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Effect of electrolytes and mixtures of surfactants on the oil-water interfacial tension and their role in formation of microemulsions
Effect of Electrolytes and Mixtures of Surfactants on the Oil-Water Interfacial Tension and Their Role in Formation of Microemulsions E. R U C K E N S T E I N AND R. K R I S H N A N State University o f New York at Buffalo, Faculty o f Engineering and Applied Sciences, Buffalo, New York 14214 Received June 11, 1979; accepted September 25, 1979 An expression for the interfacial tension of an oil-water interface in the presence of an ionic surfactant, cosurfactant, and salt is derived by coupling the Gibbs adsorption equation with a multicomponent adsorption isotherm. The adsorption isotherm includes the contribution of the electrical effects due to the surface charge generated by the ionic surfactant and to the salt. The equation derived for the interracial tension provides indications about the conditions under which microemulsions form. A comparison between an expression previously derived assuming that the interfacial tension is equal to that of an uncharged interface plus the work of charging the interface and the present equation is made. Explanations of the minima observed in the oil-water interfacial tension with respect to surfactant concentration and ionic strength are also suggested. I. INTRODUCTION
distinct characteristics concerning its action on the oil-water system: (i) ultralow interIt is well known that when surfactants facial tension below 10-2 dyn/cm can be are added to an oil-water system, they obtained (1); (ii) the interfacial tension exlower the interfacial tension and therefore hibits a minimum with respect to the conthey accumulate at the oil-water interface. centration of the surfactant (2) and a miniFor nonionic surfactants, quantitative p r e - mum with respect to ionic strength (3); (iii) dictions of the interfacial tension can be microemulsions form under certain condimade using various equations. The simplest tions (4). The scope of the present paper is among them, the Szyskowski's e q u a t i o n ; to explain these phenomena. can be derived from the Gibbs adsorption Several approaches are available to obequation and the Langmuir adsorption iso- tain the surface pressure of soluble ionized therm. More complex equations account in monolayers at the oil-water interface. addition for the interactions between the These have been amply reviewed and disadsorbed molecules. cussed by Hachisu (5). In the present paper In several technologically important ap- an expression for the interfacial tension is plications ionic surfactants are often used derived by integrating the Gibbs adsorption in conjunction with medium-chain-length equation. The procedure adopted here is alcohols (cosurfactants). For instance, a similar but simpler than that of Ref. (5). microemulsion is formed by adding sodium An adsorption isotherm of the Langmuir dodecyl sulfate (an ionic surfactant) and type which takes into account the surface hexanol to an oil-water system. An elec- charge generated by the adsorption of the trolyte is also usually present in the system. ionic surfactant is used along with the inteThe combination surfactant-cosurfac- grated Gibbs equation t o compute the detant-electrolyte cited above has several pendence of the interfacial tension on the 201 Journal of CoUoldand Interface Science, Vol. 76, No. 1, July 1980
0021-9797/80/070201° 11502.00/0 Copyright© 1980by AcademicPress,Inc. All rightsof reproductionin any formreserved.
202
RUCKENSTEIN AND KRISHNAN
bulk concentration of the surfactant and on the ionic strength. Similarly, a new multicomponent adsorption isotherm which accounts for different areas per molecule of the adsorbates (6) is used to derive an expression for the interfacial tension as a function of the bulk concentrations of surfactant and cosurfactant as well as the ionic strength. The latter equation provides the conditions under which microemulsions form. Micellization as well as the saltingout effect are suggested as possible explanations for the occurrence of the minima in the interfacial tension. II. T H E I N T E R F A C I A L
where DE is the deficiency of the R ÷ ions. The latter quantity is related to the total deficiency (D ÷) of the positive ions through D+ _
FR = F~ -
Fx = -
II.1. The Basic Equations
Ca C
D +,
[5]
Cx D +, C
[6]
and
Let us consider a completely dissociated cationic surfactant R+A - in the presence of a neutral salt X+A - and a nonionic cosurfactant Cs. F o r a planar surface, at constant temperature, the interfacial tension (y) satisfies the equation - d y = FRd/zR + Fxd/~x + FAd/xa + Fcsd/zc~ + Fo~,dt~oi,, [1] where F~ is the relative surface excess of species i with respect to water and/.~ is its electrochemical potential. The electrochemical potential /ZR at the interface can be decomposed into the sum [2]
where t ~ is the chemical part of the electrochemical potential, e is the protonic charge, and qJ is the surface potential. Because there exists a deficiency o f the R + ions in the diffuse layer, the surface excess FR = f~ [CR(z) - CR(oo)]dz differs from the surface density F~ of the surfactant monolayer on the interface itself. Here Ca(z) is the surfactant concentration at a distance z from the interface and Cg(o~) (denoted later by Ca) is its bulk concentration. Further, we have r a = r ~ - D~,
[4]
H e r e C~ is the bulk concentration of species i and C is the total bulk concentration of positive (or negative) ions. The various surface excesses can therefore be written as
TENSION
~R = / ~ + etO,
CR D+ = C____RD+" a CR + Cx C
[31
Journal of Colloid and Interface Science, Vol. 76, No, 1, July 1980
FA = D - .
[7]
Here F~ is assumed zero. The above equations are substituted in Eq. [1] to yield - d y = Frtdtz[t + F~edtO -
CR
C
D+edqJ
Cx D+kT dCx + D - k T dC C Cx C
+ rcsd/zcs.
[8]
In obtaining Eq. [8] the solution is assumed to be dilute and the relative surface excess of oil is taken to be, as usual, zero. Since the cosurfactant is nonionic no distinction is made between its surface density and its surface excess. The equality of the chemical potential in the bulk and at the interface provides kT --dCR CR
= dl.~ + edtk.
[9]
Using Eqs. [5], [9], and the definition of C ( = C a + Cx), Eq. [8] can be rewritten as kT - d y = F~dlz{~ + F~ed~b + - C × {D- -D+}dC
+ Fcsd/zcs.
[10]
INTERFACIAL TENSION Describing the electrical double layer b y the G o u y - C h a p m a n theory, the expressions for D + and D - can be written as (7)
1 / 2ECkT \1/2
D+ = ~ ~ )
[1 -- e -eo/2kv] [11]
and
l[2eCkT |\ D-
=
[ee*/2kT- 1].
[121
J
H e r e • is the dielectric constant of water. The concentrations are e x p r e s s e d in molecules per cubic centimeter. Assuming that the surfactant molecules are c o m p l e t e l y dissociated, we h a v e o - = e F t . B e c a u s e (F~ = D + + D - ) , the surface charge density cr b e c o m e s =
- -
Substituting Eqs. one obtains
sinh
.
[13]
[11]-[131 in Eq.
[10],
- d y = rRd/xa ' " + ( ~2 e C k T ) v2 sinh ( ~-~T ) dO
+
~
203
where T0 is the o i l - w a t e r interfacial tension free of surfactant, cosurfactant, and electrolyte and o- is calculated at the final value of C.
H.2. Relations Between Interracial Tension and the Bulk Concentrations In order to c o m p u t e the interfacial tension T as a function of the bulk concentration of the surface active c o m p o n e n t s , specific forms for the adsorption isotherms must be coupled with Eq. [16]. When a single surfactant is present a L a n g m u i r isotherm is used for this purpose. When a cosurfactant is present along with a surfactant, new adsorption isotherms (6) are derived I in which we neglect, for the sake of simplicity, the noncoulombic interactions b e t w e e n the adsorbed molecules but we account for the different areas per molecule of the surfactant and cosurfactant. Denoting b y Pl the n u m b e r of sites occupied by a surfactant molecule and by P2 those occupied b y a cosurfactant molecule, the chemical component of the electrochemical potentials can be written as (see Appendix)
,fie 2
tz~=
/~R,O "
+ plkT In + Fcsdtzcs.
