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Synergism in binary mixtures of surfactants
Synergism in Binary Mixtures of Surfactants I. Theoretical Analysis X I Y U A N H U A 1 AND M I L T O N J. R O S E N Department of Chemistry, Brooklyn College, City University of New York, Brooklyn, New York 11210 R e c e i v e d J a n u a r y 19, 1982; a c c e p t e d F e b r u a r y 1 l, 1982
Nonideal solution theory is used to derive equations for the conditions under which synergism can exist in aqueous binary mixtures of surfactants. For synergism in surface tension reduction efficiency, /3, the experimentally obtained molecular interaction parameter at the aqueous solution/air interface, must be negative and Iln (C°/C~)I must be <]/3], where C O and C O are the solution phase molar concentrations of surfactants 1 and 2, respectively, required to produce a given surface tension (reduction). For synergism in mixed micelle formation, ~s~, the molecular interaction parameter in mixed miceUe, must be negative and Iln (C~/C~)I must be
aC12 = X C ° exp[/3(1
- X ) 2]
[la]
(1 - a)C12 = (I - X ) C ° exp[/3X 2]
[lb]
o~C~ = X M C ~ exp[BM(1 -- XM) 2]
[2a]
(1 -
,~)c~
fraction o f s u r f a c t a n t 1 in the total surfactant in the surface phase; C °, C °, Cz2 = solution phase c o n c e n t r a t i o n s o f pure surfactants 1 a n d 2 a n d m i x e d surfactant, respectively, required to p r o d u c e a given surface tension (reduction); X M = m o l e fraction o f surfacrant 1 in the m i x e d micelle o f surfactants 1 a n d 2; C ~ , C ~ , C ~ = critical micelle concentrations ( C M C ) o f pure surfactants 1 a n d 2, a n d m i x e d surfactant, respectively; a n d t3, t3M = empirical parameters m e a s u r i n g the deviation f r o m nonideality a n d related to the m o l e c u l a r interaction between the two surfactants in the m i x e d m o n o l a y e r at the a q u e o u s solution/air interface a n d in the mixed micelle, respectively. T h e p a r a m e t e r s , / 3 a n d /3M are related to the activity coefficients, f l a n d f ~ , o f surfactant 1 in the m i x e d m o n o l a y e r a n d m i x e d micelle, respectively, by the relationship:
= (1 - x M ) c ~
× exp[flM(X~a)2],
[2b]
where a = m o l e fraction o f surfactant in the total m i x e d surfactant in solution; X = m o l e Visiting scholar, Ministry of Light Industries, People's Republic of China.
f l : expl3(1 - X ) 2
[3]
f ~ = expt3M(l - X M)2.
[41
212 0021-9797/82/110212-08502.00/0 Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and InterfaceScience,Vol. 90, No. 1, November 1982
213
SYNERGISM IN BINARY MIXTURES OF SURFACTANTS SYNERGISM IN SURFACE TENSION REDUCTION EFFICIENCY
The efficiency o f surface tension reduction by a surfactant has been defined (3) as the solution phase concentration to produce a given surface tension (reduction). In a mixture of surfactants, it can be defined as the total concentration of mixed surfactant required to produce a given surface tension (reduction). Synergism is present in the mixture if it can attain a given surface tension (reduction) at a total mixed concentration lower than that required of both components of the mixture• The point o f m a x i m u m synergism is where the lowest total concentration of mixed surfactant is required to attain a given surface tension (reduction). F r o m Eqs. [ 1a], In C~2 = l n X -
dx do~
2
[1ll
F r o m Eqs. [la] and [lb], we obtain:
a
XC ° exp[3(1 c~
1 -
-
X) 2
(1 - X ) C ° e x p [ 3 X 2]
[12]
Rearranging and taking logs, lnX-
In (1 - X ) + In (1 - ~) - In + In
C I0 /C
02 +
3(1
-
2X) = 0.
From which, 1 dX
--,--.3
X da
1
C
(1-X)
dX da
(1 -
a)
1
dX
[5]
dX
x(1 - x ) c~(1 - ~)[l - 23X(1 - X ) ]
[13]
Combining [ 11 ] and [ 13],
=lnX-lna+3(1-X)
2.
In a + 3(1 - X ) 2 < 0.
[8]
1
C Oexp[3(1 - X ) 2] C °2exp3X2 = 1,
[7]
--0.
From Eq. [5], 1 dX Xda
1
23(1
- X)
[ldX123(l_x)dX " d,~
o~
dX
~
"
[9]
Since C~2 can never be 0, when
dC~2/da will
[14]
where ~* is the mole fraction o f surfactant 1 in the total surfactant in the solution phase at the point of m a x i m u m synergism. Relationship [14] states that, for a m i n i m u m or m a x i m u m to exist in the Cla versus a curve, the mole fraction o f either surfactant in the total surface-active material in the surface phase must equal its mole fraction in the total surface-active material in the solution phase. Combining Eqs. [ 14] and [7], and recalling that (1 - X ) 2 is always a positive number, we conclude that, for synergism to exist, 3 must be negative. Since a = X at the m i n i m u m , Eq. [12] reduces to
When synergism exists, there will be a m i n i m u m in the C12 versus a curve and m a x i m u m synergism will be attained where the Cl2 versus ~ curve shows a minimum; mathematically, that
dG2 dc~
X = a*,
[61
The condition for synergism, therefore, is:
=C12
X a [ l - 23X(1 - X)]
Od
In C12 - In C O
da
or
d.
dCl2
[101
---23~=0,
and
d In CI2 da
23(1-x) dX=0
In a +1nO °+3(l-x)
lnX-
1
1 dX
be 0
from which, In
(C°/C °) =
-fl(1 - 2X).
