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Synergism in binary mixtures of surfactants
Synergism in Binary Mixtures of Surfactants V. Two-Phase Liquid-Liquid Systems at Low Surfactant Concentrations MILTON J. ROSEN AND DENNIS S. M U R P H Y Department of Chemistry, Brooklyn College, The City University of New York, Brooklyn, New York 11210 Received May 29, 1985; accepted July 25, 1985 Nonideal solution theory has been used to derive equations that predict interfacial concentrations and conditions for synergism in binary mixtures of surfactants in liquid-liquid systems at low surfactant concentrations. Equations were derived that allow prediction of synergism in three phenomena: (1) interfacial tension reduction efficiency; (2) mixed micelle formation; and (3) interfacial tension reduction effectiveness. These equations involve the experimentally determined parameters, /3[L and/3~L, related to the molecular interactions between the two surfactants in the mixed monolayer at the liquid-liquid interface and in the mixed micelles, respectively. The experimental data needed to determine whether a liquid-liquid binary surfactant system is capable of synergism in the three areas mentioned above are: (1) the interfacial tension vs log total concentration curves of the individual surfactants in the vicinity of their critical micelle concentrations (CMC); and (2) the interfacial tension vs log total concentration curve of at least one mixture in the vicinity of its CMC. q~, the ratio of the volume of the oil phase to the volume of the aqueous phase in each system investigated, and K, the partition coefficient of each surfactant, drop out of the pertinent equations provided that their values are the same in the mixed system and in the pure surfactant systems. This is generally the case in systems containing low concentrations of surfactants. Two systems were investigated: Ct2H25(OC2H4)7OH-CI2H25SO~Na ° and C12H25N~(CH2CrHs)(CH3)CH2COO°-C12H25 SO3eNa~ in heptane-water mixtures. In the systems studied, theoretical predictions are in good agreement with experimental results. © 1986AcademicPress,Inc. INTRODUCTION
THEORY
Recent work in this laboratory has used nonideal solution theory to predict surface concentrations (1) and synergism with respect to surface tension reduction efficiency (2-4), mixed micelle formation (2-4), and surface tension reduction effectiveness (4, 5), all at the aqueous solution-air interface. In the present paper, nonideal solution theory is used to derive equations that allow predictions of interfacial concentrations in twophase liquid-liquid systems and synergism with respect to interfacial reduction efficiency, mixed micelle formation, and interfacial reduction effectiveness. Two systems were investigated to demonstrate the validity of the theory.
In a system of two liquid phases and two surfactants (each after partition), the chemical potential of surfactant 1 at the interphase of the two liquids can be expressed as #1,I = #0,I(n) -[-
RTlnfI,IXI,I,
[1]
where Ul,i is the chemical potential of surfactant 1 in the mixture at the interface, #°~(m is the chemical potential of surfactant 1 in the interracial region, after partition and is a function of the interracial pressure, 17 (=~/o _ "ri, where 7 ° is the interracial tension between the two pure liquids), f~a is the activity coefficient of surfactant 1 in the mixture in the interfacial region, and X~,~is the mole fraction of surfactant 1 in the mixed surfactant in the interracial
224 0021-9797/86 $3.00 Copyright© 1986by AcademicPress,Inc. Allrightsof reproductionin any formreserved.
Journalof Colloidand InterfaceScience,VoL 110,No. 1, March 1986
SYNERGISM
IN B I N A R Y
MIXTURES
region. The chemical potential, #l,w, of surfactant 1 in the mixture in the water phase, after partition, can be expressed as #~,w = #°,w + R T In a~,w
[2]
= #°w + RTlnf~,wC~,w,
OF SURFACTANTS,
225
V
from which, A,IXI,I --- fl,wCl,wlC o1,wfo1,w"
[8]
By the same line of reasoning, one can obtain a similar relation for surfactant 2 in the mixture:
where tZ],wis the standard chemical potential f2aX2,, = a2,w " C 2,w/ , C 2,wa O r o2,w [9] ofsurfactant 1 in water (with the standard state defined as 1 M solution, but behaving ideally), and since, with only two surfactants, X2,I = (1 fl,w is the activity coefficient of surfactant 1 in - XI,I), [9] can be expressed as the mixture after partition, and Cl,w is the A , I ( 1 - - X I , I ) -_ f2,wC2,w/C2,wf 0 0 2,w. [ 10] concentration of surfactant 1 in the same sysWhen the concentrations of surfactants in tem on the molarity scale. When equilibrium is reached, the chemical the bulk phases are low, ratios f l , w / f ° w and potentials of any given solute, in all phases, f2,w/f°,w of [8] and [10], respectively, can be are equal. Therefore, Eq. [ 1] is equal to Eq. taken as equal to 1, and [8] and [10] become [2], which gives fl,I x , , I = C l , w / C ° , w
o
o
[3]
= RTln f'wCl'w
~l,I(II) - - ~ l , w
UI,IXI,I
J~,I(1
Since Xla = 1, o
/-tl,w = ]£1,I = #l,I(II).
[5]
In this case, #l,w = #~,w + R T l n
a°w
0 0 0 = #l,w + R T In fl,wCl,w,
[6]
where a°,w is the water phase activity of pure surfactant 1, after partition, required to produce the same interfacial, 1I, as that of the mixture; C°w is its water-phase molar concentration; and f ° w is its water-phase activity coefficient. From [5] and [6], 0 0 /Al,I(II) = ~ l , w + R T l n
and
•
For a system of the same two phases containing only surfactant 1 (after partition) and at the same interfacial pressure, II, as the mixture, at equilibrium: o # l , w = /d'l,I = Ul,I(I1) -]- R T l n f l a X l , i . [4]
- XI,I) = C2,w/C2,w, o
o
-
o
#l,w
o o = R T In Cl,wfl,w.
Since [3] and [7] are equal, RTlnfl'wC~'w - R T l n Cl,wfl,w o o A,iS~a
12]
where fill is a measure of the deviation from ideality in the interfacial region in the mixture and is related to the molecular interactions between surfactants 1 and 2 in that region. With these approximations, [ 11] and [12] become Cl,w XI,I" exp{fl~L(1 -- X,,I) 2} - C0w
[15]
and (1 - XI,I) ° exp{fl[e(Xla) 2}
# 1,1(II)
[
respectively. The activity coefficients of each surfactant at the interface (3q,i and3~a) can be approximated using the second term of the Margules expansions (the first term being equal to zero) (6), as f l , I = exp{fl[c(1 - X I , I ) 2} [13] and f2,I = exp{3[c(Xj,i)2}, [ 14]
0 0 C l , w f 1,w
or
[ 1 1]
_ C2,w [16] CO,w,
[7] respectively. The partition coefficient for surfactant 1 can be expressed as 1£1 = C1,B/CI,w,
[17]
Journal of Colloidand InterfaceScience, Vol. 110, No. 1, March 1986
226
ROSEN AND MURPHY
where C1, b is the concentration of surfactant 1 in the oil phase of the two-phase system after partition and Cl,w is the concentration of surfactant 1 in the water phase after partition. The total concentration of surfactant 1, Cl,t, in the mixed system is then: Cl,t =
Cl,w Vw -t- C 1,BVB
Vt
,
[18]
where Vw, Vb, and V~are the volumes of the water phase, the oil phase, and the entire system, respectively. From [ 18], C1,B =
Cl ,t gt - CI,w Vw
lib
[19]
(1 - Xl,x)2
[20]
If q~ = Vd Vw, then Vt = 4)Vw + Vw and Vb = q~Vw. Substitution into [20] then yields = F1CI,t,
[21]
where FI = (4~ + 1)/(K14~ + 1), the fraction of surfactant 1 in the aqueous phase of the surfactant mixture. Similarly,
F2C2,t,
C2,w = ~ ~ ) l - - , 2 , t =
[22]
where Fa = (4~ + 1)/(Kz4) + 1), the fraction of surfactant 2 in the aqueous phase of the surfactant mixture,
1~ C°t = F1Cl,t, o o ]
[23]
'
o o =(~K2t~2 ~.o~'~-+011I Cot =FuCE,t, ] '
[261
[27]
Since, Ci,t = Cl2,to/1, where C12,t is the total concentration of surfactant (1 and 2 combined) in the mixed surfactant system and a~ is the mole fraction of surfactant 1 in the mixed surfactant, substitution into [27] gives
3tL =
ln( Cl2,tot l f l/ C°,tF°Xl,t ) (1 -- Xl,i) 2
[28]
When 4~ = 4~° and K 1 = K 0, [28] becomes
t3[L = ln( C12,t°q/C°,tXl,l) ( 1 - XI,I)Z
[29]
q~ocan be made equal to 4~by using the same volume ratio Vb/Vw, in the mixed surfactant system as in the pure surfactant 1 system. Considering the low suffactant concentrations normally used, K ° is most probably equal to K~. In order to evaluate Xt,i, [25] is divided by [26], yielding (Xl,l)21n(FlCl2,tal/F°C°,tXl,I) ( 1 -- X 1,I)21n(F2Cl2,t( 1 - cq )/F°C°,t ( 1 - X1, I))
where F ° = (4~° + 1)/(K% ° + 1), the fraction of surfactant 1 in the aqueous phase of the system containing only surfactant 1, C°'w
C2't = ~22 (1 - Xl,i)C°,t ° exp{3[L(X,,i)2}.
VBCI,w
C l ' w = ( KI VBVt+ Vw)Ci,t.
1~11
and
filL = ln( Cl'tFl/ C°l'tF°Xl"O
Cl,t Vt - Cl,w Vw
Cl,w = ~ ~ / t - ~ l , t
[251
From [25],
from which,
C°'w = ( ~ - 1
Fo Cl,t = ~ Xl,lC?,t" exp{fl•L(1 -- XI,I)2},
Fo
,
and substitution of [19] into [17] gives K1 =
where F ° = (q~o+ 1)/(KOch02+ 1), the fraction of surfactant 2 in the aqueous phase of the system containing only surfactant 2. Here, again, the superscript o refers to systems containing only one surfactant. Substitution of [21] and [231 into [15], and of [221 and [24] into [16], respectively, followed by rearrangement gives
[24]
JournalofColloidandInterfaceScience,Vol. 110,No. 1, March1986
= 1
[30]
following the insertion of Cl,t = Cl2,toq and C2,t = C12,t(1 - oq). When q~ = q~o = 4P, KI = K~I, and/(2 =/~2, then [30] can be expressed as
S Y N E R G I S M IN BINARY M I X T U R E S OF SURFACTANTS,
(XLl )21n(Clzaoq/ C°,tXLO (1 -- X l , i ) 2 1 n [ C l 2 , t ( 1
-
,
og1 ) / C 0 , t ( l - X I , I ) ]
= 1.