[14]
Because the third t e r m on the right-hand side can be written as (f*o [Oo-/OC]dO)dC, Eq. [14] b e c o m e s - d 3 , = F~dtz; + Fc~d/zcs
r~pl
[17]
(N - F~pl - Fcsp2) and ~ C s ---- /J,'Cs,O
+ p2kT In
FcsP2 (N - F~pl - FcsP2)
[18]
H e r e the subscript 0 refers to the standard
+ ~(q,,C)dq, +
- ~ dq, dC.
[151
Since the sum of the last two t e r m s of Eq. [15] leads to d/dC f*o o-dq~, Eq. [15] leads finally, through integration, to the expression
Y=Yo-
f : F~d/~ o-d~,
[16]
1 Two different paths of integration are possible in carrying out the integrals with respect to the chemical potentials in Eq. [16]. In the first, the first integral is carried out from tz~ = -oo to tz~ = tz~ while keeping t~cs = -o~; then the second integral is performed between - 2 and /~cs keeping /z~ at its final value. The second, alternate path results if the order of integration is changed, performing the second integral at /z~ = - ~ and then the first integral keeping ~cs at its final value. The adsorption isotherms must be such that the expression obtained for the interfacial tension should be independent of the path of integration. Journal of Colloid and Interface Science, Vol. 76, No. I, July 1980
204
RUCKENSTEIN AND KRISHNAN
chemical potentials and N is the inverse of the area occupied by a site. The area of a site is equal to the greatest c o m m o n divisor of the areas occupied by one molecule of surfactant and one molecule of cosurfactant. F r o m the condition o f equilibrium for the surfactant one can write • bulk /aVR,0
Eliminating the surface charge density between Eqs. [13] and [25], Eq. [25] can be rewritten as Y = To - N k T In (1 + (K1CRe-e~/kT) 1/m + (K2Ccs) 1/p') -
/
8~C(k____T)~ 7re 2
-l- k T In Cn = IZ~,o" + p l k T
x In
F~p1 + e~b. [19] (N - F~p, - FcsP2)
The electrical contribution to the electrochemical potential o f the surfactant in the bulk is zero, since the electrical potential far away from the surface is zero. Equation [19] can be rewritten as CR =
(F~p0mee*lkr Ka(N-
F~pl - FcsP2)p'
,
[20]
where g l = exp[(/X~,o bulk -
" IZa,o)/kT].
[21]
Similarly, the equality of chemical potentials of the cosurfactant in the bulk and the interface yields
FcsP~ (N - F~p, - Fc~p~)
[22]
K 2 ( N - F ~ o l - FcsP2)p'
In order to better understand the role played by the ionic surfactant, a simple case in which the cosurfactant is absent is first considered. In this case, Eq. [16] reduces to y = yo - f r~dl~a - l o'dtk.
t4=~,,o+kTln(
r_',. ]
N - F~]'
Consequently, Cc~ =
III. I N T E R F A C I A L T E N S I O N I N T H E P R E S E N C E OF A SINGLE IONIC SURFACTANT
[26]
For a Langmuir-type adsorption the chemical potential/x~ is given by
&bulk+ kT In Cc~ =/Xcs,o CS,0 + p z k T In
The interracial tension can be calculated as a function o f t h e surfactant concentration, cosurfactant concentration, and the total ionic strength from Eqs. [25'], [13], [20], and [23].
,
[23]
[27]
where N is equal to the surface density at saturation. Substituting Eq. [27] in [26], one obtains T = To - N k T ln N + N k T
Where K2
bulk - IZc~,o)/kT]. -=--exp[(/Zcs.0
[24]
F o r given concentrations of the surfactant and cosurfactant and given values o f the adsorption constants (N, K,, K2, p , andp2), the surface densities (F~ and Fc~) can be computed by trial and error from Eqs. [20] and [23] along With Eq. [131. Using Eqs. [17], [18], [20], and [23], Eq. [16], becomes Y = To - N k T In (1 + (K1Cae-e*/kr) 'Iv' + (K2Ccs)VP0 - ~ o'dtO. [25] 3 Journal of Colloid and Interface Science, Vol. 76, No, 1, July 1980
x In (N - F~) - [ o-dqJ. [28] The condition of thermodynamic equilibrium between the surfactant in the bulk and at the interface provides the equation t! tZR,o + k T In
V~ (N - F~)
+ eqJ
= /aVR,0 ..bu,k + k T In CR,
[29]
which can be rewritten as CR --
b0Fk (N - F ~
e e~mr.
[30]
INTERFACIALTENSION
205
Hence, if the two strong inequalities mentioned above are satisfied, Eq. [38'] is agood approximation of the more rigorous equaConsequently, tion, Eq. [33]. In principle, there are other conditions in which Eq. [33] reduces to NCR Fh = [32] Eq. [38']. For instance, this happens when (Ca + boe e*/kr) eO/kT ~ 1 and Ca/bo >> 1. Numerical comSubstituting Eq. [32] in Eq. [28], one obtains putations have been carried out to identify whether there are conditions outside the T = To - NkT limiting situations mentioned above under bo e-e*/kr o'dqJ. [33] which Eq. [38'] constitutes a satisfactory approximation of Eq. [33]. The interfacial tension in the presence of This equation in combination with the relaan ionic surfactant (without a cosurfactant) tion between the surface potential and the is plotted vs the logarithm of the surfactant surface charge (Eq. [13]) allows one to comconcentration (molecules/cmz) in Fig. 1 for pute y as a function of the concentration CR. different values of N and in Fig. 2 for For uncharged surfactant molecules and various values of b0. The solid lines are the the same values of Ca, b0, and N, Eq. [33] results obtained from Eq. [33]; the dashed leads to lines correspond to Eq. [38']. Figures 1 and 2 show clearly that Eq. [38'] does not "Yuneharged = y o - N k T l n ( l + Cb0a ) . [34] approximate in a satisfactory manner Eq. [33] over the entire range of surfactant Eliminating To between Eqs. [33] and [34], concentration. one obtains IV. INTERFACIALTENSIONIN THE PRESENCE Y = Yunchar~ea + NeqJ + NkT OF SURFACTANT,COSURFACTANT, AND SALT Here
bo = e("k0-"~?0~)/kr.
(
[31]
1 + (CR/b°)) - I o-dry. [35] (CR/bo)
The interfacial tension in the presence of an ionic surfactant and cosurfactant is When the surface potential is small such plotted in Fig. 3 vs the log concentration that etk/kT ~ 1, Eq. [35] yields of surfactant for various salt concentrations. At low concentrations, as the concentration "Y -~ Yuncharged + Netb of the surfactant in the bulk increases, _ Net~ - I o'&k. [361 more molecules of surfactant are adsorbed at the interface and the interfacial tension 1 + CR/bo decreases. Because the surfactant moleIf, in addition, CR/bo ~ 1, Eq. [36] becomes cules are ionic the interfacial charge also increases. An increasing surface charge y=Yuneharged+Net~-~o--lord~b. [37] opposes the adsorption of the surfactant but favors that of the cosurfactant. In Fig. 4 the interfacial tension is plotted The latter equation can be rewritten in the vs the log of added salt concentration. form, already suggested in Ref. (8), Table I lists, in a particular case, the Y = ')/uncharged -~- F~e~b - [ o'd~b [38] contributions of the terms J Q1 = - N k T In (1 + (KiCrte-e*/kT) ~/m x In e~7/ff +
or
Y -~ •uncharged
~-
I ~bd~.
[38']
+
(K2Ccs)
)
llp~
and
Q2 = -
I
trd~b
Journal of Colloid and Interface Science, Vol. 76, No, 1, July 1980
206
RUCKENSTEIN AND KRISHNAN
44
32
.