[151
Since 0 < X < 1 and/3 must be negative Journal of Colloid and InterfaceScience, VoL 90, No, 1, November 1982
214
HUA AND ROSEN
for synergism to exist, relationship [ 15] states that a second condition for synergism is that ]ln (C°/C°)[ must be <18[. When surfactant 1 is more surface-active than surfactant l, i.e., C o < C °, then a* and X will be >0.5 at the point of m a x i m u m synergism; when surfactant I is less surface-active than 2, a* and X will be <0.5 at that point. The m i n i m u m total mixed surfactant concentration in the solution phase, Ct2,min , r e q u i r e d to produce a given surface tension (reduction), and the individual solution concentrations of surfactants 1 and 2, US1 and Cs2, respectively, at that point of m a x i m u m synergism in surface tension reduction efficiency, can be calculated from Eqs. [5] and [15]. Since a* = X a t that point, Eq. [5] reduces to: In Ct2,min --~ In C O + 8(1 - X ) 2 .
2BM M Ct2,min = C 1M
×exp[BM(flM--InCM/cM] 2] 28 M
CM=
In ( c ° / c °) + 8 = a*. 28
M) + 8 M
a*=xM=ln(C~/C2
[16]
From Eq. [15], X =
as micelle formation at total mixed surfactant concentrations in the solution phase lower than the CMCs of both surfactants in the mixture. From Eqs. [2a] and [2b], relationships analogous to those derived above can be obtained for synergism in mixed micelle formation. For synergism in this respect to exist: (i) 8 M must be negative; and (ii) rln (C~/C~)I must be
[17]
In (Cl~/C~) + 8 M 28u CM × exp [ ( ~ M - In
C~/C~'~ 2"] [231 ] J
28 M
Substituting [ 17] into [ 16],
and CsM = C M 12,rnin -- C M SI
In C12,min
C°/C°) 2
= In C o + 8(8 - In
[18]
or C l 2 m i n = C°l
exp[8( f l - l n
IJ
Thus,
]n (C°,/C °) + 8 C o
× exp[~( 8
[24]
where C M st and C s~ are the individual phase concentrations of surfactants 1 and 2 required to produce the m i n i m u m CMC value.
C?/C°2~21
V
[19]
C s l -= ot*Ct2,min =
[22]
] .]'
28 -ln-C°/C°]21 28
} .J
[201
and CS2 = Ct2,min -- C s t .
[21]
SYNERGISM IN M I X E D MICELLE F O R M A T I O N
Synergism in mixed micelle formation in binary mixtures of surfactants can be defined Journal of Colloid and Interface Science, VoL 90, No. 1, November 1982
SYNERGISM IN SURFACE TENSION R E D U C T I O N EFFECTIVENESS
The effectiveness of surface tension reduction has been defined (4) as the surface tension reduction attained at the CMC. Synergism in surface tension reduction effectiveness is therefore present in an aqueous binary mixture of surfactants when the solution at the CMC of any mixture reaches a surface tension lower than that attained at the CMCs of both surfactants comprising the mixture. The chemical potential, u s, of surfactant 1 in the surface phase of the aqueous mixture can be written (5):
~s = ~ + RTln Xfl -
')'12-41,
SYNERGISM -3.5
IN BINARY
MIXTURES
OF SURFACTANTS
(al
-3.6
.0
3.7
).9
-
-3.8
(b)
I
).7
3.9
-
2 15
N_-4. (..) ×
om_4.
~.5i I
-4.2
0.4i
-4.3
0.3
-4.4
0.2
-4.5
0.1 [
-4.6
I 0.2
I
I 0.4
1
I 0.6
Q
[
I 0.8
I 1.0
0
0.2
0.4
Ct
0.6
0.8
1.0
FIG. 1. (a) Log of total mixed surfactant concentration, C~2, to produce a given surface tension (reduction) vs mole fraction, a, of surfactant 1 in the total surfactant in the solution phase; Iln C°dC°21 = 1.7. (1) fl = - 1, (2) [3 = -2, (3) fl = -3.2, (4) fl = -4. *Experimental points for Ct2H2sSO4Na/ CIzHzs(OCzH4)8OH mixtures at 25°C; (In C°t/C°I = 1.7; fl = -3.2; surface tension reduction, 7r = 36.7 mN m -I (6). (b) Mole fraction, X, of surfactant 1 in the total surfactant in the surface phase vs mole fraction, cg of surfactant 1 in the total surfactant in the solution phase. where A I is the surface area o c c u p i e d by a m o l e o f surfactant 1 in the m i x e d m o n o l a y e r , fl its activity coefficient there, a n d %2 the surface tension o f the solution. T h e s t a n d a r d state for surfactant 1 in the surface phase is a hypothetical m o n o l a y e r o f p u r e surfactant 1 at zero surface tension. At the C M C o f the mixture, S
/q,~ = lug + R T In Xffl,o
-
TI2,cAI,c
C~o~.
C ~ = u~ + RTln
Xff~,¢ - %2,~A1,~.
C ~ + ~q,cA°,o
[28]
Al,c = A ° c .
[291
C o m b i n i n g [27], [28], a n d [29], -/,2,~ =A0,--~ l n L
C~
/ + %'°"
[301
Similarly,
[26]
At equilibrium,/zsx = u~,~. Therefore, f r o m Eqs. [25] a n d [26],
eo + R T l n /*l,~
/30 = R T l n ~1,c
where A°c is the surface area occupied by a m o l e o f surfactant 1 in a m o n o l a y e r of p u r e surfactant 1 at its C M C . W e a s s u m e that the area occupied by a m o l e of surfactant 1 in the m i x e d m o n o l a y e r at the C M C o f the m i x t u r e is the s a m e as that in a m o n o l a y e r o f the pure surfactant at its C M C , i.e.,
[25]
where the subscript, c, refers to values at the C M C o f the mixture. T h e c h e m i c a l potential, #~., o f surfactant 1 in the solution phase at the C M C o f the m i x t u r e is: uet,~ = Uq° + R T l n
# l~,c. - -
[27]
F o r an a q u e o u s solution o f p u r e surfactant 1 at its C M C , Eq. [27] reduces to:
r~2,c-
R T In F C ~ (-1 - X¢)f2,c
0
A2,c
L C~(1-~)
+v2,c
[31]
where f2.o is the activity coefficient of surfactant 2 in the m i x e d m o n o l a y e r at the C M C a n d A°~ its surface a r e a / m o l e at its C M C . W h e n %,~ < y2,c, the c o n d i t i o n for synergism in surface tension reduction effectiveness is 7n,o - % , < 0 or Journal of Colloid and Interface Science, Vol. 90, No. 1, November 1982
216
HUA AND ROSEN
-3.3 - 3.4
~
-3.5
1.0
-3.6
0.91
-3.7
o.e !
