- Email: [email protected]
Mass transfer analysis in a microemulsion system
Ceff@ids and Surplus,
64 (1992) 197-205
197
Elsevier Science Publishers B.V.,Amsterdam
Mass transfer analysis in a mi~ro~mulsion system K.Anandakrishran”,
P. Neogi” and SE. Fribergb
“Department of Chemical Engineering, Universityof Missouri-Rolla. Rolla. MO 65401, USA bDepartmen t of Chemistry, Clar~son Uniuersity,Pots~am, NY 13676, USA (Received 12 July 1991;accepted 7 January 1992) Abstract Previous work on the effects of layering H water-in-oil (w/o) microemulsion on water has been d~ument~. The concentration profiles recorded there are analyzed to obtain the fluxes in the interior and the adsorption at the interfaces. Adsorption at the interface shows a lack of equilibrium locally and the fluxes suggest that local equilibrium may not exist in the interior. Keywords:
Mass transfer; microcmulsion system; phase equilibria.
Introduction
Contacting experiments are performed in oilwater-surfactant-cosurfactant systems by layering one phase on top of another and monitoring the resulting phase changes volumetrically. Such information provides fundamental understanding of the transport process, and finds applications in enhanced oil recovery using surfactants f_.j, as well as in separation systems [ZJ. Specifically, liquid crystals are very often formed transiently in such systems and this can affect strongly the efficiency of the process Cl]. The mass transfer is very slow in general, taking months to complete in a test tube [Z]. The existing analyses follow a standard diffusion path analysis [3,4], and show that the rates can vary from being adsorption controlled to being diffusion controlled in a manner which is not fully understood at present. Such effects are aliogether ignored in the simulations of surfactantaided oil recovery which assume local equilibrium
ca C~rrespf~n~e~ce to: P. Neogi,Dept. of Chemical Enginee~ng, University of Missouri-Rolla, Rolla, MO 65401, USA. 0166-6622/92/$05.00
0 1992 -
In our previous work we went beyond volumetric measurements to measure also the concentration profiles [1,6]. The intent was to evaluate quantitatively the two mechanisms of adsorption and diffusion. This also allows one to assess how good the condition of local equilibrium is, as well as to determine how the corrections to it are to be made when required. The main restriction we impose is that in the entire analysis no particular model for constitutive equations nor any assumption regarding controlling mechanisms is to be made. The present work is the analysis of our previous data [1,6] under such constraints. The reason behind such severity in our analysis is that simple diffusion-path analysis does not explain all of the results as discussed in the Appendix. Only a limited amount of data could be analyzed. The key feature in the observations was that when a w/o microemulsion was contacted with brine such that a dilute w/o microemulsion was to result at equilibrium, lameilar liquid crystals were formed transiently. What was expected to be straightforward imbibition of brine turned out to involve lamellar liquid crystals which did not occur in the chase diagram in the immediate vicinity of the compositions of the two phases contacted.
Elsevier Science Publishers B.V. All rights reserved.
I YS
Formulation
A brief description of the previous experiments is given first. Tubes were made of seven sections of 1 cm each. Brine was layered first, then the w/o mi~roemu~sion was layered over the brine. One of these was observed for phase vo!umes as they changed with time. The others were broken into seven parts, and one such tube was broken every three days. The concentrations were analyzed in each section. The question now arises as to what the treated data could yield. The rigorous form that preliminary models [3.4] would take would be that of multicomponent diffusion [7,8-J, i.e. the fluxes of multicomponent diffusion, containing multicomponent diffusivities as parameters, are substituted into the conservation equations, which are then solved and fitted to the concentration profiles measured experimentally. The vatues of multicomponent diffusivities are those which give the best fit. This approach could be generalized to include multicomponent adsorption at interfaces. The problem with the multicomponent diffusivities, I),j, is that they are applicable only to specific systems. Let 1, 2 and 3 denote water, pentanol and surfactant respectively, then Di3 will change if decanol is substituted for pentanol, or a pentanol-decano1 mixture is used, etc. Another method is to use the generalized Stefan-Maxwell equation [9] which uses binary diffusjvities, fiijs to describe fluxes. These are of course the basic building blocks but the problem with this approach is that in firs, I and 3 have a structure which varies with the amount of 2. Thus 6,, without reference to 2 is meaningless. Then there is the question of what to do with the electrostatic effects. Because of such uncertainties, we have decided to avoid any kind of fIux (or adsorption) expression. This gives rise to a generality which, however, can be used to treat only the two-phase systems discussed below. The tube is divided into seven sections of I cm each. The origin of the _Ycoordinate is placed at the top. The sixth section then lies between x = 5 cm and s = 6 cm. It is now assumed