F °
lnX+m~+fl(1-X) [31]
Since C12,t, al, C°,t, and C°t can be determined experimentally (as can K1,/~i, K 0 ,/(2, q~, 4 °, and ~o), [31] can be solved numerically for XI,I. Once XI,I is known, [28] (or [29]) can he solved to obtain 3f~e.
Synergism in Interfacial Tension Reduction Efficiency
V
227
2-1nal<0.
[35]
And, when 4~= ~b° and K1 = K °, the condition is lnX+3LL(1-X) 2-1na~<0. [36] When synergism in interfacial reduction efficiency exists, there will be a minimum in the Clzt vs al curve and maximum synergism in this respect will be obtained where the Cl2,t vs al curve shows a minimum, i.e., dC12't = 0.
da~ The efficiency of interfacial tension reduction by surfactants in a two-phase liquid-liquid system can be defined as the total surfactant concentration in the entire system required to produce a given interfacial tension (reduction). Synergism in this respect is present in a surfactant mixture if it can attain a given interfacial tension (reduction) at a total mixed concentration lower than that required of either surfactant of the mixture by itself. The point of maximum synergism in this respect is where the lowest total concentration of mixed surfactant is required to attain a given interfacial tension (reduction). The use of Cl,t = C12,toQ and C2, t = Cl2,t(1 - a:) in [25] and [26], respectively, yields
From [34], d i n C12,t_ 1 dX
dal
F'~ vc'O . exp[fl~L(1 -- X) 2]
2fl~L(1-- X) da 1
where it is assumed that K1 does not change with o~1 at the low surfactant concentrations generally used. Since, d In Cl2,t
dC12,t da I
=
C12,t
da~
'
[37] can be expressed as dCl2,t doQ = Cl2't
F1 dX
= F1 -~-'--l,t
dX
1 Of1
[371
0
G2,tal
Xdal
O/1
[32]
23[L(1-X)~]
= 0.
Since C~2,t ~ O, and
l dX X da~
F o
C,2,t(1 - al) = F22 (1 - X)C°,t • exp[fl[L(X)2], [331 respectively, where X - Xla. From [32],
dal
F,
+3[L(1-X) 2-1noq.
[34]
Since, for synergism to be present, C~2,t must be
2all.(1
- x)
dX
dcq
= o
at the point of maximum synergism. From this,
dX
In C12,t - In C°,t = In X + In F°t
1 al
X al(1 - 2fl[LX + 2 ~" /JLLX 2~ ] "
[38]
Division of [32] by [33] followed by rearrangement gives In
0 0 F1F2C2't o o +ln°q
F2F1CI,t
+ln(1--X)--3[L
+ 23~LX--ln(l--oq)--lnX=O. Journal of Colloid and Interface Science,
[39]
VoL 110, No. 1, March 1986
228
ROSEN AND MURPHY
i F2FVCy,t o oI
Differentiating with respect to a l gives 1 a 1
1
dX
m
2
(1 -- X ) doe I q-
In F1FOCO,t] < I#~LI
dX
tr
3LL doq
and when ~b = ~b° = ~b° = ~o and K1 = K ° and /~2 = K °,
l dX - - - 0, X dal
1 + - (1-al)
X(1 - X ) ~l(1 - ~1)[1 - 23~LX + 23[LX 2] " [411
Combining [411 and [381, X cq(1 - 2(3[LX +
Substitution of [42] into [34] gives In C,2,t,min= l n ( ~ )
+ f i l L ( 1 - X) 2, [47]
where C l 2 , t , m i n is the minimum total surfactant concentration required to produce a given interfacial tension (reduction). From [44], 0
0
0
0
X = ln(Cl,tF2F1/C2,tFzF1)
23~,LX2)
+ filL = ot~ [48]
or, when ~b = q~o = q~o, K1 = K °, and K2 =K~2,
x(1 - x )
~x) :
and
[421
where a~' is the mole fraction of surfactant 1 in the total surfactant in the entire system at the point of maximum synergism in inteffacial reduction efficiency. This relation states that, at the point of maximum synergism in this respect, the mole fraction of either surfactant in the interfacial region is equal to its mole fraction in the entire system. Substitution of [42] into [36] gives the first condition for synergism in this respect as 3[L(1 -- X) 2 < 0,
23~L Substitution of [481 into [47] gives
(1 - X )
"T = X,
[49]
S = l n ( C ° ' t / C ° ' t ) + flEE = c~*.
2 b*Lb ~ X21I
or
[431
and since (1 - X) 2 is always positive, filL must be negative for synergism to exist. From [42] and [39], 0
[461
2 fiLL "
Oel(1 -- oq)[1 -- 2 3 £ L X +
(1 -
Jln~2,,1 C°,tl< Ifl°LLl"
[401
where it is assumed that K1 and K2 do not change with change in al. Equation [40] can be expressed as
dX d~l
[451
- -
Cl2,t,rnin
X exp
-
C 0l,,Fl0 - F1 3~,L
FLL
0
0
0
23~.L
0
2
_] J
[5o1 and, when ~b = 05° = ~b°, K1 = K °, and/<2 = I~2, [50] becomes
C12,,,=i. = CO,,
¢o, r# L
× expl~LLL
~
_1.1
[511
Synergism in Mixed Micelle Formation Synergism in mixed micelle formation in binary mixtures of surfactants in a two-phase liquid-liquid system is present when the CMC l of the mixture, Clz,t, M is lower than the
0
F 2 F 1 C I , t __
In F~t~2cO,t
--3[L(1 -- 2X).
[44]
Since 0 < X < 1 and 3[L must be negative, the second condition for synergism in interradial tension reduction efficiency is Jdutnal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
1 The CMC of the system is the total molar concentration of surfactant in the entire system at the point of intersection of the two almost linear portions of the interfacial tension vs log total (i.e., in the entire system) concentration curve in the region where miceUization occurs in the system.
SYNERGISM IN BINARY MIXTURES OF SURFACTANTS, V CMC of either surfactant comprising the mixture by itself in a similar liquid-liquid system. From Eqs. [32] and [33], relationships analogous to those derived for synergism with respect to interfacial tension reduction efficiency can be obtained for synergism in mixed micelle formation. For synergism to exist in this respect when q5 = ~b° = q~o, K1 = K °, and K2 K°: (i) flLML must be negative, and (ii) [ln C,,VC2,t[ M M must be <[flLMLI,where the superscript, M, refers to micelle formation, iliaL is the experimentally determined parameter related to molecular interaction between the two surfactants in the mixed micelles in the system, and C~t and C2,~tare the CMC's of individual surfactants 1 and 2, respectively, by themselves in similar two-phase, liquid-liquid systems, flLMLis evaluated from equations analogous to Eqs. [28] and [29]. At the point of maximum synergism in this respect, =
M
0
M
229
Synergism in InterfaciaI Tension Reduction Effectiveness The effectiveness of the interfacial tension reduction in a two-phase, liquid-liquid system can be defined as the interfacial tension reduction attained at the CMC of the system. Synergism in interfacial tension reduction effectiveness is, therefore, present in a liquidliquid, binary surfactant system when the mixed surfactant system can attain a lower interfacial tension at its CMC, C M m , than either surfactant comprising the mixture can attain at its respective CMC, CI,M or C~,,. The basic equations, analogous to [32] and [33], above, for adsorption at the liquid-liquid interface at the CMC, C~,t, of the system are cIM2't°Q =
t O I r [ wO ~"~ l't°exp[fl~L(1
FI
and,
F0
c M , t ( 1 -- ~1 ) = F22
(1 -
- X) 2]
[56]
X)C°,t. exp[fl[L(X)2].
0
a*, M = X M = ln(CI,tF2FI/C2,tF1F2) + timL
[57]
2ML
The linear portion of the interfacial tension,
[521 7i, vs In C Oof each individual surfactant can or, when ~b = ~b° = ~b°, K 1 = K °, and K2 = xo:
o~,,M = xM = ln(C~dC~,) + fl~L 2flLML
,
[531
hypothetically be extended to concentration values above the CMC to yield interfacial tension values equal to the interfacial tension, "YCMC,m , of the mixture at its CMC, yielding the relationships "}/CMC, 12,t = $11 71-
and
Slln C°,t
[58]
and
M
C 12,t,min -
M 0 CI,tFI Fl
"YCMC,12,t= S~ + S21n C°,t,
× exp{flLML[flLML - - ln( C~tF°12fl~LF2/CMF, t ,F°)]2 ] J' [54] and, when ~b = q~o = q~0, K1 = K °, and/£2 = K °,
[59]
where $1 and $2 are the slopes of the 3' vs In C O plots of the individual surfactants and S't and S~ are the hypothetical -r intercepts of the same plots. From [56] and [58], 7CMC,12,t = Srl
+ SI{lnIF ~ cM,tal] - /3[L(I- x)z} . [60]
M M Cl2,t,min = C 1 , t
X exp{flML[-/~LML-ln(C~t/C~)']2~
3J
[551
When synergism in interfacial tension reduction effectiveness exists, there will be a miniJournal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
230
ROSEN AND MURPHY
mum in the matically;
~CMC,12,t
VS a 1 curve, or mathe-
d'YcMc,12,t = O.