-3.8
0.7
- 3.9
0.6
o -4,0
X
0.5
-4.1!
0.4
-4.2
0.5
3 2
0.2 0.1
-4.4 I
0
I
I
0.2
I
0.4
[
I
I
0.6
I
I
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
(2
FIG. 2. (a) Log of the CMC, C ~ o f the mixture vs mole fraction, c~, of surfactant 1 in the total surfactant in the solution phase; Iln C~/C~I = 1.9. (1) (3M = - 1 , (2) #M = - 2 , (3) #M = - 2 . 6 , (4) ~tM = - 3 . 5 , (5) #r~ = - 4 . *Experimental points for C12H25SO4Na/Ct2H~s(OC2H4)8OH mixtures at 25°C; ]ln C~/C~I = 1.9; #~a = - 2 . 6 (6). (b) Mole fraction, X M, of surfactant 1 in the m i x e d micelle vs mole fraction, c~, o f surfactant 1 in the total surfactant in the solution phase.
R T l gC~.Xc.f~c] n L
' ] < 0.
t321
Since RT/A°,c is always > 0, relationship [32] reduces to in - -
c~a
or
-<0
cMa In (fl,c'Xc) < In C---~- "
[33]
APPLICATION TO HYPOTHETICAL A N D R E A L SYSTEMS
From Eqs. [2a] and [4],
C,~.
In C - ~ = In X ~a + tiM(1 - XM) 2 =
In X M + In f ? .
Surface Tension Reduction Efficiency [34]
From Eqs. [33] and [34], the condition for synergism in surface tension reduction effectiveness is: Yl,c" Xc < fM. X M "
[351
Relationship [35] states that the condition for synergism in surface tension reduction effecJournal of Colloid and Interface Science, Vol. 90, No. 1, November
tiveness is that, at the CMC of the mixture, the activity of each surfactant in the mixed monolayer at the aqueous solution/air interface be lower than its activity in the mixed micelle. This means that, when the mole fraction of surfactant in the mixed monolayer at the aqueous solution/air interface equals that in the mixed miceUe, the value of/5 must be smaller than the value of tim for synergism to exist.
1982
The conditions for synergism in this respect are: (i) /~ is negative; (ii) ]In (C°/C°){ < I/~L.Figure la shows plots of log C~2 versus a for some hypothetical systems where Iln (C°/C°)l = 1.7 and/3 is varied from - 1 to - 4. Values of Cz2 were calculated for various values of a and fl, at fixed values of C o and C O= 2.8 × l 0 - 4 and 5.0 × 10 -s mol dm -3, respectively, by solving the relation-
217
SYNERGISM IN BINARY MIXTURES OF SURFACTANTS TABLE I Calculated and ExpefimentaP Maximum Synergism Values, C~z,~and System
o?*
C~2SO4Na/Cs(EO)4
C12SO4Na/Cs(EO)6 Cj2SO4Na/Cs(EOh2 C1gN(CH3)3C1/CI2(EO)sin 0.0024 N NaCI C2oN(CH3)sCI/CI2(EO)s
0.48 0.43 0.5 l 0.53 0.51
CP~,in (eale) (mol dm -3)
3.43 × 3.55 X 3.04 × 4.20 × 3.13 ×
10-3 10-3 10-3 10-5 10-5
Cl2.~ (talc)~ (tool dln ~a)
0.24
C~2SO4Na/CI2(EO)s in 0.5 M NaC1
M Cl2,mi~
4.19 × 10-5
C~2,,~i. (expfl) (mol dm -3)
3.49 X 3.33 × 3.03 × 4.25 × 3.14 ×
(at c~ = 10-3 (at a = 10-3 (at a = 10-5 (at a = 10-5 (at a =
10 -3
0.5) 0.4) 0.5) 0.5) 0.5)
Cn,~a~ (exptl)b (mol dm -3)
4.57 × 10-5 (at a = 0.5)
Data from Ref. (6). bFor 7r = 36.7 mN m 1.
ship: x ( 1 - ~ ) c ° exp[~(1 - x ) 2]
(1 -
X)aC°2 exp[flX 2]
= 1,
[36]
obtained b y dividing Eq. [la] by [lb], iteratively for X, a n d t h e n substituting in Eq. [la] to obtain the value o f C~2. It shows t h e l a c k o f s y n e r g i s m in t h e s y s t e m (curve 1) w h e n [In (C°/C°)l is >13[ a n d the increasing degree o f synergism as [3[ b e c o m e s larger c o m p a r e d to Iln (C°/C°)[. S o m e experimental points for the system CI2HzsSO4Na/CIzH25(OC2H4)sOH in 0.5 M NaC1, where C o a n d C o have the same respective values and/~ = - 3 . 2 , taken f r o m the data o f L a n g e a n d Beck (6), are s h o w n for c o m p a r i s o n with curve 3, w h i c h has the same value of/3. Figure l b shows plots o f X vs a for the s a m e four hypothetical systems. T h e 45 ° line cuts the curves at the points o f m a x i m u m synergism, where a* = X.