K. Anandakrishrari
ct al./Cnllnids
Surfaces
64 (1992)
197-205
that the concentration obtained in the sixth section, which is an average concentration in that section, can be approximated as the concentration at the mid-point in the sixth section, x = 5.5 cm. Further, the total volume does not change with time, so the approximation that the density p is a constant is appropriate. The continuity equation [8] becomes
which on integration
becomes
PO, = c(r)
(2)
where v, is the mass average velocity. But at _Y= 0, for the phase at the top, v, = 0, and it is also zero for the phase at the bottom at x = 8 cm. Thus it is zero throughout, that is, c(rf = 0. However, PO,
=
i:
i= 1
(3)
p1i.r
where nix are the mass fluxes. Thus for constant one has
where rii = rri/p. The conservation aPi -=dt
are
& dx
or at constant
p
&
aivi
dt
equations
p,
-
-
2s
where 1~‘i= pi/p, the -:deight fraction. Equation (6) can be considered as a first-order differential equation for iti, provided dwi/dp is knowJn as a function of _Yat a particular time t. Indeed, it is known from the experimental data and can be approximated as the backward difference (7) where k denotes time and IZ the location in space. Now, instead of finite di~erencing the right-hand
K. Anaudakrishran ef al./Colloids Surfaces 64 (1992) 197-205
side in Eqn (61, fii, is expressed as a po~ynomiai. Let the phase at the top cover the top four layers (or N layers). Then on substitution of the polynomial into Eqn (61, one has four resulting equations at x=0.5, 1.5, 2.5, 3.5 (N equations). However, tiiX also has a condition that it is zero at x = 0. Hence, there are five conditions (N + 1 conditions~ and these can be used to evaluate completely a 4thorder polynomial (Nth-order polynomial). This process can also be used for the bottom phase. However, a section which contains the interface cannot be included. The tii, values are obtained at the interface by extrapolation. Using the polynomial, one has at the interface, moving with a velocity u,*, d iti_~ -
V:
A ~0;
=
ffi
(8)
where d( } denotes ( ) of the top phase less that in the bottom phase and pki is the rate of adsorption. This method cannot be used in a three-phase system, which occurs quite frequently. The reason is that one condition is needed to connect some feature from the end of the first phase to the beginning of the second. No such condition appears to be availabie without the use of modeling assumptions, which are to be avoided here. As a result, only one data set from Ref. [l] is used, i.e. the set designated “1” in Fig. l(E) of Ref. [l] where a w/o microemulsion is contacted with pure water, to give rise to w/o m~~roemulsions at eqtiilibrium. The time evolution is given in Figs 2(Aj-2(C) and 3 of Ref. [I] (see Appendix). The chemicals used are sodium dodecylsulfate, water and n-pentanol. Only filX and it,, are computed from the data and ii,, is computed from Eqn (4). The individual velocities are vix
2= jlix/\i’i
(9)
Results and discussion
Fluxes of water (ti,,) and surfa~tant (&,f are shown in Figs 1 and 2. Of course the flux of pentanol is -(fi ix + &). The .two velocities, uiX and u3X, are shown in Figs 3 and 4. From Eqn (9),
Fig. I. The variations in the fluxes of water as a function of position, at different times. The times are in days and ri, is in centimeters per day and pfi,, is the true flux. The left-hand side is the w/o microemulsion and the right-hand side is the aqueous phase. The fluxes appear discontiniIous at the interface (6, 12 and IS days). At longer times (36. 42 and 48 days) the third phase appears and the system is analyzabIe only in the w/o microemulsion phase as shown. A11 broken lines arc extrapolations using the theory.
Figures 1-4 can be divided into two time ranges. The first range is that of relatively short times where only two phases (w/o microemulsions on the Ieft and aqueous phase on the right in the figures) exist. The system is fully analyzable. Note the discontinuity at the interface in the fluxes due to the interfacial adsorption. At moderate times a third phase, lamellar liquid crystals, appears at the interphase and proceeds to engulf the aqueous phase. This range is not fully analyzable, both because of the appearance of the third phase and because the liquid crystal phase and the aqueous phases cover such small ranges of x (space) that the concentratjon data are very limited. Still, sufficient data are available for the w/o microemulsion systems and those results are shown. u2* = - fwr /w2)vtX
- (w,/w~)v,,.
K. drlorlrlokrisl,rct,t
Fig. 2. Same as Fig. I csccpt A reversal in fluxes takes microcm~isian phase.
that surfactant long
place a!
iluxcs arc shown. times in the w/o
L
I
Fig. 3. The vclocitics
I
I
I
I
(cm day- *) of water for the data in Fig. I.
et n/./Colloids
Surfaces
64 ( 1992) 197-205
Fig. 4. The velocities of surfxtant for the data in Fig. 2. Note that the velocities of water and surfactant (Figs 3 atld 4) appear to have opposite signs.