Taking natural logs and differentiating [64] and [65] with respect to al, we obtain d In C~,t
dal
dal
Thus, d'YcMC,12,t =da,
Xdotll dX
S'( 1 +dlndaxClM2,~t
-1 +[ 1
- a--i-
~ -
2fl[aL(1 --
xM)q dX M J d~,
[66]
and +2fl[L(1--X)~al)=0,
[61] dlnCM,t=
dal
where it is assumed that KI and ff do not change with change in al. Since Sj 4= 0:
dX X dal
1 + dci M n ,~
al
dal
dal
and d l n cM,t = ( 1
dal
)dX - 2fl~.L(1 -- X)
1
dal
al [62]
Following the same line of reasoning, one can derive, from [61] and [59], d In CM,t
+
X1 2fl~LX) dda
1 -
1,
al
[63]
where it is assumed that /(2 and ~b do not change with change in a~. Using the basic equations of nonideal solution theory, equations analogous to [32] and [33] can be derived for mixed micelle formation in a two-phase, liquid-liquid system: = F° y,
1 -- X M
l
2flMLXM doz---T' dXU [67]
1
1
1 ------Y- 2/~[LX
1 -- X u
~--2fl[L(1--X) x----~--2flML(1--xM) 1 = 1 2¢~tLXM
[68] which reduces to X = X M.
dal =
Ol 1
respectively, where it is assumed that K~ and /(2 do not change with change to a~. [621 From Eqs. [66], [62], [67], and [63], we obtain
+2fl[.L(1--X) d X = 0
(1~
E
_ 1
-1
1 --
[69]
Relationship [69] states that when a minimum or a maximum occurs in the "~CMC,12,t vs al curve, the composition of the mixed monolayer at the liquid-liquid interface must equal the composition of the mixed micelles of the system. Combining [56], [64], and [69], and taking natural logs yields In C ~ - In C°,t = (/3~.L --/~ML)(1 -- X ) z [70]
C~tXM. exp[fl~L(1 -- xM) 21 [64] and combining [57], [65], and [69] gives In C~, - In C°,t = (fill -- flML)(X)2. [71]
and
Substituting [58] into [70] gives
C12,t(1 - al)
S'I + Slln C~t - ~CMC,12,t = F ° C ~ ( 1 - x M ) • exp[3~L(X~)2]. F~ Journal of Colloid andlnterface Science,
[651
VoL 110, No. 1, March 1986
= S~(t~L -- ¢~Mo(1 -- X ) 2.
SYNERGISM
IN B I N A R Y M I X T U R E S
Since, o
"rCMC,I,t = S'j + Slln CI,t,M where 7cuc,~,t o is the interfacial tension in a pure surfactant 1 system at the critical micelle concentration for the entire system, 0 ~CMC, I , t -
7CMC,12,t = S l ( / ~ L
--
/3LML)(1- - X ) . [72]
By the same line of reasonsing, using [59] in [71]:
OF SURFACTANTS,
V
231
When '~CMC, o I,t > '~CMC,2,t, o since S~ and $2 are always negative, the smallest negative value of/3~L --/3ME, equal to 0 0 ~CMC, I,t - - ~CMC,2,t
Sl will be obtained when X = O. When ~CMC, o 1,t 0 < 7cMc,z,t, the smallest negative value of/3OL -- /3LMLwill equal 0 0 "YCMC, I,t - - ~CMC,2,t
0 ']/CMC,2,t - - '~CMC,12,t = S 2 ( f l [ L - - /~LML)(X) 2,
-$2
[73] where 7CMC,2,t o is the interracial tension in a pure surfactant 2 system at the critical micelle concentration for the entire system. Substituting [73] into [72] gives 0 0 "YCMC, I,t - - "YCMC,2,t
= [~3~c -/3ML][S~(1 -- X) 2 - $2(X)2]. [74] Conditions for the existence of synergism in interfacial tension reduction effectiveness. These are determined from the above in the following fashion: 1. From [58] and [59], since the slopes $1 and Sz are negative, the larger the value of C°l,t or C2°,t, the smaller will be the value of "YcMe,lZ,t. Under ideal conditions, i.e., when /3f~L= 0, no synergism exists. From [56] and [57], in order for C°t or C°,t to be larger than under ideal conditions,/3[L must be negative. Therefore, for synergism in interfacial tension reduction effectiveness to be present in the system, /~L must be negative. 2. Synergism in this respect is present in a 0 0 system when 'YCMC,12,t < q/CMC, I,t, "YcMc,2,t. From [72] and [73], since the slopes S~ and $2 are always negative and the quantities (I X) 2 and X 2 are always positive, then the value of(/3~L -- g3L~L)must be negative for synergism in interfacial tension reduction effectiveness to be present. 3. From [74],
obtained when X = 1. Therefore, for synergism in this respect to exist, the (negative) value of /~[L - - flLML m u s t b e >]('YCMC, o o I,t -- 'YCMC,2,t)/S], where S is the slope of the '~i vs In Ct plot of the individual surfactant having the larger interfacial tension value at its CMC. In summary, the conditions for the existence of synergism in interfacial tension reduction effectiveness, when the partition coefficients (Kt and K2) and the volume ration (q~) do not change with al, are 2. ~ L -- ~ML < 0 0 3. rt~L - #~L[ > I ( ~ c ~ c , ~ , t - ~cMc,2,0/sl.
Conditions at the point of maximum synergism. Experimental evidence has been provided that at the liquid/air interface, at the point of maximum synergism in surface tension reduction effectiveness, the concentrations of the two surfactants in the mixed monolayer are approximately equal (i.e., XI X2 ~ 0.5), (5). Assuming that this is true at the liquid-liquid interface, i.e., X = 0.5, and combining this with [64], [65], and [69] gives a*'a =
F2F°C1M, t FzFOC~t , + F1Fz0 C I , tM •
[76]
-
/ ~ L - - /~ML =
0 0 "YCMC,I,t -- ")'CMC,2,t $1(1 - X) 2 - 8 2 X 2 "
[75]
When q~ = q~o = ~2o, K~ = K °, and/£2 = K °, this becomes ~,,E _
C~,
C~ + C~'
[77]
where a *'E is the mole fraction of surfactant 1 in the total mixed surfactant solution at the Journal of Colloidand InterfaceScience, Vol. 110, No. 1, March 1986
232
ROSEN AND MURPHY
point of maximum synergism in this respect. Introducing [76] into [64] or [65] yields oM
M,, = FF2FICI,t +~F1F2C2,tl.exp__, C12't
2FIF2
L
J
4 [78]
where C~;* is the critical micelle concentration for the entire system at the point of maximum synergism in interfacial tension reduction effectiveness. When 4~ = 4 ° = 40, K1 = K °, and/(2 = K °, [83] becomes
M,
Cl2;t -
+ c2, t 2 .exp - ~ - .
[79]
Finally, from [72] and [73], and using X1 = X M = 0.5, we obtain the relationship * 0 "YCMC,12,t = "]/CMC, I(2),t - -
S 1(2)(/~LL 4
- - /~ML) '
[801
where Y~MC,IZ,tis the lowest interfacial tension attainable at the CMC of any mixture of the two components. Equation [80] states that the amount of reduction of the interfacial tension of a surfactant at its CMC as a result of interaction with a second surfactant depends upon the interfacial concentration of the surfactant, as indicated by S, and upon the difference between the tendencies of the mixture to adsorb at the interface and to form micelles (3[L -- 3~L)- The greater the interfacial concentration of the surfactant and the greater the adsorption tendency relative to the micellization tendency, the greater will be the reduction in the interfacial tension at the CMC. Best agreement between calculated "YCMC,12,t * values and experimental values is obtained when the smaller of the two y°~c,l(2), t values for the individual surfactants is used in [80]. EXPERIMENTAL
Materials C12H25(OC2H4)7OH(CI2EOv), >98% purity as indicated by gas chromatography (Nikko Chemical Co., Tokyo, Japan). Sodium doJournal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
decane sulfonate (C12SO3eNa®), >99% purity (Research Plus, Bayonne, N. J.). N-Dodecyl-N-benzyl-N-methylglycine, C12HzsN ®(CHzC6Hs)(CH3)CH2COO®(ClzBMG), >98% purity and synthesized in this laboratory (7). Before being used for interfacial tension measurements, aqueous solutions of the surfactants (in water that had first been deionized, and then distilled twice; the last time through a 1-m-high Vigreaux column with quartz condenser and receiver) were further purified by repeated passage (8) through minicolumns (SEP-PAK C18 Cartridge, Waters Assoc., Milford, Mass.) of octadecylsilanized silica gel to remove any traces of impurities more interfacial active than the parent compound. The oil phase used throughout was n-heptane (Fisher Scientific) that was purified by passage through a 31 × 3.4-cm column (Fisher and Porter Co.) of silica gel 922 (Will Corp.) that had been heated at 120°C for 3 h before use. The UV absorbance of the treated n-heptane (1-cm cell) was <0.015 at 255 nm, with 95% ethanol used as a blank.
Interfacial Tension Measurements All interracial tension measurements were made by the spinning drop technique using a Model 500 Spinning Drop Interfacial Tensiometer (University of Texas). Readings were taken at 0.5 h intervals until three consecutive readings coincided. All readings were done at 25.0°C. The density of water was taken as 1.000 g]cm 3 and that of heptane as 0.684 g/cm 3.
Partition Coefficients One hundred milliliters of the desired surfactant solution was overlayed with 50 ml of heptane. This two-phase system was then allowed to stand for 1 month after which equilibrium was achieved (9). Shaking was avoided so as to not form an emulsion which may cause an erroneous partition coefficient value. The partition coefficient of pure C12H25(OC2H4)vOH was obtained by weighing the
233
SYNERGISM IN BINARY M I X T U R E S OF SURFACTANTS, V TABLE I 26
CI2EO]~ Cl2S03No
Interfacial Tension Equilibrium Times System
Time to equilibrium (hr)
CI2SO3Na C12EO7
10 ~
~14
7~
:1o
CI2BMG
½a
CI2EO7_CI2SO3Na
lb ]b
C~2BMG-C~2SO3Na
--18 X
6 2i
-6,0
,
,
-5.0
-4.0
At all concentrations studied. b At all mole fractions studied.