Synergism in Mixed Micelle Formation T h e conditions for synergism here are: (i) /~M is negative; (ii) tln C~/C~I < l/3MI. Figure 2a shows plots o f log C ~ versus for s o m e h y p o t h e t i c a l systems where [ln C1M/C~I = 1.9 a n d tim is varied f r o m - 1 to
- 4 . Values o f C ~ were calculated for various values o f a a n d t3M, at fixed values o f C ~ = 4.1 × 10-4 a n d C2M = 6.0 × 10 -5 m o l d m -3, by solving the relationship: XM(1 - ~)C1~ exp[/3M(1 -- XM) 2] (I -- xM)aC~ exp[/3~a(XM) ~] = 1, [37] obtained by dividing Eq. [2a] by [2b] iteratively for X M, and then substituting in Eq. [2a] to obtain the value o f C ~. Here, curve 1, where /3 -- - 1 , shows n o synergism a n d curve 2, where/3 = - 2 , a barely perceptible m i n i m u m . Experimental points for the system C1zH25SO4Na/CL2H2~(OCzH4)8OH in 0.5 M NaC1 (6), where C ~ a n d C ~ have the same respective values a n d / 3 ~ = - 2 . 6 , are shown for c o m p a r i s o n with curve 3, which has the same value of/3 ~. Figure 2b shows plots o f X M versus a for the same four hypothetical systems. T h e 45 ° line cuts the curves at the point o f m a x i m u m synergism, where a* = X M. Table I lists s o m e values for the points o f m a x i m u m synergism in b o t h surface tension r e d u c t i o n efficiency, C12,rnin, a n d m i x e d micelle formation, C M 12,mi,, calculated b y Eqs. [19] a n d [22], respectively, c o m p a r e d to experimental values for the same systems. Journal of Colloid and Interface Science, VoL 90, No. l, November t982
218
HUA AND ROSEN TABLE II Synergism in Surface Tension Reduction Effectivenessa "~/'12,c
a
X¢
X~
fx,~
(mN m -I)
f~X M
Synergism
System:Ca2H25SO4Na/C12H25(OC2H4)8OH in 0.5 M NaC1; tim = --2.6; 13= --3.2 0.2 0,5 0.9
0.24 0.36 0.58
0.17 0.30 0.53
0.036 0.098 0.324
0.028 0.082 0.298
33,8 32,9 31.8
NO No No
32.4 31.2 29.9
NO No Yes
30.6 29.1 27.5
Yes Yes Yes
System: hypothetical, tim = --2.6; fl = --4.0 0.2 0.5 0.9
0.27 0.39 0.57
0.17 0.30 0.53
0.032 0.085 0.276
0.028 0.082 0.298
System: hypothetical; tiM = 0.2 0.5 0.9
0.31 0.41 0.57
0.17 0.30 0.53
0.0278 0.071 0.225
--2.6; 13 = --5.0 0.0284 0.082 0.298
a Data from Ref. (6): C~ = 4.1 X 10-3 mol din-3; C~ = 6.0 X 10-3 mol dm-3; 3'~ = 30.8 mN m-J; -y~ = 35.2 mN m-I; RT/A° = 11.81 naN m-l; RT/A° = 7.89 mN m -I.
Synergism in Surface Tension Reduction Effectiveness
(xM) in
The condition for synergism here is: Y,,cx¢
<
,
(1
-
in
[
c (1
-
.) 7
= 1
[381
[35]
at the C M C o f the mixture. There are almost no data in the literature from which calculations of/3, tiM, X¢, and X u can be m a d e to verify the above relationship. Table II lists some data for a real system in which synergism of this type does not exist, together with data for some hypothetical systems in which the values o f C ~ , C ~ , A °, A °, -y~, 3'~, (i.e., the CMC, area per molecule and surface tension at the C M C for the individual surfactants) and flM are identical with those of the real system, but in which the value of 13 is m o r e negative. The values ofXc, X M, fl,¢Xc, fMxM, and ~2,¢ at various values o f a in Table II are obtained by the following procedure: (1) The relationships derived by Rubingh
(2), Journal of Colloid and Interface Science, Vol. 90, No. 1, November 1982
and In \ c M x M ] tM
--
(1 -- xM) 2
'
[39]
are used to calculate X M and tiM. TO do this, values of the C M C of the individual surfacrants, C M and C M, and the C M C s of one or m o r e mixtures o f them, C M, are obtained experimentally. Having obtained the value of/3 M, values o f X M, C 1112,and f ~ for various values of a are obtained f r o m Eq. [37], [2a], and [4], respectively. (2) The value of t is obtained f r o m equations analogous to [38] and [39], using experimental values o f C o and C o for the individual surfactants to produce a given surface tension and experimental values, C~2, for
SYNERGISM IN BINARY MIXTURES OF SURFACTANTS
one or more of these mixtures required to produce the same surface tension (1). Equations [30] and [31] are solved numerically for Xc and 3'12,~ for the same values of a as used above. To do this, the relationships (1), f l,c = t3(1 - X~)z and f2,c = ~(X~)2 are first substituted into Eqs. [30] and [3 I], and then. the values of C ~ obtained above for the same values of a, and values of A°~ and A2,c, 0 C ~, and C ~, obtained from the surface tension-concentration curves of the individual surfactants, as substituted. From the data in Table II, it is apparent that f~,~X~ must be < f ~ X M for synergism to occur. As the value of ~ becomes more negative compared to that of t3M, the synergistic effect increases. The hypothetical system in which ~ = - 4 . 0 is particularly instructive in
219
that it illustrates the fact that synergism may exist only at certain values of a. Additional work in this area is currently under way in our laboratory. REFERENCES 1. Rosen, M. J., and Hua, X. Y., J. Colloid Interface Sci. 86, 164 (1982). 2. Rubingh, D. N., in "Solution Chemistry of Surfac).ants" (K. L. Mittal, Ed.), Vol. 1, pp. 337-354. Plenum, New York, 1979. 3. Rosen, M. J., J. Amer. Oil Chemists Soc. 51, 461 (1974). 4. Rosen, M. J., J. Colloid Interface Sci. 56, 320 (1976). 5. Delay, R., Prigogine, I., and Bellemans, A., "Surface Tension and Adsorption," p. 166. John Wiley, New York, 1966. 6. Lange, H., and Beck, K. H., Kolloid Z. Z. Polym. 251, 424, (1973).
