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Liquid solubilization dynamics
Liquid Solubilization D y n a m i c s III. Use of a Quiescent Technique to Determine the Correct Boundary Condition for Solubilization of Oleic Acid in Sodium Taurodeoxycholate KEVIN T. RATERMAN, 1 KURT L. ADAMS, AND JOSEPH A. SHAEIWITZ2 Department of Chemical Engineering, University of Illinois, Urbana, Illinois 61801 Received June 27, 1983; accepted September 20, 1983 An unsteady state experimental technique has been employed to study interfacial mass transfer boundary conditions in quiescent liquid/liquid systems. This method is superior than steady state methods for identifying the specific nature of the interfacial step. Moreover, the technique lends mechanistic insight into those systems in which transient effects may dictate steady state behavior. Experimental data are compared to theoretical results derived by assuming specific boundary conditions at the interface. A constant concentration boundary condition correctly describes the intertransport of partially immiscible phases (benzaldehyde into water), and a constant flux boundary condition correctly describes the dissociation of solute dimer from one immiscible phase into another (benzoic acid from benzene to water). The method was extended to identify the underlying mechanism of the micellar solubilization of oleic acid by sodium taurodeoxycholate at 37°C, a model fatty acid digestion system. The phenomenon appears to be surfactant diffusion controlled. The effects ofpH, added NaC1 and surfactant concentration are also discussed. INTRODUCTION
Recently, much effort has been devoted to the elucidation of detergency mechanisms (14). Particular attention has been paid to model fat digestion systems in which the Solubilization of water-insoluble dietary components is presumed to be micelle-mediated. Since time scales associated with mieelle-solute equilibrium often exceed contact times associated with digestion, it has been deemed necessary to characterize this dynamic solubilization process (1). In addition, questions have been raised as to whether fatty components "leave the oil/water interface in molecular form or whether collision of the micelle with the interface is necessary" (5). Such considerations are of interest not only to physiologists but also to formulators ~of pharmaceuticals, detergents, and soil-suspending agents. To date, the majorityof studies (1-4) have I Present address: Amoco Production Research Company, Tulsa, OK 74102. 2 To.whom correspondence should be addressed.
been limited to steady state experiments in which the rate of detergency has been measured as a function of fluid flow in the micellar solution. Since mass transfer characteristics of the various experimental contacting devices were well defined, these studies provided needed insight into the diffusional dependences for these complex systems. In most instances, it was determined that solubilization rates deviated from those predicted by a diffusioncontrolled process alone. For the dissolution of fatty acids and cholesterol by sodium dodecyl sulfate (SDS) and bile salts, Shaeiwitz et aL (3) noted large resistances to transfer which were overcome when Lawrence's penetration temperature was surpassed. Presumably, this resistance is associated with the ability of the aqueous solution to penetrate the solid and, therefore, is reduced once a temperature is reached at which the solid lattice is appreciably disrupted or melted. For the dissolution of liquid linoleie acid by sodium taurodeoxycholate (NaTDC), a bile salt, Huang et aL (4) proposed a mechanism which
406 0021-9797/84 $3,00 Copyright© 1984by AcademicPress,Inc. All rightsof reproductionin any-formreserved.
Journalof Colloidand InterfaceScience.Vol.98, No. 2, April t984
LIQUID SOLUBILIZATIONDYNAMICS, III involved the diffusion, adsorption, and surface reaction of surfactant as rate-limiting steps. The solubilization of liquid oleic acid by NaTDC was shown to be related to the formation ofmicroemulsions; solubilization rates being determined by the physical properties of an interfacial liquid crystalline phase and flow conditions (2). In this paper we propose an alternate method by which the mechanisms of detergency may be determined. The method itself, involves monitoring solubilization rates under unsteady transfer conditions. Since solubilization rates are not necessarily constant during unsteady regimes, the determination of solute fluxes with respect to time can provide valuable information about interfacial boundary conditions for mass transfer. In principle, the unsteady equations governing mass transfer can be solved for boundary conditions appropriate to given physical situations and compared to experimental results. Thus, some inference can be made concerning the kinetics and mode of transport associated with the phase boundary. Steady state techniques provide no such direct insight into the interfacial mechanism because only deviations from diffusion control can be detected. Determination of interfaeial mechanisms may be of primary importance in systems for which transient events dictate steady state behavior. Moreover, the method is deemed significantly better suited for detergency studies as opposed to previous methods which only characterize interfacial resistances to mass transfer. Prominent among these were studies which utilized Lewis cells (6, 7) or laminar jets (8). The former apparatus measures the overall resistance to mass transfer under turbulent flow conditions. If bulk liquid transfer resistances are known, interfacial resistances can be back-calculated. Unfortunately, under turbulent flow conditions the physical status of the interface is unknown and therefore the complex rheology of the interface should be considered for proper data analysis. In laminar jet experiments, interfacial transport is characterized over short contact times, usually less
407
than a second. Such brief contact times, however, are far too short for the determination of transfer behavior which may be influenced by surface aging. Transient surface aging is likely for detergent systems. Finally, both of the cited methods measure only the magnitudes of interfacial resistances; therefore, no inferences can be made with respect to underlying transport mechanisms. The proposed method can overcome these deficiencies. Validation of the method is provided concerning the model transport systems, benzaldehyde/water and benzene saturated with benzoic acid/water. The technique is then applied to the oleic acid/aqueous sodium taurodeoxycholate system for the purpose of determining the interfacial kinetics of mass transfer for this model fatty acid digestion system. The effects of pH, added NaC1, and surfactant concentration on the solubilization mechanism are also elucidated for this system. EXPERIMENTAL Benzaldehyde (EK Industries), benzene (Fisher Scientific), benzoic acid (Mallinckrodt), and chloride-free sodium taurodeoxycholate (Calbiochem) were all of purity 98% or greater and were used as received. Stocks of standardized radiolabeled oleic acid were prepared by dilution of [3H]oleic acid (New England Nuclear) in pharmaceutical grade oleic acid (Mallinckrodt). A saturated solution of radiolabeled benzoic acid in benzene was prepared by dissolving [14C]benzoic acid (New England Nuclear) in benzene and adding unlabeled benzoic acid until precipitation occurred. All aqueous surfactant solutions were prepared by weight using doubly distilled deionized water. The apparatus consists of a constant-bore glass tube, 1.5 cm in diameter and 22 cm in height, which has been water-jacketed for temperature control. Fluid is removed through a stopcock and needle-valve assembly (see Fig. 1). An experiment is conducted in the following manner. Approximately 20 cm 3 of aqueous solution is allowed to come to thermal equiJournal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
408
RATERMAN, ADAMS, AND SHAEIWITZ
WATER[~ JACKET CONNECTION ORGANI PHASE--'C,.\ \ \ \
-*-AQUEOUS PHASE
\ \
~]=[] NEEDLE VALVE STOPCOCK FIG. 1. Experimentalapparatus for unsteadyquiescent solubilization studies.
librium in the apparatus. This quantity was deemed sufficient over the time periods considered to satisfy the semi-infinite approximation adopted by the theoretical analysis. Three to five cubic centimeters of the less dense phase is then gently layered upon the aqueous phase by means of a syringe. Transfer is allowed to occur for a predetermined time. Upon completion of the designated time period, the aqueous phase is quickly removed, mixed, and analyzed. Removal time is always a few seconds, much less than the duration of an experiment. For radioactively labeled materials, a Beckman LS 7000 Liquid Scintillation counter is employed. Concentrations are obtained by comparison to externally prepared calibration curves (9). Unlabeled materials are detected in the aqueous phase by means of a Waters Associates Model R-403 differential refractometer calibrated with external standards. Volume average concentrations are reported for the aqueous phase. An experiment is repeated no less than four times for a given time period. An entire data set consists of seven points; the first six corresponding to 5-min intervals, the last corresponding to a 45-min elapsed time period. The data are plotted as the change in average solute concentration (AC) versus the change in time (At), where AC is defined as the difference in concentration between the concenJournal of CoUoid and Interface Science, Vol. 98, No. 2, April 1984
tration of interest and that of the initial data point at 5 min. The At is similarly defined. This procedure is deemed sufficient to remove any error incurred through the initial layering process and uncertainties in the exact location of the interface when sampling. Best fits between experimental data and theoretical models are determined by linear regression analysis where feasible. Otherwise, best fits are determined by inspection. In all cases, this analysis scheme was found satisfactory. THEORETICAL The corresponding theoretical analysis for the preceding experimental method encompasses the following. In general, the one-dimensional unsteady diffusion equation for semiinfinite geometry is solved for several interfacial (x = 0) boundary conditions provided the initial solute concentration and the solute concentration far from the interface (x ---, ~ ) are zero in the acceptor phase. The total interfacial flux of solute is then evaluated, integrated with respect to time, and divided by the total volume of the acceptor phase to obtain the average solute concentration associated with a specified time period for transfer. In mathematical terms,
C(t) = - F
7 x x=o
where DA(OC/Ox)x=Ois the total instantaneous interfacial flow of solute. A plot of AC(t) versus At can then be generated where AC and At have been previously defined. Specifically, several physical situations can be associated with the interfacial transport process which necessitate solutions to various forms and combinations of the unsteady diffusion equation. Since it is the intent of this paper to elucidate the mechanisms of detergency, only three general classifications, pictured in Fig. 2, will be considered. These classifications are based upon whether the transport process is presumed surfactant mediated; and, if so, whether the process is rate limited
LIQUID SOLUBILIZATION DYNAMICS, III OLEIC ACID
x=0
WATER
FIG. 2. Hypotheticalmodesofoleic acid solubilizafion. by mechanistic steps associated with the incorporation of solute into the micellar phase. The first classification (Fig. 2A) assumes unfacilitated solute transfer across the phase boundary. The goveming transport equation is OC
= D -
Ot
with boundary conditions, t=O
C=O
O~
x---, oo
C = 0
Vt,
oo,
[3]
[4]
Vt.
OC _ D 02C at ~ - k"C,
[51
[6]
with interfacial boundary condition, X = 0
[2]
Ox 2 '
C=?,j=?
Solutions to Eq. [1] are found by first solving Eq. [2]. These are tabulated in Table IA for the various physical situations which pertain to boundary condition [5]. Diffusion in the donor phase is also considered. The second classification (Fig. 2B) assumes solute crosses the phase boundary in its free form, but subsequently reacts with a component of the accepting phase. In detergent processes, this situation corresponds to unmediated transfer of solute into the aqueous surfactant solution whereupon it is incorporated into a micelle not connected with the phase boundary. The homogeneous incorporation process is modeled as a first-order irreversible reaction since it is presumed that surfactant is present in excess. The governing solute transport equation is
02C
--
409
C =
Ceq
Vt.