,
-310
-2.0
Log C 12,t
amount in the initial aqueous surfactant solution and the amount in a portion of the final aqueous solution of the two-phase system. In the presence of C~2H25SO~Na®, it was assumed that this anionic surfactant, due to its charge, did not partition into the heptane at all. The partition coefficient of C12H25(OC2H4)7OH, with C 1 2 H 2 5 S O ~ Na e in the system, was determined by weighing (as in the case of pure C12H25(OC2H4)7OH), but, here the C12H25SOa~Na® original weight in the aqueous phase of the initial pure aqueous system was subtracted from the total weight obtained in the aqueous phase of the two-phase system in order to determine the weight of C 1 2 H E s ( O C 2 H 4 ) 7 O H left in the aqueous phase. The partition coefficient of pure C~2BMG was obtained by determining the concentration in the initial aqueous solution by using UV spectroscopy, e263 nm,CI2BMG = 355 (7), then determining the concentration in the aqueous phase of the final two-phase system. The partition coefficient of C~2BMG in the
FIG. 1. Interfacial tension vs log total surfactant concentration for CIaEOT, C12CO3Na, and C~2EOT-C~2SO3Na mixtures at 25.0°C; ~b = ¢0 = ¢0 = 1.23 × 10-2; heptane as the oil phase. (A) C12SO3Na, (E3) C12EO7, (Q) CI2EOTC12SO3Na, aC~2EO~= 3.75 × 10-3, (X) C12EOT-ClaEO7C12SO3Na, aq2Eo7 = 0.249.
presence of C12H25SO3°Na® was determined the same way as for pure C12BMG, since C12H25SO3°Na® does not absorb in the UV. RESULTS A N D DISCUSSION
Equilibrium Times and Partition Coefficients Times for reaching equilibrium interfacial tension for the systems studied appear in Table
-\ 11 9
~z 5
TABLE II 3i
Partition Coefficients~ System
aj
Ct2EO 7
1.0
CI2EO7-CIESO3Na CI2BMG C~2BMG-C~2SO3Na
0.465 1.0 0.557
K~tb
Kjb
0.475
0.156 8.62 × 10-3 6.93 × 10-3
"Defined as the concentration in the oil (heptane) divided by the concentration in the water; all runs done at 25.0°C. b Average of two runs.
-4:2
-3'.8
-3~.4
-3'.0
-2~.6
-2,2 \~
LogC 12,t
FIG. 2. Interfacial tension vs log total surfactant concentration for Ct2BMG, CtzSO3Na, and C~2BMGCt2SO3Na mixtures at 25.0°C, ¢ = ¢0 = 4)0 = 1.23 × 10-2; heptane as the oil phase. (×) C~2SO3Na, (®) C12BMG, (E3) CI2BMG-CI2SO3Na, ac~2aM6 = 0.0277, ( ~ ) CI2BMGC12SO3Na, a c ~ M 6 = 0.0431, (&) C]2BMG-C12SO3Na, aCI2BMG = 0.945. Journal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
234
ROSEN AND MURPHY T A B L E III Effect of Interface o n Value of f i l l System
/~
C12BMG-CI2SO3Na ~ C~2BMG-Ct2SO3Na b C12EOs-CI2SO3Na c CI2EOT-CI2SOaNa b
-5,7 -4.51 - 1.5 +0,19
Interface H20/Air HzO/Heptane H20/Air H20/Heptane
D a t a from Ref. (4). b ~b = ~b° = q~0 = 1.23 × 10-2; 25.0°C.
a
c D a t a from Ref. (10); CI2EO8 = Ct2H25(OC2H4)sOH.
I. It is interesting to note that the required equilibrium time for the C12EO7-C12SO3Na system is much less than that required for either individual surfactant comprising that mixture. The partition coefficients for the systems studied are listed in Table II. The t3 values calculated using Eq. [28] and those calculated using Eq. [29] differ by less than 0.1% in all cases. This indicates that the values K~ and K1 are numerically close enough to each other to make valid the assumption that setting K ° equal to KI leads to insignificant error. Since this assumption leads to insignificant error, it follows that the assumption that K1 does not change with change in OL1 also leads to insignificant error. At the low surfactant concentrations involved, the presence of a second surfactant would not be expected to cause a significant change in K °.
in the absolute value of the molecular interaction parameters, indicating weaker interaction between the two surfactants in the presence of heptane. An explanation is suggested by the data in Table IV, which shows the interfacial areas per surfactant molecule, as calculated from the slopes of the surface (or interfacial) tension log-concentration curves. Areas per surfactant molecule are larger at the heptane/aqueous solution interface than at the air/aqueous solution interface, indicating that heptane molecules are intercalated between the alkyl chains of the surfactant molecules at the former interface, increasing the distance between them and consequently decreasing interaction between them. The /3~L value at 25°C for C12BMGC12SO3Na is -3.66. The 13Mvalue at 25°C for the same system is -5.0 (4). It appears, therefore, that there is also decreased interaction between the surfactant molecules in the mixed micelle in the presence ofheptane. Here, again, this may be due to heptane molecules between the alkyl chains of the surfactants; in this case, in the mixed micelle. Interfacial Reduction Efficiency. The conditions for synergism in this respect are: (i) /~[L< 0 and (ii) [ln(C°,t/C°,t)] < I t ' L l . Data for the two systems studied are shown in Table V. The C12EOv-ClzSO3Na system does not satisfy either condition for synergism since fl~L
Values of the Interaction Parameters, t3~Land/3~L Based upon the interfacial tension data in Figs. 1 and 2, and using Eqs. [28], [29] and their analogs for mixed micelle formation, ~r M 13LL and 13LL values at 25 o C were calculated for the two systems studied. ¢/~Lvalues are listed in Table III, flL~Lvalues are shown in Table VI. Table III also lists t" values at 25°C at the aqueous solution/air interface for similar systems. It is apparent from the listed values that the replacement of air by heptane at the aqueous solution interface results in a decrease Journal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
T A B L E IV Effect of Interface o n Interfacial Area per Surfactant Molecule Area (A)2
System
H20/ Air interface
H20/ Heptanff interface
C12BMG CI/SO3Na C12EO7
56 a 56 a 57 b
66 57 78
D a t a from Ref. (4). b D a t e from Ref. (11). c W i t h 4P = 4~° = 1.23 × 10-2; 25.0°C. a
235
SYNERGISM IN BINARY MIXTURES OF SURFACTANTS, V TABLE V Synergism in Interfacial Tension Reduction Efficiencya Sy~em Property
CI2EOT-CI2SO3Na
C12BMG-C12SOaNa
]ln(C°,t/C2,t)l(at "~1,mN m -~) ~[e (average) Synergism predicted by theory Synergism obtained experimentally a* (calculated) Cl2,t,~i, (calculated)
4.97 (8.0) +0.19 No No ---
3.96 (7.5) --4.51 Yes Yes 0.939 1.22 × 10 -4 M 1.19 X 10-4 M
Cl2,t ( e x p e r i m e n t a l )
b
--
a~ = ~0 = ¢o = 1.23 X 10-2; oil = heptane; 25.0°C. b At a ~ , ~ = 0.945.
= + 0 . 1 9 > 0 a n d [ln(C°,dC°,t)[ = 4.97 > [/3ILl = 0.19. T h e C12BMG-C12SO3Na syst e m meets b o t h c o n d i t i o n s for synergism. T h e r e is g o o d a g r e e m e n t between p r e d i c t e d synergism a n d e x p e r i m e n t a l results and, in the C12BMG-C12SO3Na system where synergism exists, between the calculated C 1 2 , t , m i n value and that obtained experimentally. Mixed micelle formation. T h e c o n d i t i o n s for synergism in this respect are: (i) BLML< 0 a n d (ii) [ln(C~/C~t)[ < [3LML[.T a b l e V I lists d a t a for the two systems studied. T h e CtESO3Na system meets the first condition, b u t
fails the second. T h e C12BMG-CI2SOaNa syst e m meets b o t h conditions. As in interfacial tension r e d u c t i o n , above, there is g o o d a r g u m e n t between p r e d i c t e d synergism a n d calculated values based u p o n t h e o r y a n d e x p e r i m e n t a l results. Interracial reduction effectiveness. T h e conditions for synergism in this respect are s u m m a r i z e d in the theoretical section. D a t a for the systems studied are shown in T a b l e VII. T h e C~2EOv-CxESO3Na system fails all three conditions. T h e C~2BMG-Ct2SO3Na system meets all three conditions. Again, there is g o o d
TABLE VI Synergism in Mixed Micelle Formationa System Property
[ln(C~t/C~)l 3~L (average) Synergism predicted by theory Synergism obtained experimentally a *'M (calculated) CMm,mi. (calculated) C~,t (experimental)b
CI2EOT-CI2SO3Na
3.83 -0.19 No NO
CI2BMG-Cj2SO3Na
3.11 -3.66 Yes
Yes 0.925 2.89 × 10 -4 M 2.69 X 10-4 M
1.23 × 10-z; oil = heptane; 25.0°C. b a~ua,e = 0.945.
a ~b = ~0~0 =
Journal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
236
ROSEN AND MURPHY TABLE VII
ACKNOWLEDGMENT
Synergism in Interfacial Tension Reduction Effectiveness a
The material is based upon work supported in part by the National Science Foundation under Grant CPE8317134.
System Property
o l,t "}'cIvlC, 0
(S0 '~CMC,2,t(82) ~[L -- ~LML
CuEOT-CI2SO3Na
Ct2BMG--CIrSOaNa
1.80 (-35.36) 7.39 (-14.48) +0.388
2.00 (-35.85) 7.39 (-14.48) -0.85
~0CMC,I,t
--3'~c,2,t Synergism predicted by theory Synergism obtained experimentally c~*'E (calculated) Cl~',* (calculated) C~,t (experimental) b * (calculated, "~CMC,12,t naN m -m) "YcM¢,I2,, (experimental, naN m -m)
0.388
0.375
No
Yes
No ----
Yes 0.043 1.39 × 10-3 M 1.53 × 10-3 M
--
0.80
--
1.58
a q~ = 4~o= ~bo= 1.23 × 10-2; oil = heptane; 25.0°C. t, c ~ ¢ = 0.0431. agreement between predicted synergism and e x p e r i m e n t a l results.