Journal of Colloid and Interface Science, Vol. 90, No. 1, November 1982
Nonideal solution theory is used to derive equations for the conditions under which synergism can exist in aqueous binary mixtures of surfactants. For synergism in surface tension reduction efficiency, /3, the experimentally obtained molecular interaction parameter at the aqueous solution/air interface, must be negative and Iln (C°/C~)I must be <]/3], where C O and C O are the solution phase molar concentrations of surfactants 1 and 2, respectively, required to produce a given surface tension (reduction). For synergism in mixed micelle formation, ~s~, the molecular interaction parameter in mixed miceUe, must be negative and Iln (C~/C~)I must be
aC12 = X C ° exp[/3(1
- X ) 2]
[la]
(1 - a)C12 = (I - X ) C ° exp[/3X 2]
[lb]
o~C~ = X M C ~ exp[BM(1 -- XM) 2]
[2a]
(1 -
,~)c~
fraction o f s u r f a c t a n t 1 in the total surfactant in the surface phase; C °, C °, Cz2 = solution phase c o n c e n t r a t i o n s o f pure surfactants 1 a n d 2 a n d m i x e d surfactant, respectively, required to p r o d u c e a given surface tension (reduction); X M = m o l e fraction o f surfacrant 1 in the m i x e d micelle o f surfactants 1 a n d 2; C ~ , C ~ , C ~ = critical micelle concentrations ( C M C ) o f pure surfactants 1 a n d 2, a n d m i x e d surfactant, respectively; a n d t3, t3M = empirical parameters m e a s u r i n g the deviation f r o m nonideality a n d related to the m o l e c u l a r interaction between the two surfactants in the m i x e d m o n o l a y e r at the a q u e o u s solution/air interface a n d in the mixed micelle, respectively. T h e p a r a m e t e r s , / 3 a n d /3M are related to the activity coefficients, f l a n d f ~ , o f surfactant 1 in the m i x e d m o n o l a y e r a n d m i x e d micelle, respectively, by the relationship:
= (1 - x M ) c ~
× exp[flM(X~a)2],
[2b]
where a = m o l e fraction o f surfactant in the total m i x e d surfactant in solution; X = m o l e Visiting scholar, Ministry of Light Industries, People's Republic of China.
f l : expl3(1 - X ) 2
[3]
f ~ = expt3M(l - X M)2.
[41
212 0021-9797/82/110212-08502.00/0 Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and InterfaceScience,Vol. 90, No. 1, November 1982
213
SYNERGISM IN BINARY MIXTURES OF SURFACTANTS SYNERGISM IN SURFACE TENSION REDUCTION EFFICIENCY
The efficiency o f surface tension reduction by a surfactant has been defined (3) as the solution phase concentration to produce a given surface tension (reduction). In a mixture of surfactants, it can be defined as the total concentration of mixed surfactant required to produce a given surface tension (reduction). Synergism is present in the mixture if it can attain a given surface tension (reduction) at a total mixed concentration lower than that required of both components of the mixture• The point o f m a x i m u m synergism is where the lowest total concentration of mixed surfactant is required to attain a given surface tension (reduction). F r o m Eqs. [ 1a], In C~2 = l n X -
dx do~
2
[1ll
F r o m Eqs. [la] and [lb], we obtain:
a
XC ° exp[3(1 c~
1 -
-
X) 2
(1 - X ) C ° e x p [ 3 X 2]
[12]
Rearranging and taking logs, lnX-
In (1 - X ) + In (1 - ~) - In + In
C I0 /C
02 +
3(1
-
2X) = 0.
From which, 1 dX
--,--.3
X da
1
C
(1-X)
dX da
(1 -
a)
1
dX
[5]
dX
x(1 - x ) c~(1 - ~)[l - 23X(1 - X ) ]
[13]
Combining [ 11 ] and [ 13],
=lnX-lna+3(1-X)
2.
In a + 3(1 - X ) 2 < 0.
[8]
1
C Oexp[3(1 - X ) 2] C °2exp3X2 = 1,
[7]
--0.
From Eq. [5], 1 dX Xda
1
23(1
- X)
[ldX123(l_x)dX " d,~
o~
dX
~
"
[9]
Since C~2 can never be 0, when
dC~2/da will
[14]
where ~* is the mole fraction o f surfactant 1 in the total surfactant in the solution phase at the point of m a x i m u m synergism. Relationship [14] states that, for a m i n i m u m or m a x i m u m to exist in the Cla versus a curve, the mole fraction o f either surfactant in the total surface-active material in the surface phase must equal its mole fraction in the total surface-active material in the solution phase. Combining Eqs. [ 14] and [7], and recalling that (1 - X ) 2 is always a positive number, we conclude that, for synergism to exist, 3 must be negative. Since a = X at the m i n i m u m , Eq. [12] reduces to
When synergism exists, there will be a m i n i m u m in the C12 versus a curve and m a x i m u m synergism will be attained where the Cl2 versus ~ curve shows a minimum; mathematically, that
dG2 dc~
X = a*,
[61
The condition for synergism, therefore, is:
=C12
X a [ l - 23X(1 - X)]
Od
In C12 - In C O
da
or
d.
dCl2
[101
---23~=0,
and
d In CI2 da
23(1-x) dX=0
In a +1nO °+3(l-x)
lnX-
1
1 dX
be 0
from which, In
(C°/C °) =
-fl(1 - 2X).