At very long times the system returns to a twophase state; w/o microemulsions and lamellar liquid crystals. The Buxes i:l the microemuision phase are all low 2nd are not shown. The fluxes and velocities, as well as their discontinuities across the interface, are shown in Figs 5-8 for pentanol and surfactant. Almost all the activity is confined to the interfacial region. Figure 9 shows the adsorption rates, Ri, at different times. Except for water and pentanol in the first phase, the rates of adsorption (or dcsorption) are approximately constant. The iiuxes obtained here are in keeping with ;he qualitative discussion provided earlier [l]. The veiocities in the w/o microemulsion phase show that pentanol and surfactant move towards the interface and it can be easily shown that the velocity in water is negative, i.e. water moves away from the interface. All velocities are of similar orders of magnitude, i.e. none are negligible. The striking feature of the experimental observa-
K. Anandakrishran et al./Colioids Surfam
64 (1992) 197-205
i .:
I t t I
h
Fig. 5. Fluxes of pcntanol at long times (54 and 60 days), arc shown. The left-hand side is the w/o microemulsion phase where the fluxes are small and have not been shown. The discontinuities at the interface and the fluxes in the lamellar liquid crystal phase (right-hand side) are shown. Note that the activity IS confined to the interracial layer alone.
tions is that the microemulsions do not imbibe water in a simple way. The surfactant moves to the interface where lamellar liquid crystals are formed. The liquid crystals soak up the water and then reform into droplets. It is tempting to say that the droplets move to the interface, unfold into lamellar sheets, then refold to spheres. The key result here is the discovery that this cannot happen as water and surfactant move in opposite directions. There are obvious implications in oil recovery and other extraction processes. Simulations assume that local equilibria exist [S). These are best described as macro-local equilibria and involve the assumption that mass transfer is infinitely fast. As a consequence, if overall concentrations and the temperature are known at a point in the drive, then the phase diagram yields the number of phases, their structures and compositions at that
Fig. 6. Same as Fig. 5 except that surfactant
Fig. 7. The velocities of pentanol
fluxes are shown.
from Fig. 5.
K. Anandakrishran et al./Colloids St&aces 64 (1992) 197-205
202
Fig. 8. The velocities of the surfactant from Fig. 6.
point. It is observable here and in previous work that has been summarized eisewhere [IO] that in a test tube, equilibration takes months, even years. There is another kind of local equilibrium, best
c1 SOS
0 “*O
0012 0 on08
R 0004
. 0
a
t3
a
00
-0004
-
0
Fig. 9. [ntcrfa~ial adsorption of the three species in ~entimetcrs per day. The break in the plots indicates the three-phase region not analyzed. The left-hand side refers to the w/o microemulsions with an aqueous phase, and the right-hand side to those with the lamellar liquid crystal phase.
referred to here as micro-local equilibrium, which is operative even when mass transfer is going on, According to this principle even though the entire system is not at equilibrium, every point in the system satisfies the thermodynamic equation of state, i.e. in the present system the concentrations from the top of the tube to the bottom can be traced on the phase equilibrium diagram. This is the “diffusion path” [3,4]. The contention here is that in the present case, and in many others, this micro-local equilibrium may not apply. Firstly, when there is adsorption at the interface (as here for instance) the microlocal equilibrium does not apply there: the concentrations on the two sides of the interface are not given by the two ends of a tie line. Secondly, even in the interior, micro-local equilibrium does not hold. in the study of diffusion of micelles, the simplest of the aggregates, it was found that microlocal equilibrium did not apply, in that a competitive reaction (of aggregation) was needed to describe the process [IO,111. In a w/o microemulsiou system similar to the one described here, mass transfer under constraints showed peculiarities which have been att~buted to the lack of micro-local equilibrium [12]. This is seen here in that the water velocities compare quite well with the velocities of pentanol. (This can be inferred from Fig. 2 which shows the fluxes of the surfactant at short times to be small, but in Fig. 1, the fluxes of water are large. From Eqn (4) it follows that the fluxes of pentanol are also large and opposite to those of the water.) Whereas pentanol should show no resistance to transfer out of the w/o phase, water should encounter a high resistance to transferring into the w/o phase because of its low solubility in pentanol. Yet the calculations show that their fluxes are similar in magnitude. Hence, aah r must also travel as “bulk”. Of course, these remarks are confined to the short times when the fluxes are large. Measurements of NMR self-diffusion coefficients of the solubilized phase in equilibrated systems [13,14] also suggest that the solubilized phase in a microemulsion system is effectively continuous.
203
K. Anundakr~s~ra~~ et ~~.~~a~laids Surfures 64 { 1992) 197-205
Obviously during collisions there can be an exchange in mass. The fact that collisions can be “seen” by the fast NMR methods is not surprising, but it is surprising that slow methods in nonequilibrium situations suggest the same. Seen together with the fact that the droplets have to reconstruct (diameters woulid increase and numbers decrease), whatever the details of the droplets during mass transfer (fused, fluctuating, etc.) may be, they are most likely not spherical. If this is the case then the phase diagrams would not apply in the w/o system, just as they do not apply at the interfaces. Thus, tracing the process as a path on the equilibrium diagram can be invalid. It is aIso possible that the present example is an extreme one, for as mentioned in the Appendix, the system hovers near a critical point for a long time. However, such critical points are also commonly encountered in contacting systems. In contrast, the analysis of the liquid crystal phase at large times shows no peculiarities. The region is easily divisible into a very active interfacial region and an interior region of low activity (Figs 5-8): the fluxes are such that the outer layer is first swollen with oil and then “dissolved” off as wfo microemulsion. In conclusion, the interfacial adsorption shows lack of micro-local equilibrium. In the interior, a lack of micro-local equilibrium provides the simplest unified explanation for the behavior of the fluxes in the w/o microemulsion phase, but the liquid crystal region shows no anomalous behavior. I his implies that the pros and cons of making micro- or macrolocal equilibrium assumptions need to be weighed more clearly, in that virtually all ‘%ases” and their “exceptions” seem to occur in parallel. References S.E. Friberg, Z. Ma and P. Neogi, in D.K. Smith (Ed.), Surfactant-Based Mobility Control, American Chemical Society, Washington, DC, 1988, p. 108. S.E. Friberg and P. Neogi, in J.F. Scamehorn and J.H. Harweh (Eds), Surfactant-Based Separation Processes, Marcel Dekker, Inc., New York, 1989, p. 119. P. Neogi, M. Kim and SE. Friberg, Sep. Sci. Technol., 20 (1985) 613.
