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Colloids and Surfaces A: Physicochem. Eng. Aspects 244 (2004) 113–119
Experiments and modeling of growth of myelinic figures in Aerosol OT/water system Jinhua Bai, Clarence A. Miller∗ Department of Chemical Engineering (MS 362), Rice University, P.O. Box 1892, Houston, TX 77251-1892, USA Received 23 October 2003; accepted 14 May 2004 Available online 18 August 2004
Abstract A quantitative penetration scan in a vertically oriented cell was used to determine behavior when water was brought into contact with neat Aerosol OT (AOT) at 30 ◦ C. Myelinic figures developed having roots at the initial surface of contact between the phases and length proportional to the square root of time, in agreement with previous observations for AOT and other surfactants by various workers. However, the present experiments also included measurement of positions of the interfaces between lamellar and cubic and cubic and reverse hexagonal phases, which were also proportional to the square root of time. A similar experiment in which a thin layer of AOT was contacted with a large quantity of water was also conducted. A model is presented which includes both growth of myelinic figures and diffusion in the various liquid crystalline phases. With this model and data from the two experiments effective diffusion coefficients for all three liquid crystalline phases and the composition of the myelinic figures were calculated. The former were found to be of order 10−10 m2 /s, the latter approximately 50 wt.% AOT. © 2004 Elsevier B.V. All rights reserved. Keywords: Aerosol OT; Myelinic figures; Diffusion; Dissolution; Anionic surfactants
1. Introduction The rate and manner of dissolution of surfactants in water is of considerable interest. In particular, continuing trends in household laundry processes toward decreased water usage, which requires higher surfactant concentrations, and toward lower temperatures have increased the importance of achieving rapid dissolution. One issue is whether viscous liquid crystalline phases formed during surfactant dissolution cause significant decreases in dissolution rates or trap solid particles of builders such as zeolites, preventing them from performing their functions. For isomerically pure, neat liquid nonionic surfactants below their cloud points Chen et al. [1] showed that dissolution rates were controlled by diffusion. Effective diffusion coefficients in both the initial aqueous and surfactant phases, as well as in the intermediate liquid crystalline phases which formed, were calculated by combining observations of dissolution times for surfactant drops injected into ∗
Corresponding author. Tel.: +1 713 348 4904; fax: +1 713 348 5478. E-mail address: [email protected] (C.A. Miller).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.05.004
water with measurements of the time-dependent positions of the various interfaces in a vertically oriented penetration cell in which the surfactant and water were contacted without mixing. Values of order 10−10 m2 /s were found. At temperatures well below their cloud points, drops of nonionic surfactant mixtures or commercial nonionic surfactants dissolved at rates comparable to those for the pure surfactants. However, dissolution was slower and more complex in nature at temperatures just below the cloud point [2]. Bai et al. [3] found that granules about 500 m in diameter consisting of micron-size particles of a nonswelling zeolite (approximately 70% by volume) held together by a pure, liquid nonionic surfactant below its cloud point disintegrated rapidly in water, releasing the zeolite particles. However, no significant disintegration occurred at temperatures above the cloud point. Similar behavior was observed for granules made with binary mixtures of pure nonionic surfactants. Pure ionic surfactants are typically crystalline solids or highly viscous liquid crystalline phases near ambient temperature. In the former case, significant porosity exists if crystals are packed into a penetration cell such as that used in the nonionic surfactant work, making interpretation of the
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results problematic. In the latter case, use of quantitative penetration scans is feasible if one can provide an initial surfactant region with relatively few air bubbles and a reasonably smooth surface, as required to justify the one-dimensional analysis used to interpret the results. Another complicating factor for some anionic surfactants is that dissolution is not always completely controlled by diffusion. For example, Haran et al. [4] have reported that, when contacted with water, the pure anionic surfactant Aerosol OT (AOT) exhibits myelinic figures formed by swelling of an intermediate lamellar liquid crystalline phase. Moreover, they confirmed for this system the previous finding of Buchanan et al. [5] for the pure nonionic surfactant C12 E3 at ambient temperature that such swelling is accompanied by flow of the aqueous phase through the spaces between the myelinic figures toward the roots of the myelins. As the solubility of C12 E3 in water is extremely low for these conditions, a very large quantity of water would be required for complete dissolution, which is not ordinarily observed for this surfactant in practical applications. Solubility in water is also minimal for phospholipids, whose ability to exhibit myelinic figures is well known [4,6,7]. In contrast, the solubility of AOT in water is much larger (1.4 wt.%), and complete dissolution is expected for many situations of interest. Electron micrographs of phospholipid myelins showed that the bilayers were concentric cylinders [6], and a similar structure is presumed for myelins in other surfactant systems. In this paper, we describe results of quantitative penetration experiments in vertically oriented cells in which AOT is contacted with water, both when relatively large quantities of both phases are present initially and can be considered semi-infinite in extent for purposes of analysis and when a thin layer of AOT is contacted by a large quantity of water. In the latter case, the original reverse hexagonal (H2 ) phase of AOT is completely dissolved during the time frame of the experiment. In both cases myelinic figures grow upward from the initial surface of contact. A simple model is presented which permits effective diffusion coefficients to be calculated for all phases below the initial surface of contact, namely lamellar L␣ , viscous isotropic or cubic V1 , and H2 phases. It also yields a value for composition of the myelinic figures.
2. Experimental Sodium bis(2-ethylhexyl) sulfosuccinate, also known as Aerosol OT or AOT, of purity greater than 96%, was purchased from TCI America, Portland, OR. Water used in the experiments was deionized. Linear penetration experiments were conducted using vertically oriented rectangular capillary cells (50 mm × 4 mm × 0.4 mm, Vitro Dynamics Inc., Cat. No. 2540) placed in a microscope with a vertical stage [1]. Pure AOT, which is a highly viscous H2 phase, was placed in the bottom of the cell and manipulated with glass micropipet to achieve a
nearly horizontal interface several mm above the bottom of the cell. Then the bottom of the cell was sealed and water or an aqueous solution of AOT was carefully injected from the top of the cell. Videomicroscopy was used to record the resulting behavior and especially to follow the positions of all interfaces as a function of time. The experiments were conducted in an environmental room maintained at 30 ◦ C. A novel aspect of the present work was similar experiments with only a thin layer (<2 mm) of surfactant to provide additional information on effective diffusion coefficients as discussed below.