\ , ' \, \
\\\\ \\\\
20
"x-x'x
'\\ \\\2
44
\
\
\3 \
8 2
2
\
\
\
\\
\
\
\
\\
t5.2 ~tt61.4 t7.6~.,~. Log(Surfoctant\Concentrotion in molecules/cm z) I
FIG. l. Interfacial tension vs log (concentration of surfactant). T h e values of the p a r a m e t e r s are: b0 = 1 × 1015 molecules/cm a, concentration of salt = 10 TM molecules/cm ~, 3'0 = 50 d y n / c m , and the saturation adsorption N is 1 × 1014, 3 × 1014, and 5 × 10 TM molecules/cm ~ for curves 1, 2, and 3, respectively. The solid lines represent the rigorous equation [33] and the d a s h e d lines the approximate equation [38']. In all the figures the t e m p e r a t u r e is 25°C.
to the interfacial tension. The term Qa represents the effect of the chemical part of the chemical potentials of the surfactant and cosurfactant on the interfacial tension, while
the term Q2 is the free energy of formation of the double layer. Q1 decreases (becomes more negative) with an increase of the concentration of the added salt because
\\\\\
32
\~t
i
•~ 20 t4
\\\\\ t
T.
\\\xx~X
\\ 8
"\\ %\N
21
45,2
I 46.4
\
x. I ~. ~17:6
t 48.8
LoQ (Surfactant Concentration in molecules/m =} FIG. 2. Effect of the adsorption c o n s t a n t b0 on the plot interracial tension vs log concentration of surfactant. T h e values of the p a r a m e t e r s are: 3/o = 50 dyn/cm, N = 3 × 1014 molecules/cm 2, concentration of salt = 1019 molecules/cm ~, and the values of b0 are 1 x 10 TM, 1 × 1017, and 1 × 10 TM molecules/cm ~ for curves 1, 2, and 3, respectively. Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980
INTERFAC/AL
TENSION
207
26
20 ~o 14
FIG. 3. I n t e r f a c i a l t e n s i o n vs log c o n c e n t r a t i o n of s u r f a c t a n t w h e n a c o s u r f a c t a n t is a l s o p r e s e n t . T h e v a l u e s o f t h e p a r a m e t e r s are: K1 = 6 × 10 -18 c m 3 / m o l e c u l e , Ks = 1 × 10 -18 c m 3 / m o l e c u l e , N = 1015 m o l e c u l e s / c m 2, P l = 3, p~ = 2, the c o n c e n t r a t i o n o f t h e c o s u r f a c t a n t = 1 × 10 TMm o l e c u l e s / c m 8, 3'o = 50 d y n / c m , a n d the c o n c e n t r a t i o n s o f the s a l t are 5 x 10 TM, 5 × 10 TM, a n d 1 x 1031 m o l e c u l e s / c m 8 for c u r v e s 1, 2, a n d 3, r e s p e c t i v e l y .
the shielding of the electrical field allows more surfactant to be adsorbed. Q2 has a minimum because while the charge increases with the increase in adsorption, the surface potential increases with the charge but decreases with the salt concentration. The oil-water interfacial tension is plotted in Fig. 5 as a function of the cosurfactant concentration for various concentrations of
surfactant. Since the cosurfactant is uncharged its adsorption is not impeded by the electric field; on the contrary the competition with the surfactant for the sites of the interface is resolved in its favor. The effect of the ratio of areas per molecule for surfactant and cosurfactant on the plot interfacial tension vs log concentration of surfactant is shown in Fig. 6.
52
26
20
44
t
-o
~6,4
~7.6
t8.8
20
Log (Salt Concentration in molecules/cm5) -4(
FIG. 4. I n t e r r a c i a l t e n s i o n vs log o f s a l t c o n c e n t r a t i o n . A c o s u r f a c t a n t is p r e s e n t a l o n g w i t h a n i o n i c s u r f a c t a n t . T h e v a l u e s of t h e p a r a m e t e r s are: y0 = 50 d y n / c m , K1 = 6 × 10 -18 c m 3 / m o l e c u l e , K2 = 1 × 10 -18 c m 3 / m o l e c u l e , N = 10 TM m o l e c u l e s / c m ~, P l = 3, P2 = 2, the c o n c e n t r a t i o n o f t h e cos u r f a c t a n t = 1 × 1018 m o l e c u l e s / c m 8, a n d t h e c o n c e n t r a t i o n s o f t h e s u r f a c t a n t a r e 1 × 10 TM, 5 × 10 TM, a n d 1 x 10 TM m o l e c u l e s / c m 3 for c u r v e s 1 t h r o u g h 3, r e s p e c t i v e l y . Journal of Colloid and Interface Science, VoL 76, No. 1, July 1980
208
RUCKENSTEIN
AND KRISHNAN
TABLE I Effect of Ionic Strength on Surface Propert!es ~ - Q 1 = N k T In
C~at (molecules/cm 3)
1 1 5 1 5 1 1
×
× × × x × ×
1017 10 TM 1013 1019 10 la 1091 1022
Surface density of surfactant (1014 molecules/cm 2)
Surface density of cosurfactant (10~4 molecules/cm 2)
Surface charge (104 esu)
Surface potential (10-~ esu)
(1 + (KICRe-~'er) ~l~' + (K2Cc~)~'~) (dyn/cm)
-Q2 = f o'd0 (dyn/crn)
y = yo + Q~ + Q2 (dyn/cm)
0.778 0.789 0.831 0.871 1.038 1.533 1.895
1.916 1.908 1.876 1.846 1.721 1.350 1.078
3.73 3.78 3.99 4.18 4.98 7.36 9.09
0.478 0.473 0.456 0.439 0.376 0.214 0.102
39.43 39.62 40.30 40.97 43.85 53.83 63.08
5.66 5.72 5.94 6.14 6.83 6.98 4.49
4.91 4.66 3.76 2.89 -0.68 -10.81 - 17.57
a 3'o = 50 d y n / c m , N = 8 × 1014 m o l e c u l e s / c m 2, K1 = 6 × 10 -19 c m 3 / m o l e c u l e , Ks = 1 × 10 -I9 c m 3 / m o l e c u l e , 1013 m o l e c u l e s / c m z, a n d Csurfaetant = 10 TM m o l e c u l e s / c m 3 .
