230
Ind. Eng. Chem. Fundam. 1903, 22, 230-239
concentration of coke in catalyst when deactivation is complete; qo.* is q* corresponding to a ( ( ) = 1 R = particle diameter or gas constant S B = selectivity of B T = temperature To = temperature at the surface of catalyst particle Tof = temperature at the entrance of fixed-bed reactor t = reaction time U = velocity of reacting mixture X = dimensionless concentration of reactant A (= c A / CAO) X h = dimensionless concentration of reactant A in fixed bed q* =
(= CAO/c+d
Y = dimensionless concentration of react;ant B (= 2 = axial distance in fixed-bed reactor 2, = axial length of fixed-bed reactor P = (-mCA&A/(Toke) = (-aH) TOf/CpCAOf
@b 6r t
= DA/DB = EIRTn
tc
= E',/RF0
= E/RTof = ElfRTO = E,/RTo { = dimensionless axial distance in gxed-bed reactor (= Z/Zo) q = effectiveness factor 8 = dimensionless temperature in catalyst particle (= T / To) tf ti tp
= dimensionless temperature in fixed-bed reactor (=T / TW)
Ob K
= k2o/k10 = dimensionless radial distance (= r / R )
p = density of reacting mixture p~ = bulk density of catalyst in pc = density of catalyst Tb $A
fixed bed
= CAOkcQt/qO* = (PBkOf/PcU)ZO = R(kO/DA)1'2
Literature Cited Aris, R. "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts"; Oxford University Press: Oxford, 1975; Voi. 1. Butt, J. B. Chem. Eng. Sci. 1988. 21. 275. Corbett, W. E., Jr.; Luss, D. Chem. Eng. Sci. 1974, 2 9 , 1472. Juang, H.-D.; Weng, H.-S.; Wang, C.G. Proc. 7th I n f . Congr. Catal. Tokyo 1980, 866. Kasaoka, S.; Sakata, Y. J. Chem. Eng. Jpn. 1988, 1 , 138. Minhas, S.; Carberry, J. J. J. Catal. 1989, 74, 270. Ozawa, Y.; Bischoff, K. B. Ind. Eng. Chem. Process D e s . Dev. 1988, 7, 72. Ramachandran, P. A.; Kam, E. K. T.; Hughes, R. J. Catal. 1977, 4 8 , 177. Shadman-Yazdi, F.; Petersen, E. E. Chem. Eng. Sci. 1972, 27, 227. Smith, J. M. "Chemical Engineering Kinetics", 3rd ed.; McGraw-Hili: New York, 1970; p 501. Smith, T. C. Ind. Eng. Chem. ProcessDes. Dev. 1978, 15, 388.
Receiued f o r review December 28, 1981 Accepted January 19, 1983
Thermodynamics of Mixed Micellization. Pseudo-Phase Separation Models Robert F. Kamrath and Ellas I. Franses' School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
We present a general model of micellization of binary nonionic and binary ionic surfactant mixtures In dilute solution. For the latter we use the parameter ,f3 to describe the fraction of counterions bound to the micelle. For the former, ,f3 = 0. The degree of nonideality of mixing in the micelles is described by a dimensionless excess free energy function w ( x ) = gEl[RTx(l - x ) ] , where x is the mole fraction of component 1 in the mixed micelles. For the strictly regular solution model, w ( x ) = wo, we calculate cmc(s): monomer, micelle, and counterion inventories: micelle compositions; and second cmc and micellar pseudoghase diagrams when demixing occurs. The condition for "azeotrope" micellization, in which the micelle composition matches the overall monomer composition, is lwol 2 (1 @)In ( c 2 " I c 1 ' )where , cl' and c 2 * are the pure surfactant cmc's. The condition for micelle demixing is w o 2 2. I n comparing our model with experimental results, we observe that if w o is estimated from measured mixed cmc's it is extremely sensitive to small errors in c and to uncertainties in P. Hence, comparison with a wider body of data is needed for precisely determining the nonideality of mixing in the mixed micelles.