3. Results and interpretation The phase diagram of the AOT/water system is shown in Fig. 1 [8]. It indicates that intermediate lamellar L␣ and viscous isotropic V1 phases should form when the pure AOT H2 phase is brought into contact with water without stirring. Such behavior was, in fact, observed, as shown in Fig. 2, except that the portion of the lamellar phase above the surface of initial contact consisted of many vertically oriented, closely packed myelinic figures. The base of the myelinic figures remained at the surface of initial contact throughout the experiment. Fig. 3 shows that the length of the myelinic figures was proportional to the square root of time, a result that has been found by several groups for different systems [4,5,7,9]. Also shown by Fig. 3 is that the displacement of the phase boundaries between the various liquid crystalline phases from the surface of initial contact was proportional to the square root of time. These observations suggest that transport in the liquid crystalline phases can be modeled by ordinary diffusion. We consider the region x > 0, where x is the distance measured from the base of the myelins toward the H2 phase (downward in Fig. 2). If each phase i has a constant and uniform effective diffusivity Dei , the governing differential equation therein is given by 2 ∂ωi ∂ ωi = Dei (1) ∂t ∂x2 where ωi is surfactant mass fraction. At the phase boundaries between H2 and V1 and between V1 and L␣ , local equilibrium is assumed in accordance with Fig. 1 and conservation of mass of surfactant is imposed. When both initial
Fig. 1. Phase diagram for AOT/water system at 25 ◦ C.
J. Bai, C.A. Miller / Colloids and Surfaces A: Physicochem. Eng. Aspects 244 (2004) 113–119
115
Thickness of myelins (mm)
1.2 1 1/2
Slope = 0.106mm/min
0.8 0.6 0.4 0.2 0 0
3
Time
(a)
9
6 1/2
12
1/2
(min )
2.5
Slope=0.107mm/min1/2
Thickness of La(mm)
2 1.5 1 0.5 0 0
6
3
1/2
(b)
Time
9
12
1/2
(min )
The growth rate of V1 phase 0.25
phases are present in large amounts, the differential equation, initial condition, and these boundary conditions can be satisfied by the following similarity solution, provided that the displacement of all phase boundaries is proportional to the square root of time as observed experimentally: ωi = a1i + a2i erf ηi ,
Thickness of V1 (mm)
Fig. 2. Video frames of AOT/water semi-infinite experiment at 30 ◦ C (low magnification, the width of the cell is 4 mm).
0.2
Slope=0.0229mm/min1/2
0.15 0.1 0.05
(2)
where ηi = [x/(4Dei t)1/2 ]. One additional boundary condition is that surfactant mass fraction in the H2 phase must approach its initial value at distances far from the surface of contact, i.e., as x → +∞. So far, the analysis is similar to that used previously for penetration experiments involving pure nonionic surfactants [1]. What is new here is the condition for conservation of mass of surfactant at the base of the myelins x = 0: ∂ωlam d Delam (0) + jaq ωL1 = [εm t 1/2 ωlam (0)φ] ∂x dt (3) The first term on the left side of this equation represents the flux of surfactant to x = 0 from the portion of the lamellar phase below the myelinic figures. The second term represents surfactant dissolved in the aqueous phase, which, as discussed above, flows between the myelins to their base at x = 0. jaq is the total flux and ωL1 is the surfactant content.
0 0
(c)
3
6 1/2
Time
9
12
1/2
(min )
Fig. 3. Plots of interfacial positions for AOT/water semi-infinite experiment at 30 ◦ C.
That is, AOT could be contacted with a solution with mass fraction ωL1 of AOT instead of with pure water, as discussed below. On the right side of the equation εm t1/2 is the length of the myelins with εm a constant, ωlam (0) is surfactant mass fraction in the lamellar phase at the base of the myelins and in the myelins themselves, and φ is the fraction of the cross sectional area occupied by the myelins. Surfactant mass fraction is assumed uniform along the myelins, i.e., diffusion along the myelins is neglected, since they are of nearly uniform diameter and are forced out of the lamellar phase by the swelling process. For uniform close packed cylinders it can be shown that φ is approximately 91%, in good agreement with the value found for myelinic figures of C12 E3 [5].