[151
Since 0 < X < 1 and/3 must be negative Journal of Colloid and InterfaceScience, VoL 90, No, 1, November 1982
214
HUA AND ROSEN
for synergism to exist, relationship [ 15] states that a second condition for synergism is that ]ln (C°/C°)[ must be <18[. When surfactant 1 is more surface-active than surfactant l, i.e., C o < C °, then a* and X will be >0.5 at the point of m a x i m u m synergism; when surfactant I is less surface-active than 2, a* and X will be <0.5 at that point. The m i n i m u m total mixed surfactant concentration in the solution phase, Ct2,min , r e q u i r e d to produce a given surface tension (reduction), and the individual solution concentrations of surfactants 1 and 2, US1 and Cs2, respectively, at that point of m a x i m u m synergism in surface tension reduction efficiency, can be calculated from Eqs. [5] and [15]. Since a* = X a t that point, Eq. [5] reduces to: In Ct2,min --~ In C O + 8(1 - X ) 2 .
2BM M Ct2,min = C 1M
×exp[BM(flM--InCM/cM] 2] 28 M
CM=
In ( c ° / c °) + 8 = a*. 28
M) + 8 M
a*=xM=ln(C~/C2
[16]
From Eq. [15], X =
as micelle formation at total mixed surfactant concentrations in the solution phase lower than the CMCs of both surfactants in the mixture. From Eqs. [2a] and [2b], relationships analogous to those derived above can be obtained for synergism in mixed micelle formation. For synergism in this respect to exist: (i) 8 M must be negative; and (ii) rln (C~/C~)I must be
[17]
In (Cl~/C~) + 8 M 28u CM × exp [ ( ~ M - In
C~/C~'~ 2"] [231 ] J
28 M
Substituting [ 17] into [ 16],
and CsM = C M 12,rnin -- C M SI
In C12,min
C°/C°) 2
= In C o + 8(8 - In
[18]
or C l 2 m i n = C°l
exp[8( f l - l n
IJ
Thus,
]n (C°,/C °) + 8 C o
× exp[~( 8
[24]
where C M st and C s~ are the individual phase concentrations of surfactants 1 and 2 required to produce the m i n i m u m CMC value.
C?/C°2~21
V
[19]
C s l -= ot*Ct2,min =
[22]
] .]'
28 -ln-C°/C°]21 28
} .J
[201
and CS2 = Ct2,min -- C s t .
[21]
SYNERGISM IN M I X E D MICELLE F O R M A T I O N
Synergism in mixed micelle formation in binary mixtures of surfactants can be defined Journal of Colloid and Interface Science, VoL 90, No. 1, November 1982
SYNERGISM IN SURFACE TENSION R E D U C T I O N EFFECTIVENESS
The effectiveness of surface tension reduction has been defined (4) as the surface tension reduction attained at the CMC. Synergism in surface tension reduction effectiveness is therefore present in an aqueous binary mixture of surfactants when the solution at the CMC of any mixture reaches a surface tension lower than that attained at the CMCs of both surfactants comprising the mixture. The chemical potential, u s, of surfactant 1 in the surface phase of the aqueous mixture can be written (5):
~s = ~ + RTln Xfl -
')'12-41,
SYNERGISM -3.5
IN BINARY
MIXTURES
OF SURFACTANTS
(al
-3.6
.0
3.7
).9
-
-3.8
(b)
I
).7
3.9
-
2 15
N_-4. (..) ×
om_4.
~.5i I
-4.2
0.4i
-4.3
0.3
-4.4
0.2
-4.5
0.1 [
-4.6
I 0.2
I
I 0.4
1
I 0.6
Q
[
I 0.8
I 1.0
0
0.2
0.4
Ct
0.6
0.8
1.0
FIG. 1. (a) Log of total mixed surfactant concentration, C~2, to produce a given surface tension (reduction) vs mole fraction, a, of surfactant 1 in the total surfactant in the solution phase; Iln C°dC°21 = 1.7. (1) fl = - 1, (2) [3 = -2, (3) fl = -3.2, (4) fl = -4. *Experimental points for Ct2H2sSO4Na/ CIzHzs(OCzH4)8OH mixtures at 25°C; (In C°t/C°I = 1.7; fl = -3.2; surface tension reduction, 7r = 36.7 mN m -I (6). (b) Mole fraction, X, of surfactant 1 in the total surfactant in the surface phase vs mole fraction, cg of surfactant 1 in the total surfactant in the solution phase. where A I is the surface area o c c u p i e d by a m o l e o f surfactant 1 in the m i x e d m o n o l a y e r , fl its activity coefficient there, a n d %2 the surface tension o f the solution. T h e s t a n d a r d state for surfactant 1 in the surface phase is a hypothetical m o n o l a y e r o f p u r e surfactant 1 at zero surface tension. At the C M C o f the mixture, S
/q,~ = lug + R T In Xffl,o
-
TI2,cAI,c
C~o~.
C ~ = u~ + RTln
Xff~,¢ - %2,~A1,~.
C ~ + ~q,cA°,o
[28]
Al,c = A ° c .
[291
C o m b i n i n g [27], [28], a n d [29], -/,2,~ =A0,--~ l n L
C~
/ + %'°"
[301
Similarly,
[26]
At equilibrium,/zsx = u~,~. Therefore, f r o m Eqs. [25] a n d [26],
eo + R T l n /*l,~
/30 = R T l n ~1,c
where A°c is the surface area occupied by a m o l e o f surfactant 1 in a m o n o l a y e r of p u r e surfactant 1 at its C M C . W e a s s u m e that the area occupied by a m o l e of surfactant 1 in the m i x e d m o n o l a y e r at the C M C o f the m i x t u r e is the s a m e as that in a m o n o l a y e r o f the pure surfactant at its C M C , i.e.,
[25]
where the subscript, c, refers to values at the C M C o f the mixture. T h e c h e m i c a l potential, #~., o f surfactant 1 in the solution phase at the C M C o f the m i x t u r e is: uet,~ = Uq° + R T l n
# l~,c. - -
[27]
F o r an a q u e o u s solution o f p u r e surfactant 1 at its C M C , Eq. [27] reduces to:
r~2,c-
R T In F C ~ (-1 - X¢)f2,c
0
A2,c
L C~(1-~)
+v2,c
[31]
where f2.o is the activity coefficient of surfactant 2 in the m i x e d m o n o l a y e r at the C M C a n d A°~ its surface a r e a / m o l e at its C M C . W h e n %,~ < y2,c, the c o n d i t i o n for synergism in surface tension reduction effectiveness is 7n,o - % , < 0 or Journal of Colloid and Interface Science, Vol. 90, No. 1, November 1982
216
HUA AND ROSEN
-3.3 - 3.4
~
-3.5
1.0
-3.6
0.91
-3.7
o.e !