[7]
Solutions to Eq. [1] based upon Eqs. [3], [4], [6], and [7] are given in Table IB for several values of k", the reaction rate constant. The final classification (Fig. 2C) pertains to detergency processes exclusively. Namely,
TABLE IA C(t): Various Non-Reaction BoundaryConditions Boundary condition [5]
Solution to Eq. [1]
Comment
C = Co
2CoA ( Dt) I/z C(t) = - ' 7 k'-~l
[IA-1]
Constant concentration donor phase pure
j =k
0 (t) = --~ kA t
[IA-2]
Constant flux
[IA-3]
Equilibrium at interface; donor phase (I) initiallyat concentration C1~ (10)
Ci = mCll
Jl
=
jn
=
C(t) k'Clo
2mC1~4 (_~)1/2 V[m + (Dj/DIt) 1/2]
~(t) = k,Cl~a f t
V
do
[k,2t\
[
expt---~) erfc t -
k,~),/z
dt [IA-4]
Donorphase(I) initiallyat Cl=; irreversiblereaction at interface (11)
Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
410
RATERMAN, ADAMS, AND SHAEIWlTZ TABLE IB C'(t): IrreversibleHomogeneousReaction
Value k"
Solution to Eq. [1]
Comment
Moderate to small
C(t)
CoA(Dt)m [ ~ + V (k"t)l/2 + 2 (k t) v2 erf(k"t)~12 1 exp(-k,,t)J7 [IB-1] Homogenousreaction
Large
C(t)
C°A(k"D)l/2 V t
[IB-21
solubilization is assumed to be a three-step surfactant mediated process at or near the oil/ water interface. Specifically, the three steps are transport of free surfactant to the interface; heterogenous interaction of solute and surfactant m o n o m e r to form mixed aggregate; and, subsequent transport of the mixed aggregate away from the phase boundary. The transport equations are
02CB
or B
Ot
-
DB
[8]
~
Ox 2 '
and OCM
02CM
[91
O-"-t- = DM OX 2 '
with boundary conditions, t=0
CR=CBoo, CM=O
0
[10]
x ~
oo
x = o
Fast reaction or large concentration of reactant in acceptor phase (10)
CB = C s ~ , CM = O
v t,
[11]
--JB = njM = interracial reaction.
[12] where Ca is the free surfactant micelle concentration and CM is the mixed micelle concentration. Equations [8] and [9] are coupled through boundary condition [12] when the interfacial reaction is reversible and are not when the interracial reaction is irreversible. The flux of solute is equivalent to the flux of mixed micelles times a stoichiometric coefficient corresponding to the average number of solute molecules incorporated per mixed micelle. Pertinent solutions to Eq. [1 ] for the above, assuming DB = DM, are found in Table IC. All interfacial reactions are considered firstorder or pseudo-first-order since previous findings (1) indicate the solubilization of oleic acid by NaTDC to be consistent with first-
TABLE IC C'(t): ReversibleReaction Boundary Condition Boundary condition [14]
--ja = Jm = k ' C s o - k " C M o
Solution to Eq. [1]
C(t)
=
Comment
k'CB~t ~ exp (k' V [(k + k ) [ +k")2D] / t
- J B = JM =
k'CBo
=~
Jo e x p ~ - ) effc k' -~
Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
Reversible interfacial reaction (9)
\1/27
dt
[IC-2]
Irreversible interfacial reaction
411
LIQUID SOLUBILIZATION DYNAMICS, III i
order kinetics over the pertinent range of surfactant concentrations. RESULTS
The preceeding methodology was tested for reliability with respect to the model mass transfer systems of benzaldehyde/water and benzene saturated with benzoic acid/water. In the former system, benzaldehyde, as a neat fluid, was contacted with water at 25 °C. Since interfacial resistance to mass transfer can be deemed negligible (12), equilibrium can be assumed with benzaldehyde present at its saturation concentration near the aqueous phase boundary. The expected average benzaldehyde concentration versus time dependence in the aqueous phase would correspond to solution one (Table IA) in which C is a function of t 1/2. A plot of AC versus A(t 1/2) appears in Fig. 3 for benzaldehyde/H20 at 25°C. The correlation coefficient, r, for the linear fit is 0.99. In the latter system, benzene saturated with benzoic acid was contacted with water at 25 ° C. In benzene, benzoic acid exists primarily as dimer; whereas in water, benzoic acid exists almost exclusively as m o n o m e r (13). Thus, as benzoic acid crosses the benzene/water interface, the d i m e r irreversibly dissociates to monomer. The dissociation process is modeled in the aqueous phase as a first-order homogenous or heterogenous reaction dependent upon whether or not the reaction is associated with the phase boundary. Equations [IB- 1] and [IB-2] of Table IB refer to the homogenous
-2 8 %
%
F
o
L ....
--2~.~0 IO o
i
i
~
4_
o
5
I0
AI
I
(min)
15
20
I
25
FIG. 4. Average concentration of benzoic acid in water versus t at 25°C.
reaction case. The fourth entry in Table IA corresponds to the heterogenous reaction case. Figure 4 is a plot of 2xC versus At for the transfer of benzoic aid between benzene and water. The correlation coefficient for the linear fit is 0.99. The method was also applied to systems of oleic acid contacted with water at 37°C. The effects of added NaC1, altered pH, added sodium taurodeoxycholate, and NaC1 in conjunction with NaTDC in the aqueous phase were determined. Figure 5 indicates the effect of added NaC1 (0.0, 0.15, and 0.5 M) on the unsteady flux of oleic acid into water. Generally, with increasing salt concentrations fluxes were diminished. The average flux was approximately 1.0 × 10 . 7 g/cm 2 sec over the time period of
i
IE
....
i
t
o oleic/H20 r=0,995 oleic/0,15 M r =0,989 u oleic/0,5 M r=0,982
, . . ,
×
'<3
0
I0 20 A(I I/2) (secI/2}
FIG. 3. Average concentration ofbenzaldehyde in water versus t m at 25°C.
I0
20 30 A(t w2) Bed/z)
40
FIG. 5. Average concentration of oleic acid in water versus #/2; effect of increasing NaC1. Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
412
R A T E R M A N , ADAMS, A N D SHAEIWITZ
interest. In each instance the average concent --a--- 0 M NaCI, 2% NaTDC r i ---~--- 0,075 M NoCI, 2% NoTDC ~_ tration dependence with time was best fit by 15 -.-o-- 0,15 M NoCl, 2% NoTDC //..I//~'3"~ a ?/2 functionality. For plots of AC versus ~. ~ 0,5 M NaCI, 2% NaTDC i" "a~/ A(t w2) regression coefficients were equal to or greater than 0.98. Figure 6 shows the effect of altered pH (2.2 .~/J or 7.7) on the unsteady flux of oleic acid into %o solutions of phosphate-buffered saline (PBS), 5 //,/~,/'f'~/ ionic strength 0.15 M (14). At a p H 2.2 the average c o n c e n t r a t i o n d e p e n d e n c e in the I{/" I [ I I I I _L_I aqueous phase was best fit by a t u2 function300 600 900 1200 1500 1800 2100 2400 A t (sec) ality (r = 0.99). However, at pH 7.7 the data were highly nonlinear on a AC versus A(t 1/2) FIG. 7. Average c o n c e n t r a t i o n o f o]eic acid in 2 w t % plot. A Marangoni-type instability and the N a T D C versus t; effect of increasing NaC1. formation of an opaque interracial film were noted; the instability dying out within 5 to 10 min after initiation of an experiment. After phosphate-buffered 2% N a T D C solution was 15 min the flux of oleic acid into the pH 7.7 reduced from 7.7 to 2.2, unsteady flux values PBS solution was essentially zero. Average decreased. Marangoni-type instabilities and fluxes into either PBS solution were no greater interfacial films, however, were not noted for than 1.0 X 10 -7 g/cm 2 sec. Average c o n c e n the pH 7.7 solution. Generally, average fluxes trations of oleic acid in the p H 7.7 solutions, over the time course of an experiment for however, were significantly higher than those aqueous solutions of 2% N a T D C were an orfor the pH 2.2 solutions over the time period der of magnitude larger than for solutions of interest. without added surfactant. In all instances the Figures 7 and 8, respectively, indicate the best fit for A~ as function of time was obtained effects of added NaC1 and altered pH on the when the interfacial flux of solute was given unsteady flux ofoleic acid into 2 wt% N a T D C as solutions at 37°C. As previously, with inJlx=0 = k l + k2 e-aat, [13] creased NaC1 concentration (0.0 to 0.5 M ) the unsteady flux of oleic acid into the aqueous where kl, k2, and/3 are constants. Values for phase decreased. Likewise, as the pH of the these parameters are provided in Table II.
i
~ 15
i
u f
/ o
I0
i i
---.a-- oleic/PBS pH 7,4 I =0.15 ---~--- oleic/O,15M NaCI r=0.989 ---¢--- oleic/PBS pH 2,2 r=0,987 I=0,15
/
/
o 2%NaTDC, PBS (pH=7,6 I=O,15M) A 2% NaTDC, PBS (pH=2.2 I =OJ5MI
30
- ~ - - - -...-o-- - -
/
/~'/
20
×
x <
I0
20 30 A (1I/2) ($e¢I/2)
40
FIG. 6. Average concentration of oleic acid in water versus tin; effect of altered p H (I = 0.15 M, PBS). Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
I0
I
500
I
600
I
900
I
1200
I
1500
I
1800
I
2100
I
2400
At (sec)
FIG. 8. Average concentration of oleic acid in 2 wt% N a T D C versus t; effect o f altered p H (I = 0.15 M, PBS).