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An overall balance for the entire surfactant-containing region including the myelins requires that the flux of aqueous solution entering from the bulk L1 phase be equal to the rate of increase in volume of the myelins: d jaq = [εm t 1/2 φ] (4) dt
Substitution of Eqs. (2) and (4) into Eq. (3) yields εm φ(a1lam − ωL1 ) Delam 1/2 a2lam = π 2
Delam (m2 /s)
DeV1 (m2 /s)
1.2 × 10−10 1.31 × 10−10 1.44 × 10−10 2.0 × 10−10 3.0 × 10−10
1.59 1.70 1.82 2.20 2.63
× × × × ×
10−10 10−10 10−10 10−10 10−10
DeH2 (m2 /s)
Myelin mass fraction of AOT
1.94 × 10−11 3.56 × 10−11 5.6 × 10−11 1.47 × 10−10 2.87 × 10−10
0.483 0.497 0.513 0.562 0.617
(5)
Suppose that it is desired to predict the rate of growth of myelins and all intermediate phases when pure AOT is contacted with water, starting with knowledge of all effective diffusion coefficients and the phase diagram of Fig. 1. Eq. (1) and its solution Eq. (2) apply for all three phases below the surface of contact. The two interfaces between them must have positions εi t1/2 to be consistent with the similarity solution. Thus, there are nine unknowns to be determined from the boundary conditions: three a1i , three a2i , two εi and εm . Five equations in these unknowns can be obtained from the boundary condition for x → +∞ and the known compositions of each phase at the two interfaces between liquid crystalline phases. Three more equations come from surfactant mass balances at the these interfaces and from Eq. (5). The last boundary condition involves the mechanism for growth of myelinic figures. Let us assume that their rate of growth is proportional to the osmotic pressure difference p between the bulk aqueous solution and the lamellar phase at the base of the myelins (x = 0), a quantity which is independent of time for the present similarity solution. Let us further assume that the growth rate is inversely proportional to myelin length. For example, contact between the closely packed myelins likely provides resistance to the swelling process. With these assumptions, we find dεm t 1/2 k p = (6) dt (εm t 1/2 ) where k is a constant incorporating the various proportionality factors. This equation is readily simplified to εm = (2k p)1/2
Table 1 Diffusivity values and myelin composition for assumed values of De in lamellar phase (AOT/water semi-infinite experiment)
(7)
The osmotic pressure difference depends on ωL1 and the composition a1lam of the lamellar phase at x = 0 but does not introduce any new unknowns. Thus, if k can be estimated, the equations can be solved to determine the concentration distributions and rates of interfacial motion. In contrast to this hypothetical situation, we do not in our present experiments know k (or the product k p, which may be considered a single unknown), or the three diffusivities Dei . However, we can measure interfacial positions as a function of time and obtain ε1 , ε2 and εm . That is, we now have ten unknowns (including k p) with the same nine equations. Consequently, the problem cannot be solved
completely using results of the semi-infinite penetration experiment alone. What can be done is to solve for the effective diffusivities of the H2 and V1 phases and for myelin composition with an assumed value of effective diffusivity of the L␣ phase. Table 1 presents the results of such calculations for the AOT data of Fig. 3 where ωL1 = 0, indicating that plausible values of all the parameters can be found for a range of values for Delam slightly larger than 10−10 m2 /s. For pure nonionic surfactants, it was also found that the semi-infinite penetration experiment was insufficient to determine all effective diffusivities. In that case, an additional measurement of the time required for a surfactant drop of known initial diameter to completely dissolve provided additional information that allowed the diffusivities to be calculated [1]. This approach clearly cannot be used here, since pure AOT is highly viscous and not readily formed into small drops. A suitable alternative is to carry out penetration experiments starting with a thin layer of AOT and a large amount of water and continuing until time t∗ when the H2 phase disappears completely. Since after some initial time period the concentration profile in the H2 phase departs from that of the similarity solution owing to the no flux boundary condition at the bottom of the cell, a numerical solution of the governing equations can be used with the measured value of t∗ to find values for all the effective diffusivities and myelin composition, i.e., to select the best row of values from Table 1. Details of the numerical procedure may be found elsewhere [10]. Measurements of positions of the various interfaces as a function of time for such a thin layer experiment for AOT are shown in Fig. 4 along with the results of simulations using De values of 1.31, 1.70 and 0.36 × 10−10 m2 /s for L␣ , V1 , and H2 phases, respectively. Agreement is good. It should be noted that the myelinic figures were far enough from the bottom of the cell that their base remained at the initial contact surface within experimental error and that their growth rate was virtually the same as in the semi-infinite experiment. Presumably, surfactant mass fraction at the base of the myelins also remained nearly constant at the calculated value of 49.7 wt.%. However, it would change eventually if the experiment were continued for a much longer time. Similar experiments were performed in which pure AOT was contacted with an aqueous solution containing 1 wt.% AOT instead of water, i.e., ωL1 = 0.01. The growth rates of the myelinic figures and of the L␣ and V1 phases were the
La/V1Interface position(mm)
J. Bai, C.A. Miller / Colloids and Surfaces A: Physicochem. Eng. Aspects 244 (2004) 113–119 2.4
1.6 Exp. Value Numerical solution 0.8
0 0
V1/H2 interface position(mm)
(a)
5000
10000
15000
Time(sec)
2.4
1.6 Exp. Value Numerical solution 0.8
117
Thus, both our experiments and the model indicate that the small amount of surfactant in the water flowing between the myelins has a minimal effect on myelin growth. There are limitations in applying this method to other systems. For example, even though the rate of growth of myelinic figures of phosphatidylcholine (PC) has been measured [7,9] and the phase behavior is well known, neither the effective diffusivity in the lamellar phase nor myelin composition can be calculated from penetration experiments. Since the lamellar phase is the only liquid crystalline phase present, there is no time t∗ for disappearance of another phase in the thin layer experiment as in the AOT case. Nevertheless, if osmotic pressure difference were known as a function of myelin composition, it might be possible to estimate De for the lamellar phase from a thin layer experiment which was continued until myelin growth rate deviated significantly from proportionality to the square root of time. Of course, myelin composition might also change once this
0 0
(b)
5000
10000
15000
Time(sec)
Fig. 4. Plots of interfacial positions for AOT/water thin layer experiment and comparison with simulation.
same as in Figs. 3 and 4 within experimental error. These results are discussed below.
4. Discussion The model of the preceding section is similar in some respects to that suggested by Buchanan et al. [5]. Key differences are that we have included diffusion in the liquid crystalline phases below the base of the myelins and combined results of the semi-infinite and thin layer penetration experiments. As a result, we are able to estimate the effective diffusivities. Moreover, we have not assumed that the myelins are fully swollen, i.e., have the composition of the lamellar phase in equilibrium with the L1 phase (12 wt.% for AOT). Instead, myelin composition is calculated as part of the data analysis. Haran et al. [4] have reported that using a dilute solution of AOT (of order 1 wt.%) instead of water as the initial aqueous phase caused the myelinic figures to grow more slowly in the initial experimental period (5–8 min). As indicated above, we did not observe such an effect in our longer experiments (2–3 h). We have no explanation for this discrepancy. Taking ωL1 = 0.01 instead of zero for the experiment shown in Fig. 3 yields a modified Table 1 from which the row best fitting the data of Fig. 4 is selected. Effective diffusivities De of 1.31, 1.72 and 0.39 × 10−10 m2 /s were found in this manner for L␣ , V1 , and H2 phases, respectively. The myelin surfactant content was 50.3 wt.%. These values are only slightly different from those given above for ωL1 = 0.
Fig. 5. (a) Video frame of our AOT myelins (high magnification, 2 h and 2 min after contact), (b) PC myelins from the literature.