+
Introduction Surfactants are key components in many technologies, such as lubricant formulation, mineral ore flotation, enhanced oil recovery, and in many consumer products. Surfactants can control surface and interfacial tensions, wettability, interfacial hydrodynamics, and dispersion stability and rheology. Their properties stem from adsorption at surfaces and interfaces and from the formation of equilibrium and nonequilibrium microstructures. The most important microstructures formed by surfactants and either water or hydrocarbons are micelles and lyotropic liquid crystals. Much work has been accomplished on the thermodynamics of micellization of single surfactants (Murray and 0196-4313/83/ 1022-0230$01.50/0
Hartley, 1935; Hall and Pethica, 1967; Mukerjee, 1975; Bidner et al., 1976; Israelachvili et al., 1977;Tanford, 1980, to name but a few). Much less has been done on mixed surfactants, which are more commercially significant. Mixed micelles, which contain more than one type of surfactant, exist in equilibrium with unassociated surfactant monomers. Their composition depends on the cmc's (critical micellization concentrations) of the single surfactants and the overall composition of the solution. Shinoda (1954) calculated the mixed cmc's of binary mixtures of potassium soaps by assuming that the surfactants mix ideally in the micelles. The so-predicted cmc's were less accurate the more the soaps differed in chain length. Lange and Beck (1973) ignored ionic effects and 0 1983 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
assumed ideal mixing in the mixed micelles in calculating the cmc’s of nonionic-nonionic, ionic-ionic, and ionicnonionic surfactant mixtures. For the last case they found negative deviations from ideal mixing as did Rubingh (1979). Mixed cmc’s were also calculated by Mysels and Otter (1961), Moroi et al. (1974, 1975a, and 1975b), Clint (1975), Shinoda and Nomura (1980), and Funasaki and Hada (1980). In the latter two papers and in others (Mukerjee and Mysels, 1975; Mukerjee and Yang, 1976; Perron et al., 1981), positive deviations from ideal mixing were found in mixtures of hydrocarbon surfactants with fluorocarbon surfactants. These authors have suggested that demixing may occur, i.e., that two types of mixed micelles coexist at equilibrium. In previous models, except Clint’s and Rubingh’s, monomer concentrations and micelle compositions at concentrations other than the cmc were not calculated. These parameters are crucial in subsequently estimating properties of the mixed surfactant solutions. Moreover, only the one-parameter regular solution model was used. In this paper we present a general thermodynamic model, which is valid for nonionic as well as ionic surfactants and can account in principle for any nonideality. In this and most previous models, the “pseudo-phase separation” approximation of micellization (Stainsby and Alexander, 1950) is used. We present a general model and then specialize in the regular solution model to obtain a comparison with available experimental results. Hall and Pethica (1967) have used the formal small systems thermodynamics approach and Franses et al. (1977) and Kamrath and Franses (1982) have used the mass-action model of mixed micellization.
Theory Summary of Model. For binary ionic and nonionic surfactants in water the micellization equilibria are respectively
I
231
I
1-PHASE
I I
10.
.o
1 .o
.5
X Figure 1. Representation of one- and two-pseudo-phase regions for binary mixtures of Surfactants for the regular solution model. The broken curve is the spinodal. The solid curve is the binodal. z is the micellar pseudo-phase mole fraction and w is the nonideality parameter wo.
where ci* is the cmc of i and cM+ is the concentration of the counterion. Based on these premises, we have derived the following. Above the mixed cmc the monomer concentrations are
xM1- + (1- x)M2- + /3M+ + (MlAM2,1-xMB)(1-@)(la) xM1 + (1- x)M2 + (MI,XM~,I-A
Ob)
where M i , M2-, M1, and M2 are surfactant monomers, M+ is the counterion, and x is the mole fraction of component 1in the mixed micelles. The latter model can also apply to micelles in nonaqueous solvents. The counterion binding fraction /3 (0 < /3 < 1) is taken to be constant, because then the dependence of the cmc on the counterion concentration can be adequately described (Elworthy and Mysels, 1966; Mukerjee, 1967), and, moreover, because a more physically realistic model (Stigter, 1974) is too complex. In the ionic model, which is described in more detail in Kamrath (1981) we use (i) material balance equations for the monomers 1and 2 and for the counterion, where we include the concentration c, of a salt with a common ion; (ii) equality of chemical potentials of components 1 and 2 and counterion in solution and in the mixed micelles; (iii) ideal mixing of monomers and micelles with the solvent and nonideal mixing in the micelles; and (iv) no micelles before the cmc (the pseudo-phase separation approximation). We choose the following formal expression for the excess free energy of mixing, from which we can calculate the activity coefficients of the monomers in the mixed micelles according to standard solution thermodynamics (Prausnitz, 1969) g&) = RTx(1 - x)w(x) (2)
where the exponential terms equal the activity coefficients yl(x) and y2(x). The mole fraction x* at the mixed cmc c* is independent of the salt concentration c, and is given by the following nonlinear equation
The reference chemical potential of component i (i = 1, 2) is
If there is no added salt, we use eq 7 with c*(c, = 0) c*. For nonionic mixtures (eq lb) we use eq 3 through 8 with
(1 - 2x*)w(x*) + x*(l -
1 - x*
where a is the mole fraction of component 1 in the overall mixture. The cmc c*(c,) is found from 1 c*(c,) [c* (c,)
-
+ cs]@
a yl(x*) (Cl*)(l+@)
+
(1- a) y2(x*)(c2*)(l
+@)
(7) which yields
(8)
232
I d . Eng. Chem. Fundam., Vol. 22, No. 2, 1983
a I .5
%
cu y1.0 %
u
t
’s .o
.o
.2
.L(
.6
1 .o
.8
.o
.2
.L(
ALPHA 2.0
1.5
.6
.8
1 .o
.8
1 .o
FlLPHA
I
1
I
1
c1
*!
%
cu
cu
y1.0 *!
y1.0 %
u
u
.5
1
.o
.2
.L(
.6
ALPHA
RLPHA
Figure 2. Plots of the mixed cmc c*/cz* vs. a for the mixed ionic pseudo-phase separation model in the absence of added salt with values of cz*/cI* and j3 as follows: (a) 1.0 and 0.6;(b) 2.0 and 0.5; (c) 1.0 and 1.0; and (d) 2.0 and 1.0. The values of w o are as labeled.