413
LIQUID SOLUBILIZATION DYNAMICS, III TABLE II Parameter Fits for Lumped Parameter Model NaTDC (wt%)
[NaC1] (M)
klA/ V (g/cm3 sec)
k2A/ V (g/cm3 see)
1~ (sec-l)
Q
0.5 0.75 1.0 2.0 4.0
0.0 0.0 0.0 0.0 0.0
2.22 x 10 -9 3.44 X 10 -9 1.42 X 10-8 1.42 × 10-8 1.42 X 10-8
3.10 2.72 2.46 2.46 2.46
X 10-8 X 10 -7 X 10 -7 X 10 -7 X 10 -7
1.20 X 10 -3 2.00 X 10 -3 1.25 × 10-a 1.25 X 10-3 1.25 X 10 -3
5.18 26.35 18.17 9.51 5.14
2.0 2.0 2.0
0.075 0.15 0.50
1.20 X 10-8 1.41 X 10-s 2.00 X 10-8
2.35 X 10 -7 1.46 X 10 -7 9.09 X 10-s
1.50 × 10 -3 1.30 X 10-3 1.25 X 10-3
9.13 5.97 3.98
2.0 2.0
pH 7.7 pH 2.2
3.33 × 10-8 3.20 X 10 8
3.07 X 10-7 2.18 X 10 -7
1.30 × 10-3 1.25 X 10-3
11.67 8.53
Finally, the effect o f i n c r e a s e d N a T D C cont e n t in the a q u e o u s phase f r o m 0.5 to 4.0% is s h o w n in Fig. 9. W i t h i n c r e a s e d s u r f a c t a n t c o n c e n t r a t i o n u n s t e a d y fluxes increase; but, however, reach a l i m i t at a p p r o x i m a t e l y 1% a d d e d N a T D C . As before, best fits for A ~ as a f u n c t i o n o f At are o b t a i n e d w h e n t h e interfacial flux is given b y Eq. [13]. Values for p e r t i n e n t p a r a m e t e r s are given in T a b l e II. DISCUSSION As was earlier indicated, the initial experi m e n t s with the systems b e n z a l d e h y d e / w a t e r a n d benzene saturated with benzoic a c i d / w a t e r were designed to d e t e r m i n e t h e validity o f the e x p e r i m e n t a l a n d theoretical m e t h o d o l o g y . In the f o r m e r case, in w h i c h a p u r e d o n o r phase was c o n t a c t e d with a n a c c e p t o r phase initially donor-free, the average c o n c e n t r a t i o n s h o u l d v a r y with t h e square r o o t o f time. F i g u r e 3 shows t h a t the average c o n c e n t r a t i o n o f benza l d e h y d e in w a t e r does i n d e e d v a r y with the square r o o t o f time. I n the latter system, in which solute d i m e r s dissociate irreversibly u p o n e n t e r i n g the a c c e p t o r phase, the average c o n c e n t r a t i o n d e p e n d e n c e is d i c t a t e d b y t h e physical l o c a t i o n o f the first-order d e c o m p o s i t i o n reaction. F i g u r e 4 indicates t h a t t h e d a t a are best fit b y a l i n e a r - t i m e d e p e n d e n c e which c o r r e s p o n d s to a h o m o g e n o u s d e c o m -
p o s i t i o n r e a c t i o n with a large r e a c t i o n - r a t e c o n s t a n t (Eq. [IB-2]). Hence, the m e t h o d is sufficiently sensitive to elucidate the n a t u r e o f the dissociative r e a c t i o n a n d also its a p p a r e n t m a g n i t u d e . W e , therefore, feel c o n f i d e n t to a p p l y this m e t h o d to m a s s t r a n s p o r t systems in which interfacial transfer m e c h a n i s m s have been p r e v i o u s l y u n c h a r a c t e r i z e d . T h e m e t h o d was initially applied to systems o f oleic acid transferring to a q u e o u s solutions o f varying NaC1 c o n t e n t a n d p H (Figs. 5 a n d 6) respectively. Best fits were o b t a i n e d f o r / x ~ expressed as a f u n c t i o n o f the square r o o t o f time. T h e o n l y n o t a b l e e x c e p t i o n was the case for p H 7.7, ionic strength 0.15 M. A t 1/2 de-
o 15
'~
b,5~7o Bile ( n o s a i l ) '
v 1,00 % ~] 2 . 0 0 %
4>
. . . . ....
~ /
/
I
I
D
5
O~
~
I 600
1200
1800
I 2400
AI (sec)
FIG. 9. Average concentration of oleic acid in NaTDC solution versus t; effect of increasing NaTDC concentration. Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
414
RATERMAN,
ADAMS, AND SHAEIWITZ
pendence is consistent with a constant concentration boundary condition for which the donor phase is pure and at or near equilibrium with the acceptor phase at the phase boundary (Eq. [IA-1]). The evident effect of increased NaC1 concentration in the aqueous phase was to reduce the average flux of oleic acid into that phase. According to Eq. [IA-1], this trend would be noted if either the equilibrium concentration of solute, Co, or the effective diffusivity of solute in the acceptor phase, D, diminished with increased NaC1 content. Shankland (15), however, reports that NaC1 has very little effect on the equilibrium solubility of oleic acid in water. To the contrary, effective diffusivity studies ofoleic acid in water suggest that solute diffusivities do indeed diminish with increased NaC1 levels (9). Presumably, this phenomenon is due in part to the formation of sodium oleate aggregates whose diffusivities are considerably less than oleic acid monomer. The CMC for sodium oleate is approximately 0.001 M or 3.0 X 10 -4 g/ml (15). The equilibrium solubility ofoleic acid in water is about 2.0 X 10-3 g/ml (9). Thus, the formation of sodium oleate is not unlikely provided sufficient amounts of counterion are present. Such is the case as the NaCI content is raised in the aqueous phase. The constant concentration boundary condition model is also consistent with the altered pH data. Here, however, equilibrium solubilities rather than effective diffusivities are most affected. The solubilities of oleic acid at high and low pH are 4.0 X 10-3 and 1.5 X 10-3 g/ml, respectively (9). Since the pKa of oleic acid is about 4.7 (15), the increased solubility and thereby increased initial flux of oleic acid in water agrees well with the ability of oleic acid to become ionized to oleate ion at high pH's. In fact, at pH 7.7 the formation of sodium oleate, a surface active agent (16), is adequate to induce Marangoni-type instabilities at the phase boundary; the instabilities arising from localized gradients of surfactant in the plane of the interface. The concommitant agitation of the interfacial region is Journal of Colloid and Interface Science. Vol. 98, No. 2, April 1984
sufficiently violent to significantly increase the initial flux of oleic acid. In time, however, adequate amounts of oleate are generated to form an interfacial film which dampens the instabilities and thereby the flux of oleic acid into the aqueous media. At a pH of 2.2 the amount of oleate present is presumably insufficient for the development of Marangoni instabilities. As expected, low pH data obey the proposed constant concentration boundary condition model for the transport process. Noticeably, the average flux at pH 2.2 is less than that for the analogous 0.15 Madded NaC1 solution whose pH is approximately 5.5. This observation is consistent with the evidently synergistic effects of decreased solubilities at low pH and decreased diffusivities arising from aggregate formation. Hence, for systems of oleic acid/water the interfacial transfer process can be described by a constant concentration boundary condition model. Moreover, the effects of NaC1 and pH on oleic acid transfer can be ascribed to the variations of phenomenological coefficients appearing in the model. These variations are apparently linked to the formation of sodium oleate, a surfactant. The behavior of oleic acid/water systems with added NaTDC is significantly different from the above as indicated by Figs. 7-9. Of initial note is the fact that average fluxes for NaTDC solutions are generally an order of magnitude larger than average fluxes for solutions without added surfactant. This observation implies that the solubilization of oleic acid is accomplished primarily through a surfactant-mediated mechanism. Figure 10 depicts the AC versus time dependence for oleic acid/2% NaTDC, no salt at 37°C and for several surfactant-mediated models. These models, expressions for which are found in Tables IB and IC, primarily postulate the micellar incorporation of solute as the overall ratecontrolling step ofsolubilization. Clearly, that model, which expresses the interfacial flux of oleic acid as Eq. [13], provides the best data fit. Equation [ 13], however, was derived as a
415
LIQUID SOLUBILIZATION DYNAMICS, III ,
,5
2% BILE (NO SALT)
j , / f
,
(iii) Incorporation of surfactant into a mixed aggregate in conjunction with oleic acid
/~//
•/(-" //
KR %
Io <]
CBi ~ CMi,
.
5
~'//2"o EXPERIMENTAL DATA f / ~ R S I B L E MOMOGENEOUS RXN
/ ~ ~ B L E O~
HETEROGENEgUSR×N
- - L U M P E D PARAMETER MODEL I I I I I I I I 500 600 900 1200 1500 1800 2100 2400
At (see)
FIG. 10. Predicted average concentration of solute in
acceptorphaseversustime for severalsurfactantmediated solubilizationmodels. result of the data and not from a previously proposed mechanism. Nevertheless, the actual incorporation of oleic acid into the surfactant phase, whether it be a heterogenous or homogenous process, apparently does not solely control the overall rate of solubilization. Dynamic solubilization studies for these solutions indicate that at sufficiently low surfactant concentrations (~<2 wt%) the solubilization process seems to be diffusion-controlled; the flux versus Reynolds-number dependence being approximately that predicted for the diffusion limit of mixed micelles (1). At higher NaTDC concentrations additional studies (2) show that the solubilization rate is controlled by the formation of a liquid crys± talline interfacial phase. Here, too, the amount of the interfacial phase formed is apparently diffusion limited. As a consequence, it is reasonable to expect that any formulation which produces Eq. [13] must involve diffusion of micelles as the overall rate-controlling step of the solubilization mechanism. With this in mind, the overall incorporation mechanism can be reduced to a five-step process. The steps are (i) Surfactant diffusion to the phase boundary kB
Ca ~ CBS,
[14]
(ii) Adsorption ofsurfactant to the interface
[ 16]
Here, CM refers to mixed micelles on a solutefree basis, since the solubilization process is assumed to be exclusively surfactant mediated. An increased solute capacity per micelle is reflected as an increased surfactant efficiency (i.e., fewer surfactant monomers required to solubilize a given quantity of solute). (iv) Desorption of mixed aggregate Kd
CMi ~ CMS,
[17]
(v) Diffusion of mixed aggregate away from the interface kM
c ~ ~ c~.
[18]
The above rate constants, kM and kB, are mass transfer coefficients and refer only to physical diffusion. R e m a i n i n g constants describe physical equilibria. The above description, however, does not imply exact stoichiometry. Since surfactant adsorbs to interfaces primarily as monomer, the equilibria between micelle and monomer is presumed implicit within step ii. Oleic acid is also presumed to be present in excess near the phase boundary. The model has little significance at or below the CMC for the surfactant of interest. The variation of surfactant concentration with respect to time near the interface can be described by dCBs v~
dt
-
k~(CB
-
CBs)
-- k M A ( C M s --
CM), [19]
where Vi is the apparent interfacial volume. If all reaction steps are in rapid equilibrium as previous information indicates, then CMs is related to Cas by
Ka
CBs ~ C~i,
[15]
CMS =
QCBs,
[20]
Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
416
RATERMAN,
ADAMS, AND SHAEIWlTZ
where Q is the overall equilibrium constant for steps (ii) through (iv). Hence, Eq. [ 19] can be solved to give
CBs = K - ( K - CBso)e-eat.