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deviation began, a factor which could be incorporated into the (numerical) solution of the governing equations. Another consideration is that the analysis neglects transfer of surfactant from the outer surfaces of the myelins into the aqueous phase flowing toward their base. This neglect is entirely appropriate when solubility of the surfactant is very low, as for PC, C12 E3 , and similar lipophilic surfactants. However, it may not be as good for AOT and other ionic surfactants, which have higher solubilities. And when solubility is very large, e.g., in the C12 E5 /water system at ambient temperature where the L1 /L␣ phase boundary occurs at approximately 50 wt.% surfactant, diffusion dominates and no indication of myelinic figures is seen [1,5]. Since local equilibrium exists at the interface, there is no osmotic driving force there of the type incorporated in the above model. Indeed, experiments suggest that the myelins in the pure AOT system are not as clearly defined as in the PC and C12 E3 systems. Fig. 5a is a video frame taken from our experiments, while Fig. 5b shows more well developed myelins from the PC/water system [7]. Moreover, Temgire et al. [11] noted that, in contrast to the situation for PC myelins, the concentric bilayer internal structure of AOT myelins was not clearly evident in scanning electron micrographs. It is also noteworthy that well defined myelins were observed when a lamellar phase made with AOT and NaCl brine was contacted with NaCl brine [12]. Addition of NaCl reduces the solubility of AOT in the L1 phase [13] and thus would be expected to promote development of myelinic figures at the expense of diffusion of surfactant into the L1 phase. It appears that while some diffusion of surfactant into the aqueous phase may have occurred in our experiments described above, swelling and formation of myelinic figures dominated. Evidence for this conclusion is that if the data of Figs. 3 and 4 are analyzed assuming no myelins, bi-
Table 2 Calculated diffusivities of intermediate phases of 100% AOT at 30 ◦ C assuming all the interfaces are in equilibrium and no convection Delam (m2 /s)
DeL1 (m2 /s)
DeV1 (m2 /s)
DeH2 (m2 /s)
0.9 × 10−10 1.2 × 10−10 2.0 × 10−10
8.4 × 10−9 3.0 × 10−8 1.8 × 10−7
1.2 × 10−10 1.8 × 10−10 3.3 × 10−10
1.9 × 10−12 0.4 × 10−10 6.0 × 10−10
nary diffusion in all phases and planar interfaces, the values found for DeL1 are of order 10−8 m2 /s, far above the reasonable range, for plausible values of DL␣ of order 10−10 m2 /s (Table 2). This result indicates that convection is present in the L1 phase, which is consistent with the existence of flow of water to the roots of the myelins. We rule out the existence of significant flow due to natural convection since the experiment was conducted in a cell only 400 m thick in a room where temperature was maintained at 30 ◦ C. In addition, the top of the cell was sealed to prevent evaporation, which could promote natural convection. Reducing the osmotic driving force for myelin growth should also decrease the role of swelling and increase that of diffusion. In experiments where a lamellar phase containing only 34 wt.% AOT was contacted with water, we saw little evidence of myelinic figures (Fig. 6). Analysis of semi-infinite and thin layer experiments assuming no myelins and diffusion control yielded reasonable values of diffusivity of the L1 phase (Table 3), indicating that extensive convection was absent. Finally, it should be emphasized that whether and how myelinic figures develop depend on the structure of the lamellar phase at the scale of microns, e.g., the number and type of defects. Buchanan et al. [12] found in their work with the AOT/NaCl brine system that no myelins formed if
Fig. 6. Video frame of 34% AOT/water semi-infinite experiment at 30 ◦ C (2 min after contact).
J. Bai, C.A. Miller / Colloids and Surfaces A: Physicochem. Eng. Aspects 244 (2004) 113–119 Table 3 Calculated diffusivities of intermediate phases of 34% AOT at 30 ◦ C assuming a diffusion controlled process Delam (m2 /s) 10−10
1.1 × 1.2 × 10−10 1.31 × 10−10 1.44 × 10−10
119
of the myelinic figures which grew from the initial surface of contact toward the aqueous phase. Reasonable values of these quantities were obtained.
DeL1 /Delam 0.5 1.7 3.5 6.2
the lamellar phase was subjected to extensive shear before contact with brine, generating a microstructure of nearly spherical particles (onions). In our experiments with pure AOT, the lamellar phase was not present initially but nucleated near the initial surface of contact. It is not known what defect structure arose during this process, but lamellar microstructures involving concentric cylinders or tori are well known. If present, these could presumably be extruded during the swelling process. Further studies to investigate such matters would be of interest.
5. Summary Quantitative penetration experiments were conducted in the AOT/water system both with a relatively large amount of surfactant and with only a thin layer. A model has been developed which was used to obtain effective diffusivities in the L␣ , V1 , and H2 phases as well as the composition
Acknowledgements We thank Dr. John E. Wilson for discussion and Jie Yu for participating in some experiments.
References [1] B.H. Chen, C.A. Miller, P.R. Garrett, Langmuir 16 (2000) 5276. [2] B.H. Chen, C.A. Miller, P.R. Garrett, Colloids Surf. A: Physicochem. Eng. Aspects 191 (2001) 183. [3] J. Bai, C.A. Miller, J.E. Wilson, J. Surfactants Detergents 6 (2003) 7. [4] M. Haran, A. Chowdhury, C. Manohar, J. Bellare, Colloids Surf. A: Physicochem. Eng. Aspects 205 (2002) 21. [5] M. Buchanan, S.U. Egelhaaf, M.E. Cates, Langmuir 16 (2000) 3718. [6] I. Sakurai, T. Suzuki, S. Sakurai, Mol. Cryst. Liq. Cryst. 180B (1990) 305. [7] H. Dave, M. Surve, C. Manohar, J. Bellare, J. Colloid Interface Sci. 264 (2003) 76. [8] E.I. Franses, T.J. Hart, J. Colloid Interface Sci. 94 (1983) 1. [9] I. Sakurai, Y. Kawamura, Biochim. Biophys. Acta 777 (1984) 347. [10] J. Bai, Dissolution Rates of Surfactants and Granules, Ph.D. thesis, Rice University, 2003. [11] M.K. Temgire, C. Manohar, J. Bellare, Abstract for Presentation of Meeting of Electron Microscope Society of India, IIT Bombay, 2002. [12] M. Buchanan, J. Arrault, M.E. Cates, Langmuir 14 (1998) 7371. [13] O. Ghosh, C.A. Miller, J. Phys. Chem. 91 (1987) 4528.