IV
.o .O
.P
.’4
ALPHR
-
.6
.E
1.0
RLPHR
Figure 3. Micellar pseudo-phase diagrams for the mixed ionic pseudo-phase separation model with cz*/cI* = 1.0 and fl = 0.7 for (a) w,,= 2.1;(b) wo = 3.0; and (c) wo (totaldemising). The diagrams show the ranges of %/Q* and a at which there we: (I) no micelles; (11)micelles rich in 2; (111) micelles rich in 1; and (IV) micelles of both types with compositions xA and x B . The dashed curves asymptotically approach the binodals, denoted by the vertical line segments at the top of the diagrams.
/3 = 0. For nonionic surfactants, the mole fraction x above
following coupled equations
the cmc is given by
For ionic surfactants,x and cM+ are determined from the
(10)
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
Table I. Experimental cmc’s and Calculated w n and x * Values for Aqueous Mixtures of SDS and SDeSa x* (- 1%) w,( + 1%) C* IC,* w,(-1%) e* ( m M ) b x*( 0) w n( 0 ) “SDS
1.00 0.81 0.59 0.46 0.39 0.34 0.30 0.26 0.26 0.24
33.5 27.2 19.6 15.4 13.2 11.5 10.2 8.8 8.6 8.1
0.000
0.050 0.153 0.290 0.400 0.509 0.652 0.799 0.892 1.000
0.13 -0.03 0.11 0.04 -0.35 -0.36 -1.69 -1.30
0.346 0.651 0.819 0.877 0.891 0.934 0.911 0.963
0.341 0.656 0.831 0.890 0.904 0.949 0.923 0.977
0.20 0.05 0.22 0.20 -0.17 -0.05 -1.47 -0.72
0.28 0.12 0.34 0.38 0.03 0.34 -1.22 0.53
233
x * ( t 1%)
0.335 0.661 0.843 0.904 0.917 0.964 0.934 0.933
a ~ ~ s D is sthe mole fraction of SDS in the mixture; 0 is taken as 0.645;w,(i 1%) and x*( t 1%) denote values calculated if the measured mixed cmc is changed by i 1%. This gives an idea of the sensitivity of w, and x* to small errors in e*. Values determined conductimetrically a t 2 5 “C by Mysels and Otter (1961).
1 .o
I
I
i
I
.o I 2 .o
1
I
l’O
h
I
I
I
I
1
1::
I
I
,Ih I
‘
-
I
1.5
I
’/
1
3 .@
7.9
MI
cu u \ V
-3
/ -
.o .I
I
.n
7:
CT/C2*
.o
2 .o
Li .O
6 .O
8 .O
10 .o
CT/C2* Figure 4. Plots vs. total surfactant concentration, for w o = 3.0, c2*/c1* = 3.0, a = 0.5, 0 = 0.5, and no added salt of (a) monomer concentration c,/cz*; (b) c2/cz*;(c) (cl + cz)/cz*;(d) quantity cg (see text); (e) mean monomer activity; (f) counterion concentration; (9) micelle inventory of pseudo-phase B; (h) micelle inventory of pseudo-phase A, (i) mole fraction of surfactant 1in micellar pseudo-phase B; and 6 ) mole fraction of surfactant 1 in micellar pseudo-phase A.
Above the cmc, the micelle inventory c, (concentration in moles/lOOO g of solvent) is
Figure 5. Inventories and composition plots for the SDS-SDeS system with CY = 0.29, c2*/c1* = 4.136(1 refers to SDS), 0 = 0.645 (estimated from SDeS data), and w o = 0.22, as calculated (see Table I and text). Cmc data were obtained from Mysels and Otter (1961). Curves (a) through (f) correspond to the same quantities as in Figure 4;(9) micelle inventory; (h) micelle composition; arrow A points to the maximum in cz/cz*.
234
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
00 U
u
U U O O
where cl*(c,) and c2*(c,) are the cmc’s of 1 and 2 at a salt concentration c,. Equation 16 is the same as the equation for nonionic surfactants (eq 7 with /3 = 0). The conclusion is that at high salinities, mixed cmc’s of ionic surfactants can be calculated via a nonionic surfactant model. Nevertheless, x* for ionic surfactants depends on /3 (eq 6). For the regular solution model, we use w(x) = wo in the previous equations; wo = 0 denotes ideal mixing. For wo < 2 there is only one value of x* which satisfies eq 6 (Kamrath, 1981). For w o > 2, there can be as many as three values (see subsection on micelle demixing). Conditions for Azeotrope Micellization. By “azeotrope micellization” we mean that the composition of the micellar pseudophase at the cmc is the same as the composition of the total monomer inventory and hence the same as that of the overall surfactant inventory, x* = CY (Prigogine and Defay, 1954, p 464). This condition and eq 6 yield the following condition for CY = C Y A ~
00000000
For the regular solution model with wo # 0
CYAZ < 1 that for azeotropy to be possible (we take c2* > cl* without losing generality)
It readily follows from (18) and 0
2. Although azeotropy in mixed micellization has not been documented experimentally, it should be observable and could well be important in practice when the monomer composition must be maintained constant as the total surfactant concentration is increased. Micelle Demixing. General Conditions. Using standard criteria for thermodynamic stability (Prigogine and Defay, 1954),we obtain that for mixed micelles to be stable, x and w(x) must satisfy the inequality
Solving (20) (with an equal sign) we determine the boundaries between metastable and unstable mixed micelles. For w(x) = wo, a solution exists if wo > 2.