[21]
Here, K=
kBCB + kMCM kB + QkM
'
A vi
/3 = ~ (kB + QkM),
[22]
[231
and CBSOis the initial concentration of surfactant micelles near the interface. Since the flux of oleic acid is assumed proportional to the flux of mixed aggregates, the appearance with time of solute in the surfactant solution is approximately
j ~ kMQ K
kBCB e_~At kB + QkM + CBsoe-~a~]. [24]
The above expression was obtained by setting the concentration of mixed aggregates in the bulk solution, CM, to zero. Finally, since Q is always greater than zero and CBso is essentially CB, the bulk surfactant concentration, Eq. [24] reduces to a form similar to Eq. [13]. Thus, the above model predicts the observed concentration versus time dependence for the solubilizafion of oleic acid by aqueous solutions of NaTDC. Moreover, the assumptions of surfactant diffusion control and rapid solute incorporation into the micellar phase are consistent with the aforementioned dynamic solubilization findings. The model suffers, however, from the utilization of averaged or lumped parameters in the description of essentially an unsteady process. Such parameters are more appropriately applied to steady state phenomena. But, perhaps, this result is not altogether unexpected. The detergency process itself probably cannot be described accurately when solute associated and nonassociated surfactant aggregates are considered as well-defined enJournalof Colloidand InterfaceScience,Vol. 98, No. 2, April 1984
tities and thereby treated accordingly in an unsteady theoretical analysis. This approach neglects the polydisperse nature of NaTDC solutions (17) and ignores the ability of oleic acid to form aggregates under the proper experimental conditions. In addition, the experimental technique, itself, is essentially an averaging technique. Hence, it is incapable of monitoring instantaneous solute concentration profiles rather than total solute concentrations. It is also incapable of identifying the actual physical state of solute molecules. Therefore, in light of the experimental technique and the apparent complexity of species interactions, it is not unreasonable to presume that a lumped or averaged parameter model best describes the detergency process. Despite limitations, the model is nevertheless helpful in explaining observed experimental trends. When Eq. [ 13] is integrated to obtain C as a function of t, two parameters emerge as being most influential in determining the shape of a given profile. These parameters are k2A//3 V, which determines the initial steepness of a profile; and/3, which determines the profile's duration of curvature. (Values for klA/V are arbitrary, since they are dictated by the method of analysis.) From Eqs. [24] and [23]:
k2A
Q2
Vi
/3V - CB (1 + Q)---------~V '
[25]
where [3 is defined by Eq. [23]. In addition, the unsteady mass transfer coefficients, kB and kM, can be approximated by the expression ~(DI1/2
k,,M
\~-~1
.
[26]
Here, tm is one-half the average time duration of an experiment and D, the micellar diffusivity, is assumed to be 1.0 X 10 -6 cmZ/sec (18). The kM and kB are calculated to be approximately 2.0 × 10-5 cm/sec. Assuming average values for/3 and kzA//3V from Table II, Eqs. [23] and [25] can be solved in conjunction to obtain values for Q and Vi]A. Q is of order 10 and Vi/A, the penetration depth, is nearly 1.3 mm. The calculated value for the total
LIQUID SOLUBILIZATION
equilibrium constant is consistent with the observed large initial fluxes of solute, since diffusion control is minimal for very short times. Values for Q are tabulated in Table II. The above value for the penetration depth is also consistent with experimentally observed diffusion boundary layers. Examination of Table II indicates that for solutions of 2% N a T D C with added NaC1, values for Q decrease with increasing NaC1 concentration. According to Eq. [20], Q can be defined as CMs/CBs, the concentration of mixed aggregates over the concentration of solute-flee aggregates near the phase boundary. If solute-free and mixed aggregates can be described by time-independent mean aggregation numbers, then this ratio is essentially the concentration of surfactant m o n o m e r associated with mixed micelles over the concentration of surfactant m o n o m e r associated with nonmixed micelle. Therefore, a decrease in the value of Q can be attributed to either a decrease in the amount of m o n o m e r required to solubilize a given quantity of oleic acid or an increase in the stability of solute-flee aggregates (i.e., a shift in monomer-micelle equilibrium which favors the micelle). Shankland reports that the addition of simple electrolyte enables a given quantity of bile salt (NaTDC) to solubilize a greater portion of oleic acid (15). Presumably, this stems from the ability of oleic acid to act as its own surfactant in the presence of a suitable counterion. Thereby, the requirement for additional surfactant to solubilize a specified amount of oleic acid is reduced. These observations would tend to suggest the former of the above explanations as the cause for the observed trend in Q with added NaC1. Conversely, it has also been observed from ion-specific electrode experiments that aggregation begins at lower concentrations for NaTDC with added NaC1 concentration (9). This implies a greater a propensity to form aggregates and therefore, perhaps a greater micelle stability. The concommitant result would be a decrease in the amount of monomer available to adsorb to the interface and form mixed aggregate. (Monomer activity is
D Y N A M I C S , III
417
constant above the CMC, and approximately equal to the CMC.) Hence, either explanation seems plausible for the observed experimental trends. Table II also shows the effect of added surfactant on values of Q without added NaC1. Unlike the previous data, values for Q initially increase with increased concentration, but reach a maximum at approximately 0.75 wt% and begin to decrease thereafter. The decreasing trend is viewed with significance whereas the increasing trend is not. This stems from the fact that at 0.5 wt% N a T D C micellar solubilization is deemed insignificant; since aggregation for NaTDC begins at approximately 0.35 wt% (9). In accordance, a fit of C for the 0.5 wt% data versus t 1/2 gives a correlation coefficient (r) or 0.99, as would be predicted by previous data for the non-micellar aided solubilization of oleic acid. As before, explanations for decreasing trends in Q must center about the previous suppositions, either increased micelle stability or increased solubilizer effectiveness. Unfortunately, to date very little quantitative information exists on the stability of N a T D C aggregates with increasing surfactant concentration. However, since N a T D C aggregates in a stepwise fashion (19) (i.e., additional monomer preferentially associates with existing micelles rather than forming new aggregates of the same size), it is possible to construe an increased micellar stability with increased NaTDC concentration. To the contrary, it is unlikely that the trend in Q can be explained by increased solubilizer effectiveness. Previous researchers (1) have found that oleic acid solubilization is first-order with respect to NaTDC concentration. This does not imply an enhanced effectiveness at higher surfactant concentration. Neither can the observed behavior be explained by a saturation phenomenon as Fig. 9 might suggest. Comparison of absolute values for C for progressively larger surfactant concentrations indicate that average solute concentrations do indeed increase (9), although somewhat less than that predicted by Journal of ColloM and Interface Science, Vol. 98, No. 2, April 1984
418
RATERMAN, ADAMS, AND SHAEIWITZ
a first-order process. This observation, however, agrees with the trend predicted by Eq. [22] which defines the asymptotic limit of an average concentration vs time curve. Finally, the effect on Q of raising the surfactant solution pH from 2.2 to 7.7 can be found in Table II. As the solution pH is raised at constant ionic strength, the value of Q decreases. This can be explained quite unambiguously. As was previously reported, at high pH, appreciable amounts ofoleic acid are ionized to oleate ion. In the presence of counteflon, oleate ion is free to aggregate and form stable micelles; thereby, negating the need for large additional amounts of additional surfactants to stabilize a given quantity of oleic acid. The overall effect is the reduction of the amount of NaTDC required for mixed aggregates. This fact is reflected in the observed reduction of Q as the solution pH is increased. CONCLUSION
overall equilibrium constant which describes the interfacial interaction of surfactant and solute. For fatty adds, this constant is not unexpectedly a strong function of the physical status of the solute, since these acids can be ionized to surfactants themselves. The implications of surfactant diffusion control are far reaching regarding the process of fat digestion. This study and previous studies (2) indicate that the digestion process can be affected by a myriad of factors; the largest apparently being the degree of agitation within the digestive tract. At physiologic conditions (i.e., 0.9 wt% bile salt, pH ~ 6.0, and ionic strength 0.15 M) (20) the mode of solubilization can proceed in one of two ways. Under essentially quiescent conditions, uptake is controlled by the relatively slow process of diffusion. Under sufficiently agitated conditions, however, fatty acids, in conjunction with bile salts and simple electrolyte, can form interfacial microemulsions or gels which can be sheared and dispersed. Rather than diffusion, the controlling mechanism of solubilization now depends upon the physical properties of the gel and the external rate of agitation. Obviously, those factors, which promote the diffusion of greater quantities of fatty acid under quiescent conditions or which promote easily shearable gels under agitated conditions, will enhance the overall rate of fat digestion.
Thus, an experimental and theoretical methodology has been presented which is sufficiently sensitive to distinguish between various mass transfer boundary conditions for interfacial transport in liquid/liquid systems. Specifically, the methodology was applied to a model fatty acid digestion system whereby the kinetics of fatty acid incorporation into a micellar phase could be elucidated. Findings suggest that the overall rate of fatty acid solubilization is surfactant diffusion controlled. Moreover, the actual incorporation of solute A into the micellar phase can be treated as a C rapid step, essentially at equilibrium. The Ca proposed model and ensuing data also suggest CM that above the CMC the incorporation process is primarily surfactant-mediated and occurs Co at or very near the phase boundary. This is C~ in direct contrast to models which propose C~q the free-transport of fatty acids across the in- D terface whereupon they can be subsequently j incorporated into the micellar phase. In ad- k', k" dition, the effects of altered solution pH, NaC1 kl, k2 content, and surfactant concentration can be kB, kM explained by relatively small shifts in the Ka Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
APPENDIX: NOMENCLATURE
interfacial area concentration micelle concentration mixed micelle concentration average concentration of solute initial concentration of solute bulk concentration of solute equilibrium concentration of solute diffusivity flux reaction rate constants parameter fit constants mass transfer coefficients adsorption equilibrium constant
LIQUID SOLUBILIZATION DYNAMICS, Ill
KR Kd m n
Q t
V
Vi
reaction equilibrium constant desorption equilibrium constant partition coefficient stoichiometric factor overall equilibrium constant time total volume interfacial volume parameter fit constant REFERENCES
1. Stowe, L. R., and Shaeiwitz, J. A., J. CoIIoidInterface Sci. 90, 495 (1982). 2. Raterman, K. T., and Shaeiwitz, J. A., J. Colloid Interface Sci. 98, 394 (1984). 3. Shaeiwitz, J. A., Chan, A. F-C., Cussler, E. L., and Evans, D. F., J. Colloid Interface Sci. 84, 47 (1981). 4. Huang, C., Evans, D. F., and Cussler, E. L., J. Colloid Interface Sci. 82, 49 (1981). 5. Hofmann, A. F., "Fat Digestion: The Interaction of Lipid Digestion with Micellar Bile Acid Solutions" (K. Rommel and H. Goebell, Eds.), Chap. 1. Univ. Book Press, Baltimore, 1976. 6. Lewis, J. B., Chem. Eng. Sci. 3, 248 (1954). 7. Lewis, J. B., Chem. Eng. Sci. 3, 260 (1954).
419
8. Ward, W. J., and Quinn, J. A., AIChE J. 10, 155 (1964). 9. Raterman, K. T., "Mechanisms of Solubilization and Mass Transfer in Oleic Acid/Aqueous Surfactant Systems," Ph.D. dissertation, Univ. of Illinois, Urbana, 1983. 10. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., "Transport Phenomena." Wiley, New York, 1960. 11. Crank, J., "The Mathematics of Diffusion," 2nd ed. Oxford Univ. Press, Oxford, 1975. 12. Shaeiwitz, J. A., and Raterman, K. T., IEC Fundam. 21, 154 (1982). 13. Huq, A. K. M. S., and Lodhi, S. A. K., J. Phys. Chem. 70, 1354 (1966). 14. "Handbook of Biochemistry and Molecular Biology" (G. D. Fusman, Ed.), Vol. I, p. 19. Chem. Rubber Pub. Co., Cleveland, Ohio. 15. Shankland, W., J. Colloid Interface Sci. 34, 9 (1970). 16. "Microemulsions: Theory and Practice" (L. M. Prince, Ed.). Academic Press, New York, 1977. 17. Mazer, N. A., Carey, M. C., Kwasnick, R. F., and Benedek, G. B., Biochemistry 18, 3064 (1979). 18. Oh, S. Y., McDonnell, M. E., Holzbach, R. T., and Jamieson, A. M., Biochim. Biophys. Acta 488, 25 (1977). 19. Chang, Y., and Cardinal, J. R., J. Pharm. Sci. 67, 994 (1978). 20. Ganong, W. F., "Review of Medical Physiology," 7th ed., Chap. 25. Lange Med. Pub. Los Altos, Calif., 1975.
Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
Recently, much effort has been devoted to the elucidation of detergency mechanisms (14). Particular attention has been paid to model fat digestion systems in which the Solubilization of water-insoluble dietary components is presumed to be micelle-mediated. Since time scales associated with mieelle-solute equilibrium often exceed contact times associated with digestion, it has been deemed necessary to characterize this dynamic solubilization process (1). In addition, questions have been raised as to whether fatty components "leave the oil/water interface in molecular form or whether collision of the micelle with the interface is necessary" (5). Such considerations are of interest not only to physiologists but also to formulators ~of pharmaceuticals, detergents, and soil-suspending agents. To date, the majorityof studies (1-4) have I Present address: Amoco Production Research Company, Tulsa, OK 74102. 2 To.whom correspondence should be addressed.
been limited to steady state experiments in which the rate of detergency has been measured as a function of fluid flow in the micellar solution. Since mass transfer characteristics of the various experimental contacting devices were well defined, these studies provided needed insight into the diffusional dependences for these complex systems. In most instances, it was determined that solubilization rates deviated from those predicted by a diffusioncontrolled process alone. For the dissolution of fatty acids and cholesterol by sodium dodecyl sulfate (SDS) and bile salts, Shaeiwitz et aL (3) noted large resistances to transfer which were overcome when Lawrence's penetration temperature was surpassed. Presumably, this resistance is associated with the ability of the aqueous solution to penetrate the solid and, therefore, is reduced once a temperature is reached at which the solid lattice is appreciably disrupted or melted. For the dissolution of liquid linoleie acid by sodium taurodeoxycholate (NaTDC), a bile salt, Huang et aL (4) proposed a mechanism which
406 0021-9797/84 $3,00 Copyright© 1984by AcademicPress,Inc. All rightsof reproductionin any-formreserved.
Journalof Colloidand InterfaceScience.Vol.98, No. 2, April t984
LIQUID SOLUBILIZATIONDYNAMICS, III involved the diffusion, adsorption, and surface reaction of surfactant as rate-limiting steps. The solubilization of liquid oleic acid by NaTDC was shown to be related to the formation ofmicroemulsions; solubilization rates being determined by the physical properties of an interfacial liquid crystalline phase and flow conditions (2). In this paper we propose an alternate method by which the mechanisms of detergency may be determined. The method itself, involves monitoring solubilization rates under unsteady transfer conditions. Since solubilization rates are not necessarily constant during unsteady regimes, the determination of solute fluxes with respect to time can provide valuable information about interfacial boundary conditions for mass transfer. In principle, the unsteady equations governing mass transfer can be solved for boundary conditions appropriate to given physical situations and compared to experimental results. Thus, some inference can be made concerning the kinetics and mode of transport associated with the phase boundary. Steady state techniques provide no such direct insight into the interfacial mechanism because only deviations from diffusion control can be detected. Determination of interfaeial mechanisms may be of primary importance in systems for which transient events dictate steady state behavior. Moreover, the method is deemed significantly better suited for detergency studies as opposed to previous methods which only characterize interfacial resistances to mass transfer. Prominent among these were studies which utilized Lewis cells (6, 7) or laminar jets (8). The former apparatus measures the overall resistance to mass transfer under turbulent flow conditions. If bulk liquid transfer resistances are known, interfacial resistances can be back-calculated. Unfortunately, under turbulent flow conditions the physical status of the interface is unknown and therefore the complex rheology of the interface should be considered for proper data analysis. In laminar jet experiments, interfacial transport is characterized over short contact times, usually less
407
than a second. Such brief contact times, however, are far too short for the determination of transfer behavior which may be influenced by surface aging. Transient surface aging is likely for detergent systems. Finally, both of the cited methods measure only the magnitudes of interfacial resistances; therefore, no inferences can be made with respect to underlying transport mechanisms. The proposed method can overcome these deficiencies. Validation of the method is provided concerning the model transport systems, benzaldehyde/water and benzene saturated with benzoic acid/water. The technique is then applied to the oleic acid/aqueous sodium taurodeoxycholate system for the purpose of determining the interfacial kinetics of mass transfer for this model fatty acid digestion system. The effects of pH, added NaC1, and surfactant concentration on the solubilization mechanism are also elucidated for this system. EXPERIMENTAL Benzaldehyde (EK Industries), benzene (Fisher Scientific), benzoic acid (Mallinckrodt), and chloride-free sodium taurodeoxycholate (Calbiochem) were all of purity 98% or greater and were used as received. Stocks of standardized radiolabeled oleic acid were prepared by dilution of [3H]oleic acid (New England Nuclear) in pharmaceutical grade oleic acid (Mallinckrodt). A saturated solution of radiolabeled benzoic acid in benzene was prepared by dissolving [14C]benzoic acid (New England Nuclear) in benzene and adding unlabeled benzoic acid until precipitation occurred. All aqueous surfactant solutions were prepared by weight using doubly distilled deionized water. The apparatus consists of a constant-bore glass tube, 1.5 cm in diameter and 22 cm in height, which has been water-jacketed for temperature control. Fluid is removed through a stopcock and needle-valve assembly (see Fig. 1). An experiment is conducted in the following manner. Approximately 20 cm 3 of aqueous solution is allowed to come to thermal equiJournal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
408
RATERMAN, ADAMS, AND SHAEIWITZ
WATER[~ JACKET CONNECTION ORGANI PHASE--'C,.\ \ \ \
-*-AQUEOUS PHASE
\ \
~]=[] NEEDLE VALVE STOPCOCK FIG. 1. Experimentalapparatus for unsteadyquiescent solubilization studies.
librium in the apparatus. This quantity was deemed sufficient over the time periods considered to satisfy the semi-infinite approximation adopted by the theoretical analysis. Three to five cubic centimeters of the less dense phase is then gently layered upon the aqueous phase by means of a syringe. Transfer is allowed to occur for a predetermined time. Upon completion of the designated time period, the aqueous phase is quickly removed, mixed, and analyzed. Removal time is always a few seconds, much less than the duration of an experiment. For radioactively labeled materials, a Beckman LS 7000 Liquid Scintillation counter is employed. Concentrations are obtained by comparison to externally prepared calibration curves (9). Unlabeled materials are detected in the aqueous phase by means of a Waters Associates Model R-403 differential refractometer calibrated with external standards. Volume average concentrations are reported for the aqueous phase. An experiment is repeated no less than four times for a given time period. An entire data set consists of seven points; the first six corresponding to 5-min intervals, the last corresponding to a 45-min elapsed time period. The data are plotted as the change in average solute concentration (AC) versus the change in time (At), where AC is defined as the difference in concentration between the concenJournal of CoUoid and Interface Science, Vol. 98, No. 2, April 1984
tration of interest and that of the initial data point at 5 min. The At is similarly defined. This procedure is deemed sufficient to remove any error incurred through the initial layering process and uncertainties in the exact location of the interface when sampling. Best fits between experimental data and theoretical models are determined by linear regression analysis where feasible. Otherwise, best fits are determined by inspection. In all cases, this analysis scheme was found satisfactory. THEORETICAL The corresponding theoretical analysis for the preceding experimental method encompasses the following. In general, the one-dimensional unsteady diffusion equation for semiinfinite geometry is solved for several interfacial (x = 0) boundary conditions provided the initial solute concentration and the solute concentration far from the interface (x ---, ~ ) are zero in the acceptor phase. The total interfacial flux of solute is then evaluated, integrated with respect to time, and divided by the total volume of the acceptor phase to obtain the average solute concentration associated with a specified time period for transfer. In mathematical terms,
C(t) = - F
7 x x=o
where DA(OC/Ox)x=Ois the total instantaneous interfacial flow of solute. A plot of AC(t) versus At can then be generated where AC and At have been previously defined. Specifically, several physical situations can be associated with the interfacial transport process which necessitate solutions to various forms and combinations of the unsteady diffusion equation. Since it is the intent of this paper to elucidate the mechanisms of detergency, only three general classifications, pictured in Fig. 2, will be considered. These classifications are based upon whether the transport process is presumed surfactant mediated; and, if so, whether the process is rate limited
LIQUID SOLUBILIZATION DYNAMICS, III OLEIC ACID
x=0
WATER
FIG. 2. Hypotheticalmodesofoleic acid solubilizafion. by mechanistic steps associated with the incorporation of solute into the micellar phase. The first classification (Fig. 2A) assumes unfacilitated solute transfer across the phase boundary. The goveming transport equation is OC
= D -
Ot
with boundary conditions, t=O
C=O
O~
x---, oo
C = 0
Vt,
oo,
[3]
[4]
Vt.
OC _ D 02C at ~ - k"C,
[51
[6]
with interfacial boundary condition, X = 0
[2]
Ox 2 '
C=?,j=?
Solutions to Eq. [1] are found by first solving Eq. [2]. These are tabulated in Table IA for the various physical situations which pertain to boundary condition [5]. Diffusion in the donor phase is also considered. The second classification (Fig. 2B) assumes solute crosses the phase boundary in its free form, but subsequently reacts with a component of the accepting phase. In detergent processes, this situation corresponds to unmediated transfer of solute into the aqueous surfactant solution whereupon it is incorporated into a micelle not connected with the phase boundary. The homogeneous incorporation process is modeled as a first-order irreversible reaction since it is presumed that surfactant is present in excess. The governing solute transport equation is
02C
--
409
C =
Ceq
Vt.