Experiments and modeling of growth of myelinic figures in Aerosol OT/water system Jinhua Bai, Clarence A. Miller∗ Department of Chemical Engineering (MS 362), Rice University, P.O. Box 1892, Houston, TX 77251-1892, USA Received 23 October 2003; accepted 14 May 2004 Available online 18 August 2004
Abstract A quantitative penetration scan in a vertically oriented cell was used to determine behavior when water was brought into contact with neat Aerosol OT (AOT) at 30 ◦ C. Myelinic figures developed having roots at the initial surface of contact between the phases and length proportional to the square root of time, in agreement with previous observations for AOT and other surfactants by various workers. However, the present experiments also included measurement of positions of the interfaces between lamellar and cubic and cubic and reverse hexagonal phases, which were also proportional to the square root of time. A similar experiment in which a thin layer of AOT was contacted with a large quantity of water was also conducted. A model is presented which includes both growth of myelinic figures and diffusion in the various liquid crystalline phases. With this model and data from the two experiments effective diffusion coefficients for all three liquid crystalline phases and the composition of the myelinic figures were calculated. The former were found to be of order 10−10 m2 /s, the latter approximately 50 wt.% AOT. © 2004 Elsevier B.V. All rights reserved. Keywords: Aerosol OT; Myelinic figures; Diffusion; Dissolution; Anionic surfactants
1. Introduction The rate and manner of dissolution of surfactants in water is of considerable interest. In particular, continuing trends in household laundry processes toward decreased water usage, which requires higher surfactant concentrations, and toward lower temperatures have increased the importance of achieving rapid dissolution. One issue is whether viscous liquid crystalline phases formed during surfactant dissolution cause significant decreases in dissolution rates or trap solid particles of builders such as zeolites, preventing them from performing their functions. For isomerically pure, neat liquid nonionic surfactants below their cloud points Chen et al. [1] showed that dissolution rates were controlled by diffusion. Effective diffusion coefficients in both the initial aqueous and surfactant phases, as well as in the intermediate liquid crystalline phases which formed, were calculated by combining observations of dissolution times for surfactant drops injected into ∗
Corresponding author. Tel.: +1 713 348 4904; fax: +1 713 348 5478. E-mail address: [email protected] (C.A. Miller).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.05.004
water with measurements of the time-dependent positions of the various interfaces in a vertically oriented penetration cell in which the surfactant and water were contacted without mixing. Values of order 10−10 m2 /s were found. At temperatures well below their cloud points, drops of nonionic surfactant mixtures or commercial nonionic surfactants dissolved at rates comparable to those for the pure surfactants. However, dissolution was slower and more complex in nature at temperatures just below the cloud point [2]. Bai et al. [3] found that granules about 500 m in diameter consisting of micron-size particles of a nonswelling zeolite (approximately 70% by volume) held together by a pure, liquid nonionic surfactant below its cloud point disintegrated rapidly in water, releasing the zeolite particles. However, no significant disintegration occurred at temperatures above the cloud point. Similar behavior was observed for granules made with binary mixtures of pure nonionic surfactants. Pure ionic surfactants are typically crystalline solids or highly viscous liquid crystalline phases near ambient temperature. In the former case, significant porosity exists if crystals are packed into a penetration cell such as that used in the nonionic surfactant work, making interpretation of the
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results problematic. In the latter case, use of quantitative penetration scans is feasible if one can provide an initial surfactant region with relatively few air bubbles and a reasonably smooth surface, as required to justify the one-dimensional analysis used to interpret the results. Another complicating factor for some anionic surfactants is that dissolution is not always completely controlled by diffusion. For example, Haran et al. [4] have reported that, when contacted with water, the pure anionic surfactant Aerosol OT (AOT) exhibits myelinic figures formed by swelling of an intermediate lamellar liquid crystalline phase. Moreover, they confirmed for this system the previous finding of Buchanan et al. [5] for the pure nonionic surfactant C12 E3 at ambient temperature that such swelling is accompanied by flow of the aqueous phase through the spaces between the myelinic figures toward the roots of the myelins. As the solubility of C12 E3 in water is extremely low for these conditions, a very large quantity of water would be required for complete dissolution, which is not ordinarily observed for this surfactant in practical applications. Solubility in water is also minimal for phospholipids, whose ability to exhibit myelinic figures is well known [4,6,7]. In contrast, the solubility of AOT in water is much larger (1.4 wt.%), and complete dissolution is expected for many situations of interest. Electron micrographs of phospholipid myelins showed that the bilayers were concentric cylinders [6], and a similar structure is presumed for myelins in other surfactant systems. In this paper, we describe results of quantitative penetration experiments in vertically oriented cells in which AOT is contacted with water, both when relatively large quantities of both phases are present initially and can be considered semi-infinite in extent for purposes of analysis and when a thin layer of AOT is contacted by a large quantity of water. In the latter case, the original reverse hexagonal (H2 ) phase of AOT is completely dissolved during the time frame of the experiment. In both cases myelinic figures grow upward from the initial surface of contact. A simple model is presented which permits effective diffusion coefficients to be calculated for all phases below the initial surface of contact, namely lamellar L␣ , viscous isotropic or cubic V1 , and H2 phases. It also yields a value for composition of the myelinic figures.
2. Experimental Sodium bis(2-ethylhexyl) sulfosuccinate, also known as Aerosol OT or AOT, of purity greater than 96%, was purchased from TCI America, Portland, OR. Water used in the experiments was deionized. Linear penetration experiments were conducted using vertically oriented rectangular capillary cells (50 mm × 4 mm × 0.4 mm, Vitro Dynamics Inc., Cat. No. 2540) placed in a microscope with a vertical stage [1]. Pure AOT, which is a highly viscous H2 phase, was placed in the bottom of the cell and manipulated with glass micropipet to achieve a
nearly horizontal interface several mm above the bottom of the cell. Then the bottom of the cell was sealed and water or an aqueous solution of AOT was carefully injected from the top of the cell. Videomicroscopy was used to record the resulting behavior and especially to follow the positions of all interfaces as a function of time. The experiments were conducted in an environmental room maintained at 30 ◦ C. A novel aspect of the present work was similar experiments with only a thin layer (<2 mm) of surfactant to provide additional information on effective diffusion coefficients as discussed below.