P l = 3, p~ : 2, Ceosurraetant :
It is interesting to discuss the implications of the behavior of the interfacial tension of the oil-water interface in the presence of surfactant, cosurfactant, and salt on the formation of microemulsions. Figures 3, 4, 5, and 6 show that there are conditions under which this interfacial tension becomes negative. Of course, such an interface is not thermodynamically stable. For this reason 50 40 50 4
ZO
I
0
-10
20
log !cosurfactant concentration \ \ \ \ ~nmolecutes/cm3)
-20 -50 FIG. 5. I n t e r f a c i a l t e n s i o n v s log c o s u r f a c t a n t conc e n t r a t i o n . T h e v a l u e s o f t h e p a r a m e t e r s are: 3'0 = 50 d y n / c m , K1 = 6 × 10 -19 c m 3 / m o l e c u l e , K2 = 1 x 10 -19 c m a / m o l e c u l e , N = 8 x 1014 m o l e c u l e s / c m ~, P l = 3, P2 = 2, t h e c o n c e n t r a t i o n o f t h e sal[ is 1 × 10 TM m o l e c u l e s / c m 3, a n d t h e c o n c e n t r a t i o n s o f the s u r f a c t a n t are 1 × 1015, 1 × 101~, 1 × 10 is, a n d 5 x 1018 m o l e c u l e s / c m 3 for c u r v e s 1 t h r o u g h 4, respectively. Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980
the area of the interface increases, resulting in the formation of microemulsions. The increase in the interfacial area is associated with accumulation of surfactant and cosurfactant on the interface and takes place until the bulk concentrations of surfactant and cosurfactant become sufficiently low for the interfacial tension to reach a small but positive value determined by the condition of thermodynamic equilibrium (10). The ionic surfactant affects the interfacial tension through the accumulation on the interface and through the electric field generated by the charged interface. Since the presence of the charge impedes adsorption of the charged surfactant, the reduction due to accumulation is smaller than that of a hypothetical equivalent uncharged surfactant of the same bulk concentration. The electric field reduces the amount adsorbed but, as our numerical computations show, can contribute through the term - f o-d0 to a decrease of the interfacial tension as large as 6 dyn/cm. For the usual range of adsorption constants, the total decrease due to the ionic surfactant seems to be smaller than 50 dyn/cm, which is the interfacial tension of an oil-water interface free of any surfactants. For this reason a further decrease through the adsorption of a cosurfactant is generally needed for microemulsification. Figure 4 shows that microemulsions can form above a value of the ionic
INTERFACIAL
209
TENSION
52
26 2o
==
44
2 3
t5.2
46.4
17.6
t8.s'~
Log (Surfoctont Concentration in molecules/cm 3)
FIG. 6. I n t e r r a c i a l t e n s i o n v s log c o n c e n t r a t i o n o f s u r f a c t a n t . T h e v a l u e s o f t h e p a r a m e t e r s are: 3'o = 50 d y n / c m , K , = 6 x 10 -'8 c m 3 / m o l e c u l e , K2 = 1 × 10 -18 c m 3 / m o l e c u l e , c o n c e n t r a t i o n o f salt = 10 z° m o l e c u l e s / c m 3, c o n c e n t r a t i o n of c o s u r f a c t a n t = 10 TM m o l e c u l e s / c m 3, P2 = 2, N = 10 '5 m o l e c u l e s / c m 2, a n d t h e v a l u e s o f p l are 3, 4, a n d 5 o n c u r v e s 1, 2, a n d 3, r e s p e c t i v e l y .
strength. For sufficiently small ionic strengths the surface potential is sufficiently large and the contributions to the decrease of the interracial tension are the adsorption of the cosurfactant, the impeded adsorption of the surfactant, and the free energy - f (rd~ of formation of the double layer. For large ionic strengths, the major contributions are due to the adsorption of surfactant and cosurfactant; the free energy - f o-dqJ has a minor role. The detailed computations of the radius of microemulsions and of the phase equilibria in microemulsions will be published elsewhere. Two effects, micellization and salting out, which have not been included in the above theory may impede the formation of microemulsions. As the concentration of salt is increased the critical micelle concentration decreases because the coulombic repulsion between the charged head groups is increasingly shielded. Consequently, an increasing amount of surfactant molecules can aggregate as micelles (or perhaps mixed micelles) and less free molecules of surfactant (and cosurfactant if mixed micelles form) are available for adsorption. Thus a minimum of the interracial tension can occur at a given salt concentration. A minimum can also occur with respect to the
surfactant concentration when mixed micelles form. An additional effect is the decrease of the maximum solubility of the surfactant and cosurfactant in water when the salt concentration increases (salting out). The salting out in water is generally associated with a salting in in the oil phase. At sufficiently low salt concentrations the interfacial tension decreases with increasing added salt concentration for reasons already mentioned:: At high enough concentrations of salt the salting out-salting in process reduces the concentration of the surfactant per unit volume as well as the amount adsorbed on the interface. Therefore, the interfacial tension starts to increase. Thus for a given total amount of surfactant a minimum in the interfacial tension occurs with respect to the amount of added salt. When the salting out exceeds the salting in, a surfactant phase forms between water and oil (11). This middle phase contains some water and oil and probably has a liquid crystalline or a microemulsion structure. V. C O N C L U S I O N S
The oil-water interfacial tension in the presence of an ionic surfactant, a CosurJournal of Colloid and Interface Science, Vol. 76, No. 1, July 1980
210
RUCKENSTEIN AND KR1SHNAN
factant, and an electrolyte becomes negative under certain conditions. The interface is not stable in such conditions and therefore the interfacial area increases spontaneously (a microemulsion forms) until the bulk concentrations of surfactant and cosurfactant b e c o m e sufficiently low for the interfacial tension to reach a small, positive value required by the condition of thermodynamic equilibrium. Minima in the interfacial tension with respect to the salt concentration and surfactant concentration are explained in terms of micellization and salting out.
and
APPENDIX
F2 =
ADSORPTION ISOTHERMS FOR M I X T U R E S CONTAINING U N C H A R G E D SURFACTANT AND COSURFACTANT
The a d s o r p t i o n - d e s o r p t i o n equilibrium of c o m p o n e n t s j = 1 a n d j = 2 at the interface yields k l a C l O v' = k l a ~ ~ [A1] and k2aC2 0v' - k2a~ ~. [A21 Here kja are the adsorption constants, k~d are the desorption coefficients, C~ are the concentrations in the bulk phases, 0j are the fractions of sites covered, 0 is the total fractions of sites that are unoccupied, pj are the numbers of sites per adsorbed molecule, a n d j = 1, 2 refers to the surfactant and cosurfactant, respectively. Equations [A1] and [A2] can be rewritten as
where N is the inverse o f the area occupied by a site. This area is the greatest c o m m o n divisor of the areas occupied by the surfactant and cosurfactant molecules. Substituting Eqs. [A3] and [A4] in Eqs. [A6] and [A71, the adsorption isotherms are obtained as r 1=
N
(KIC1) l/p1
Pl
1 + ( K I C O llw + ( K 2 C 2 ) l/v2
(~2) N
t"K 2C 2J~l/v,' K C "~l/loz [A9] 1 + (K1C1) 1/p' + t 2 2J
( kla C1) 1/pl ~
(K1Cl)l/p~
[A31
F r o m Eqs. [A8] and [A9] the concentrations C1 and C2 can be written as 1 { C1 = ~
Flpl (N - r ~ l "-
r~p~)
}m
[A10 ]
and
1{
r22
)P AI,1
H e n c e the chemical potentials b e c o m e I ~ = lXao + p l k T
x In
(FIp0 (N - Flpl - F2p2)
[A12]
and 1~2 = lZ2o + p z k T (F2p2)
(N - FIpl - F2pz)
[A13]
REFERENCES
and 02 _ ( k 2 a C21 ~lp2 _ ,~.. t~2 C ~J~l/p2 • ] 0 \ k2a
[A4]
Because 0 = 1 - 01 - 02, from [A31 and [A4] one obtains 0 =
[A81
and
× In 01 _ O \ km
[A7]
F 2 = (Nip2)02,
1
1 + (K1C1) vp. + (K2C2) i/v'
[A51
The surface excesses are obtained from F1 = (N/pI)01
[A6]
Journalof Colloidand InterfaceScience, Vol. 76, No. 1, July 1980
1. Shah, D. O., J. Colloid Interface Sci. 37, 744 (1971). 2. Foster, W. R., J. Petrol. Technol. 25, 205 (1973). 3. Cayias, J. L., Schechter, R. S., and Wade, W. H., in "Adsorption at Interfaces" (K. L. Mittal, Ed.), ACS Symposium Series No. 8, p. 234. Amer. Chem. Soc., Washington, D. C., 1975. 4. Friberg, S. E., Buraczewska, I., and Ravey, J. C., in "Micellization, Solubilization and Microemulsions" (K. L. Mittal, Ed.), Vol. II, p. 901. Plenum, New York, 1977.
INTERFACIAL TENSION 5. Hachisu, S., J. Colloid Interface Sci. 33, 445 (1970). 6. Ruckenstein, E., and Kfishnan, R., J. Colloid Interface Sci. 76, 188 (1980). 7. Bikerman, J. J., Z. Phys. Chem. 163A, 378 (1933). 8. Levine, S., and Robinson, K., J. Phys. Chem. 76, 876 (1972).