-3.8
0.7
- 3.9
0.6
o -4,0
X
0.5
-4.1!
0.4
-4.2
0.5
3 2
0.2 0.1
-4.4 I
0
I
I
0.2
I
0.4
[
I
I
0.6
I
I
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
(2
FIG. 2. (a) Log of the CMC, C ~ o f the mixture vs mole fraction, c~, of surfactant 1 in the total surfactant in the solution phase; Iln C~/C~I = 1.9. (1) (3M = - 1 , (2) #M = - 2 , (3) #M = - 2 . 6 , (4) ~tM = - 3 . 5 , (5) #r~ = - 4 . *Experimental points for C12H25SO4Na/Ct2H~s(OC2H4)8OH mixtures at 25°C; ]ln C~/C~I = 1.9; #~a = - 2 . 6 (6). (b) Mole fraction, X M, of surfactant 1 in the m i x e d micelle vs mole fraction, c~, o f surfactant 1 in the total surfactant in the solution phase.
R T l gC~.Xc.f~c] n L
' ] < 0.
t321
Since RT/A°,c is always > 0, relationship [32] reduces to in - -
c~a
or
-<0
cMa In (fl,c'Xc) < In C---~- "
[33]
APPLICATION TO HYPOTHETICAL A N D R E A L SYSTEMS
From Eqs. [2a] and [4],
C,~.
In C - ~ = In X ~a + tiM(1 - XM) 2 =
In X M + In f ? .
Surface Tension Reduction Efficiency [34]
From Eqs. [33] and [34], the condition for synergism in surface tension reduction effectiveness is: Yl,c" Xc < fM. X M "
[351
Relationship [35] states that the condition for synergism in surface tension reduction effecJournal of Colloid and Interface Science, Vol. 90, No. 1, November
tiveness is that, at the CMC of the mixture, the activity of each surfactant in the mixed monolayer at the aqueous solution/air interface be lower than its activity in the mixed micelle. This means that, when the mole fraction of surfactant in the mixed monolayer at the aqueous solution/air interface equals that in the mixed miceUe, the value of/5 must be smaller than the value of tim for synergism to exist.
1982
The conditions for synergism in this respect are: (i) /~ is negative; (ii) ]In (C°/C°){ < I/~L.Figure la shows plots of log C~2 versus a for some hypothetical systems where Iln (C°/C°)l = 1.7 and/3 is varied from - 1 to - 4. Values of Cz2 were calculated for various values of a and fl, at fixed values of C o and C O= 2.8 × l 0 - 4 and 5.0 × 10 -s mol dm -3, respectively, by solving the relation-
217
SYNERGISM IN BINARY MIXTURES OF SURFACTANTS TABLE I Calculated and ExpefimentaP Maximum Synergism Values, C~z,~and System
o?*
C~2SO4Na/Cs(EO)4
C12SO4Na/Cs(EO)6 Cj2SO4Na/Cs(EOh2 C1gN(CH3)3C1/CI2(EO)sin 0.0024 N NaCI C2oN(CH3)sCI/CI2(EO)s
0.48 0.43 0.5 l 0.53 0.51
CP~,in (eale) (mol dm -3)
3.43 × 3.55 X 3.04 × 4.20 × 3.13 ×
10-3 10-3 10-3 10-5 10-5
Cl2.~ (talc)~ (tool dln ~a)
0.24
C~2SO4Na/CI2(EO)s in 0.5 M NaC1
M Cl2,mi~
4.19 × 10-5
C~2,,~i. (expfl) (mol dm -3)
3.49 X 3.33 × 3.03 × 4.25 × 3.14 ×
(at c~ = 10-3 (at a = 10-3 (at a = 10-5 (at a = 10-5 (at a =
10 -3
0.5) 0.4) 0.5) 0.5) 0.5)
Cn,~a~ (exptl)b (mol dm -3)
4.57 × 10-5 (at a = 0.5)
Data from Ref. (6). bFor 7r = 36.7 mN m 1.
ship: x ( 1 - ~ ) c ° exp[~(1 - x ) 2]
(1 -
X)aC°2 exp[flX 2]
= 1,
[36]
obtained b y dividing Eq. [la] by [lb], iteratively for X, a n d t h e n substituting in Eq. [la] to obtain the value o f C~2. It shows t h e l a c k o f s y n e r g i s m in t h e s y s t e m (curve 1) w h e n [In (C°/C°)l is >13[ a n d the increasing degree o f synergism as [3[ b e c o m e s larger c o m p a r e d to Iln (C°/C°)[. S o m e experimental points for the system CI2HzsSO4Na/CIzH25(OC2H4)sOH in 0.5 M NaC1, where C o a n d C o have the same respective values and/~ = - 3 . 2 , taken f r o m the data o f L a n g e a n d Beck (6), are s h o w n for c o m p a r i s o n with curve 3, w h i c h has the same value of/3. Figure l b shows plots o f X vs a for the s a m e four hypothetical systems. T h e 45 ° line cuts the curves at the points o f m a x i m u m synergism, where a* = X.