1
=
1 (1 f
2
00000000
4
-
q2 2wo
By analogy to bulk solution thermodynamics, we term these boundaries spinodals. If two micellar pseudo-phases A and B coexist at equilibrium, then their mole fractions, called binodals are given by
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
235
Table 111. Experimental cmc's and Calculated w, and x * Values for Aqueous Hydrocarbon/FluorocarbonMixturesa
p = 0.53 CYSPFO
c*(mWb
c*/c,*
0.101 0.246 0.491 0.748 1.000
31.5 32.7 36.4 40.7 35.4 30.6
1.00 1.04 1.15 1.29 1.12 0.97
0.000 0.170 0.339 0.490 0.570 0.666 0.835 1.000
25.7 27.9 32.0 35.7 37.5 37.6 33.4 30.6
0.84 0.91 1.04 1.16 1.23 1.23 1.09 1.00
0.000
p = 0.645
w&O)
x*(O)
0.91 1.76 1.66 1.70
0.049 0.070 0.516 0.930
0.68 1.76 1.72 1.70
p = 0.645
p = 0.645
w,(O)
x*(O)
w,(+l%)
0.061 0.070 0.525 0.931
1.01 2.06 1.79 2.00
0.045 0.051 0.533 0.949
1.48 2.47 1.86 2.41
0.028 0.033 0.547 0.968
0.061 0.076 0.191 0.481 0.899 0.944
1.29 2.09 1.91 1.88 2.44 2.06
0.045 0.057 0.168 0.470 0.921 0.962
1.77 2.45 2.04 1.95 2.69 2.66
0.028 0.038 0.144 0.440 0.942 0.980
x*(-l%) a. SDeS/SPFO
w,(-1%)
#*(+I%)
b. SL/SPFO
1.10 1.72 1.68 1.75 2.14 1.79
0.055 0.086 0.223 0.526 0.889 0.951
CYSpFO is the mole fraction of SPFO in the mixture. Yang (1976).
0.96 1.82 1.80 1.82 2.23 1.68
Values determined conductimetrically at 25 "C by Mukerjee and
For nonionic mixtures, 8 = 0 and c** is the maximum of these quantities. For ionic mixtures, however, because of counterion effects the second cmc (taking that surfactant 1 micellizes first) is (Kamrath, 1981) c** = (c2*)B
c2*
These equations can be solved very simply if w(x) = wo (see Kamrath, 1981), because then xB + xA = 1. The spinodal and binodal curves (xB < 0.5) are shown in Figure 1. For wo = 6, xA = 0.99745, and xB = 0.00255, which are very close to complete demixing (wo m). When w o > 2, there can be up to three solutions for x * . The physically valid micellar mole fraction is the value which (Kamrath, 1981) satisfies condition (20) and yields the smallest value of the cmc c*(c,) when used in eq 8. Second CMC. For fixed a and a total surfactant concentration above the first cmc, x varies continuously and approaches a with increasing total surfactant concentration. If, at some total concentration, x equals a binodal composition xA or xB, a second micellar pseudo-phase will form. The smallest total concentration for which this occurs is called the second cmc, c**. Above this concentration two micellar pseudo-phases of fixed compositions xA and xB coexist until, in certain cases, the critical demicellization concentration is reached (Mysels, 1978; Kamrath and Frames, 1983). Above this concentration only one type of micelles is again present. For w ( x ) = wo and nonionic surfactants
-
c**
x(l - x ) -(cl* exp[(l - X (a- x )
(1- cy)l/(l+B) [ ( l - cYp)(c2*)'+B
+ p(1 - a ) ( c l * ) ' + @ ] ~ / ( l + B ) (26)
When there is added salt, the first cmc is given by
and the second cmc by c**(c,)
[+ c,
c**(c,)
[
(::: )+"I 1"
1 - ap + 8(1- a) -
=
Conditions for Azeotrope Micellization When Demixing is Possible. If cyAz is to be stable, cyM must be outside the binodal range (xB,xA).For w ( x ) = w o > 2 and if we take c2* > cl*, we get
) ~ W -~ c2* ] exp[x2wo]) (24)
where x = xA or xB. For ionic Surfactants, we have to use w(x) = wo and x = xA in eq 10 and 11and solve for cM+ and CT = c** simultaneously. We observe that above the second cmc the monomer concentrations remain fixed for nonionic mixtures (eq 4 and 5 with 8 = 0). In contrast, they decrease for ionic mixtures because of the counterion effect, although their ratio remains constant. The micelle inventories cmA and cmBcan be found from material balance equations analogous to eq 12. Limit of Complete Demixing. As w o m, the first cmc is
-
Sample Calculations We have developed computer programs (available from the authors upon request) with which, given wo,cz*/cl*, 8, and cy, one can calculate mixed cmc's, micelle compositions, and monomer, counterion, and micelle inventories (Kamrath, 1981). In Figure 2 we plot mixed cmc's vs. cy for cz*/cl* = 1and 2, 8 = 0.5 and 1.0, and various values of w,,. If the regular solution model describes mixing adequately (see later), then from such plots w o can be estimated (see, e.g., Rubingh, 1979). We observe that particularly for w o > 2 small uncertainties in c* can lead to large uncertainties in w,,.Moreover, the estimation is also quite sensitive to the value of 8. In Figure 3 we show
238 Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
-