4 5 6 7 8 9 10 I1 12 13 14
K.H. Raney and C.A. Miller, AIChE J., 33 (1987) t79!. N. Saad, GA. Pope and K. Sepchmoori, SPE Res. Eng., 4 (1989) 7. 2. Ma, S.E. Friberg and P. Neogi, Colloids Surfaces, 33 (1988) 249. E.L. Cussler, Multicomponent Diffusion, Elsevier, New York, 1976. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, Inc., New York, 1960. E.A. Mason and L.A. Viehland, 1. Chem. Phys., 68 (1978) 3562. R.M. Weinhammer, D.F. Evans and E.L. Cussler, J. Colloid Interface Sci., 80 (1981) 357. D.F. Evans, S. Mukherjee, D.J. Mitchell and B.W. Ninham, J. Colloid Interface Sci., 93 ($983) 184. Z. Ma, S.E. Friberg and P. Neogi. AIChE J., 35 (1989) 1678. P.-G. Nilsson and B. Lindman, J. Phys. Chem., 86 (1982) 271. B. Lindman, P. Stilbs and ME. Moseley, J. Colloid Interface Sci., 83 (1981) 569,
Appendix
The phases and the progress of individual phase volumes for the system analyzed in Ref. [l] are shown schematically in Fig. 10(a). Figure IO(b) shows schematically the phase equilibrium diagram and approximate locations of samples (1, 2 and 3) which were contacted with water. The overall compositions are shown as l’, 2’ and 3’. Figures 10(a) and IO(c) pertain to the first case, 1 + water only. In Fig. 10(c) is shown the initial diffusion path. It is made up of three straight lines, the middle one being a tie-line. The bottom line near the water corner is supersaturated as it lies inside the two-phase region and gives rise to spontaneous emulsification (turbidity), as shown in Fig. 10(a), The initial flux of surfactant into the bottom (aqueous) phase is very high and solubilizes the precipitate, as shown in Fig. 10(d). Figure 10(d) is a two-phase region and is representative of the results for 6, 12 and 18 days shown in Figs 1-4. At much longer times (36-54 days) there are three phases: the oil phase, liquid crystals and the water phase, as shown in Fig. IO(e), Next, the water phase disappears (Fig. 10(f)). Eventually, the system becomes all oil-phase, l’, as in Fig. 10(b). There is a gap between Figs lO(d} and IO(e), where the liquid crystal appears in the middle of
204
K. Arm~dokrishran
et al./Colloids
Surfaces 64 (1992) 197-205
LAYER
DAYS
data
(a> voiunetric
Cl1
u
t-l
n2U
CnP
’
/
\
I
\
/
/
/ z_/
\
\
//JI\ \
/
/
Cl1 Au’h
A
i’‘\ ’
data
con-tatting
/ / 4
.’
,,,;, / -zk_::-> ,,, ------> fC r--_-_- \
--l
I ___._
I
Cc> 0 to
36
6
-
54
\
days
days
Cd> 6
(f> 54
-
18
days
+- days
Fig. IN (a) The phases and phase volumes on contacting; (b) the systers contacted; (c)--(f) the difTusion paths. Between (d) and (e) there is a ga? which could not be cxpiained on the phase diagram. The details are given in the Appendix.
the aqueous phase and the top oil/water interface disappears after some time. This we found difficult to explain on the phase equilibrium diagram. For point 2 in Fig. IO(b), the response up to Fig. IO(e)
is similar but abbreviated. For point 3, the liquid crystal phase starts immediately. The junction of the w/o and o/w phases has been shown as a continuous region. Actually there is a critical point
K. Anandakrishran
et ai.~~~iiaids
Surfaces
64 ($992)
197-205
there, the oil-water plait point. The density differences are so low there that phases cannot be separated properly. The points 1 and 2 are sufficiently to the left that initially the diffusion path lies to the left of the plait points (Fig. lo(c)) but with time it crosses over this point to move to the right. However, point 3 is too much to the right and never crosses the plait point, and never shows anomalous effects.