21 1
9. Molliet, J. L., Collie, B., and Black, W., "Surface Activity." Van Nostrand, Princeton, N. J., 1960. 10. Ruckenstein, E., Chem. Phys. Lett. 57,517 (1978); Ruckenstein, E., and Krishnan, R., J. Colloid Interface Sci. 71, 321 (1979). 11. Healy, R. N., and Reed, R. L., Soc. Petrol. Eng. J. 17, 129 (1977).
Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980
distinct characteristics concerning its action on the oil-water system: (i) ultralow interIt is well known that when surfactants facial tension below 10-2 dyn/cm can be are added to an oil-water system, they obtained (1); (ii) the interfacial tension exlower the interfacial tension and therefore hibits a minimum with respect to the conthey accumulate at the oil-water interface. centration of the surfactant (2) and a miniFor nonionic surfactants, quantitative p r e - mum with respect to ionic strength (3); (iii) dictions of the interfacial tension can be microemulsions form under certain condimade using various equations. The simplest tions (4). The scope of the present paper is among them, the Szyskowski's e q u a t i o n ; to explain these phenomena. can be derived from the Gibbs adsorption Several approaches are available to obequation and the Langmuir adsorption iso- tain the surface pressure of soluble ionized therm. More complex equations account in monolayers at the oil-water interface. addition for the interactions between the These have been amply reviewed and disadsorbed molecules. cussed by Hachisu (5). In the present paper In several technologically important ap- an expression for the interfacial tension is plications ionic surfactants are often used derived by integrating the Gibbs adsorption in conjunction with medium-chain-length equation. The procedure adopted here is alcohols (cosurfactants). For instance, a similar but simpler than that of Ref. (5). microemulsion is formed by adding sodium An adsorption isotherm of the Langmuir dodecyl sulfate (an ionic surfactant) and type which takes into account the surface hexanol to an oil-water system. An elec- charge generated by the adsorption of the trolyte is also usually present in the system. ionic surfactant is used along with the inteThe combination surfactant-cosurfac- grated Gibbs equation t o compute the detant-electrolyte cited above has several pendence of the interfacial tension on the 201 Journal of CoUoldand Interface Science, Vol. 76, No. 1, July 1980
0021-9797/80/070201° 11502.00/0 Copyright© 1980by AcademicPress,Inc. All rightsof reproductionin any formreserved.
202
RUCKENSTEIN AND KRISHNAN
bulk concentration of the surfactant and on the ionic strength. Similarly, a new multicomponent adsorption isotherm which accounts for different areas per molecule of the adsorbates (6) is used to derive an expression for the interfacial tension as a function of the bulk concentrations of surfactant and cosurfactant as well as the ionic strength. The latter equation provides the conditions under which microemulsions form. Micellization as well as the saltingout effect are suggested as possible explanations for the occurrence of the minima in the interfacial tension. II. T H E I N T E R F A C I A L
where DE is the deficiency of the R ÷ ions. The latter quantity is related to the total deficiency (D ÷) of the positive ions through D+ _
FR = F~ -
Fx = -
II.1. The Basic Equations
Ca C
D +,
[5]
Cx D +, C
[6]
and
Let us consider a completely dissociated cationic surfactant R+A - in the presence of a neutral salt X+A - and a nonionic cosurfactant Cs. F o r a planar surface, at constant temperature, the interfacial tension (y) satisfies the equation - d y = FRd/zR + Fxd/~x + FAd/xa + Fcsd/zc~ + Fo~,dt~oi,, [1] where F~ is the relative surface excess of species i with respect to water and/.~ is its electrochemical potential. The electrochemical potential /ZR at the interface can be decomposed into the sum [2]
where t ~ is the chemical part of the electrochemical potential, e is the protonic charge, and qJ is the surface potential. Because there exists a deficiency o f the R + ions in the diffuse layer, the surface excess FR = f~ [CR(z) - CR(oo)]dz differs from the surface density F~ of the surfactant monolayer on the interface itself. Here Ca(z) is the surfactant concentration at a distance z from the interface and Cg(o~) (denoted later by Ca) is its bulk concentration. Further, we have r a = r ~ - D~,
[4]
H e r e C~ is the bulk concentration of species i and C is the total bulk concentration of positive (or negative) ions. The various surface excesses can therefore be written as
TENSION
~R = / ~ + etO,
CR D+ = C____RD+" a CR + Cx C
[31
Journal of Colloid and Interface Science, Vol. 76, No, 1, July 1980
FA = D - .
[7]
Here F~ is assumed zero. The above equations are substituted in Eq. [1] to yield - d y = Frtdtz[t + F~edtO -
CR
C
D+edqJ
Cx D+kT dCx + D - k T dC C Cx C
+ rcsd/zcs.
[8]
In obtaining Eq. [8] the solution is assumed to be dilute and the relative surface excess of oil is taken to be, as usual, zero. Since the cosurfactant is nonionic no distinction is made between its surface density and its surface excess. The equality of the chemical potential in the bulk and at the interface provides kT --dCR CR
= dl.~ + edtk.
[9]
Using Eqs. [5], [9], and the definition of C ( = C a + Cx), Eq. [8] can be rewritten as kT - d y = F~dlz{~ + F~ed~b + - C × {D- -D+}dC
+ Fcsd/zcs.
[10]
INTERFACIAL TENSION Describing the electrical double layer b y the G o u y - C h a p m a n theory, the expressions for D + and D - can be written as (7)
1 / 2ECkT \1/2
D+ = ~ ~ )
[1 -- e -eo/2kv] [11]
and
l[2eCkT |\ D-
=
[ee*/2kT- 1].
[121
J
H e r e • is the dielectric constant of water. The concentrations are e x p r e s s e d in molecules per cubic centimeter. Assuming that the surfactant molecules are c o m p l e t e l y dissociated, we h a v e o - = e F t . B e c a u s e (F~ = D + + D - ) , the surface charge density cr b e c o m e s =
- -
Substituting Eqs. one obtains
sinh
.
[13]
[11]-[131 in Eq.
[10],
- d y = rRd/xa ' " + ( ~2 e C k T ) v2 sinh ( ~-~T ) dO
+
~
203
where T0 is the o i l - w a t e r interfacial tension free of surfactant, cosurfactant, and electrolyte and o- is calculated at the final value of C.
H.2. Relations Between Interracial Tension and the Bulk Concentrations In order to c o m p u t e the interfacial tension T as a function of the bulk concentration of the surface active c o m p o n e n t s , specific forms for the adsorption isotherms must be coupled with Eq. [16]. When a single surfactant is present a L a n g m u i r isotherm is used for this purpose. When a cosurfactant is present along with a surfactant, new adsorption isotherms (6) are derived I in which we neglect, for the sake of simplicity, the noncoulombic interactions b e t w e e n the adsorbed molecules but we account for the different areas per molecule of the surfactant and cosurfactant. Denoting b y Pl the n u m b e r of sites occupied by a surfactant molecule and by P2 those occupied b y a cosurfactant molecule, the chemical component of the electrochemical potentials can be written as (see Appendix)
,fie 2
tz~=
/~R,O "
+ plkT In + Fcsdtzcs.
[14]
Because the third t e r m on the right-hand side can be written as (f*o [Oo-/OC]dO)dC, Eq. [14] b e c o m e s - d 3 , = F~dtz; + Fc~d/zcs
r~pl
[17]
(N - F~pl - Fcsp2) and ~ C s ---- /J,'Cs,O
+ p2kT In
FcsP2 (N - F~pl - FcsP2)
[18]
H e r e the subscript 0 refers to the standard
+ ~(q,,C)dq, +
- ~ dq, dC.