micellar pseudo-phase diagrams for w o ranging from 2.1, at which the binodals differ little, to w o (complete demixing). In Figure 4 we illustrate how concentrations change when there is demixing with ionic surfactant mixtures. We notice that both monomer concentrations show a maximum at the first and second cmc. These maxima are due to both counterion and mixture effects and are also observed with ionic surfactants when there is no demixing (see next section and Kamrath, 1981). These maxima can be important in interpretation of conductivity and monomer activity data. For nonionic surfactants only one maximum is observed; it is important in understanding adsorption maxima and tension minima (Clint, 1975; Franses et al., 1977). The mean monomer activity c* = [(cl + C ~ ) C ~ + ] increases . ~ / ~ above the first and the second cmc of the mixture, as it does for single surfactants above their cmc (Kamrath, 1981). The parameter cB, however, which is defined as cg E [(cl + c2)cM+B]l/(l+@) (30)
8 I1 P
remains constant above the second cmc when there is demixing (Kale et al., 1980). When there is no demixing, cB increases slowly but steadily above the cmc. These results should also have implications on interpreting data of surface and interfacial tensions, which probably reflect mean monomer activities. Comparison with Experimental Results To apply the model presented here, one needs to estimate the function w ( x ) from experimental data. To illustrate the method, we will test whether the regular solution model applies to available experimental results, which are mostly in the form c* = f ( a ) . We first calculate by an iteration procedure the value of w o which corresponds to each pair (a,c*). In Table I we present calculated results for mixtures of sodium dodecyl sulfate (SDS) and sodium decyl sulfate (SDeS). To estimate the uncertainty in w o arising from the uncertainty in c* we varied the cmc by f l % . The w o varies from -0.35 to 0.03 for Q ~ D S= 0.509 and from -1.30 to 0.53 for QSDS = 0.892. In Table I1 we show that w o and x* are also substantially sensitive to the value of @. This sensitivity indicates that, nonionic micellization models are inadequate for describing the micellar mixing of ionic surfactants. Although w ois so sensitive to errors in the measured cmc and to uncertainties in 0,a clear trend is evident in the results of Table I. For systems rich in SDeS (LYSDS < 0.5), wois slightly larger than zero; i.e., the micellar mixing behavior deviates positively from ideality. Systems rich in SDS exhibit negative deviations from ideality. Moreover, because wovaries significantlywith compositions,the strictly regular solution pseudo-phase separation model with @ = constant seems to be inadequate for describing the mixing behavior accurately in these mixed micelles. Nevertheless, it appears that in this mixture the deviations from ideality are only slight. This is expected because the surfactant molecules are quite similar (Shinoda, 1954; Demchenko and Shapoval, 1969). In Figure 5 we have plotted the monomer, micelle, and counterion inventories and the micelle compositions for aSDS = 0.29 with the parameters w o = 0.22 and 0 = 0.645. The SDS monomer concentration curve (a) exhibits a maximum at the mixed cmc (cT/c2* = 0.46) and the SDeS monomer concentration curve (b) has a broad maximum at cT/c2* of about 1.6. The micelle composition curve decreases from SDS rich (xSDS = 0.83) at the cmc to SDeS-rich (xSDS = 0.29).
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983 237
Table V. Analysis of Differential Conductance Dataa for Aqueous SDSlSPFO Mixtures a. ExDerimental cmc’s and Calculated w, and x* Values
p = 0.53 CYSDS
c*(mM)
30.6 20.8 12.5 8.1
0.0 0.2 0.5 1.0
c*/c,*
w,(O)
x*(O)
1.00 0.68 0.41 0.26
1.28 1.53
0.808 0.970
p = 0.645 w,(-1%) 1.91 b
P = 0.645
p = 0.645
x*(-l%)
w,(O)
x*(O)
0.916 b
2.15 b
0.935 b
wo(+l%) x*(+l%) 2.46 b
0.954 b
b. Comparison of the Measured Second Inflection Point in the Differential Conductance Curve for aSDS= 0.2 and Maxima in the Calculated SPFO Monomer Concentration Curves for Various Values of p and w , c,/c,* for the max normalized 2nd wo c*/c,*(calcd) infl point concn of SPFO monomer P
0.53 0.645 0.645 0.645 0.645
1.28 1.50 1.91 2.15 2.46
0.68 0.66 0.67 0.68 0.69
1.18 i 1.18 f 1.18 f 1.18 ?r 1.18 i
0.06 0.06 0.06 0.06 0.06
1.25 1.26 1.14 1.12 1.13
a Mukerjee and Yang (1976); the experimental mixed cmc c* was taken as the first inflection point in the differential conThere is no value of w , for which the calculated and the experimental cmc’s match. ductance curve. 1 .o
9
.8
c*
i
CDC
4
.6
X .Lt
C*
i
1
.2
I
1.5
1.0
/
*
i
\
.o
/’
-
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cu
i cu
\ V
\ 0
#I
.o
u
u
-
.5
.5
J
a
.@
.o
1 .a
2 .@ CT/C2r
3
.@
.-n Li .O
.o
1 .o
2.0
a 3.0
11.0
CT/C2i
Figure 6. Inventories and composition plots for the SDS-SPFO system with CQDS = 0.20, c~*/c?*= 3.778,@ = 0.645,and w o = 1.5. Data were obtained from Mukerjee and Yang (1976).The curves are labeled the same as in Figure 5.