Consequently, we feel that the anomalies in the regio:n between Figs 10(d) and 10(e) have to do with the fact that the system is close to critical. Not only that, but one could also suggest that the approach to the plait point in Figs 10(b) and IO(c) is also expected to be anomalous. This is because the oil/water interface in Fig. IO(a) is thick, a feature which is associated with critical phenomena.
64 (1992) 197-205
197
Elsevier Science Publishers B.V.,Amsterdam
Mass transfer analysis in a mi~ro~mulsion system K.Anandakrishran”,
P. Neogi” and SE. Fribergb
“Department of Chemical Engineering, Universityof Missouri-Rolla. Rolla. MO 65401, USA bDepartmen t of Chemistry, Clar~son Uniuersity,Pots~am, NY 13676, USA (Received 12 July 1991;accepted 7 January 1992) Abstract Previous work on the effects of layering H water-in-oil (w/o) microemulsion on water has been d~ument~. The concentration profiles recorded there are analyzed to obtain the fluxes in the interior and the adsorption at the interfaces. Adsorption at the interface shows a lack of equilibrium locally and the fluxes suggest that local equilibrium may not exist in the interior. Keywords:
Mass transfer; microcmulsion system; phase equilibria.
Introduction
Contacting experiments are performed in oilwater-surfactant-cosurfactant systems by layering one phase on top of another and monitoring the resulting phase changes volumetrically. Such information provides fundamental understanding of the transport process, and finds applications in enhanced oil recovery using surfactants f_.j, as well as in separation systems [ZJ. Specifically, liquid crystals are very often formed transiently in such systems and this can affect strongly the efficiency of the process Cl]. The mass transfer is very slow in general, taking months to complete in a test tube [Z]. The existing analyses follow a standard diffusion path analysis [3,4], and show that the rates can vary from being adsorption controlled to being diffusion controlled in a manner which is not fully understood at present. Such effects are aliogether ignored in the simulations of surfactantaided oil recovery which assume local equilibrium
ca C~rrespf~n~e~ce to: P. Neogi,Dept. of Chemical Enginee~ng, University of Missouri-Rolla, Rolla, MO 65401, USA. 0166-6622/92/$05.00
0 1992 -
In our previous work we went beyond volumetric measurements to measure also the concentration profiles [1,6]. The intent was to evaluate quantitatively the two mechanisms of adsorption and diffusion. This also allows one to assess how good the condition of local equilibrium is, as well as to determine how the corrections to it are to be made when required. The main restriction we impose is that in the entire analysis no particular model for constitutive equations nor any assumption regarding controlling mechanisms is to be made. The present work is the analysis of our previous data [1,6] under such constraints. The reason behind such severity in our analysis is that simple diffusion-path analysis does not explain all of the results as discussed in the Appendix. Only a limited amount of data could be analyzed. The key feature in the observations was that when a w/o microemulsion was contacted with brine such that a dilute w/o microemulsion was to result at equilibrium, lameilar liquid crystals were formed transiently. What was expected to be straightforward imbibition of brine turned out to involve lamellar liquid crystals which did not occur in the chase diagram in the immediate vicinity of the compositions of the two phases contacted.
Elsevier Science Publishers B.V. All rights reserved.
I YS
Formulation
A brief description of the previous experiments is given first. Tubes were made of seven sections of 1 cm each. Brine was layered first, then the w/o mi~roemu~sion was layered over the brine. One of these was observed for phase vo!umes as they changed with time. The others were broken into seven parts, and one such tube was broken every three days. The concentrations were analyzed in each section. The question now arises as to what the treated data could yield. The rigorous form that preliminary models [3.4] would take would be that of multicomponent diffusion [7,8-J, i.e. the fluxes of multicomponent diffusion, containing multicomponent diffusivities as parameters, are substituted into the conservation equations, which are then solved and fitted to the concentration profiles measured experimentally. The vatues of multicomponent diffusivities are those which give the best fit. This approach could be generalized to include multicomponent adsorption at interfaces. The problem with the multicomponent diffusivities, I),j, is that they are applicable only to specific systems. Let 1, 2 and 3 denote water, pentanol and surfactant respectively, then Di3 will change if decanol is substituted for pentanol, or a pentanol-decano1 mixture is used, etc. Another method is to use the generalized Stefan-Maxwell equation [9] which uses binary diffusjvities, fiijs to describe fluxes. These are of course the basic building blocks but the problem with this approach is that in firs, I and 3 have a structure which varies with the amount of 2. Thus 6,, without reference to 2 is meaningless. Then there is the question of what to do with the electrostatic effects. Because of such uncertainties, we have decided to avoid any kind of fIux (or adsorption) expression. This gives rise to a generality which, however, can be used to treat only the two-phase systems discussed below. The tube is divided into seven sections of I cm each. The origin of the _Ycoordinate is placed at the top. The sixth section then lies between x = 5 cm and s = 6 cm. It is now assumed