Journal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
REFERENCES 1. Rosen, M. J., and Hua, X. Y., J. Colloid Interface Sci. 86, 164 (1982). 2. Hua, X. Y., and Rosen, M. J., J. Colloid Interface Sci. 90, 212 (1982). 3. Rosen, M. J., and Hua, X. Y., J. Amer, Oil Chem. Soc. 59, 582 (1982). 4. Rosen, M. J., and Zhu, B. Y., J. Colloidlnterface Sci. 99, 427 (1984). 5. Zhu, B. Y., and Rosen, M. J., J. Colloid Interface Sci. 99, 435 (1984). 6. Fried, V., Blukis, U., and Hameka, H. F., "Physical Chemistry," pp. 224, 252. Macmillan, Co., New York, 1977. 7. Dahanayake, M., and Rosen, M. J., in "Structure/ Performance Relationships in Surfactants" (M. J. Rosen, Ed.), ACS Symposium Series 253. Amer. Chem. Soc., Washington, D. C., 1984. 8. Rosen, M. J., J. ColloidlnterfaceSci. 79; 587 (1981). 9. Warr, G. G., Grieser, F., and Healy, T. W., J. Phys. Chem. 87, 4520 (t983). 10. Rosen, M. J., and Zhao, F., J. Colloid Interface Sci. 95, 443 (1983). 11. Rosen, M. J., Cohen, A. W., Dahanayake, M., and Hua, X. Y., J. Phys. Chem. 86, 541 (1982).
THEORY
Recent work in this laboratory has used nonideal solution theory to predict surface concentrations (1) and synergism with respect to surface tension reduction efficiency (2-4), mixed micelle formation (2-4), and surface tension reduction effectiveness (4, 5), all at the aqueous solution-air interface. In the present paper, nonideal solution theory is used to derive equations that allow predictions of interfacial concentrations in twophase liquid-liquid systems and synergism with respect to interfacial reduction efficiency, mixed micelle formation, and interfacial reduction effectiveness. Two systems were investigated to demonstrate the validity of the theory.
In a system of two liquid phases and two surfactants (each after partition), the chemical potential of surfactant 1 at the interphase of the two liquids can be expressed as #1,I = #0,I(n) -[-
RTlnfI,IXI,I,
[1]
where Ul,i is the chemical potential of surfactant 1 in the mixture at the interface, #°~(m is the chemical potential of surfactant 1 in the interracial region, after partition and is a function of the interracial pressure, 17 (=~/o _ "ri, where 7 ° is the interracial tension between the two pure liquids), f~a is the activity coefficient of surfactant 1 in the mixture in the interfacial region, and X~,~is the mole fraction of surfactant 1 in the mixed surfactant in the interracial
224 0021-9797/86 $3.00 Copyright© 1986by AcademicPress,Inc. Allrightsof reproductionin any formreserved.
Journalof Colloidand InterfaceScience,VoL 110,No. 1, March 1986
SYNERGISM
IN B I N A R Y
MIXTURES
region. The chemical potential, #l,w, of surfactant 1 in the mixture in the water phase, after partition, can be expressed as #~,w = #°,w + R T In a~,w
[2]
= #°w + RTlnf~,wC~,w,
OF SURFACTANTS,
225
V
from which, A,IXI,I --- fl,wCl,wlC o1,wfo1,w"
[8]
By the same line of reasoning, one can obtain a similar relation for surfactant 2 in the mixture:
where tZ],wis the standard chemical potential f2aX2,, = a2,w " C 2,w/ , C 2,wa O r o2,w [9] ofsurfactant 1 in water (with the standard state defined as 1 M solution, but behaving ideally), and since, with only two surfactants, X2,I = (1 fl,w is the activity coefficient of surfactant 1 in - XI,I), [9] can be expressed as the mixture after partition, and Cl,w is the A , I ( 1 - - X I , I ) -_ f2,wC2,w/C2,wf 0 0 2,w. [ 10] concentration of surfactant 1 in the same sysWhen the concentrations of surfactants in tem on the molarity scale. When equilibrium is reached, the chemical the bulk phases are low, ratios f l , w / f ° w and potentials of any given solute, in all phases, f2,w/f°,w of [8] and [10], respectively, can be are equal. Therefore, Eq. [ 1] is equal to Eq. taken as equal to 1, and [8] and [10] become [2], which gives fl,I x , , I = C l , w / C ° , w
o
o
[3]
= RTln f'wCl'w
~l,I(II) - - ~ l , w
UI,IXI,I
J~,I(1
Since Xla = 1, o
/-tl,w = ]£1,I = #l,I(II).
[5]
In this case, #l,w = #~,w + R T l n
a°w
0 0 0 = #l,w + R T In fl,wCl,w,
[6]
where a°,w is the water phase activity of pure surfactant 1, after partition, required to produce the same interfacial, 1I, as that of the mixture; C°w is its water-phase molar concentration; and f ° w is its water-phase activity coefficient. From [5] and [6], 0 0 /Al,I(II) = ~ l , w + R T l n
and
•
For a system of the same two phases containing only surfactant 1 (after partition) and at the same interfacial pressure, II, as the mixture, at equilibrium: o # l , w = /d'l,I = Ul,I(I1) -]- R T l n f l a X l , i . [4]
- XI,I) = C2,w/C2,w, o
o
-
o
#l,w
o o = R T In Cl,wfl,w.
Since [3] and [7] are equal, RTlnfl'wC~'w - R T l n Cl,wfl,w o o A,iS~a
12]
where fill is a measure of the deviation from ideality in the interfacial region in the mixture and is related to the molecular interactions between surfactants 1 and 2 in that region. With these approximations, [ 11] and [12] become Cl,w XI,I" exp{fl~L(1 -- X,,I) 2} - C0w
[15]
and (1 - XI,I) ° exp{fl[e(Xla) 2}
# 1,1(II)
[
respectively. The activity coefficients of each surfactant at the interface (3q,i and3~a) can be approximated using the second term of the Margules expansions (the first term being equal to zero) (6), as f l , I = exp{fl[c(1 - X I , I ) 2} [13] and f2,I = exp{3[c(Xj,i)2}, [ 14]
0 0 C l , w f 1,w
or
[ 1 1]
_ C2,w [16] CO,w,
[7] respectively. The partition coefficient for surfactant 1 can be expressed as 1£1 = C1,B/CI,w,
[17]
Journal of Colloidand InterfaceScience, Vol. 110, No. 1, March 1986
226
ROSEN AND MURPHY
where C1, b is the concentration of surfactant 1 in the oil phase of the two-phase system after partition and Cl,w is the concentration of surfactant 1 in the water phase after partition. The total concentration of surfactant 1, Cl,t, in the mixed system is then: Cl,t =
Cl,w Vw -t- C 1,BVB
Vt
,
[18]
where Vw, Vb, and V~are the volumes of the water phase, the oil phase, and the entire system, respectively. From [ 18], C1,B =
Cl ,t gt - CI,w Vw
lib
[19]
(1 - Xl,x)2
[20]
If q~ = Vd Vw, then Vt = 4)Vw + Vw and Vb = q~Vw. Substitution into [20] then yields = F1CI,t,
[21]
where FI = (4~ + 1)/(K14~ + 1), the fraction of surfactant 1 in the aqueous phase of the surfactant mixture. Similarly,
F2C2,t,
C2,w = ~ ~ ) l - - , 2 , t =
[22]
where Fa = (4~ + 1)/(Kz4) + 1), the fraction of surfactant 2 in the aqueous phase of the surfactant mixture,
1~ C°t = F1Cl,t, o o ]
[23]
'
o o =(~K2t~2 ~.o~'~-+011I Cot =FuCE,t, ] '
[261
[27]
Since, Ci,t = Cl2,to/1, where C12,t is the total concentration of surfactant (1 and 2 combined) in the mixed surfactant system and a~ is the mole fraction of surfactant 1 in the mixed surfactant, substitution into [27] gives
3tL =
ln( Cl2,tot l f l/ C°,tF°Xl,t ) (1 -- Xl,i) 2
[28]
When 4~ = 4~° and K 1 = K 0, [28] becomes
t3[L = ln( C12,t°q/C°,tXl,l) ( 1 - XI,I)Z
[29]
q~ocan be made equal to 4~by using the same volume ratio Vb/Vw, in the mixed surfactant system as in the pure surfactant 1 system. Considering the low suffactant concentrations normally used, K ° is most probably equal to K~. In order to evaluate Xt,i, [25] is divided by [26], yielding (Xl,l)21n(FlCl2,tal/F°C°,tXl,I) ( 1 -- X 1,I)21n(F2Cl2,t( 1 - cq )/F°C°,t ( 1 - X1, I))
where F ° = (4~° + 1)/(K% ° + 1), the fraction of surfactant 1 in the aqueous phase of the system containing only surfactant 1, C°'w
C2't = ~22 (1 - Xl,i)C°,t ° exp{3[L(X,,i)2}.
VBCI,w
C l ' w = ( KI VBVt+ Vw)Ci,t.
1~11
and
filL = ln( Cl'tFl/ C°l'tF°Xl"O
Cl,t Vt - Cl,w Vw
Cl,w = ~ ~ / t - ~ l , t
[251
From [25],
from which,
C°'w = ( ~ - 1
Fo Cl,t = ~ Xl,lC?,t" exp{fl•L(1 -- XI,I)2},
Fo
,
and substitution of [19] into [17] gives K1 =
where F ° = (q~o+ 1)/(KOch02+ 1), the fraction of surfactant 2 in the aqueous phase of the system containing only surfactant 2. Here, again, the superscript o refers to systems containing only one surfactant. Substitution of [21] and [231 into [15], and of [221 and [24] into [16], respectively, followed by rearrangement gives
[24]
JournalofColloidandInterfaceScience,Vol. 110,No. 1, March1986
= 1
[30]
following the insertion of Cl,t = Cl2,toq and C2,t = C12,t(1 - oq). When q~ = q~o = 4P, KI = K~I, and/(2 =/~2, then [30] can be expressed as
S Y N E R G I S M IN BINARY M I X T U R E S OF SURFACTANTS,
(XLl )21n(Clzaoq/ C°,tXLO (1 -- X l , i ) 2 1 n [ C l 2 , t ( 1
-
,
og1 ) / C 0 , t ( l - X I , I ) ]
= 1.
F °
lnX+m~+fl(1-X) [31]
Since C12,t, al, C°,t, and C°t can be determined experimentally (as can K1,/~i, K 0 ,/(2, q~, 4 °, and ~o), [31] can be solved numerically for XI,I. Once XI,I is known, [28] (or [29]) can he solved to obtain 3f~e.
Synergism in Interfacial Tension Reduction Efficiency
V
227
2-1nal<0.
