Langmuir 1992,8, 691-709
691
The Cubic Phase Region in the System Didodecyldimethylammonium Bromide-Water-Styrene Pelle Strom* and David M. Anderson+ Physical Chemistry 1, University of Lund, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden Received April 14,1991.In Final Form: September 4, 1991 The determination of five separate one-phase regions, corresponding to five distinct bicontinuous cubic phase microstructures, in the phase diagram of DDAB (didodecyldimethylammoniumbromide)-waterstyrene at 20 "C, provides a unique opportunity to study progressions between distinct cubic phases. A thorough phase behavior study of this region is reported, involving hundreds of samples and aided by the NMR self-diffusion technique and small-angle X-ray scattering (SAXS) technique. Two-phase samples have been found in each of the four two-phase regions separating these different monophasic regions, as well as the required three-phase samples involving cubic, lamellar, water-rich, and microemulsion phases. The structural sequence for the four cubic regions starting at lowest water content and increasing, identified by reference to the minimal surface dividing the infinitely connected bilayer, is proposed to be G (Ia3d) D (Pn3m) P (Im3m) C(P)(Im3m) (Neovius' surface). The structure of the fifth cubic phase is unknown, but it is shown that the minimal surface describing this structure has a low value of y = -(2/.lr)(S/V2/3)3/~~, more specifically y C 1.7and probably y C 1.5, where the dimensionless area and the Euler characteristicare measured over any representativepatch of surface. We conjecture that the "C(D)" minimal surface describes this cubic phase. A free energy theory based on surfactant chain lengths and head group areas givesquanitativeagreementwith the observed phase boundaries and with latticeparameters reported in the literature with cyclohexane and octane as hydrophobe. This progression can also be qualitativelyunderstood in terms of simple arguments focusing on the mean curvature of the polar-apolar interface and does not require the introduction of a second curvature free energy contribution such as a Gaussian curvature term; a more precise interpretation is that the mean curvature term far outweighs the "difference curvature" term. The same two theoretical approaches also explain the strong slope of the phase boundaries with respect to the lines of constant w@er content, the "slowing down" of the lattice parameter increase as the phase boundaries are approached, and the tell-tale jump in the monolayer mean curvature across each of the two-phase regions. The water self-diffusion rate plotted as a functionof water volume fraction falls on five curves corresponding to the five one-phase regions, with discontinuities at the phase boundaries. These discontinuitiesfacilitate precise structural comparisons between different microstructures. A quantitative fit of the water self-diffusion is obtained, using the results of recent solutionsto the diffusion equation in cubic phase models, and the results show that the interconnected-rod description is best at low water and the parallel or H-surface description is best at high water. Polymerizationof styrene was performed in samples from the first four cubic phase regions, and SAXS,deuterium NMR (on samples containing DzO), and water self-diffusion rates for the polymerized samples were indistinguishable from unpolymerized samples.
-
-
-
Introduction The surfactant didodecyldimethylammonium bromide (DDAB) is insoluble in both water and hydrocarbons and is therefore located a t the interface between water and hydrocarbon in ternary DDAB-water-hydrophobe systems. The presence of two hydrocarbon chains on each surfactant molecule generally results in microstructures in which the interfacial surfactant-rich film has zero or reversed mean curvature, i.e. curvature toward water. Of current interest is the relatively large cubic phase area that appears with many different hydrophobic compounds.' The microstructure in eachof those cubic phases which have been analyzed is presently described by a triply periodic minimal surface over which an oil-swollen surfactant bilayer is d r a ~ e d . ~ s Over the past 5 years there have been different proposals as to which space groups the DDAB cubic phases belong to, and furthermore, once the space group is rigorously established, there is still the question as to the identity of + Present address: Biomaterials DeDartment. SUNY Buffalo, Buffalo, NY 14214. (1) Fontell, K.; Jansson, M. h o g . Colloid Polym. Sci. 1988, 76,169. (2) Barois, P.; S. T.: Ninham, B.; Dowline, . Hyde. - T. Lanamuir 1990, 6, 1136-1140. (3) Radiman, S.; Toprakcioglu, C.; Faragi, A. R. J . Phys. (Paris) 1990, 51. 1501.