K. Anandakrishrari
ct al./Cnllnids
Surfaces
64 (1992)
197-205
that the concentration obtained in the sixth section, which is an average concentration in that section, can be approximated as the concentration at the mid-point in the sixth section, x = 5.5 cm. Further, the total volume does not change with time, so the approximation that the density p is a constant is appropriate. The continuity equation [8] becomes
which on integration
becomes
PO, = c(r)
(2)
where v, is the mass average velocity. But at _Y= 0, for the phase at the top, v, = 0, and it is also zero for the phase at the bottom at x = 8 cm. Thus it is zero throughout, that is, c(rf = 0. However, PO,
=
i:
i= 1
(3)
p1i.r
where nix are the mass fluxes. Thus for constant one has
where rii = rri/p. The conservation aPi -=dt
are
& dx
or at constant
p
&
aivi
dt
equations
p,
-
-
2s
where 1~‘i= pi/p, the -:deight fraction. Equation (6) can be considered as a first-order differential equation for iti, provided dwi/dp is knowJn as a function of _Yat a particular time t. Indeed, it is known from the experimental data and can be approximated as the backward difference (7) where k denotes time and IZ the location in space. Now, instead of finite di~erencing the right-hand
K. Anaudakrishran ef al./Colloids Surfaces 64 (1992) 197-205
side in Eqn (61, fii, is expressed as a po~ynomiai. Let the phase at the top cover the top four layers (or N layers). Then on substitution of the polynomial into Eqn (61, one has four resulting equations at x=0.5, 1.5, 2.5, 3.5 (N equations). However, tiiX also has a condition that it is zero at x = 0. Hence, there are five conditions (N + 1 conditions~ and these can be used to evaluate completely a 4thorder polynomial (Nth-order polynomial). This process can also be used for the bottom phase. However, a section which contains the interface cannot be included. The tii, values are obtained at the interface by extrapolation. Using the polynomial, one has at the interface, moving with a velocity u,*, d iti_~ -
V:
A ~0;
=
ffi
(8)
where d( } denotes ( ) of the top phase less that in the bottom phase and pki is the rate of adsorption. This method cannot be used in a three-phase system, which occurs quite frequently. The reason is that one condition is needed to connect some feature from the end of the first phase to the beginning of the second. No such condition appears to be availabie without the use of modeling assumptions, which are to be avoided here. As a result, only one data set from Ref. [l] is used, i.e. the set designated “1” in Fig. l(E) of Ref. [l] where a w/o microemulsion is contacted with pure water, to give rise to w/o m~~roemulsions at eqtiilibrium. The time evolution is given in Figs 2(Aj-2(C) and 3 of Ref. [I] (see Appendix). The chemicals used are sodium dodecylsulfate, water and n-pentanol. Only filX and it,, are computed from the data and ii,, is computed from Eqn (4). The individual velocities are vix
2= jlix/\i’i
(9)
Results and discussion
Fluxes of water (ti,,) and surfa~tant (&,f are shown in Figs 1 and 2. Of course the flux of pentanol is -(fi ix + &). The .two velocities, uiX and u3X, are shown in Figs 3 and 4. From Eqn (9),
Fig. I. The variations in the fluxes of water as a function of position, at different times. The times are in days and ri, is in centimeters per day and pfi,, is the true flux. The left-hand side is the w/o microemulsion and the right-hand side is the aqueous phase. The fluxes appear discontiniIous at the interface (6, 12 and IS days). At longer times (36. 42 and 48 days) the third phase appears and the system is analyzabIe only in the w/o microemulsion phase as shown. A11 broken lines arc extrapolations using the theory.
Figures 1-4 can be divided into two time ranges. The first range is that of relatively short times where only two phases (w/o microemulsions on the Ieft and aqueous phase on the right in the figures) exist. The system is fully analyzable. Note the discontinuity at the interface in the fluxes due to the interfacial adsorption. At moderate times a third phase, lamellar liquid crystals, appears at the interphase and proceeds to engulf the aqueous phase. This range is not fully analyzable, both because of the appearance of the third phase and because the liquid crystal phase and the aqueous phases cover such small ranges of x (space) that the concentratjon data are very limited. Still, sufficient data are available for the w/o microemulsion systems and those results are shown. u2* = - fwr /w2)vtX
- (w,/w~)v,,.
K. drlorlrlokrisl,rct,t
Fig. 2. Same as Fig. I csccpt A reversal in fluxes takes microcm~isian phase.
that surfactant long
place a!
iluxcs arc shown. times in the w/o
L
I
Fig. 3. The vclocitics
I
I
I
I
(cm day- *) of water for the data in Fig. I.
et n/./Colloids
Surfaces
64 ( 1992) 197-205
Fig. 4. The velocities of surfxtant for the data in Fig. 2. Note that the velocities of water and surfactant (Figs 3 atld 4) appear to have opposite signs.