[35]
And, when 4~= ~b° and K1 = K °, the condition is lnX+3LL(1-X) 2-1na~<0. [36] When synergism in interfacial reduction efficiency exists, there will be a minimum in the Clzt vs al curve and maximum synergism in this respect will be obtained where the Cl2,t vs al curve shows a minimum, i.e., dC12't = 0.
da~ The efficiency of interfacial tension reduction by surfactants in a two-phase liquid-liquid system can be defined as the total surfactant concentration in the entire system required to produce a given interfacial tension (reduction). Synergism in this respect is present in a surfactant mixture if it can attain a given interfacial tension (reduction) at a total mixed concentration lower than that required of either surfactant of the mixture by itself. The point of maximum synergism in this respect is where the lowest total concentration of mixed surfactant is required to attain a given interfacial tension (reduction). The use of Cl,t = C12,toQ and C2, t = Cl2,t(1 - a:) in [25] and [26], respectively, yields
From [34], d i n C12,t_ 1 dX
dal
F'~ vc'O . exp[fl~L(1 -- X) 2]
2fl~L(1-- X) da 1
where it is assumed that K1 does not change with o~1 at the low surfactant concentrations generally used. Since, d In Cl2,t
dC12,t da I
=
C12,t
da~
'
[37] can be expressed as dCl2,t doQ = Cl2't
F1 dX
= F1 -~-'--l,t
dX
1 Of1
[371
0
G2,tal
Xdal
O/1
[32]
23[L(1-X)~]
= 0.
Since C~2,t ~ O, and
l dX X da~
F o
C,2,t(1 - al) = F22 (1 - X)C°,t • exp[fl[L(X)2], [331 respectively, where X - Xla. From [32],
dal
F,
+3[L(1-X) 2-1noq.
[34]
Since, for synergism to be present, C~2,t must be
2all.(1
- x)
dX
dcq
= o
at the point of maximum synergism. From this,
dX
In C12,t - In C°,t = In X + In F°t
1 al
X al(1 - 2fl[LX + 2 ~" /JLLX 2~ ] "
[38]
Division of [32] by [33] followed by rearrangement gives In
0 0 F1F2C2't o o +ln°q
F2F1CI,t
+ln(1--X)--3[L
+ 23~LX--ln(l--oq)--lnX=O. Journal of Colloid and Interface Science,
[39]
VoL 110, No. 1, March 1986
228
ROSEN AND MURPHY
i F2FVCy,t o oI
Differentiating with respect to a l gives 1 a 1
1
dX
m
2
(1 -- X ) doe I q-
In F1FOCO,t] < I#~LI
dX
tr
3LL doq
and when ~b = ~b° = ~b° = ~o and K1 = K ° and /~2 = K °,
l dX - - - 0, X dal
1 + - (1-al)
X(1 - X ) ~l(1 - ~1)[1 - 23~LX + 23[LX 2] " [411
Combining [411 and [381, X cq(1 - 2(3[LX +
Substitution of [42] into [34] gives In C,2,t,min= l n ( ~ )
+ f i l L ( 1 - X) 2, [47]
where C l 2 , t , m i n is the minimum total surfactant concentration required to produce a given interfacial tension (reduction). From [44], 0
0
0
0
X = ln(Cl,tF2F1/C2,tFzF1)
23~,LX2)
+ filL = ot~ [48]
or, when ~b = q~o = q~o, K1 = K °, and K2 =K~2,
x(1 - x )
~x) :
and
[421
where a~' is the mole fraction of surfactant 1 in the total surfactant in the entire system at the point of maximum synergism in inteffacial reduction efficiency. This relation states that, at the point of maximum synergism in this respect, the mole fraction of either surfactant in the interfacial region is equal to its mole fraction in the entire system. Substitution of [42] into [36] gives the first condition for synergism in this respect as 3[L(1 -- X) 2 < 0,
23~L Substitution of [481 into [47] gives
(1 - X )
"T = X,
[49]
S = l n ( C ° ' t / C ° ' t ) + flEE = c~*.
2 b*Lb ~ X21I
or
[431
and since (1 - X) 2 is always positive, filL must be negative for synergism to exist. From [42] and [39], 0
[461
2 fiLL "
Oel(1 -- oq)[1 -- 2 3 £ L X +
(1 -
Jln~2,,1 C°,tl< Ifl°LLl"
[401
where it is assumed that K1 and K2 do not change with change in al. Equation [40] can be expressed as
dX d~l
[451
- -
Cl2,t,rnin
X exp
-
C 0l,,Fl0 - F1 3~,L
FLL
0
0
0
23~.L
0
2
_] J
[5o1 and, when ~b = 05° = ~b°, K1 = K °, and/<2 = I~2, [50] becomes
C12,,,=i. = CO,,
¢o, r# L
× expl~LLL
~
_1.1
[511
Synergism in Mixed Micelle Formation Synergism in mixed micelle formation in binary mixtures of surfactants in a two-phase liquid-liquid system is present when the CMC l of the mixture, Clz,t, M is lower than the
0
F 2 F 1 C I , t __
In F~t~2cO,t
--3[L(1 -- 2X).
[44]
Since 0 < X < 1 and 3[L must be negative, the second condition for synergism in interradial tension reduction efficiency is Jdutnal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
1 The CMC of the system is the total molar concentration of surfactant in the entire system at the point of intersection of the two almost linear portions of the interfacial tension vs log total (i.e., in the entire system) concentration curve in the region where miceUization occurs in the system.
SYNERGISM IN BINARY MIXTURES OF SURFACTANTS, V CMC of either surfactant comprising the mixture by itself in a similar liquid-liquid system. From Eqs. [32] and [33], relationships analogous to those derived for synergism with respect to interfacial tension reduction efficiency can be obtained for synergism in mixed micelle formation. For synergism to exist in this respect when q5 = ~b° = q~o, K1 = K °, and K2 K°: (i) flLML must be negative, and (ii) [ln C,,VC2,t[ M M must be <[flLMLI,where the superscript, M, refers to micelle formation, iliaL is the experimentally determined parameter related to molecular interaction between the two surfactants in the mixed micelles in the system, and C~t and C2,~tare the CMC's of individual surfactants 1 and 2, respectively, by themselves in similar two-phase, liquid-liquid systems, flLMLis evaluated from equations analogous to Eqs. [28] and [29]. At the point of maximum synergism in this respect, =
M
0
M
229
Synergism in InterfaciaI Tension Reduction Effectiveness The effectiveness of the interfacial tension reduction in a two-phase, liquid-liquid system can be defined as the interfacial tension reduction attained at the CMC of the system. Synergism in interfacial tension reduction effectiveness is, therefore, present in a liquidliquid, binary surfactant system when the mixed surfactant system can attain a lower interfacial tension at its CMC, C M m , than either surfactant comprising the mixture can attain at its respective CMC, CI,M or C~,,. The basic equations, analogous to [32] and [33], above, for adsorption at the liquid-liquid interface at the CMC, C~,t, of the system are cIM2't°Q =
t O I r [ wO ~"~ l't°exp[fl~L(1
FI
and,
F0
c M , t ( 1 -- ~1 ) = F22
(1 -
- X) 2]
[56]
X)C°,t. exp[fl[L(X)2].
0
a*, M = X M = ln(CI,tF2FI/C2,tF1F2) + timL
[57]
2ML
The linear portion of the interfacial tension,
[521 7i, vs In C Oof each individual surfactant can or, when ~b = ~b° = ~b°, K 1 = K °, and K2 = xo:
o~,,M = xM = ln(C~dC~,) + fl~L 2flLML
,
[531
hypothetically be extended to concentration values above the CMC to yield interfacial tension values equal to the interfacial tension, "YCMC,m , of the mixture at its CMC, yielding the relationships "}/CMC, 12,t = $11 71-
and
Slln C°,t
[58]
and
M
C 12,t,min -
M 0 CI,tFI Fl
"YCMC,12,t= S~ + S21n C°,t,
× exp{flLML[flLML - - ln( C~tF°12fl~LF2/CMF, t ,F°)]2 ] J' [54] and, when ~b = q~o = q~0, K1 = K °, and/£2 = K °,
[59]
where $1 and $2 are the slopes of the 3' vs In C O plots of the individual surfactants and S't and S~ are the hypothetical -r intercepts of the same plots. From [56] and [58], 7CMC,12,t = Srl
+ SI{lnIF ~ cM,tal] - /3[L(I- x)z} . [60]
M M Cl2,t,min = C 1 , t
X exp{flML[-/~LML-ln(C~t/C~)']2~
3J
[551
When synergism in interfacial tension reduction effectiveness exists, there will be a miniJournal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
230
ROSEN AND MURPHY
mum in the matically;
~CMC,12,t
VS a 1 curve, or mathe-
d'YcMc,12,t = O.
Taking natural logs and differentiating [64] and [65] with respect to al, we obtain d In C~,t
dal
dal
Thus, d'YcMC,12,t =da,
Xdotll dX
S'( 1 +dlndaxClM2,~t
-1 +[ 1
- a--i-
~ -
2fl[aL(1 --
xM)q dX M J d~,
[66]
and +2fl[L(1--X)~al)=0,
[61] dlnCM,t=
dal
where it is assumed that KI and ff do not change with change in al. Since Sj 4= 0:
dX X dal
1 + dci M n ,~
al
dal
dal
and d l n cM,t = ( 1
dal
)dX - 2fl~.L(1 -- X)
1
dal
al [62]
Following the same line of reasoning, one can derive, from [61] and [59], d In CM,t
+
X1 2fl~LX) dda
1 -
1,
al
[63]
where it is assumed that /(2 and ~b do not change with change in a~. Using the basic equations of nonideal solution theory, equations analogous to [32] and [33] can be derived for mixed micelle formation in a two-phase, liquid-liquid system: = F° y,
1 -- X M
l
2flMLXM doz---T' dXU [67]
1
1
1 ------Y- 2/~[LX
1 -- X u
~--2fl[L(1--X) x----~--2flML(1--xM) 1 = 1 2¢~tLXM
[68] which reduces to X = X M.
dal =
Ol 1
respectively, where it is assumed that K~ and /(2 do not change with change to a~. [621 From Eqs. [66], [62], [67], and [63], we obtain
+2fl[.L(1--X) d X = 0
(1~
E
_ 1
-1
1 --
[69]
Relationship [69] states that when a minimum or a maximum occurs in the "~CMC,12,t vs al curve, the composition of the mixed monolayer at the liquid-liquid interface must equal the composition of the mixed micelles of the system. Combining [56], [64], and [69], and taking natural logs yields In C ~ - In C°,t = (/3~.L --/~ML)(1 -- X ) z [70]
C~tXM. exp[fl~L(1 -- xM) 21 [64] and combining [57], [65], and [69] gives In C~, - In C°,t = (fill -- flML)(X)2. [71]
and
Substituting [58] into [70] gives
C12,t(1 - al)
S'I + Slln C~t - ~CMC,12,t = F ° C ~ ( 1 - x M ) • exp[3~L(X~)2]. F~ Journal of Colloid andlnterface Science,
[651
VoL 110, No. 1, March 1986
= S~(t~L -- ¢~Mo(1 -- X ) 2.