[7]
Solutions to Eq. [1] based upon Eqs. [3], [4], [6], and [7] are given in Table IB for several values of k", the reaction rate constant. The final classification (Fig. 2C) pertains to detergency processes exclusively. Namely,
TABLE IA C(t): Various Non-Reaction BoundaryConditions Boundary condition [5]
Solution to Eq. [1]
Comment
C = Co
2CoA ( Dt) I/z C(t) = - ' 7 k'-~l
[IA-1]
Constant concentration donor phase pure
j =k
0 (t) = --~ kA t
[IA-2]
Constant flux
[IA-3]
Equilibrium at interface; donor phase (I) initiallyat concentration C1~ (10)
Ci = mCll
Jl
=
jn
=
C(t) k'Clo
2mC1~4 (_~)1/2 V[m + (Dj/DIt) 1/2]
~(t) = k,Cl~a f t
V
do
[k,2t\
[
expt---~) erfc t -
k,~),/z
dt [IA-4]
Donorphase(I) initiallyat Cl=; irreversiblereaction at interface (11)
Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
410
RATERMAN, ADAMS, AND SHAEIWlTZ TABLE IB C'(t): IrreversibleHomogeneousReaction
Value k"
Solution to Eq. [1]
Comment
Moderate to small
C(t)
CoA(Dt)m [ ~ + V (k"t)l/2 + 2 (k t) v2 erf(k"t)~12 1 exp(-k,,t)J7 [IB-1] Homogenousreaction
Large
C(t)
C°A(k"D)l/2 V t
[IB-21
solubilization is assumed to be a three-step surfactant mediated process at or near the oil/ water interface. Specifically, the three steps are transport of free surfactant to the interface; heterogenous interaction of solute and surfactant m o n o m e r to form mixed aggregate; and, subsequent transport of the mixed aggregate away from the phase boundary. The transport equations are
02CB
or B
Ot
-
DB
[8]
~
Ox 2 '
and OCM
02CM
[91
O-"-t- = DM OX 2 '
with boundary conditions, t=0
CR=CBoo, CM=O
0
[10]
x ~
oo
x = o
Fast reaction or large concentration of reactant in acceptor phase (10)
CB = C s ~ , CM = O
v t,
[11]
--JB = njM = interracial reaction.
[12] where Ca is the free surfactant micelle concentration and CM is the mixed micelle concentration. Equations [8] and [9] are coupled through boundary condition [12] when the interfacial reaction is reversible and are not when the interracial reaction is irreversible. The flux of solute is equivalent to the flux of mixed micelles times a stoichiometric coefficient corresponding to the average number of solute molecules incorporated per mixed micelle. Pertinent solutions to Eq. [1 ] for the above, assuming DB = DM, are found in Table IC. All interfacial reactions are considered firstorder or pseudo-first-order since previous findings (1) indicate the solubilization of oleic acid by NaTDC to be consistent with first-
TABLE IC C'(t): ReversibleReaction Boundary Condition Boundary condition [14]
--ja = Jm = k ' C s o - k " C M o
Solution to Eq. [1]
C(t)
=
Comment
k'CB~t ~ exp (k' V [(k + k ) [ +k")2D] / t
- J B = JM =
k'CBo
=~
Jo e x p ~ - ) effc k' -~
Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
Reversible interfacial reaction (9)
\1/27
dt
[IC-2]
Irreversible interfacial reaction
411
LIQUID SOLUBILIZATION DYNAMICS, III i
order kinetics over the pertinent range of surfactant concentrations. RESULTS
The preceeding methodology was tested for reliability with respect to the model mass transfer systems of benzaldehyde/water and benzene saturated with benzoic acid/water. In the former system, benzaldehyde, as a neat fluid, was contacted with water at 25 °C. Since interfacial resistance to mass transfer can be deemed negligible (12), equilibrium can be assumed with benzaldehyde present at its saturation concentration near the aqueous phase boundary. The expected average benzaldehyde concentration versus time dependence in the aqueous phase would correspond to solution one (Table IA) in which C is a function of t 1/2. A plot of AC versus A(t 1/2) appears in Fig. 3 for benzaldehyde/H20 at 25°C. The correlation coefficient, r, for the linear fit is 0.99. In the latter system, benzene saturated with benzoic acid was contacted with water at 25 ° C. In benzene, benzoic acid exists primarily as dimer; whereas in water, benzoic acid exists almost exclusively as m o n o m e r (13). Thus, as benzoic acid crosses the benzene/water interface, the d i m e r irreversibly dissociates to monomer. The dissociation process is modeled in the aqueous phase as a first-order homogenous or heterogenous reaction dependent upon whether or not the reaction is associated with the phase boundary. Equations [IB- 1] and [IB-2] of Table IB refer to the homogenous
-2 8 %
%
F
o
L ....
--2~.~0 IO o
i
i
~
4_
o
5
I0
AI
I
(min)
15
20
I
25
FIG. 4. Average concentration of benzoic acid in water versus t at 25°C.
reaction case. The fourth entry in Table IA corresponds to the heterogenous reaction case. Figure 4 is a plot of 2xC versus At for the transfer of benzoic aid between benzene and water. The correlation coefficient for the linear fit is 0.99. The method was also applied to systems of oleic acid contacted with water at 37°C. The effects of added NaC1, altered pH, added sodium taurodeoxycholate, and NaC1 in conjunction with NaTDC in the aqueous phase were determined. Figure 5 indicates the effect of added NaC1 (0.0, 0.15, and 0.5 M) on the unsteady flux of oleic acid into water. Generally, with increasing salt concentrations fluxes were diminished. The average flux was approximately 1.0 × 10 . 7 g/cm 2 sec over the time period of
i
IE
....
i
t
o oleic/H20 r=0,995 oleic/0,15 M r =0,989 u oleic/0,5 M r=0,982
, . . ,
×
'<3
0
I0 20 A(I I/2) (secI/2}
FIG. 3. Average concentration ofbenzaldehyde in water versus t m at 25°C.
I0
20 30 A(t w2) Bed/z)
40
FIG. 5. Average concentration of oleic acid in water versus #/2; effect of increasing NaC1. Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
412
R A T E R M A N , ADAMS, A N D SHAEIWITZ
interest. In each instance the average concent --a--- 0 M NaCI, 2% NaTDC r i ---~--- 0,075 M NoCI, 2% NoTDC ~_ tration dependence with time was best fit by 15 -.-o-- 0,15 M NoCl, 2% NoTDC //..I//~'3"~ a ?/2 functionality. For plots of AC versus ~. ~ 0,5 M NaCI, 2% NaTDC i" "a~/ A(t w2) regression coefficients were equal to or greater than 0.98. Figure 6 shows the effect of altered pH (2.2 .~/J or 7.7) on the unsteady flux of oleic acid into %o solutions of phosphate-buffered saline (PBS), 5 //,/~,/'f'~/ ionic strength 0.15 M (14). At a p H 2.2 the average c o n c e n t r a t i o n d e p e n d e n c e in the I{/" I [ I I I I _L_I aqueous phase was best fit by a t u2 function300 600 900 1200 1500 1800 2100 2400 A t (sec) ality (r = 0.99). However, at pH 7.7 the data were highly nonlinear on a AC versus A(t 1/2) FIG. 7. Average c o n c e n t r a t i o n o f o]eic acid in 2 w t % plot. A Marangoni-type instability and the N a T D C versus t; effect of increasing NaC1. formation of an opaque interracial film were noted; the instability dying out within 5 to 10 min after initiation of an experiment. After phosphate-buffered 2% N a T D C solution was 15 min the flux of oleic acid into the pH 7.7 reduced from 7.7 to 2.2, unsteady flux values PBS solution was essentially zero. Average decreased. Marangoni-type instabilities and fluxes into either PBS solution were no greater interfacial films, however, were not noted for than 1.0 X 10 -7 g/cm 2 sec. Average c o n c e n the pH 7.7 solution. Generally, average fluxes trations of oleic acid in the p H 7.7 solutions, over the time course of an experiment for however, were significantly higher than those aqueous solutions of 2% N a T D C were an orfor the pH 2.2 solutions over the time period der of magnitude larger than for solutions of interest. without added surfactant. In all instances the Figures 7 and 8, respectively, indicate the best fit for A~ as function of time was obtained effects of added NaC1 and altered pH on the when the interfacial flux of solute was given unsteady flux ofoleic acid into 2 wt% N a T D C as solutions at 37°C. As previously, with inJlx=0 = k l + k2 e-aat, [13] creased NaC1 concentration (0.0 to 0.5 M ) the unsteady flux of oleic acid into the aqueous where kl, k2, and/3 are constants. Values for phase decreased. Likewise, as the pH of the these parameters are provided in Table II.
i
~ 15
i
u f
/ o
I0
i i
---.a-- oleic/PBS pH 7,4 I =0.15 ---~--- oleic/O,15M NaCI r=0.989 ---¢--- oleic/PBS pH 2,2 r=0,987 I=0,15
/
/
o 2%NaTDC, PBS (pH=7,6 I=O,15M) A 2% NaTDC, PBS (pH=2.2 I =OJ5MI
30
- ~ - - - -...-o-- - -
/
/~'/
20
×
x <
I0
20 30 A (1I/2) ($e¢I/2)
40
FIG. 6. Average concentration of oleic acid in water versus tin; effect of altered p H (I = 0.15 M, PBS). Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
I0
I
500
I
600
I
900
I
1200
I
1500
I
1800
I
2100
I
2400
At (sec)
FIG. 8. Average concentration of oleic acid in 2 wt% N a T D C versus t; effect o f altered p H (I = 0.15 M, PBS).
413
LIQUID SOLUBILIZATION DYNAMICS, III TABLE II Parameter Fits for Lumped Parameter Model NaTDC (wt%)
[NaC1] (M)
klA/ V (g/cm3 sec)
k2A/ V (g/cm3 see)
1~ (sec-l)
Q
0.5 0.75 1.0 2.0 4.0
0.0 0.0 0.0 0.0 0.0
2.22 x 10 -9 3.44 X 10 -9 1.42 X 10-8 1.42 × 10-8 1.42 X 10-8
3.10 2.72 2.46 2.46 2.46
X 10-8 X 10 -7 X 10 -7 X 10 -7 X 10 -7
1.20 X 10 -3 2.00 X 10 -3 1.25 × 10-a 1.25 X 10-3 1.25 X 10 -3
5.18 26.35 18.17 9.51 5.14
2.0 2.0 2.0
0.075 0.15 0.50
1.20 X 10-8 1.41 X 10-s 2.00 X 10-8
2.35 X 10 -7 1.46 X 10 -7 9.09 X 10-s
1.50 × 10 -3 1.30 X 10-3 1.25 X 10-3
9.13 5.97 3.98
2.0 2.0
pH 7.7 pH 2.2
3.33 × 10-8 3.20 X 10 8
3.07 X 10-7 2.18 X 10 -7
1.30 × 10-3 1.25 X 10-3
11.67 8.53
Finally, the effect o f i n c r e a s e d N a T D C cont e n t in the a q u e o u s phase f r o m 0.5 to 4.0% is s h o w n in Fig. 9. W i t h i n c r e a s e d s u r f a c t a n t c o n c e n t r a t i o n u n s t e a d y fluxes increase; but, however, reach a l i m i t at a p p r o x i m a t e l y 1% a d d e d N a T D C . As before, best fits for A ~ as a f u n c t i o n o f At are o b t a i n e d w h e n t h e interfacial flux is given b y Eq. [13]. Values for p e r t i n e n t p a r a m e t e r s are given in T a b l e II. DISCUSSION As was earlier indicated, the initial experi m e n t s with the systems b e n z a l d e h y d e / w a t e r a n d benzene saturated with benzoic a c i d / w a t e r were designed to d e t e r m i n e t h e validity o f the e x p e r i m e n t a l a n d theoretical m e t h o d o l o g y . In the f o r m e r case, in w h i c h a p u r e d o n o r phase was c o n t a c t e d with a n a c c e p t o r phase initially donor-free, the average c o n c e n t r a t i o n s h o u l d v a r y with t h e square r o o t o f time. F i g u r e 3 shows t h a t the average c o n c e n t r a t i o n o f benza l d e h y d e in w a t e r does i n d e e d v a r y with the square r o o t o f time. I n the latter system, in which solute d i m e r s dissociate irreversibly u p o n e n t e r i n g the a c c e p t o r phase, the average c o n c e n t r a t i o n d e p e n d e n c e is d i c t a t e d b y t h e physical l o c a t i o n o f the first-order d e c o m p o s i t i o n reaction. F i g u r e 4 indicates t h a t t h e d a t a are best fit b y a l i n e a r - t i m e d e p e n d e n c e which c o r r e s p o n d s to a h o m o g e n o u s d e c o m -
p o s i t i o n r e a c t i o n with a large r e a c t i o n - r a t e c o n s t a n t (Eq. [IB-2]). Hence, the m e t h o d is sufficiently sensitive to elucidate the n a t u r e o f the dissociative r e a c t i o n a n d also its a p p a r e n t m a g n i t u d e . W e , therefore, feel c o n f i d e n t to a p p l y this m e t h o d to m a s s t r a n s p o r t systems in which interfacial transfer m e c h a n i s m s have been p r e v i o u s l y u n c h a r a c t e r i z e d . T h e m e t h o d was initially applied to systems o f oleic acid transferring to a q u e o u s solutions o f varying NaC1 c o n t e n t a n d p H (Figs. 5 a n d 6) respectively. Best fits were o b t a i n e d f o r / x ~ expressed as a f u n c t i o n o f the square r o o t o f time. T h e o n l y n o t a b l e e x c e p t i o n was the case for p H 7.7, ionic strength 0.15 M. A t 1/2 de-
o 15
'~
b,5~7o Bile ( n o s a i l ) '
v 1,00 % ~] 2 . 0 0 %
4>
. . . . ....