At very long times the system returns to a twophase state; w/o microemulsions and lamellar liquid crystals. The Buxes i:l the microemuision phase are all low 2nd are not shown. The fluxes and velocities, as well as their discontinuities across the interface, are shown in Figs 5-8 for pentanol and surfactant. Almost all the activity is confined to the interfacial region. Figure 9 shows the adsorption rates, Ri, at different times. Except for water and pentanol in the first phase, the rates of adsorption (or dcsorption) are approximately constant. The iiuxes obtained here are in keeping with ;he qualitative discussion provided earlier [l]. The veiocities in the w/o microemulsion phase show that pentanol and surfactant move towards the interface and it can be easily shown that the velocity in water is negative, i.e. water moves away from the interface. All velocities are of similar orders of magnitude, i.e. none are negligible. The striking feature of the experimental observa-
K. Anandakrishran et al./Colioids Surfam
64 (1992) 197-205
i .:
I t t I
h
Fig. 5. Fluxes of pcntanol at long times (54 and 60 days), arc shown. The left-hand side is the w/o microemulsion phase where the fluxes are small and have not been shown. The discontinuities at the interface and the fluxes in the lamellar liquid crystal phase (right-hand side) are shown. Note that the activity IS confined to the interracial layer alone.
tions is that the microemulsions do not imbibe water in a simple way. The surfactant moves to the interface where lamellar liquid crystals are formed. The liquid crystals soak up the water and then reform into droplets. It is tempting to say that the droplets move to the interface, unfold into lamellar sheets, then refold to spheres. The key result here is the discovery that this cannot happen as water and surfactant move in opposite directions. There are obvious implications in oil recovery and other extraction processes. Simulations assume that local equilibria exist [S). These are best described as macro-local equilibria and involve the assumption that mass transfer is infinitely fast. As a consequence, if overall concentrations and the temperature are known at a point in the drive, then the phase diagram yields the number of phases, their structures and compositions at that
Fig. 6. Same as Fig. 5 except that surfactant
Fig. 7. The velocities of pentanol
fluxes are shown.
from Fig. 5.
K. Anandakrishran et al./Colloids St&aces 64 (1992) 197-205
202
Fig. 8. The velocities of the surfactant from Fig. 6.
point. It is observable here and in previous work that has been summarized eisewhere [IO] that in a test tube, equilibration takes months, even years. There is another kind of local equilibrium, best
c1 SOS
0 “*O
0012 0 on08
R 0004
. 0
a
t3
a
00
-0004
-
0
Fig. 9. [ntcrfa~ial adsorption of the three species in ~entimetcrs per day. The break in the plots indicates the three-phase region not analyzed. The left-hand side refers to the w/o microemulsions with an aqueous phase, and the right-hand side to those with the lamellar liquid crystal phase.
referred to here as micro-local equilibrium, which is operative even when mass transfer is going on, According to this principle even though the entire system is not at equilibrium, every point in the system satisfies the thermodynamic equation of state, i.e. in the present system the concentrations from the top of the tube to the bottom can be traced on the phase equilibrium diagram. This is the “diffusion path” [3,4]. The contention here is that in the present case, and in many others, this micro-local equilibrium may not apply. Firstly, when there is adsorption at the interface (as here for instance) the microlocal equilibrium does not apply there: the concentrations on the two sides of the interface are not given by the two ends of a tie line. Secondly, even in the interior, micro-local equilibrium does not hold. in the study of diffusion of micelles, the simplest of the aggregates, it was found that microlocal equilibrium did not apply, in that a competitive reaction (of aggregation) was needed to describe the process [IO,111. In a w/o microemulsiou system similar to the one described here, mass transfer under constraints showed peculiarities which have been att~buted to the lack of micro-local equilibrium [12]. This is seen here in that the water velocities compare quite well with the velocities of pentanol. (This can be inferred from Fig. 2 which shows the fluxes of the surfactant at short times to be small, but in Fig. 1, the fluxes of water are large. From Eqn (4) it follows that the fluxes of pentanol are also large and opposite to those of the water.) Whereas pentanol should show no resistance to transfer out of the w/o phase, water should encounter a high resistance to transferring into the w/o phase because of its low solubility in pentanol. Yet the calculations show that their fluxes are similar in magnitude. Hence, aah r must also travel as “bulk”. Of course, these remarks are confined to the short times when the fluxes are large. Measurements of NMR self-diffusion coefficients of the solubilized phase in equilibrated systems [13,14] also suggest that the solubilized phase in a microemulsion system is effectively continuous.