SYNERGISM
IN B I N A R Y M I X T U R E S
Since, o
"rCMC,I,t = S'j + Slln CI,t,M where 7cuc,~,t o is the interfacial tension in a pure surfactant 1 system at the critical micelle concentration for the entire system, 0 ~CMC, I , t -
7CMC,12,t = S l ( / ~ L
--
/3LML)(1- - X ) . [72]
By the same line of reasonsing, using [59] in [71]:
OF SURFACTANTS,
V
231
When '~CMC, o I,t > '~CMC,2,t, o since S~ and $2 are always negative, the smallest negative value of/3~L --/3ME, equal to 0 0 ~CMC, I,t - - ~CMC,2,t
Sl will be obtained when X = O. When ~CMC, o 1,t 0 < 7cMc,z,t, the smallest negative value of/3OL -- /3LMLwill equal 0 0 "YCMC, I,t - - ~CMC,2,t
0 ']/CMC,2,t - - '~CMC,12,t = S 2 ( f l [ L - - /~LML)(X) 2,
-$2
[73] where 7CMC,2,t o is the interracial tension in a pure surfactant 2 system at the critical micelle concentration for the entire system. Substituting [73] into [72] gives 0 0 "YCMC, I,t - - "YCMC,2,t
= [~3~c -/3ML][S~(1 -- X) 2 - $2(X)2]. [74] Conditions for the existence of synergism in interfacial tension reduction effectiveness. These are determined from the above in the following fashion: 1. From [58] and [59], since the slopes $1 and Sz are negative, the larger the value of C°l,t or C2°,t, the smaller will be the value of "YcMe,lZ,t. Under ideal conditions, i.e., when /3f~L= 0, no synergism exists. From [56] and [57], in order for C°t or C°,t to be larger than under ideal conditions,/3[L must be negative. Therefore, for synergism in interfacial tension reduction effectiveness to be present in the system, /~L must be negative. 2. Synergism in this respect is present in a 0 0 system when 'YCMC,12,t < q/CMC, I,t, "YcMc,2,t. From [72] and [73], since the slopes S~ and $2 are always negative and the quantities (I X) 2 and X 2 are always positive, then the value of(/3~L -- g3L~L)must be negative for synergism in interfacial tension reduction effectiveness to be present. 3. From [74],
obtained when X = 1. Therefore, for synergism in this respect to exist, the (negative) value of /~[L - - flLML m u s t b e >]('YCMC, o o I,t -- 'YCMC,2,t)/S], where S is the slope of the '~i vs In Ct plot of the individual surfactant having the larger interfacial tension value at its CMC. In summary, the conditions for the existence of synergism in interfacial tension reduction effectiveness, when the partition coefficients (Kt and K2) and the volume ration (q~) do not change with al, are 2. ~ L -- ~ML < 0 0 3. rt~L - #~L[ > I ( ~ c ~ c , ~ , t - ~cMc,2,0/sl.
Conditions at the point of maximum synergism. Experimental evidence has been provided that at the liquid/air interface, at the point of maximum synergism in surface tension reduction effectiveness, the concentrations of the two surfactants in the mixed monolayer are approximately equal (i.e., XI X2 ~ 0.5), (5). Assuming that this is true at the liquid-liquid interface, i.e., X = 0.5, and combining this with [64], [65], and [69] gives a*'a =
F2F°C1M, t FzFOC~t , + F1Fz0 C I , tM •
[76]
-
/ ~ L - - /~ML =
0 0 "YCMC,I,t -- ")'CMC,2,t $1(1 - X) 2 - 8 2 X 2 "
[75]
When q~ = q~o = ~2o, K~ = K °, and/£2 = K °, this becomes ~,,E _
C~,
C~ + C~'
[77]
where a *'E is the mole fraction of surfactant 1 in the total mixed surfactant solution at the Journal of Colloidand InterfaceScience, Vol. 110, No. 1, March 1986
232
ROSEN AND MURPHY
point of maximum synergism in this respect. Introducing [76] into [64] or [65] yields oM
M,, = FF2FICI,t +~F1F2C2,tl.exp__, C12't
2FIF2
L
J
4 [78]
where C~;* is the critical micelle concentration for the entire system at the point of maximum synergism in interfacial tension reduction effectiveness. When 4~ = 4 ° = 40, K1 = K °, and/(2 = K °, [83] becomes
M,
Cl2;t -
+ c2, t 2 .exp - ~ - .
[79]
Finally, from [72] and [73], and using X1 = X M = 0.5, we obtain the relationship * 0 "YCMC,12,t = "]/CMC, I(2),t - -
S 1(2)(/~LL 4
- - /~ML) '
[801
where Y~MC,IZ,tis the lowest interfacial tension attainable at the CMC of any mixture of the two components. Equation [80] states that the amount of reduction of the interfacial tension of a surfactant at its CMC as a result of interaction with a second surfactant depends upon the interfacial concentration of the surfactant, as indicated by S, and upon the difference between the tendencies of the mixture to adsorb at the interface and to form micelles (3[L -- 3~L)- The greater the interfacial concentration of the surfactant and the greater the adsorption tendency relative to the micellization tendency, the greater will be the reduction in the interfacial tension at the CMC. Best agreement between calculated "YCMC,12,t * values and experimental values is obtained when the smaller of the two y°~c,l(2), t values for the individual surfactants is used in [80]. EXPERIMENTAL
Materials C12H25(OC2H4)7OH(CI2EOv), >98% purity as indicated by gas chromatography (Nikko Chemical Co., Tokyo, Japan). Sodium doJournal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
decane sulfonate (C12SO3eNa®), >99% purity (Research Plus, Bayonne, N. J.). N-Dodecyl-N-benzyl-N-methylglycine, C12HzsN ®(CHzC6Hs)(CH3)CH2COO®(ClzBMG), >98% purity and synthesized in this laboratory (7). Before being used for interfacial tension measurements, aqueous solutions of the surfactants (in water that had first been deionized, and then distilled twice; the last time through a 1-m-high Vigreaux column with quartz condenser and receiver) were further purified by repeated passage (8) through minicolumns (SEP-PAK C18 Cartridge, Waters Assoc., Milford, Mass.) of octadecylsilanized silica gel to remove any traces of impurities more interfacial active than the parent compound. The oil phase used throughout was n-heptane (Fisher Scientific) that was purified by passage through a 31 × 3.4-cm column (Fisher and Porter Co.) of silica gel 922 (Will Corp.) that had been heated at 120°C for 3 h before use. The UV absorbance of the treated n-heptane (1-cm cell) was <0.015 at 255 nm, with 95% ethanol used as a blank.
Interfacial Tension Measurements All interracial tension measurements were made by the spinning drop technique using a Model 500 Spinning Drop Interfacial Tensiometer (University of Texas). Readings were taken at 0.5 h intervals until three consecutive readings coincided. All readings were done at 25.0°C. The density of water was taken as 1.000 g]cm 3 and that of heptane as 0.684 g/cm 3.
Partition Coefficients One hundred milliliters of the desired surfactant solution was overlayed with 50 ml of heptane. This two-phase system was then allowed to stand for 1 month after which equilibrium was achieved (9). Shaking was avoided so as to not form an emulsion which may cause an erroneous partition coefficient value. The partition coefficient of pure C12H25(OC2H4)vOH was obtained by weighing the
233
SYNERGISM IN BINARY M I X T U R E S OF SURFACTANTS, V TABLE I 26
CI2EO]~ Cl2S03No
Interfacial Tension Equilibrium Times System
Time to equilibrium (hr)
CI2SO3Na C12EO7
10 ~
~14
7~
:1o
CI2BMG
½a
CI2EO7_CI2SO3Na
lb ]b
C~2BMG-C~2SO3Na
--18 X
6 2i
-6,0
,
,
-5.0
-4.0
At all concentrations studied. b At all mole fractions studied.
,
-310
-2.0
Log C 12,t
amount in the initial aqueous surfactant solution and the amount in a portion of the final aqueous solution of the two-phase system. In the presence of C~2H25SO~Na®, it was assumed that this anionic surfactant, due to its charge, did not partition into the heptane at all. The partition coefficient of C12H25(OC2H4)7OH, with C 1 2 H 2 5 S O ~ Na e in the system, was determined by weighing (as in the case of pure C12H25(OC2H4)7OH), but, here the C12H25SOa~Na® original weight in the aqueous phase of the initial pure aqueous system was subtracted from the total weight obtained in the aqueous phase of the two-phase system in order to determine the weight of C 1 2 H E s ( O C 2 H 4 ) 7 O H left in the aqueous phase. The partition coefficient of pure C~2BMG was obtained by determining the concentration in the initial aqueous solution by using UV spectroscopy, e263 nm,CI2BMG = 355 (7), then determining the concentration in the aqueous phase of the final two-phase system. The partition coefficient of C~2BMG in the
FIG. 1. Interfacial tension vs log total surfactant concentration for CIaEOT, C12CO3Na, and C~2EOT-C~2SO3Na mixtures at 25.0°C; ~b = ¢0 = ¢0 = 1.23 × 10-2; heptane as the oil phase. (A) C12SO3Na, (E3) C12EO7, (Q) CI2EOTC12SO3Na, aC~2EO~= 3.75 × 10-3, (X) C12EOT-ClaEO7C12SO3Na, aq2Eo7 = 0.249.
presence of C12H25SO3°Na® was determined the same way as for pure C12BMG, since C12H25SO3°Na® does not absorb in the UV. RESULTS A N D DISCUSSION
Equilibrium Times and Partition Coefficients Times for reaching equilibrium interfacial tension for the systems studied appear in Table
-\ 11 9
~z 5
TABLE II 3i
Partition Coefficients~ System
aj
Ct2EO 7
1.0
CI2EO7-CIESO3Na CI2BMG C~2BMG-C~2SO3Na
0.465 1.0 0.557
K~tb
Kjb
0.475
0.156 8.62 × 10-3 6.93 × 10-3
"Defined as the concentration in the oil (heptane) divided by the concentration in the water; all runs done at 25.0°C. b Average of two runs.