~ /
/
I
I
D
5
O~
~
I 600
1200
1800
I 2400
AI (sec)
FIG. 9. Average concentration of oleic acid in NaTDC solution versus t; effect of increasing NaTDC concentration. Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
414
RATERMAN,
ADAMS, AND SHAEIWITZ
pendence is consistent with a constant concentration boundary condition for which the donor phase is pure and at or near equilibrium with the acceptor phase at the phase boundary (Eq. [IA-1]). The evident effect of increased NaC1 concentration in the aqueous phase was to reduce the average flux of oleic acid into that phase. According to Eq. [IA-1], this trend would be noted if either the equilibrium concentration of solute, Co, or the effective diffusivity of solute in the acceptor phase, D, diminished with increased NaC1 content. Shankland (15), however, reports that NaC1 has very little effect on the equilibrium solubility of oleic acid in water. To the contrary, effective diffusivity studies ofoleic acid in water suggest that solute diffusivities do indeed diminish with increased NaC1 levels (9). Presumably, this phenomenon is due in part to the formation of sodium oleate aggregates whose diffusivities are considerably less than oleic acid monomer. The CMC for sodium oleate is approximately 0.001 M or 3.0 X 10 -4 g/ml (15). The equilibrium solubility ofoleic acid in water is about 2.0 X 10-3 g/ml (9). Thus, the formation of sodium oleate is not unlikely provided sufficient amounts of counterion are present. Such is the case as the NaCI content is raised in the aqueous phase. The constant concentration boundary condition model is also consistent with the altered pH data. Here, however, equilibrium solubilities rather than effective diffusivities are most affected. The solubilities of oleic acid at high and low pH are 4.0 X 10-3 and 1.5 X 10-3 g/ml, respectively (9). Since the pKa of oleic acid is about 4.7 (15), the increased solubility and thereby increased initial flux of oleic acid in water agrees well with the ability of oleic acid to become ionized to oleate ion at high pH's. In fact, at pH 7.7 the formation of sodium oleate, a surface active agent (16), is adequate to induce Marangoni-type instabilities at the phase boundary; the instabilities arising from localized gradients of surfactant in the plane of the interface. The concommitant agitation of the interfacial region is Journal of Colloid and Interface Science. Vol. 98, No. 2, April 1984
sufficiently violent to significantly increase the initial flux of oleic acid. In time, however, adequate amounts of oleate are generated to form an interfacial film which dampens the instabilities and thereby the flux of oleic acid into the aqueous media. At a pH of 2.2 the amount of oleate present is presumably insufficient for the development of Marangoni instabilities. As expected, low pH data obey the proposed constant concentration boundary condition model for the transport process. Noticeably, the average flux at pH 2.2 is less than that for the analogous 0.15 Madded NaC1 solution whose pH is approximately 5.5. This observation is consistent with the evidently synergistic effects of decreased solubilities at low pH and decreased diffusivities arising from aggregate formation. Hence, for systems of oleic acid/water the interfacial transfer process can be described by a constant concentration boundary condition model. Moreover, the effects of NaC1 and pH on oleic acid transfer can be ascribed to the variations of phenomenological coefficients appearing in the model. These variations are apparently linked to the formation of sodium oleate, a surfactant. The behavior of oleic acid/water systems with added NaTDC is significantly different from the above as indicated by Figs. 7-9. Of initial note is the fact that average fluxes for NaTDC solutions are generally an order of magnitude larger than average fluxes for solutions without added surfactant. This observation implies that the solubilization of oleic acid is accomplished primarily through a surfactant-mediated mechanism. Figure 10 depicts the AC versus time dependence for oleic acid/2% NaTDC, no salt at 37°C and for several surfactant-mediated models. These models, expressions for which are found in Tables IB and IC, primarily postulate the micellar incorporation of solute as the overall ratecontrolling step ofsolubilization. Clearly, that model, which expresses the interfacial flux of oleic acid as Eq. [13], provides the best data fit. Equation [ 13], however, was derived as a
415
LIQUID SOLUBILIZATION DYNAMICS, III ,
,5
2% BILE (NO SALT)
j , / f
,
(iii) Incorporation of surfactant into a mixed aggregate in conjunction with oleic acid
/~//
•/(-" //
KR %
Io <]
CBi ~ CMi,
.
5
~'//2"o EXPERIMENTAL DATA f / ~ R S I B L E MOMOGENEOUS RXN
/ ~ ~ B L E O~
HETEROGENEgUSR×N
- - L U M P E D PARAMETER MODEL I I I I I I I I 500 600 900 1200 1500 1800 2100 2400
At (see)
FIG. 10. Predicted average concentration of solute in
acceptorphaseversustime for severalsurfactantmediated solubilizationmodels. result of the data and not from a previously proposed mechanism. Nevertheless, the actual incorporation of oleic acid into the surfactant phase, whether it be a heterogenous or homogenous process, apparently does not solely control the overall rate of solubilization. Dynamic solubilization studies for these solutions indicate that at sufficiently low surfactant concentrations (~<2 wt%) the solubilization process seems to be diffusion-controlled; the flux versus Reynolds-number dependence being approximately that predicted for the diffusion limit of mixed micelles (1). At higher NaTDC concentrations additional studies (2) show that the solubilization rate is controlled by the formation of a liquid crys± talline interfacial phase. Here, too, the amount of the interfacial phase formed is apparently diffusion limited. As a consequence, it is reasonable to expect that any formulation which produces Eq. [13] must involve diffusion of micelles as the overall rate-controlling step of the solubilization mechanism. With this in mind, the overall incorporation mechanism can be reduced to a five-step process. The steps are (i) Surfactant diffusion to the phase boundary kB
Ca ~ CBS,
[14]
(ii) Adsorption ofsurfactant to the interface
[ 16]
Here, CM refers to mixed micelles on a solutefree basis, since the solubilization process is assumed to be exclusively surfactant mediated. An increased solute capacity per micelle is reflected as an increased surfactant efficiency (i.e., fewer surfactant monomers required to solubilize a given quantity of solute). (iv) Desorption of mixed aggregate Kd
CMi ~ CMS,
[17]
(v) Diffusion of mixed aggregate away from the interface kM
c ~ ~ c~.
[18]
The above rate constants, kM and kB, are mass transfer coefficients and refer only to physical diffusion. R e m a i n i n g constants describe physical equilibria. The above description, however, does not imply exact stoichiometry. Since surfactant adsorbs to interfaces primarily as monomer, the equilibria between micelle and monomer is presumed implicit within step ii. Oleic acid is also presumed to be present in excess near the phase boundary. The model has little significance at or below the CMC for the surfactant of interest. The variation of surfactant concentration with respect to time near the interface can be described by dCBs v~
dt
-
k~(CB
-
CBs)
-- k M A ( C M s --
CM), [19]
where Vi is the apparent interfacial volume. If all reaction steps are in rapid equilibrium as previous information indicates, then CMs is related to Cas by
Ka
CBs ~ C~i,
[15]
CMS =
QCBs,
[20]
Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
416
RATERMAN,
ADAMS, AND SHAEIWlTZ
where Q is the overall equilibrium constant for steps (ii) through (iv). Hence, Eq. [ 19] can be solved to give
CBs = K - ( K - CBso)e-eat.
[21]
Here, K=
kBCB + kMCM kB + QkM
'
A vi
/3 = ~ (kB + QkM),
[22]
[231
and CBSOis the initial concentration of surfactant micelles near the interface. Since the flux of oleic acid is assumed proportional to the flux of mixed aggregates, the appearance with time of solute in the surfactant solution is approximately
j ~ kMQ K
kBCB e_~At kB + QkM + CBsoe-~a~]. [24]
The above expression was obtained by setting the concentration of mixed aggregates in the bulk solution, CM, to zero. Finally, since Q is always greater than zero and CBso is essentially CB, the bulk surfactant concentration, Eq. [24] reduces to a form similar to Eq. [13]. Thus, the above model predicts the observed concentration versus time dependence for the solubilizafion of oleic acid by aqueous solutions of NaTDC. Moreover, the assumptions of surfactant diffusion control and rapid solute incorporation into the micellar phase are consistent with the aforementioned dynamic solubilization findings. The model suffers, however, from the utilization of averaged or lumped parameters in the description of essentially an unsteady process. Such parameters are more appropriately applied to steady state phenomena. But, perhaps, this result is not altogether unexpected. The detergency process itself probably cannot be described accurately when solute associated and nonassociated surfactant aggregates are considered as well-defined enJournalof Colloidand InterfaceScience,Vol. 98, No. 2, April 1984
tities and thereby treated accordingly in an unsteady theoretical analysis. This approach neglects the polydisperse nature of NaTDC solutions (17) and ignores the ability of oleic acid to form aggregates under the proper experimental conditions. In addition, the experimental technique, itself, is essentially an averaging technique. Hence, it is incapable of monitoring instantaneous solute concentration profiles rather than total solute concentrations. It is also incapable of identifying the actual physical state of solute molecules. Therefore, in light of the experimental technique and the apparent complexity of species interactions, it is not unreasonable to presume that a lumped or averaged parameter model best describes the detergency process. Despite limitations, the model is nevertheless helpful in explaining observed experimental trends. When Eq. [ 13] is integrated to obtain C as a function of t, two parameters emerge as being most influential in determining the shape of a given profile. These parameters are k2A//3 V, which determines the initial steepness of a profile; and/3, which determines the profile's duration of curvature. (Values for klA/V are arbitrary, since they are dictated by the method of analysis.) From Eqs. [24] and [23]:
k2A
Q2
Vi
/3V - CB (1 + Q)---------~V '
[25]
where [3 is defined by Eq. [23]. In addition, the unsteady mass transfer coefficients, kB and kM, can be approximated by the expression ~(DI1/2
k,,M
\~-~1
.