Figure 7. Calculated results for the same system aa in Figure 6,but with w0 = 2.15. The curves are labeled the same aa in Figure 4. CDC stands for critical demicellization concentration, at which emA = 0; see text.
We have used our model to calculate w ovalues for three binary hydrocarbon/fluorocarbon surfactant systems which were studied by Mukerjee and Yang (1976) to test for the suggested possibility of demixing. In Table I11 we present wovalues for the SDeS-sodium perfluorooctanoate (SPFO) and the sodium laurate (SL)-SPFO systems. The values of w o range from 0.91 to 2.69. We show the sensitivity of w o and x* to 0 for the SDeS-SPFO system in Table IV. If the regular solution model applies, w omust be larger than 2 for demixing to occur. Since w o is highly sensitive to 0 and the uncertainty in the measured mixed cmc, it is unclear whether demixing occurs. We observe
that for both systems w o is about 1.0 for micelles rich in hydrocarbon surfactant. This indicates that fluorocarbon surfactants are more soluble in hydrocarbon-surfactantmicelles than vice versa. Since w o changes with composition, the regular solution model does not completely describe the mixing in these micelles. Hence, the more general demixing criterion (20) should be applied, once the function w ( x ) is known. In Table V we show similar results for SDS-SPFO mixtures. We have determined the experimental mixed cmc’s from the first inflection points of differential conductance vs. mean concentration curves reported by
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Mukerjee and Yang (1976). For CQDS = 0.2, the calculated values of wo for p = 0.53 and p = 0.645 range from 1.28 to 2.46. Hence, demixing is likely but again not clearly established. For CYSDS= 0.5 and /3 = 0.53, the calculated w o is 1.53. For CYSDS= 0.5 and /3 = 0.645, there is no value of w ofor which the calculated mixed cmc is the same as the measured mixed cmc. The differential conductance curves for the SDS-SPFO mixtures exhibit a second inflection point, which is absent in similar curves for SDS-SDeS mixtures. Mukerjee and Yang (1976) have suggested that the second inflection point is evidence that a second type of mixed micelle rich in SPFO appears and that two types of micelles coexist at concentrations above the second inflection point. We have calculated monomer, micelle, and counterion inventories and micelle composition for the CYSDS= 0.2 system to investigate the possible significance of the second inflection point. In Figures 6 and 7 we have used the values wo = 1.5 and wo = 2.15 for CYSDS = 0.2 and p = 0.645. The SPFO monomer concentration curve (b) exhibits a maximum which is much sharper than the maximum in the SDeS monomer concentration curve shown in Figure 5. Furthermore, for w o = 1.5 the maximum occurs close, to within 0.02 of cT/c2*,to the concentration which for w o = 2.15 corresponds to the calculated second cmc c**/c2* = 1.12 (see Table Vb). The second inflection point could, therefore, be due simply to the abrupt maximum in the SPFO monomer ion concentration and not necessarily to micelle demixing. Hence, although micelle demixing is quite plausible, it is not proven by the existence of these breaks. A detailed conductivity model, or a more direct experimental method, or both, are needed to resolve the issue. For w o = 1.5, there is clearly a transition from micelles rich in SDS to micelles rich in SPFO as the total concentration rises, but there is always’onlyone type of micelles present. For w o = 2.15, there are two types of micelles present above the second cmc (c**/c2*= 1.12). Moreover, for these conditions the calculated inventory of SDS-rich micelles becomes 0 at cT/c2* = 1.90. This concentration corresponds to demicellization, i.e., transition from two types to one type of mixed micelles. Such a phenomenon was postulated by Mysels (1978);see Kamrath and Franses (1983).
Discussion and Conclusions We have developed a general pseudo-phase separation model for micellization of binary ionic surfactanta with the same head group and the same counterion. The model applies also to nonionic surfactants. The parameters calculated from such a model can be used in predicting adsorption and other thermodynamic or transport properties of mixed surfactant solutions. In principle, any nonideality of mixing can be formally described by the function w ( x ) . In practice, however, no more than a few parameters can be reliably determined from experimental results on cmc’s as one can judge from the uncertainties indicated in Tables I through IV. The conclusion is that more extensive data, such as compositions of mixed micelles and monomer activities, are needed to establish the function w ( x ) . The general model described in this paper has pointed toward several interesting nonideality phenomena which have not yet been documented experimentally. We specifically refer to azeotropic micellization, metastable mixed micelles, and demicellization. We feel that by choosing mixtures with proper values of cz*/cl*, 8, and w ( x ) these and other phenomena can be observed and lead to ap-
plications in surfactant formulations for detergency, wetting, etc. For a more accurate description of mixed micellization, one has to consider nonideal mixing with the solvent and possible changes of the counterion binding parameter p with micelle composition and especially with ionic strength. Then more detailed physical descriptions of the micelle double layer should be used (e.g., Stigter, 1974). Moreover, because micelle sizes are in fact finite and distributed, they should be accounted for in mass action models of mixed micellization (Kamrath and Franses 1982).