-4:2
-3'.8
-3~.4
-3'.0
-2~.6
-2,2 \~
LogC 12,t
FIG. 2. Interfacial tension vs log total surfactant concentration for Ct2BMG, CtzSO3Na, and C~2BMGCt2SO3Na mixtures at 25.0°C, ¢ = ¢0 = 4)0 = 1.23 × 10-2; heptane as the oil phase. (×) C~2SO3Na, (®) C12BMG, (E3) CI2BMG-CI2SO3Na, ac~2aM6 = 0.0277, ( ~ ) CI2BMGC12SO3Na, a c ~ M 6 = 0.0431, (&) C]2BMG-C12SO3Na, aCI2BMG = 0.945. Journal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
234
ROSEN AND MURPHY T A B L E III Effect of Interface o n Value of f i l l System
/~
C12BMG-CI2SO3Na ~ C~2BMG-Ct2SO3Na b C12EOs-CI2SO3Na c CI2EOT-CI2SOaNa b
-5,7 -4.51 - 1.5 +0,19
Interface H20/Air HzO/Heptane H20/Air H20/Heptane
D a t a from Ref. (4). b ~b = ~b° = q~0 = 1.23 × 10-2; 25.0°C.
a
c D a t a from Ref. (10); CI2EO8 = Ct2H25(OC2H4)sOH.
I. It is interesting to note that the required equilibrium time for the C12EO7-C12SO3Na system is much less than that required for either individual surfactant comprising that mixture. The partition coefficients for the systems studied are listed in Table II. The t3 values calculated using Eq. [28] and those calculated using Eq. [29] differ by less than 0.1% in all cases. This indicates that the values K~ and K1 are numerically close enough to each other to make valid the assumption that setting K ° equal to KI leads to insignificant error. Since this assumption leads to insignificant error, it follows that the assumption that K1 does not change with change in OL1 also leads to insignificant error. At the low surfactant concentrations involved, the presence of a second surfactant would not be expected to cause a significant change in K °.
in the absolute value of the molecular interaction parameters, indicating weaker interaction between the two surfactants in the presence of heptane. An explanation is suggested by the data in Table IV, which shows the interfacial areas per surfactant molecule, as calculated from the slopes of the surface (or interfacial) tension log-concentration curves. Areas per surfactant molecule are larger at the heptane/aqueous solution interface than at the air/aqueous solution interface, indicating that heptane molecules are intercalated between the alkyl chains of the surfactant molecules at the former interface, increasing the distance between them and consequently decreasing interaction between them. The /3~L value at 25°C for C12BMGC12SO3Na is -3.66. The 13Mvalue at 25°C for the same system is -5.0 (4). It appears, therefore, that there is also decreased interaction between the surfactant molecules in the mixed micelle in the presence ofheptane. Here, again, this may be due to heptane molecules between the alkyl chains of the surfactants; in this case, in the mixed micelle. Interfacial Reduction Efficiency. The conditions for synergism in this respect are: (i) /~[L< 0 and (ii) [ln(C°,t/C°,t)] < I t ' L l . Data for the two systems studied are shown in Table V. The C12EOv-ClzSO3Na system does not satisfy either condition for synergism since fl~L
Values of the Interaction Parameters, t3~Land/3~L Based upon the interfacial tension data in Figs. 1 and 2, and using Eqs. [28], [29] and their analogs for mixed micelle formation, ~r M 13LL and 13LL values at 25 o C were calculated for the two systems studied. ¢/~Lvalues are listed in Table III, flL~Lvalues are shown in Table VI. Table III also lists t" values at 25°C at the aqueous solution/air interface for similar systems. It is apparent from the listed values that the replacement of air by heptane at the aqueous solution interface results in a decrease Journal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
T A B L E IV Effect of Interface o n Interfacial Area per Surfactant Molecule Area (A)2
System
H20/ Air interface
H20/ Heptanff interface
C12BMG CI/SO3Na C12EO7
56 a 56 a 57 b
66 57 78
D a t a from Ref. (4). b D a t e from Ref. (11). c W i t h 4P = 4~° = 1.23 × 10-2; 25.0°C. a
235
SYNERGISM IN BINARY MIXTURES OF SURFACTANTS, V TABLE V Synergism in Interfacial Tension Reduction Efficiencya Sy~em Property
CI2EOT-CI2SO3Na
C12BMG-C12SOaNa
]ln(C°,t/C2,t)l(at "~1,mN m -~) ~[e (average) Synergism predicted by theory Synergism obtained experimentally a* (calculated) Cl2,t,~i, (calculated)
4.97 (8.0) +0.19 No No ---
3.96 (7.5) --4.51 Yes Yes 0.939 1.22 × 10 -4 M 1.19 X 10-4 M
Cl2,t ( e x p e r i m e n t a l )
b
--
a~ = ~0 = ¢o = 1.23 X 10-2; oil = heptane; 25.0°C. b At a ~ , ~ = 0.945.
= + 0 . 1 9 > 0 a n d [ln(C°,dC°,t)[ = 4.97 > [/3ILl = 0.19. T h e C12BMG-C12SO3Na syst e m meets b o t h c o n d i t i o n s for synergism. T h e r e is g o o d a g r e e m e n t between p r e d i c t e d synergism a n d e x p e r i m e n t a l results and, in the C12BMG-C12SO3Na system where synergism exists, between the calculated C 1 2 , t , m i n value and that obtained experimentally. Mixed micelle formation. T h e c o n d i t i o n s for synergism in this respect are: (i) BLML< 0 a n d (ii) [ln(C~/C~t)[ < [3LML[.T a b l e V I lists d a t a for the two systems studied. T h e CtESO3Na system meets the first condition, b u t
fails the second. T h e C12BMG-CI2SOaNa syst e m meets b o t h conditions. As in interfacial tension r e d u c t i o n , above, there is g o o d a r g u m e n t between p r e d i c t e d synergism a n d calculated values based u p o n t h e o r y a n d e x p e r i m e n t a l results. Interracial reduction effectiveness. T h e conditions for synergism in this respect are s u m m a r i z e d in the theoretical section. D a t a for the systems studied are shown in T a b l e VII. T h e C~2EOv-CxESO3Na system fails all three conditions. T h e C~2BMG-Ct2SO3Na system meets all three conditions. Again, there is g o o d
TABLE VI Synergism in Mixed Micelle Formationa System Property
[ln(C~t/C~)l 3~L (average) Synergism predicted by theory Synergism obtained experimentally a *'M (calculated) CMm,mi. (calculated) C~,t (experimental)b
CI2EOT-CI2SO3Na
3.83 -0.19 No NO
CI2BMG-Cj2SO3Na
3.11 -3.66 Yes
Yes 0.925 2.89 × 10 -4 M 2.69 X 10-4 M
1.23 × 10-z; oil = heptane; 25.0°C. b a~ua,e = 0.945.
a ~b = ~0~0 =
Journal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
236
ROSEN AND MURPHY TABLE VII
ACKNOWLEDGMENT
Synergism in Interfacial Tension Reduction Effectiveness a
The material is based upon work supported in part by the National Science Foundation under Grant CPE8317134.
System Property
o l,t "}'cIvlC, 0
(S0 '~CMC,2,t(82) ~[L -- ~LML
CuEOT-CI2SO3Na
Ct2BMG--CIrSOaNa
1.80 (-35.36) 7.39 (-14.48) +0.388
2.00 (-35.85) 7.39 (-14.48) -0.85
~0CMC,I,t
--3'~c,2,t Synergism predicted by theory Synergism obtained experimentally c~*'E (calculated) Cl~',* (calculated) C~,t (experimental) b * (calculated, "~CMC,12,t naN m -m) "YcM¢,I2,, (experimental, naN m -m)
0.388
0.375
No
Yes
No ----
Yes 0.043 1.39 × 10-3 M 1.53 × 10-3 M
--
0.80
--
1.58
a q~ = 4~o= ~bo= 1.23 × 10-2; oil = heptane; 25.0°C. t, c ~ ¢ = 0.0431. agreement between predicted synergism and e x p e r i m e n t a l results.
Journal of Colloid and Interface Science, Vol. 110, No. 1, March 1986
REFERENCES 1. Rosen, M. J., and Hua, X. Y., J. Colloid Interface Sci. 86, 164 (1982). 2. Hua, X. Y., and Rosen, M. J., J. Colloid Interface Sci. 90, 212 (1982). 3. Rosen, M. J., and Hua, X. Y., J. Amer, Oil Chem. Soc. 59, 582 (1982). 4. Rosen, M. J., and Zhu, B. Y., J. Colloidlnterface Sci. 99, 427 (1984). 5. Zhu, B. Y., and Rosen, M. J., J. Colloid Interface Sci. 99, 435 (1984). 6. Fried, V., Blukis, U., and Hameka, H. F., "Physical Chemistry," pp. 224, 252. Macmillan, Co., New York, 1977. 7. Dahanayake, M., and Rosen, M. J., in "Structure/ Performance Relationships in Surfactants" (M. J. Rosen, Ed.), ACS Symposium Series 253. Amer. Chem. Soc., Washington, D. C., 1984. 8. Rosen, M. J., J. ColloidlnterfaceSci. 79; 587 (1981). 9. Warr, G. G., Grieser, F., and Healy, T. W., J. Phys. Chem. 87, 4520 (t983). 10. Rosen, M. J., and Zhao, F., J. Colloid Interface Sci. 95, 443 (1983). 11. Rosen, M. J., Cohen, A. W., Dahanayake, M., and Hua, X. Y., J. Phys. Chem. 86, 541 (1982).