[26]
Here, tm is one-half the average time duration of an experiment and D, the micellar diffusivity, is assumed to be 1.0 X 10 -6 cmZ/sec (18). The kM and kB are calculated to be approximately 2.0 × 10-5 cm/sec. Assuming average values for/3 and kzA//3V from Table II, Eqs. [23] and [25] can be solved in conjunction to obtain values for Q and Vi]A. Q is of order 10 and Vi/A, the penetration depth, is nearly 1.3 mm. The calculated value for the total
LIQUID SOLUBILIZATION
equilibrium constant is consistent with the observed large initial fluxes of solute, since diffusion control is minimal for very short times. Values for Q are tabulated in Table II. The above value for the penetration depth is also consistent with experimentally observed diffusion boundary layers. Examination of Table II indicates that for solutions of 2% N a T D C with added NaC1, values for Q decrease with increasing NaC1 concentration. According to Eq. [20], Q can be defined as CMs/CBs, the concentration of mixed aggregates over the concentration of solute-flee aggregates near the phase boundary. If solute-free and mixed aggregates can be described by time-independent mean aggregation numbers, then this ratio is essentially the concentration of surfactant m o n o m e r associated with mixed micelles over the concentration of surfactant m o n o m e r associated with nonmixed micelle. Therefore, a decrease in the value of Q can be attributed to either a decrease in the amount of m o n o m e r required to solubilize a given quantity of oleic acid or an increase in the stability of solute-flee aggregates (i.e., a shift in monomer-micelle equilibrium which favors the micelle). Shankland reports that the addition of simple electrolyte enables a given quantity of bile salt (NaTDC) to solubilize a greater portion of oleic acid (15). Presumably, this stems from the ability of oleic acid to act as its own surfactant in the presence of a suitable counterion. Thereby, the requirement for additional surfactant to solubilize a specified amount of oleic acid is reduced. These observations would tend to suggest the former of the above explanations as the cause for the observed trend in Q with added NaC1. Conversely, it has also been observed from ion-specific electrode experiments that aggregation begins at lower concentrations for NaTDC with added NaC1 concentration (9). This implies a greater a propensity to form aggregates and therefore, perhaps a greater micelle stability. The concommitant result would be a decrease in the amount of monomer available to adsorb to the interface and form mixed aggregate. (Monomer activity is
D Y N A M I C S , III
417
constant above the CMC, and approximately equal to the CMC.) Hence, either explanation seems plausible for the observed experimental trends. Table II also shows the effect of added surfactant on values of Q without added NaC1. Unlike the previous data, values for Q initially increase with increased concentration, but reach a maximum at approximately 0.75 wt% and begin to decrease thereafter. The decreasing trend is viewed with significance whereas the increasing trend is not. This stems from the fact that at 0.5 wt% N a T D C micellar solubilization is deemed insignificant; since aggregation for NaTDC begins at approximately 0.35 wt% (9). In accordance, a fit of C for the 0.5 wt% data versus t 1/2 gives a correlation coefficient (r) or 0.99, as would be predicted by previous data for the non-micellar aided solubilization of oleic acid. As before, explanations for decreasing trends in Q must center about the previous suppositions, either increased micelle stability or increased solubilizer effectiveness. Unfortunately, to date very little quantitative information exists on the stability of N a T D C aggregates with increasing surfactant concentration. However, since N a T D C aggregates in a stepwise fashion (19) (i.e., additional monomer preferentially associates with existing micelles rather than forming new aggregates of the same size), it is possible to construe an increased micellar stability with increased NaTDC concentration. To the contrary, it is unlikely that the trend in Q can be explained by increased solubilizer effectiveness. Previous researchers (1) have found that oleic acid solubilization is first-order with respect to NaTDC concentration. This does not imply an enhanced effectiveness at higher surfactant concentration. Neither can the observed behavior be explained by a saturation phenomenon as Fig. 9 might suggest. Comparison of absolute values for C for progressively larger surfactant concentrations indicate that average solute concentrations do indeed increase (9), although somewhat less than that predicted by Journal of ColloM and Interface Science, Vol. 98, No. 2, April 1984
418
RATERMAN, ADAMS, AND SHAEIWITZ
a first-order process. This observation, however, agrees with the trend predicted by Eq. [22] which defines the asymptotic limit of an average concentration vs time curve. Finally, the effect on Q of raising the surfactant solution pH from 2.2 to 7.7 can be found in Table II. As the solution pH is raised at constant ionic strength, the value of Q decreases. This can be explained quite unambiguously. As was previously reported, at high pH, appreciable amounts ofoleic acid are ionized to oleate ion. In the presence of counteflon, oleate ion is free to aggregate and form stable micelles; thereby, negating the need for large additional amounts of additional surfactants to stabilize a given quantity of oleic acid. The overall effect is the reduction of the amount of NaTDC required for mixed aggregates. This fact is reflected in the observed reduction of Q as the solution pH is increased. CONCLUSION
overall equilibrium constant which describes the interfacial interaction of surfactant and solute. For fatty adds, this constant is not unexpectedly a strong function of the physical status of the solute, since these acids can be ionized to surfactants themselves. The implications of surfactant diffusion control are far reaching regarding the process of fat digestion. This study and previous studies (2) indicate that the digestion process can be affected by a myriad of factors; the largest apparently being the degree of agitation within the digestive tract. At physiologic conditions (i.e., 0.9 wt% bile salt, pH ~ 6.0, and ionic strength 0.15 M) (20) the mode of solubilization can proceed in one of two ways. Under essentially quiescent conditions, uptake is controlled by the relatively slow process of diffusion. Under sufficiently agitated conditions, however, fatty acids, in conjunction with bile salts and simple electrolyte, can form interfacial microemulsions or gels which can be sheared and dispersed. Rather than diffusion, the controlling mechanism of solubilization now depends upon the physical properties of the gel and the external rate of agitation. Obviously, those factors, which promote the diffusion of greater quantities of fatty acid under quiescent conditions or which promote easily shearable gels under agitated conditions, will enhance the overall rate of fat digestion.
Thus, an experimental and theoretical methodology has been presented which is sufficiently sensitive to distinguish between various mass transfer boundary conditions for interfacial transport in liquid/liquid systems. Specifically, the methodology was applied to a model fatty acid digestion system whereby the kinetics of fatty acid incorporation into a micellar phase could be elucidated. Findings suggest that the overall rate of fatty acid solubilization is surfactant diffusion controlled. Moreover, the actual incorporation of solute A into the micellar phase can be treated as a C rapid step, essentially at equilibrium. The Ca proposed model and ensuing data also suggest CM that above the CMC the incorporation process is primarily surfactant-mediated and occurs Co at or very near the phase boundary. This is C~ in direct contrast to models which propose C~q the free-transport of fatty acids across the in- D terface whereupon they can be subsequently j incorporated into the micellar phase. In ad- k', k" dition, the effects of altered solution pH, NaC1 kl, k2 content, and surfactant concentration can be kB, kM explained by relatively small shifts in the Ka Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984
APPENDIX: NOMENCLATURE
interfacial area concentration micelle concentration mixed micelle concentration average concentration of solute initial concentration of solute bulk concentration of solute equilibrium concentration of solute diffusivity flux reaction rate constants parameter fit constants mass transfer coefficients adsorption equilibrium constant
LIQUID SOLUBILIZATION DYNAMICS, Ill
KR Kd m n
Q t
V
Vi
reaction equilibrium constant desorption equilibrium constant partition coefficient stoichiometric factor overall equilibrium constant time total volume interfacial volume parameter fit constant REFERENCES
1. Stowe, L. R., and Shaeiwitz, J. A., J. CoIIoidInterface Sci. 90, 495 (1982). 2. Raterman, K. T., and Shaeiwitz, J. A., J. Colloid Interface Sci. 98, 394 (1984). 3. Shaeiwitz, J. A., Chan, A. F-C., Cussler, E. L., and Evans, D. F., J. Colloid Interface Sci. 84, 47 (1981). 4. Huang, C., Evans, D. F., and Cussler, E. L., J. Colloid Interface Sci. 82, 49 (1981). 5. Hofmann, A. F., "Fat Digestion: The Interaction of Lipid Digestion with Micellar Bile Acid Solutions" (K. Rommel and H. Goebell, Eds.), Chap. 1. Univ. Book Press, Baltimore, 1976. 6. Lewis, J. B., Chem. Eng. Sci. 3, 248 (1954). 7. Lewis, J. B., Chem. Eng. Sci. 3, 260 (1954).
419
8. Ward, W. J., and Quinn, J. A., AIChE J. 10, 155 (1964). 9. Raterman, K. T., "Mechanisms of Solubilization and Mass Transfer in Oleic Acid/Aqueous Surfactant Systems," Ph.D. dissertation, Univ. of Illinois, Urbana, 1983. 10. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., "Transport Phenomena." Wiley, New York, 1960. 11. Crank, J., "The Mathematics of Diffusion," 2nd ed. Oxford Univ. Press, Oxford, 1975. 12. Shaeiwitz, J. A., and Raterman, K. T., IEC Fundam. 21, 154 (1982). 13. Huq, A. K. M. S., and Lodhi, S. A. K., J. Phys. Chem. 70, 1354 (1966). 14. "Handbook of Biochemistry and Molecular Biology" (G. D. Fusman, Ed.), Vol. I, p. 19. Chem. Rubber Pub. Co., Cleveland, Ohio. 15. Shankland, W., J. Colloid Interface Sci. 34, 9 (1970). 16. "Microemulsions: Theory and Practice" (L. M. Prince, Ed.). Academic Press, New York, 1977. 17. Mazer, N. A., Carey, M. C., Kwasnick, R. F., and Benedek, G. B., Biochemistry 18, 3064 (1979). 18. Oh, S. Y., McDonnell, M. E., Holzbach, R. T., and Jamieson, A. M., Biochim. Biophys. Acta 488, 25 (1977). 19. Chang, Y., and Cardinal, J. R., J. Pharm. Sci. 67, 994 (1978). 20. Ganong, W. F., "Review of Medical Physiology," 7th ed., Chap. 25. Lange Med. Pub. Los Altos, Calif., 1975.
Journal of Colloid and Interface Science, Vol. 98, No. 2, April 1984