3. Results and interpretation The phase diagram of the AOT/water system is shown in Fig. 1 [8]. It indicates that intermediate lamellar L␣ and viscous isotropic V1 phases should form when the pure AOT H2 phase is brought into contact with water without stirring. Such behavior was, in fact, observed, as shown in Fig. 2, except that the portion of the lamellar phase above the surface of initial contact consisted of many vertically oriented, closely packed myelinic figures. The base of the myelinic figures remained at the surface of initial contact throughout the experiment. Fig. 3 shows that the length of the myelinic figures was proportional to the square root of time, a result that has been found by several groups for different systems [4,5,7,9]. Also shown by Fig. 3 is that the displacement of the phase boundaries between the various liquid crystalline phases from the surface of initial contact was proportional to the square root of time. These observations suggest that transport in the liquid crystalline phases can be modeled by ordinary diffusion. We consider the region x > 0, where x is the distance measured from the base of the myelins toward the H2 phase (downward in Fig. 2). If each phase i has a constant and uniform effective diffusivity Dei , the governing differential equation therein is given by 2 ∂ωi ∂ ωi = Dei (1) ∂t ∂x2 where ωi is surfactant mass fraction. At the phase boundaries between H2 and V1 and between V1 and L␣ , local equilibrium is assumed in accordance with Fig. 1 and conservation of mass of surfactant is imposed. When both initial
Fig. 1. Phase diagram for AOT/water system at 25 ◦ C.
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115
Thickness of myelins (mm)
1.2 1 1/2
Slope = 0.106mm/min
0.8 0.6 0.4 0.2 0 0
3
Time
(a)
9
6 1/2
12
1/2
(min )
2.5
Slope=0.107mm/min1/2
Thickness of La(mm)
2 1.5 1 0.5 0 0
6
3
1/2
(b)
Time
9
12
1/2
(min )
The growth rate of V1 phase 0.25
phases are present in large amounts, the differential equation, initial condition, and these boundary conditions can be satisfied by the following similarity solution, provided that the displacement of all phase boundaries is proportional to the square root of time as observed experimentally: ωi = a1i + a2i erf ηi ,
Thickness of V1 (mm)
Fig. 2. Video frames of AOT/water semi-infinite experiment at 30 ◦ C (low magnification, the width of the cell is 4 mm).
0.2
Slope=0.0229mm/min1/2
0.15 0.1 0.05
(2)
where ηi = [x/(4Dei t)1/2 ]. One additional boundary condition is that surfactant mass fraction in the H2 phase must approach its initial value at distances far from the surface of contact, i.e., as x → +∞. So far, the analysis is similar to that used previously for penetration experiments involving pure nonionic surfactants [1]. What is new here is the condition for conservation of mass of surfactant at the base of the myelins x = 0: ∂ωlam d Delam (0) + jaq ωL1 = [εm t 1/2 ωlam (0)φ] ∂x dt (3) The first term on the left side of this equation represents the flux of surfactant to x = 0 from the portion of the lamellar phase below the myelinic figures. The second term represents surfactant dissolved in the aqueous phase, which, as discussed above, flows between the myelins to their base at x = 0. jaq is the total flux and ωL1 is the surfactant content.
0 0
(c)
3
6 1/2
Time
9
12
1/2
(min )
Fig. 3. Plots of interfacial positions for AOT/water semi-infinite experiment at 30 ◦ C.
That is, AOT could be contacted with a solution with mass fraction ωL1 of AOT instead of with pure water, as discussed below. On the right side of the equation εm t1/2 is the length of the myelins with εm a constant, ωlam (0) is surfactant mass fraction in the lamellar phase at the base of the myelins and in the myelins themselves, and φ is the fraction of the cross sectional area occupied by the myelins. Surfactant mass fraction is assumed uniform along the myelins, i.e., diffusion along the myelins is neglected, since they are of nearly uniform diameter and are forced out of the lamellar phase by the swelling process. For uniform close packed cylinders it can be shown that φ is approximately 91%, in good agreement with the value found for myelinic figures of C12 E3 [5].
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An overall balance for the entire surfactant-containing region including the myelins requires that the flux of aqueous solution entering from the bulk L1 phase be equal to the rate of increase in volume of the myelins: d jaq = [εm t 1/2 φ] (4) dt
Substitution of Eqs. (2) and (4) into Eq. (3) yields εm φ(a1lam − ωL1 ) Delam 1/2 a2lam = π 2
Delam (m2 /s)
DeV1 (m2 /s)
1.2 × 10−10 1.31 × 10−10 1.44 × 10−10 2.0 × 10−10 3.0 × 10−10
1.59 1.70 1.82 2.20 2.63
× × × × ×
10−10 10−10 10−10 10−10 10−10
DeH2 (m2 /s)
Myelin mass fraction of AOT
1.94 × 10−11 3.56 × 10−11 5.6 × 10−11 1.47 × 10−10 2.87 × 10−10
0.483 0.497 0.513 0.562 0.617
(5)
Suppose that it is desired to predict the rate of growth of myelins and all intermediate phases when pure AOT is contacted with water, starting with knowledge of all effective diffusion coefficients and the phase diagram of Fig. 1. Eq. (1) and its solution Eq. (2) apply for all three phases below the surface of contact. The two interfaces between them must have positions εi t1/2 to be consistent with the similarity solution. Thus, there are nine unknowns to be determined from the boundary conditions: three a1i , three a2i , two εi and εm . Five equations in these unknowns can be obtained from the boundary condition for x → +∞ and the known compositions of each phase at the two interfaces between liquid crystalline phases. Three more equations come from surfactant mass balances at the these interfaces and from Eq. (5). The last boundary condition involves the mechanism for growth of myelinic figures. Let us assume that their rate of growth is proportional to the osmotic pressure difference p between the bulk aqueous solution and the lamellar phase at the base of the myelins (x = 0), a quantity which is independent of time for the present similarity solution. Let us further assume that the growth rate is inversely proportional to myelin length. For example, contact between the closely packed myelins likely provides resistance to the swelling process. With these assumptions, we find dεm t 1/2 k p = (6) dt (εm t 1/2 ) where k is a constant incorporating the various proportionality factors. This equation is readily simplified to εm = (2k p)1/2