[151
Since the sum of the last two t e r m s of Eq. [15] leads to d/dC f*o o-dq~, Eq. [15] leads finally, through integration, to the expression
Y=Yo-
f : F~d/~ o-d~,
[16]
1 Two different paths of integration are possible in carrying out the integrals with respect to the chemical potentials in Eq. [16]. In the first, the first integral is carried out from tz~ = -oo to tz~ = tz~ while keeping t~cs = -o~; then the second integral is performed between - 2 and /~cs keeping /z~ at its final value. The second, alternate path results if the order of integration is changed, performing the second integral at /z~ = - ~ and then the first integral keeping ~cs at its final value. The adsorption isotherms must be such that the expression obtained for the interfacial tension should be independent of the path of integration. Journal of Colloid and Interface Science, Vol. 76, No. I, July 1980
204
RUCKENSTEIN AND KRISHNAN
chemical potentials and N is the inverse of the area occupied by a site. The area of a site is equal to the greatest c o m m o n divisor of the areas occupied by one molecule of surfactant and one molecule of cosurfactant. F r o m the condition o f equilibrium for the surfactant one can write • bulk /aVR,0
Eliminating the surface charge density between Eqs. [13] and [25], Eq. [25] can be rewritten as Y = To - N k T In (1 + (K1CRe-e~/kT) 1/m + (K2Ccs) 1/p') -
/
8~C(k____T)~ 7re 2
-l- k T In Cn = IZ~,o" + p l k T
x In
F~p1 + e~b. [19] (N - F~p, - FcsP2)
The electrical contribution to the electrochemical potential o f the surfactant in the bulk is zero, since the electrical potential far away from the surface is zero. Equation [19] can be rewritten as CR =
(F~p0mee*lkr Ka(N-
F~pl - FcsP2)p'
,
[20]
where g l = exp[(/X~,o bulk -
" IZa,o)/kT].
[21]
Similarly, the equality of chemical potentials of the cosurfactant in the bulk and the interface yields
FcsP~ (N - F~p, - Fc~p~)
[22]
K 2 ( N - F ~ o l - FcsP2)p'
In order to better understand the role played by the ionic surfactant, a simple case in which the cosurfactant is absent is first considered. In this case, Eq. [16] reduces to y = yo - f r~dl~a - l o'dtk.
t4=~,,o+kTln(
r_',. ]
N - F~]'
Consequently, Cc~ =
III. I N T E R F A C I A L T E N S I O N I N T H E P R E S E N C E OF A SINGLE IONIC SURFACTANT
[26]
For a Langmuir-type adsorption the chemical potential/x~ is given by
&bulk+ kT In Cc~ =/Xcs,o CS,0 + p z k T In
The interracial tension can be calculated as a function o f t h e surfactant concentration, cosurfactant concentration, and the total ionic strength from Eqs. [25'], [13], [20], and [23].
,
[23]
[27]
where N is equal to the surface density at saturation. Substituting Eq. [27] in [26], one obtains T = To - N k T ln N + N k T
Where K2
bulk - IZc~,o)/kT]. -=--exp[(/Zcs.0
[24]
F o r given concentrations of the surfactant and cosurfactant and given values o f the adsorption constants (N, K,, K2, p , andp2), the surface densities (F~ and Fc~) can be computed by trial and error from Eqs. [20] and [23] along With Eq. [131. Using Eqs. [17], [18], [20], and [23], Eq. [16], becomes Y = To - N k T In (1 + (K1Cae-e*/kr) 'Iv' + (K2Ccs)VP0 - ~ o'dtO. [25] 3 Journal of Colloid and Interface Science, Vol. 76, No, 1, July 1980
x In (N - F~) - [ o-dqJ. [28] The condition of thermodynamic equilibrium between the surfactant in the bulk and at the interface provides the equation t! tZR,o + k T In
V~ (N - F~)
+ eqJ
= /aVR,0 ..bu,k + k T In CR,
[29]
which can be rewritten as CR --
b0Fk (N - F ~
e e~mr.
[30]
INTERFACIALTENSION
205
Hence, if the two strong inequalities mentioned above are satisfied, Eq. [38'] is agood approximation of the more rigorous equaConsequently, tion, Eq. [33]. In principle, there are other conditions in which Eq. [33] reduces to NCR Fh = [32] Eq. [38']. For instance, this happens when (Ca + boe e*/kr) eO/kT ~ 1 and Ca/bo >> 1. Numerical comSubstituting Eq. [32] in Eq. [28], one obtains putations have been carried out to identify whether there are conditions outside the T = To - NkT limiting situations mentioned above under bo e-e*/kr o'dqJ. [33] which Eq. [38'] constitutes a satisfactory approximation of Eq. [33]. The interfacial tension in the presence of This equation in combination with the relaan ionic surfactant (without a cosurfactant) tion between the surface potential and the is plotted vs the logarithm of the surfactant surface charge (Eq. [13]) allows one to comconcentration (molecules/cmz) in Fig. 1 for pute y as a function of the concentration CR. different values of N and in Fig. 2 for For uncharged surfactant molecules and various values of b0. The solid lines are the the same values of Ca, b0, and N, Eq. [33] results obtained from Eq. [33]; the dashed leads to lines correspond to Eq. [38']. Figures 1 and 2 show clearly that Eq. [38'] does not "Yuneharged = y o - N k T l n ( l + Cb0a ) . [34] approximate in a satisfactory manner Eq. [33] over the entire range of surfactant Eliminating To between Eqs. [33] and [34], concentration. one obtains IV. INTERFACIALTENSIONIN THE PRESENCE Y = Yunchar~ea + NeqJ + NkT OF SURFACTANT,COSURFACTANT, AND SALT Here
bo = e("k0-"~?0~)/kr.
(
[31]
1 + (CR/b°)) - I o-dry. [35] (CR/bo)
The interfacial tension in the presence of an ionic surfactant and cosurfactant is When the surface potential is small such plotted in Fig. 3 vs the log concentration that etk/kT ~ 1, Eq. [35] yields of surfactant for various salt concentrations. At low concentrations, as the concentration "Y -~ Yuncharged + Netb of the surfactant in the bulk increases, _ Net~ - I o'&k. [361 more molecules of surfactant are adsorbed at the interface and the interfacial tension 1 + CR/bo decreases. Because the surfactant moleIf, in addition, CR/bo ~ 1, Eq. [36] becomes cules are ionic the interfacial charge also increases. An increasing surface charge y=Yuneharged+Net~-~o--lord~b. [37] opposes the adsorption of the surfactant but favors that of the cosurfactant. In Fig. 4 the interfacial tension is plotted The latter equation can be rewritten in the vs the log of added salt concentration. form, already suggested in Ref. (8), Table I lists, in a particular case, the Y = ')/uncharged -~- F~e~b - [ o'd~b [38] contributions of the terms J Q1 = - N k T In (1 + (KiCrte-e*/kT) ~/m x In e~7/ff +
or
Y -~ •uncharged
~-
I ~bd~.
[38']
+
(K2Ccs)
)
llp~
and
Q2 = -
I
trd~b
Journal of Colloid and Interface Science, Vol. 76, No, 1, July 1980
206
RUCKENSTEIN AND KRISHNAN
44
32
.
\ , ' \, \
\\\\ \\\\
20
"x-x'x
'\\ \\\2
44
\
\
\3 \
8 2
2
\
\
\
\\
\
\
\
\\
t5.2 ~tt61.4 t7.6~.,~. Log(Surfoctant\Concentrotion in molecules/cm z) I
FIG. l. Interfacial tension vs log (concentration of surfactant). T h e values of the p a r a m e t e r s are: b0 = 1 × 1015 molecules/cm a, concentration of salt = 10 TM molecules/cm ~, 3'0 = 50 d y n / c m , and the saturation adsorption N is 1 × 1014, 3 × 1014, and 5 × 10 TM molecules/cm ~ for curves 1, 2, and 3, respectively. The solid lines represent the rigorous equation [33] and the d a s h e d lines the approximate equation [38']. In all the figures the t e m p e r a t u r e is 25°C.
to the interfacial tension. The term Qa represents the effect of the chemical part of the chemical potentials of the surfactant and cosurfactant on the interfacial tension, while
the term Q2 is the free energy of formation of the double layer. Q1 decreases (becomes more negative) with an increase of the concentration of the added salt because
\\\\\
32
\~t
i
•~ 20 t4
\\\\\ t
T.