Acknowledgment We are grateful to the School of Chemical Engineering and the Computer Center of Purdue University, and to the Research Corporation, for partially supporting this work. This paper was presented in part at the AIChE 1981 Summer National Meeting in Detroit, MI.
Nomenclature A = micelle rich in surfactant 1 B = micelle rich in surfactant 2 c = concentration, molal c* = critical micellization concentration (crnc)in the absence of added salt c*(c,) = cmc in the presence of salt of concentration c, c** = second cmc c**(c,) = second cmc when salt is present c* = mean monomer activity c, = monomer concentration of component i cl* = cmc of component 1 c, = inventory of the surfactant in the micelle pseudo-phase cmA,cmB= inventories of micelles rich in surfactant 1 and 2 c, = concentration of added salt with a common counterion cT = total concentration,or inventory,in mo1/1000 g of solvent cB = quantity defined in Eq 30 gE = excess Gibbs free energy per mole M+ = surfactant counterion M 1 = surfactant monomer of component 1 R = gas constant T = temperature, K w ( x ) = excess free energy of mixing w o = excess free energy of mixing for the strictly regular solution model x = mole fraction of surfactant 1 in the micelle x* = micellar mole fraction at the cmc xA,xB = binodal compositions of coexisting micelles (xB < 0.5) xsp-, xsp+ = spinodal compositions Greek Letters
mole fraction of 1 in a binary surfactant mixture mole fraction for azeotrope micellization /3 = micelle counterion binding parameter y, = activity coefficient of component j lo = reference chemical potential pl,,* = chemical potential of the surfactant in a micelle of component i a =
(YAZ =
Literature Cited Bdner. M. S.; Larson, R. G.; Scrlven, L. E. Lat. Am. J . Chem. Eng. A@. Chem. 1978, 6,1-32. Clint, J. H. J . Chem. Soc., Faraday Trans. 11075, 7 1 , 1327-1334. Demchenko. P. A.; Shapoval. 8. S. ColM J . USSR 1969, 31. 273-276. Elworthy, P. H.;Mysels, K. J. J . CoiM Interface Sc/. 1968, 21, 331-347. Franses, E. I.; BMner. M. S.; Scrlven. L. E. I n “Micellizetlon, Sohrblllzatlon. and Mlcroemulsions”; Vol. 2, MMl, K. L., Ed.; Plenum Press: New York, 1977; pp 655-876. Funasakl, N.; Hade, S. J . Phys. Chem. 1080, 84. 736-744. Hall, D. 0.;Pethlca, B. A. I n ”Nonionic Surfactants”; Schick, M. J., Ed., Marcel Dekker: New York, 1967; pp 516-557.
Xnd. Eng. Chem. Fundam. 1983,22, 239-249 Israelachvili, J. N.; Mitdrell, D. J.; Nlnham, B. W. J . Chem. Soc., Faraday Trans.2 1978, 72, 1525-1588. Kale, K. M.; Cussler, E. L.; Evans, D. F. J . Phys. Chem. 1980, 84.593-598. Kamrath, R. F. M.S. Thesls, Pwdue University, 1981. Kamrath. R. F.; Frames, E. I. submmed for publication In J . Colbid Interface Scl. (1982). Kamrath, R. F.; Fransea, E. I. I n Proceedingsof International Symposium of Surfactants In Solution, Lund, Sweden, June 27July 2. 1982, MHtai, K. L., Ed.; to be published by Plenum Press, 1983. Lange, H.;Beck, K. H. KoHOMZ.-Z. Polym. 1973, 251, 424-431. Morol. Y.; NlshikMo, N.; Matuura, R. J . ColbH Interface Scl. 1974. 46, 111-1 17. Morol, Y.; NishikMo, N.; Matuura. R. J . colldd Interface Sci. 19751, 50, 344-351. Moroi, Y.; NishkMo, N.; Salto. M.; Matuura, R. J . CoIbH Interface Sci. 1975b, 52. 358-383. Mukerjee, P. A&. ColbH Interface Sci. 1987, 1, 241-275. Mukerjee, P. I n “Physical Chemistry: Enriching Topics from ColioM and Surface Science”; IUPAC, van Olphen, H.;Mysets, K. J., Ed.; Theorex: La Wla, CA. 1975; pp 135-153. Mukerjee, P.; Mysels, K. J. I n “Colloidal Dlsperslons and Micellar Behavior”; MHtai, K. L.. Ed.; ACS Symposium Serbs No. 9, American Chemical Society. Washington, DC, 1975; pp 239-252.