Table 1 Diffusivity values and myelin composition for assumed values of De in lamellar phase (AOT/water semi-infinite experiment)
(7)
The osmotic pressure difference depends on ωL1 and the composition a1lam of the lamellar phase at x = 0 but does not introduce any new unknowns. Thus, if k can be estimated, the equations can be solved to determine the concentration distributions and rates of interfacial motion. In contrast to this hypothetical situation, we do not in our present experiments know k (or the product k p, which may be considered a single unknown), or the three diffusivities Dei . However, we can measure interfacial positions as a function of time and obtain ε1 , ε2 and εm . That is, we now have ten unknowns (including k p) with the same nine equations. Consequently, the problem cannot be solved
completely using results of the semi-infinite penetration experiment alone. What can be done is to solve for the effective diffusivities of the H2 and V1 phases and for myelin composition with an assumed value of effective diffusivity of the L␣ phase. Table 1 presents the results of such calculations for the AOT data of Fig. 3 where ωL1 = 0, indicating that plausible values of all the parameters can be found for a range of values for Delam slightly larger than 10−10 m2 /s. For pure nonionic surfactants, it was also found that the semi-infinite penetration experiment was insufficient to determine all effective diffusivities. In that case, an additional measurement of the time required for a surfactant drop of known initial diameter to completely dissolve provided additional information that allowed the diffusivities to be calculated [1]. This approach clearly cannot be used here, since pure AOT is highly viscous and not readily formed into small drops. A suitable alternative is to carry out penetration experiments starting with a thin layer of AOT and a large amount of water and continuing until time t∗ when the H2 phase disappears completely. Since after some initial time period the concentration profile in the H2 phase departs from that of the similarity solution owing to the no flux boundary condition at the bottom of the cell, a numerical solution of the governing equations can be used with the measured value of t∗ to find values for all the effective diffusivities and myelin composition, i.e., to select the best row of values from Table 1. Details of the numerical procedure may be found elsewhere [10]. Measurements of positions of the various interfaces as a function of time for such a thin layer experiment for AOT are shown in Fig. 4 along with the results of simulations using De values of 1.31, 1.70 and 0.36 × 10−10 m2 /s for L␣ , V1 , and H2 phases, respectively. Agreement is good. It should be noted that the myelinic figures were far enough from the bottom of the cell that their base remained at the initial contact surface within experimental error and that their growth rate was virtually the same as in the semi-infinite experiment. Presumably, surfactant mass fraction at the base of the myelins also remained nearly constant at the calculated value of 49.7 wt.%. However, it would change eventually if the experiment were continued for a much longer time. Similar experiments were performed in which pure AOT was contacted with an aqueous solution containing 1 wt.% AOT instead of water, i.e., ωL1 = 0.01. The growth rates of the myelinic figures and of the L␣ and V1 phases were the
La/V1Interface position(mm)
J. Bai, C.A. Miller / Colloids and Surfaces A: Physicochem. Eng. Aspects 244 (2004) 113–119 2.4
1.6 Exp. Value Numerical solution 0.8
0 0
V1/H2 interface position(mm)
(a)
5000
10000
15000
Time(sec)
2.4
1.6 Exp. Value Numerical solution 0.8
117
Thus, both our experiments and the model indicate that the small amount of surfactant in the water flowing between the myelins has a minimal effect on myelin growth. There are limitations in applying this method to other systems. For example, even though the rate of growth of myelinic figures of phosphatidylcholine (PC) has been measured [7,9] and the phase behavior is well known, neither the effective diffusivity in the lamellar phase nor myelin composition can be calculated from penetration experiments. Since the lamellar phase is the only liquid crystalline phase present, there is no time t∗ for disappearance of another phase in the thin layer experiment as in the AOT case. Nevertheless, if osmotic pressure difference were known as a function of myelin composition, it might be possible to estimate De for the lamellar phase from a thin layer experiment which was continued until myelin growth rate deviated significantly from proportionality to the square root of time. Of course, myelin composition might also change once this
0 0
(b)
5000
10000
15000
Time(sec)
Fig. 4. Plots of interfacial positions for AOT/water thin layer experiment and comparison with simulation.
same as in Figs. 3 and 4 within experimental error. These results are discussed below.
4. Discussion The model of the preceding section is similar in some respects to that suggested by Buchanan et al. [5]. Key differences are that we have included diffusion in the liquid crystalline phases below the base of the myelins and combined results of the semi-infinite and thin layer penetration experiments. As a result, we are able to estimate the effective diffusivities. Moreover, we have not assumed that the myelins are fully swollen, i.e., have the composition of the lamellar phase in equilibrium with the L1 phase (12 wt.% for AOT). Instead, myelin composition is calculated as part of the data analysis. Haran et al. [4] have reported that using a dilute solution of AOT (of order 1 wt.%) instead of water as the initial aqueous phase caused the myelinic figures to grow more slowly in the initial experimental period (5–8 min). As indicated above, we did not observe such an effect in our longer experiments (2–3 h). We have no explanation for this discrepancy. Taking ωL1 = 0.01 instead of zero for the experiment shown in Fig. 3 yields a modified Table 1 from which the row best fitting the data of Fig. 4 is selected. Effective diffusivities De of 1.31, 1.72 and 0.39 × 10−10 m2 /s were found in this manner for L␣ , V1 , and H2 phases, respectively. The myelin surfactant content was 50.3 wt.%. These values are only slightly different from those given above for ωL1 = 0.
Fig. 5. (a) Video frame of our AOT myelins (high magnification, 2 h and 2 min after contact), (b) PC myelins from the literature.
