridge Univ. Press, New York, 1939. 13. Honig, J. M., in "The Solid-Gas Interface" (E. Alison Flood, Ed.), Vol. 1, pp. 371-396. Dekker, New York, 1967.
REFERENCES
14. Langmuir, I., J. Amer. Chem. Soc. 38, 2267 (1916); 40, 1361 (1918).
1. Cash, R. L., Cayais, J. L., Fournier, G., Jacobson, J. K., Schares, T., Schechter, R. S.,
15. LeGrand, D. G., and Gaines, G. L., Jr., J. Colloid Interface Sci. 42, 181 (1973).
Journal of Colloid and Interface Science, Vol. 64, No. 2, April 1978