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Mukerjee, P.; Yang, A. Y. S. J . Phys. Chem. 1978, 80, 1388-1390. Murray, R. C.; Hartley, 0.S. Trans. Farahy Soc. 1935, 31, 183-189. Mysels, K. J. J . Couold Interface Sci. 1978, 66, 331-334. Mysels, K. J.; Otter, R. J. J . Colbid Sci. 1981, 16, 482-473, 474-480. Perron, G.; Delisl, R.; Davldson, I.; Qenereux, S.; Desnoyers, J. E. J . Col/oH Interface Sci. 198i, 79, 432-442. Prausnk, J. M. “Molecular Thermodynamics of FiuibPhase Equlllbrla”;Prentice-Hail: Englewood Cllffs. NJ, 1989; pp 193, 289. Prigogine. I.; Defay, R. “ChemicalThermodynamics”;Longsmans (;reen and Co.: London, 1954. Rubingh, D. N. In ”Solution Chemlstry of Surfactants”; Vol. I , M h i , K. L., Ed.; Plenum Press: New York, 1979 pp 337-354. Shinoda. K. J . fhys. Chem. 1954, 58, 541-544. Shinoda, K.; Nomura, T. J . Phys. Chem. 1880, 84, 385-369. Stainsby. G.; Alexander, A. E. Trans. Faraday Soc. 1950, 46, 587-597. Stlgter, D. J . Colloid Interface Sci. 1974, 50. 473-482. Tanford, C. “the Hydrophoblc Effect”, 2nd ed.; Wiiey: New York, 1980.
Received for reuiew October 5, 1981 Revised manuscript receiued November 15, 1982 Accepted January 28, 1983
Sterically Hindered Amines for CO, Removal from Gases Guldo Sartorl’ and David W. Savage Corporate Research, Exxon Research and Engineering Company, Linden, New Jersey 07036
Steric hindrance and basicity are shown to control C0,-amine reactions. In aqueous amino alcohols, steric hlndrance is the dominant factor giving high thermodynamic capacity and fast absorption rates at high C02 loadings. Introducing steric hindrance by a bulky substituent adjacent to the amino group lowers the stability of the carbamate formed by C02-amine reactlon. Reduced carbamate stabiiity allows thermodynamic COP loadings to exceed those attainable with conventional, stable-carbamateamines. Lowering carbamate stability also leads to high free-amine concentration in solution; therefore fast amIne-CO, reaction rates are obtained despite some reduction of the rate constant owing to steric interference. Hindered amines show capacity and absorption rate advantages over conventional amines for COPremoval from gases by absorption in aqueous amine solutions and amine-promoted hot potassium carbonate. Cyclic capacity broadening of 20-40% and absorption rate increases up to 100% or more are possible with certain hindered amines.
Introduction Most commercial processes for the bulk removal of COz from gaseous streams involve the use of amines, usually amino alcohols (Kohland Reisenfeld, 1979). Amines are used either as an aqueous solution or as promoters for aqueous potassium carbonate solution. The choice of type of process depends primarily on the partial pressure of C02 in the feed gas and on the level of C02 desired in the treated gas (Tennyson and Schaaf, 1978). Amine-based processes are generally used at C02 partial pressures in the feed gas up to 1W200 psia. At higher pressures physical absorption in polar organic solvents may be preferred. The common aqueous amino alcohol processes (monoethanolamine (MEA), diethanolamine (DEA), diisopropanolamine (DIPA) and &@’-hydroxyaminoethylether (DGA))d show limited thdrmodynamic capacity to absorb C02. This can be seen in Figure 1, which presents vapor-liquid equilibrium data measured at 40 “C (Isaacs et al., 1974; Kent and Eisenberg, 1976; Lee et al., 1976). The reluctance of the common amino alcohols to load up with C02 much beyond 0.5 mol of C02/mol of amine can be attributed to the rather stable carbamates formed with amines in which the amino nitrogen is attached to a primary alkyl group C02 + 2RR’NH + RR’NCOO- + RR’N’H, (1) 0196-4313/83/1022-0239$01.50/0
When carbamate formation is the only reaction (eq l),the maximum loading is limited by stoichiometry to 0.5 mol/mol of amine. A certain amount of carbamate hydrolysis occurs with all amines so that even with MEA and DEA the loading may exceed 0.5, particularly at high pressures. Hydrolysis generates free amine which can react with additional C02, thus allowing the loading to exceed 0.5. A different limiting behavior is shown by tertiary amines which are unable to form carbamates. Aqueous tertiary amine reaction with C02 leads to formation of the bicarbonate ion C02 + R’R’’R”N
3
HCOB-
7
+ R’R’’R’’N+H
(2)
and stoichiometric absorption of COz can now reach 1.0 mol of C02/mol of amine. This is well supported by available experimental data, as shown in Figure 1 for methyldiethanolamine, MDEA (Jou et al., 1981). While the high C02pickup possible with tertiary amines is very attractive in practice, the low rates of COz absorption in tertiary amine solutions may limit their use. This paper describes the discovery of a new class of amines-sterically hindered amines-which approach the high thermodynamic capacity of 1 mol of C02/mol of amine combined with absorption rates comparable to those @ 1983 American Chemical Society