Synergism in Mixed Micelle Formation T h e conditions for synergism here are: (i) /~M is negative; (ii) tln C~/C~I < l/3MI. Figure 2a shows plots o f log C ~ versus for s o m e h y p o t h e t i c a l systems where [ln C1M/C~I = 1.9 a n d tim is varied f r o m - 1 to
- 4 . Values o f C ~ were calculated for various values o f a a n d t3M, at fixed values o f C ~ = 4.1 × 10-4 a n d C2M = 6.0 × 10 -5 m o l d m -3, by solving the relationship: XM(1 - ~)C1~ exp[/3M(1 -- XM) 2] (I -- xM)aC~ exp[/3~a(XM) ~] = 1, [37] obtained by dividing Eq. [2a] by [2b] iteratively for X M, and then substituting in Eq. [2a] to obtain the value o f C ~. Here, curve 1, where /3 -- - 1 , shows n o synergism a n d curve 2, where/3 = - 2 , a barely perceptible m i n i m u m . Experimental points for the system C1zH25SO4Na/CL2H2~(OCzH4)8OH in 0.5 M NaC1 (6), where C ~ a n d C ~ have the same respective values a n d / 3 ~ = - 2 . 6 , are shown for c o m p a r i s o n with curve 3, which has the same value of/3 ~. Figure 2b shows plots o f X M versus a for the same four hypothetical systems. T h e 45 ° line cuts the curves at the point o f m a x i m u m synergism, where a* = X M. Table I lists s o m e values for the points o f m a x i m u m synergism in b o t h surface tension r e d u c t i o n efficiency, C12,rnin, a n d m i x e d micelle formation, C M 12,mi,, calculated b y Eqs. [19] a n d [22], respectively, c o m p a r e d to experimental values for the same systems. Journal of Colloid and Interface Science, VoL 90, No. l, November t982
218
HUA AND ROSEN TABLE II Synergism in Surface Tension Reduction Effectivenessa "~/'12,c
a
X¢
X~
fx,~
(mN m -I)
f~X M
Synergism
System:Ca2H25SO4Na/C12H25(OC2H4)8OH in 0.5 M NaC1; tim = --2.6; 13= --3.2 0.2 0,5 0.9
0.24 0.36 0.58
0.17 0.30 0.53
0.036 0.098 0.324
0.028 0.082 0.298
33,8 32,9 31.8
NO No No
32.4 31.2 29.9
NO No Yes
30.6 29.1 27.5
Yes Yes Yes
System: hypothetical, tim = --2.6; fl = --4.0 0.2 0.5 0.9
0.27 0.39 0.57
0.17 0.30 0.53
0.032 0.085 0.276
0.028 0.082 0.298
System: hypothetical; tiM = 0.2 0.5 0.9
0.31 0.41 0.57
0.17 0.30 0.53
0.0278 0.071 0.225
--2.6; 13 = --5.0 0.0284 0.082 0.298
a Data from Ref. (6): C~ = 4.1 X 10-3 mol din-3; C~ = 6.0 X 10-3 mol dm-3; 3'~ = 30.8 mN m-J; -y~ = 35.2 mN m-I; RT/A° = 11.81 naN m-l; RT/A° = 7.89 mN m -I.
Synergism in Surface Tension Reduction Effectiveness
(xM) in
The condition for synergism here is: Y,,cx¢
<
,
(1
-
in
[
c (1
-
.) 7
= 1
[381
[35]
at the C M C o f the mixture. There are almost no data in the literature from which calculations of/3, tiM, X¢, and X u can be m a d e to verify the above relationship. Table II lists some data for a real system in which synergism of this type does not exist, together with data for some hypothetical systems in which the values o f C ~ , C ~ , A °, A °, -y~, 3'~, (i.e., the CMC, area per molecule and surface tension at the C M C for the individual surfactants) and flM are identical with those of the real system, but in which the value of 13 is m o r e negative. The values ofXc, X M, fl,¢Xc, fMxM, and ~2,¢ at various values o f a in Table II are obtained by the following procedure: (1) The relationships derived by Rubingh
(2), Journal of Colloid and Interface Science, Vol. 90, No. 1, November 1982
and In \ c M x M ] tM
--
(1 -- xM) 2
'
[39]
are used to calculate X M and tiM. TO do this, values of the C M C of the individual surfacrants, C M and C M, and the C M C s of one or m o r e mixtures o f them, C M, are obtained experimentally. Having obtained the value of/3 M, values o f X M, C 1112,and f ~ for various values of a are obtained f r o m Eq. [37], [2a], and [4], respectively. (2) The value of t is obtained f r o m equations analogous to [38] and [39], using experimental values o f C o and C o for the individual surfactants to produce a given surface tension and experimental values, C~2, for
SYNERGISM IN BINARY MIXTURES OF SURFACTANTS
one or more of these mixtures required to produce the same surface tension (1). Equations [30] and [31] are solved numerically for Xc and 3'12,~ for the same values of a as used above. To do this, the relationships (1), f l,c = t3(1 - X~)z and f2,c = ~(X~)2 are first substituted into Eqs. [30] and [3 I], and then. the values of C ~ obtained above for the same values of a, and values of A°~ and A2,c, 0 C ~, and C ~, obtained from the surface tension-concentration curves of the individual surfactants, as substituted. From the data in Table II, it is apparent that f~,~X~ must be < f ~ X M for synergism to occur. As the value of ~ becomes more negative compared to that of t3M, the synergistic effect increases. The hypothetical system in which ~ = - 4 . 0 is particularly instructive in
219
that it illustrates the fact that synergism may exist only at certain values of a. Additional work in this area is currently under way in our laboratory. REFERENCES 1. Rosen, M. J., and Hua, X. Y., J. Colloid Interface Sci. 86, 164 (1982). 2. Rubingh, D. N., in "Solution Chemistry of Surfac).ants" (K. L. Mittal, Ed.), Vol. 1, pp. 337-354. Plenum, New York, 1979. 3. Rosen, M. J., J. Amer. Oil Chemists Soc. 51, 461 (1974). 4. Rosen, M. J., J. Colloid Interface Sci. 56, 320 (1976). 5. Delay, R., Prigogine, I., and Bellemans, A., "Surface Tension and Adsorption," p. 166. John Wiley, New York, 1966. 6. Lange, H., and Beck, K. H., Kolloid Z. Z. Polym. 251, 424, (1973).
Journal of Colloid and Interface Science, Vol. 90, No. 1, November 1982