\\\xx~X
\\ 8
"\\ %\N
21
45,2
I 46.4
\
x. I ~. ~17:6
t 48.8
LoQ (Surfactant Concentration in molecules/m =} FIG. 2. Effect of the adsorption c o n s t a n t b0 on the plot interracial tension vs log concentration of surfactant. T h e values of the p a r a m e t e r s are: 3/o = 50 dyn/cm, N = 3 × 1014 molecules/cm 2, concentration of salt = 1019 molecules/cm ~, and the values of b0 are 1 x 10 TM, 1 × 1017, and 1 × 10 TM molecules/cm ~ for curves 1, 2, and 3, respectively. Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980
INTERFAC/AL
TENSION
207
26
20 ~o 14
FIG. 3. I n t e r f a c i a l t e n s i o n vs log c o n c e n t r a t i o n of s u r f a c t a n t w h e n a c o s u r f a c t a n t is a l s o p r e s e n t . T h e v a l u e s o f t h e p a r a m e t e r s are: K1 = 6 × 10 -18 c m 3 / m o l e c u l e , Ks = 1 × 10 -18 c m 3 / m o l e c u l e , N = 1015 m o l e c u l e s / c m 2, P l = 3, p~ = 2, the c o n c e n t r a t i o n o f t h e c o s u r f a c t a n t = 1 × 10 TMm o l e c u l e s / c m 8, 3'o = 50 d y n / c m , a n d the c o n c e n t r a t i o n s o f the s a l t are 5 x 10 TM, 5 × 10 TM, a n d 1 x 1031 m o l e c u l e s / c m 8 for c u r v e s 1, 2, a n d 3, r e s p e c t i v e l y .
the shielding of the electrical field allows more surfactant to be adsorbed. Q2 has a minimum because while the charge increases with the increase in adsorption, the surface potential increases with the charge but decreases with the salt concentration. The oil-water interfacial tension is plotted in Fig. 5 as a function of the cosurfactant concentration for various concentrations of
surfactant. Since the cosurfactant is uncharged its adsorption is not impeded by the electric field; on the contrary the competition with the surfactant for the sites of the interface is resolved in its favor. The effect of the ratio of areas per molecule for surfactant and cosurfactant on the plot interfacial tension vs log concentration of surfactant is shown in Fig. 6.
52
26
20
44
t
-o
~6,4
~7.6
t8.8
20
Log (Salt Concentration in molecules/cm5) -4(
FIG. 4. I n t e r r a c i a l t e n s i o n vs log o f s a l t c o n c e n t r a t i o n . A c o s u r f a c t a n t is p r e s e n t a l o n g w i t h a n i o n i c s u r f a c t a n t . T h e v a l u e s of t h e p a r a m e t e r s are: y0 = 50 d y n / c m , K1 = 6 × 10 -18 c m 3 / m o l e c u l e , K2 = 1 × 10 -18 c m 3 / m o l e c u l e , N = 10 TM m o l e c u l e s / c m ~, P l = 3, P2 = 2, the c o n c e n t r a t i o n o f t h e cos u r f a c t a n t = 1 × 1018 m o l e c u l e s / c m 8, a n d t h e c o n c e n t r a t i o n s o f t h e s u r f a c t a n t a r e 1 × 10 TM, 5 × 10 TM, a n d 1 x 10 TM m o l e c u l e s / c m 3 for c u r v e s 1 t h r o u g h 3, r e s p e c t i v e l y . Journal of Colloid and Interface Science, VoL 76, No. 1, July 1980
208
RUCKENSTEIN
AND KRISHNAN
TABLE I Effect of Ionic Strength on Surface Propert!es ~ - Q 1 = N k T In
C~at (molecules/cm 3)
1 1 5 1 5 1 1
×
× × × x × ×
1017 10 TM 1013 1019 10 la 1091 1022
Surface density of surfactant (1014 molecules/cm 2)
Surface density of cosurfactant (10~4 molecules/cm 2)
Surface charge (104 esu)
Surface potential (10-~ esu)
(1 + (KICRe-~'er) ~l~' + (K2Cc~)~'~) (dyn/cm)
-Q2 = f o'd0 (dyn/crn)
y = yo + Q~ + Q2 (dyn/cm)
0.778 0.789 0.831 0.871 1.038 1.533 1.895
1.916 1.908 1.876 1.846 1.721 1.350 1.078
3.73 3.78 3.99 4.18 4.98 7.36 9.09
0.478 0.473 0.456 0.439 0.376 0.214 0.102
39.43 39.62 40.30 40.97 43.85 53.83 63.08
5.66 5.72 5.94 6.14 6.83 6.98 4.49
4.91 4.66 3.76 2.89 -0.68 -10.81 - 17.57
a 3'o = 50 d y n / c m , N = 8 × 1014 m o l e c u l e s / c m 2, K1 = 6 × 10 -19 c m 3 / m o l e c u l e , Ks = 1 × 10 -I9 c m 3 / m o l e c u l e , 1013 m o l e c u l e s / c m z, a n d Csurfaetant = 10 TM m o l e c u l e s / c m 3 .
P l = 3, p~ : 2, Ceosurraetant :
It is interesting to discuss the implications of the behavior of the interfacial tension of the oil-water interface in the presence of surfactant, cosurfactant, and salt on the formation of microemulsions. Figures 3, 4, 5, and 6 show that there are conditions under which this interfacial tension becomes negative. Of course, such an interface is not thermodynamically stable. For this reason 50 40 50 4
ZO
I
0
-10
20
log !cosurfactant concentration \ \ \ \ ~nmolecutes/cm3)
-20 -50 FIG. 5. I n t e r f a c i a l t e n s i o n v s log c o s u r f a c t a n t conc e n t r a t i o n . T h e v a l u e s o f t h e p a r a m e t e r s are: 3'0 = 50 d y n / c m , K1 = 6 × 10 -19 c m 3 / m o l e c u l e , K2 = 1 x 10 -19 c m a / m o l e c u l e , N = 8 x 1014 m o l e c u l e s / c m ~, P l = 3, P2 = 2, t h e c o n c e n t r a t i o n o f t h e sal[ is 1 × 10 TM m o l e c u l e s / c m 3, a n d t h e c o n c e n t r a t i o n s o f the s u r f a c t a n t are 1 × 1015, 1 × 101~, 1 × 10 is, a n d 5 x 1018 m o l e c u l e s / c m 3 for c u r v e s 1 t h r o u g h 4, respectively. Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980
the area of the interface increases, resulting in the formation of microemulsions. The increase in the interfacial area is associated with accumulation of surfactant and cosurfactant on the interface and takes place until the bulk concentrations of surfactant and cosurfactant become sufficiently low for the interfacial tension to reach a small but positive value determined by the condition of thermodynamic equilibrium (10). The ionic surfactant affects the interfacial tension through the accumulation on the interface and through the electric field generated by the charged interface. Since the presence of the charge impedes adsorption of the charged surfactant, the reduction due to accumulation is smaller than that of a hypothetical equivalent uncharged surfactant of the same bulk concentration. The electric field reduces the amount adsorbed but, as our numerical computations show, can contribute through the term - f o-d0 to a decrease of the interfacial tension as large as 6 dyn/cm. For the usual range of adsorption constants, the total decrease due to the ionic surfactant seems to be smaller than 50 dyn/cm, which is the interfacial tension of an oil-water interface free of any surfactants. For this reason a further decrease through the adsorption of a cosurfactant is generally needed for microemulsification. Figure 4 shows that microemulsions can form above a value of the ionic
INTERFACIAL
209
TENSION
52
26 2o
==
44
2 3
t5.2
46.4
17.6
t8.s'~
Log (Surfoctont Concentration in molecules/cm 3)
FIG. 6. I n t e r r a c i a l t e n s i o n v s log c o n c e n t r a t i o n o f s u r f a c t a n t . T h e v a l u e s o f t h e p a r a m e t e r s are: 3'o = 50 d y n / c m , K , = 6 x 10 -'8 c m 3 / m o l e c u l e , K2 = 1 × 10 -18 c m 3 / m o l e c u l e , c o n c e n t r a t i o n o f salt = 10 z° m o l e c u l e s / c m 3, c o n c e n t r a t i o n of c o s u r f a c t a n t = 10 TM m o l e c u l e s / c m 3, P2 = 2, N = 10 '5 m o l e c u l e s / c m 2, a n d t h e v a l u e s o f p l are 3, 4, a n d 5 o n c u r v e s 1, 2, a n d 3, r e s p e c t i v e l y .