203
K. Anundakr~s~ra~~ et ~~.~~a~laids Surfures 64 { 1992) 197-205
Obviously during collisions there can be an exchange in mass. The fact that collisions can be “seen” by the fast NMR methods is not surprising, but it is surprising that slow methods in nonequilibrium situations suggest the same. Seen together with the fact that the droplets have to reconstruct (diameters woulid increase and numbers decrease), whatever the details of the droplets during mass transfer (fused, fluctuating, etc.) may be, they are most likely not spherical. If this is the case then the phase diagrams would not apply in the w/o system, just as they do not apply at the interfaces. Thus, tracing the process as a path on the equilibrium diagram can be invalid. It is aIso possible that the present example is an extreme one, for as mentioned in the Appendix, the system hovers near a critical point for a long time. However, such critical points are also commonly encountered in contacting systems. In contrast, the analysis of the liquid crystal phase at large times shows no peculiarities. The region is easily divisible into a very active interfacial region and an interior region of low activity (Figs 5-8): the fluxes are such that the outer layer is first swollen with oil and then “dissolved” off as wfo microemulsion. In conclusion, the interfacial adsorption shows lack of micro-local equilibrium. In the interior, a lack of micro-local equilibrium provides the simplest unified explanation for the behavior of the fluxes in the w/o microemulsion phase, but the liquid crystal region shows no anomalous behavior. I his implies that the pros and cons of making micro- or macrolocal equilibrium assumptions need to be weighed more clearly, in that virtually all ‘%ases” and their “exceptions” seem to occur in parallel. References S.E. Friberg, Z. Ma and P. Neogi, in D.K. Smith (Ed.), Surfactant-Based Mobility Control, American Chemical Society, Washington, DC, 1988, p. 108. S.E. Friberg and P. Neogi, in J.F. Scamehorn and J.H. Harweh (Eds), Surfactant-Based Separation Processes, Marcel Dekker, Inc., New York, 1989, p. 119. P. Neogi, M. Kim and SE. Friberg, Sep. Sci. Technol., 20 (1985) 613.
4 5 6 7 8 9 10 I1 12 13 14
K.H. Raney and C.A. Miller, AIChE J., 33 (1987) t79!. N. Saad, GA. Pope and K. Sepchmoori, SPE Res. Eng., 4 (1989) 7. 2. Ma, S.E. Friberg and P. Neogi, Colloids Surfaces, 33 (1988) 249. E.L. Cussler, Multicomponent Diffusion, Elsevier, New York, 1976. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, Inc., New York, 1960. E.A. Mason and L.A. Viehland, 1. Chem. Phys., 68 (1978) 3562. R.M. Weinhammer, D.F. Evans and E.L. Cussler, J. Colloid Interface Sci., 80 (1981) 357. D.F. Evans, S. Mukherjee, D.J. Mitchell and B.W. Ninham, J. Colloid Interface Sci., 93 ($983) 184. Z. Ma, S.E. Friberg and P. Neogi. AIChE J., 35 (1989) 1678. P.-G. Nilsson and B. Lindman, J. Phys. Chem., 86 (1982) 271. B. Lindman, P. Stilbs and ME. Moseley, J. Colloid Interface Sci., 83 (1981) 569,
Appendix
The phases and the progress of individual phase volumes for the system analyzed in Ref. [l] are shown schematically in Fig. 10(a). Figure IO(b) shows schematically the phase equilibrium diagram and approximate locations of samples (1, 2 and 3) which were contacted with water. The overall compositions are shown as l’, 2’ and 3’. Figures 10(a) and IO(c) pertain to the first case, 1 + water only. In Fig. 10(c) is shown the initial diffusion path. It is made up of three straight lines, the middle one being a tie-line. The bottom line near the water corner is supersaturated as it lies inside the two-phase region and gives rise to spontaneous emulsification (turbidity), as shown in Fig. 10(a), The initial flux of surfactant into the bottom (aqueous) phase is very high and solubilizes the precipitate, as shown in Fig. 10(d). Figure 10(d) is a two-phase region and is representative of the results for 6, 12 and 18 days shown in Figs 1-4. At much longer times (36-54 days) there are three phases: the oil phase, liquid crystals and the water phase, as shown in Fig. IO(e), Next, the water phase disappears (Fig. 10(f)). Eventually, the system becomes all oil-phase, l’, as in Fig. 10(b). There is a gap between Figs lO(d} and IO(e), where the liquid crystal appears in the middle of
204
K. Arm~dokrishran
et al./Colloids
Surfaces 64 (1992) 197-205
LAYER
DAYS
data
(a> voiunetric
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u
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CnP
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/
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\
/
/
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6
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Fig. IN (a) The phases and phase volumes on contacting; (b) the systers contacted; (c)--(f) the difTusion paths. Between (d) and (e) there is a ga? which could not be cxpiained on the phase diagram. The details are given in the Appendix.
the aqueous phase and the top oil/water interface disappears after some time. This we found difficult to explain on the phase equilibrium diagram. For point 2 in Fig. IO(b), the response up to Fig. IO(e)
is similar but abbreviated. For point 3, the liquid crystal phase starts immediately. The junction of the w/o and o/w phases has been shown as a continuous region. Actually there is a critical point
K. Anandakrishran
et ai.~~~iiaids
Surfaces
64 ($992)
197-205
there, the oil-water plait point. The density differences are so low there that phases cannot be separated properly. The points 1 and 2 are sufficiently to the left that initially the diffusion path lies to the left of the plait points (Fig. lo(c)) but with time it crosses over this point to move to the right. However, point 3 is too much to the right and never crosses the plait point, and never shows anomalous effects.
Consequently, we feel that the anomalies in the regio:n between Figs 10(d) and 10(e) have to do with the fact that the system is close to critical. Not only that, but one could also suggest that the approach to the plait point in Figs 10(b) and IO(c) is also expected to be anomalous. This is because the oil/water interface in Fig. IO(a) is thick, a feature which is associated with critical phenomena.