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deviation began, a factor which could be incorporated into the (numerical) solution of the governing equations. Another consideration is that the analysis neglects transfer of surfactant from the outer surfaces of the myelins into the aqueous phase flowing toward their base. This neglect is entirely appropriate when solubility of the surfactant is very low, as for PC, C12 E3 , and similar lipophilic surfactants. However, it may not be as good for AOT and other ionic surfactants, which have higher solubilities. And when solubility is very large, e.g., in the C12 E5 /water system at ambient temperature where the L1 /L␣ phase boundary occurs at approximately 50 wt.% surfactant, diffusion dominates and no indication of myelinic figures is seen [1,5]. Since local equilibrium exists at the interface, there is no osmotic driving force there of the type incorporated in the above model. Indeed, experiments suggest that the myelins in the pure AOT system are not as clearly defined as in the PC and C12 E3 systems. Fig. 5a is a video frame taken from our experiments, while Fig. 5b shows more well developed myelins from the PC/water system [7]. Moreover, Temgire et al. [11] noted that, in contrast to the situation for PC myelins, the concentric bilayer internal structure of AOT myelins was not clearly evident in scanning electron micrographs. It is also noteworthy that well defined myelins were observed when a lamellar phase made with AOT and NaCl brine was contacted with NaCl brine [12]. Addition of NaCl reduces the solubility of AOT in the L1 phase [13] and thus would be expected to promote development of myelinic figures at the expense of diffusion of surfactant into the L1 phase. It appears that while some diffusion of surfactant into the aqueous phase may have occurred in our experiments described above, swelling and formation of myelinic figures dominated. Evidence for this conclusion is that if the data of Figs. 3 and 4 are analyzed assuming no myelins, bi-
Table 2 Calculated diffusivities of intermediate phases of 100% AOT at 30 ◦ C assuming all the interfaces are in equilibrium and no convection Delam (m2 /s)
DeL1 (m2 /s)
DeV1 (m2 /s)
DeH2 (m2 /s)
0.9 × 10−10 1.2 × 10−10 2.0 × 10−10
8.4 × 10−9 3.0 × 10−8 1.8 × 10−7
1.2 × 10−10 1.8 × 10−10 3.3 × 10−10
1.9 × 10−12 0.4 × 10−10 6.0 × 10−10
nary diffusion in all phases and planar interfaces, the values found for DeL1 are of order 10−8 m2 /s, far above the reasonable range, for plausible values of DL␣ of order 10−10 m2 /s (Table 2). This result indicates that convection is present in the L1 phase, which is consistent with the existence of flow of water to the roots of the myelins. We rule out the existence of significant flow due to natural convection since the experiment was conducted in a cell only 400 m thick in a room where temperature was maintained at 30 ◦ C. In addition, the top of the cell was sealed to prevent evaporation, which could promote natural convection. Reducing the osmotic driving force for myelin growth should also decrease the role of swelling and increase that of diffusion. In experiments where a lamellar phase containing only 34 wt.% AOT was contacted with water, we saw little evidence of myelinic figures (Fig. 6). Analysis of semi-infinite and thin layer experiments assuming no myelins and diffusion control yielded reasonable values of diffusivity of the L1 phase (Table 3), indicating that extensive convection was absent. Finally, it should be emphasized that whether and how myelinic figures develop depend on the structure of the lamellar phase at the scale of microns, e.g., the number and type of defects. Buchanan et al. [12] found in their work with the AOT/NaCl brine system that no myelins formed if
Fig. 6. Video frame of 34% AOT/water semi-infinite experiment at 30 ◦ C (2 min after contact).
J. Bai, C.A. Miller / Colloids and Surfaces A: Physicochem. Eng. Aspects 244 (2004) 113–119 Table 3 Calculated diffusivities of intermediate phases of 34% AOT at 30 ◦ C assuming a diffusion controlled process Delam (m2 /s) 10−10
1.1 × 1.2 × 10−10 1.31 × 10−10 1.44 × 10−10
119
of the myelinic figures which grew from the initial surface of contact toward the aqueous phase. Reasonable values of these quantities were obtained.
DeL1 /Delam 0.5 1.7 3.5 6.2
the lamellar phase was subjected to extensive shear before contact with brine, generating a microstructure of nearly spherical particles (onions). In our experiments with pure AOT, the lamellar phase was not present initially but nucleated near the initial surface of contact. It is not known what defect structure arose during this process, but lamellar microstructures involving concentric cylinders or tori are well known. If present, these could presumably be extruded during the swelling process. Further studies to investigate such matters would be of interest.
5. Summary Quantitative penetration experiments were conducted in the AOT/water system both with a relatively large amount of surfactant and with only a thin layer. A model has been developed which was used to obtain effective diffusivities in the L␣ , V1 , and H2 phases as well as the composition
Acknowledgements We thank Dr. John E. Wilson for discussion and Jie Yu for participating in some experiments.
References [1] B.H. Chen, C.A. Miller, P.R. Garrett, Langmuir 16 (2000) 5276. [2] B.H. Chen, C.A. Miller, P.R. Garrett, Colloids Surf. A: Physicochem. Eng. Aspects 191 (2001) 183. [3] J. Bai, C.A. Miller, J.E. Wilson, J. Surfactants Detergents 6 (2003) 7. [4] M. Haran, A. Chowdhury, C. Manohar, J. Bellare, Colloids Surf. A: Physicochem. Eng. Aspects 205 (2002) 21. [5] M. Buchanan, S.U. Egelhaaf, M.E. Cates, Langmuir 16 (2000) 3718. [6] I. Sakurai, T. Suzuki, S. Sakurai, Mol. Cryst. Liq. Cryst. 180B (1990) 305. [7] H. Dave, M. Surve, C. Manohar, J. Bellare, J. Colloid Interface Sci. 264 (2003) 76. [8] E.I. Franses, T.J. Hart, J. Colloid Interface Sci. 94 (1983) 1. [9] I. Sakurai, Y. Kawamura, Biochim. Biophys. Acta 777 (1984) 347. [10] J. Bai, Dissolution Rates of Surfactants and Granules, Ph.D. thesis, Rice University, 2003. [11] M.K. Temgire, C. Manohar, J. Bellare, Abstract for Presentation of Meeting of Electron Microscope Society of India, IIT Bombay, 2002. [12] M. Buchanan, J. Arrault, M.E. Cates, Langmuir 14 (1998) 7371. [13] O. Ghosh, C.A. Miller, J. Phys. Chem. 91 (1987) 4528.