strength. For sufficiently small ionic strengths the surface potential is sufficiently large and the contributions to the decrease of the interracial tension are the adsorption of the cosurfactant, the impeded adsorption of the surfactant, and the free energy - f (rd~ of formation of the double layer. For large ionic strengths, the major contributions are due to the adsorption of surfactant and cosurfactant; the free energy - f o-dqJ has a minor role. The detailed computations of the radius of microemulsions and of the phase equilibria in microemulsions will be published elsewhere. Two effects, micellization and salting out, which have not been included in the above theory may impede the formation of microemulsions. As the concentration of salt is increased the critical micelle concentration decreases because the coulombic repulsion between the charged head groups is increasingly shielded. Consequently, an increasing amount of surfactant molecules can aggregate as micelles (or perhaps mixed micelles) and less free molecules of surfactant (and cosurfactant if mixed micelles form) are available for adsorption. Thus a minimum of the interracial tension can occur at a given salt concentration. A minimum can also occur with respect to the
surfactant concentration when mixed micelles form. An additional effect is the decrease of the maximum solubility of the surfactant and cosurfactant in water when the salt concentration increases (salting out). The salting out in water is generally associated with a salting in in the oil phase. At sufficiently low salt concentrations the interfacial tension decreases with increasing added salt concentration for reasons already mentioned:: At high enough concentrations of salt the salting out-salting in process reduces the concentration of the surfactant per unit volume as well as the amount adsorbed on the interface. Therefore, the interfacial tension starts to increase. Thus for a given total amount of surfactant a minimum in the interfacial tension occurs with respect to the amount of added salt. When the salting out exceeds the salting in, a surfactant phase forms between water and oil (11). This middle phase contains some water and oil and probably has a liquid crystalline or a microemulsion structure. V. C O N C L U S I O N S
The oil-water interfacial tension in the presence of an ionic surfactant, a CosurJournal of Colloid and Interface Science, Vol. 76, No. 1, July 1980
210
RUCKENSTEIN AND KR1SHNAN
factant, and an electrolyte becomes negative under certain conditions. The interface is not stable in such conditions and therefore the interfacial area increases spontaneously (a microemulsion forms) until the bulk concentrations of surfactant and cosurfactant b e c o m e sufficiently low for the interfacial tension to reach a small, positive value required by the condition of thermodynamic equilibrium. Minima in the interfacial tension with respect to the salt concentration and surfactant concentration are explained in terms of micellization and salting out.
and
APPENDIX
F2 =
ADSORPTION ISOTHERMS FOR M I X T U R E S CONTAINING U N C H A R G E D SURFACTANT AND COSURFACTANT
The a d s o r p t i o n - d e s o r p t i o n equilibrium of c o m p o n e n t s j = 1 a n d j = 2 at the interface yields k l a C l O v' = k l a ~ ~ [A1] and k2aC2 0v' - k2a~ ~. [A21 Here kja are the adsorption constants, k~d are the desorption coefficients, C~ are the concentrations in the bulk phases, 0j are the fractions of sites covered, 0 is the total fractions of sites that are unoccupied, pj are the numbers of sites per adsorbed molecule, a n d j = 1, 2 refers to the surfactant and cosurfactant, respectively. Equations [A1] and [A2] can be rewritten as
where N is the inverse o f the area occupied by a site. This area is the greatest c o m m o n divisor of the areas occupied by the surfactant and cosurfactant molecules. Substituting Eqs. [A3] and [A4] in Eqs. [A6] and [A71, the adsorption isotherms are obtained as r 1=
N
(KIC1) l/p1
Pl
1 + ( K I C O llw + ( K 2 C 2 ) l/v2
(~2) N
t"K 2C 2J~l/v,' K C "~l/loz [A9] 1 + (K1C1) 1/p' + t 2 2J
( kla C1) 1/pl ~
(K1Cl)l/p~
[A31
F r o m Eqs. [A8] and [A9] the concentrations C1 and C2 can be written as 1 { C1 = ~
Flpl (N - r ~ l "-
r~p~)
}m
[A10 ]
and
1{
r22
)P AI,1
H e n c e the chemical potentials b e c o m e I ~ = lXao + p l k T
x In
(FIp0 (N - Flpl - F2p2)
[A12]
and 1~2 = lZ2o + p z k T (F2p2)
(N - FIpl - F2pz)
[A13]
REFERENCES
and 02 _ ( k 2 a C21 ~lp2 _ ,~.. t~2 C ~J~l/p2 • ] 0 \ k2a
[A4]
Because 0 = 1 - 01 - 02, from [A31 and [A4] one obtains 0 =
[A81
and
× In 01 _ O \ km
[A7]
F 2 = (Nip2)02,
1
1 + (K1C1) vp. + (K2C2) i/v'
[A51
The surface excesses are obtained from F1 = (N/pI)01
[A6]
Journalof Colloidand InterfaceScience, Vol. 76, No. 1, July 1980
1. Shah, D. O., J. Colloid Interface Sci. 37, 744 (1971). 2. Foster, W. R., J. Petrol. Technol. 25, 205 (1973). 3. Cayias, J. L., Schechter, R. S., and Wade, W. H., in "Adsorption at Interfaces" (K. L. Mittal, Ed.), ACS Symposium Series No. 8, p. 234. Amer. Chem. Soc., Washington, D. C., 1975. 4. Friberg, S. E., Buraczewska, I., and Ravey, J. C., in "Micellization, Solubilization and Microemulsions" (K. L. Mittal, Ed.), Vol. II, p. 901. Plenum, New York, 1977.
INTERFACIAL TENSION 5. Hachisu, S., J. Colloid Interface Sci. 33, 445 (1970). 6. Ruckenstein, E., and Kfishnan, R., J. Colloid Interface Sci. 76, 188 (1980). 7. Bikerman, J. J., Z. Phys. Chem. 163A, 378 (1933). 8. Levine, S., and Robinson, K., J. Phys. Chem. 76, 876 (1972).
21 1
9. Molliet, J. L., Collie, B., and Black, W., "Surface Activity." Van Nostrand, Princeton, N. J., 1960. 10. Ruckenstein, E., Chem. Phys. Lett. 57,517 (1978); Ruckenstein, E., and Krishnan, R., J. Colloid Interface Sci. 71, 321 (1979). 11. Healy, R. N., and Reed, R. L., Soc. Petrol. Eng. J. 17, 129 (1977).
Journal of Colloid and Interface Science, Vol. 76, No. 1, July 1980