the minimal surface and thus to the toDologv of the microstr~cture.'-~In a related system, with a ijxture of single- and double-tailed quaternary ammonium surfactants turned to approximately zero preferred mean curvature of the monolayer and with considerable oil present, there is even a cubic phase which has been rather convincingly shown to be described by a monolayer of surfactant conforming not to a minimal surface but rather to a surface of constant, nonzero mean curvature.6 In the particular case of DDAB-water-hydrocarbon, until fairly recently it was tacitly assumed that the whole cubic phase region conformed to one space group and one s t r ~ c t u r e(note: ~ ? ~ we consider that a variation in lattice parameter does not in itself constitute a change in structure; a true change in structure implies a phase transition). In 1988 Fontell was the first to propose a change of structure within the cubic phase region in DDAB-water-hexene, at the point where the water content exceeds about 60% .l This has then been followed up by Radlinska et al.7 and Barois et a1.2who showed, with help (4) Anderson, D. M. PhD Thesis, University of Minnesota, 1986. See
also Anderson, D. M.; Nitache, J. C. C.; Davis, H. T.;Scriven, L. E. Adu.
Chem. Phys. 1990, 77,337. (5) Fontell, K.; Ceglia, A.; Lindman, B.; Ninham, B. W. Acta Chem. Scand., Ser. A 1986, A40, 241. (6) Mdler, J. D.; Radiman, S.;De Vallera, A,; Toprakcioglu, C. Physica B 1989,156,398.
0743-7463/92/2408-0691$03.00/0 0 1992 American Chemical Society
Strom and Anderson
692 Langmuir, Vol. 8, No. 2, 1992
of specific heat and small angle X-ray scattering (SAXS) measurements, that the cubic phase DDAB-water-cyclohexane contains at least two, possibly more, different cubic phases separated by narrow two-phase regions; these data were consistent with a transition from a structure described by the Schwarz "D" minimal surface to one described by the Schwarz "P" minimal surface, at about 50 wt 7% water content. Our original motivation for choosing styrene as the hydrophobe was that the cubic phase could then be polymerized, resulting in a microporous material with a triply periodic pore space, and exhibiting monodisperse pores.8~~ Indeed, we present SAXS and NMR self-diffusion data in this paper showing that these cubic phases can be polymerized with retention of the triply periodic structure. Interestingly, we found that when styrene is used as the hydrophobe, the cubic phase region extends over a huge range in water content, from about 9 wt 7% to 80 wt 7% water. The transitions observed by Fontell and that observed by Radlinska and Barois are observed, and in fact there are exactly five distinct structures within the cubic phase region, separated of course by two- and threephase regions conforming to the laws of first-order phase transitions. There is no mystery to the observation that the lowest water content cubic phases are also those that contain the highest levels of hydrophobe and the highest hydrophobe: surfactant ratios, for it is at low water volume fractions that the highest (reversed) curvatures are required. However, there are two mechanisms by which the presence of a third, hydrophobic component such as styrene can affect the mean curvature a t the polar-apolar interfacg, and distinguishing between these two possibilities is not yet possible although some results bearing on this question will be given in this work. In terms of the u / A L packing parameter of the Ninham and Israelachvili school, the increase of hydrophobe can increase u, the bilayer volume per surfactant molecule including hydrophobe.ll This is simply the usual "wedge" effect, increasing the preferred mean curvature at the polar-apolar interface. On the other hand, without affecting the preferred mean curvature, it can affect the actual monolayer curvature by forming a separate layer of pure hydrophobe between the ends of the surfactant chains of the bilayer, thus increasing L in the formula (H)= 0 ~ ? / y L(see eq 7 below; also ref 12). In the latter case the bilayer width, computed from SAXS lattice parameters, shows a composition dependence, which at the hydrophobe:DDAB ratios for the cubic phase treated in this paper would be always less than 5 A, even at the extremes of styrene content, making it a difficult question to settle experimentally. The propensity for styrene to penetrate deep into the palisade layer in this system would tend to favor the first mechanism. This large penetration is due to two effects. First of all, the aromatic ring is well-known to sandwich between cationic head groups (see, e.g., ref 10) because of its electron density distribution. And second, the ?r bond of the vinyl group makes styrene more polar. To a first approximation, both effects are (7) Radlinska, E.; Hyde, S. T.; Ninham, B. Langmuir 1989,5, 1427. (8) Anderson, D. M.; Strom, P. Microemulsion and Liquid Crystals; Polymer Association Structures; ACS Symposium Series 384, El-Nokaly, M. A., Ed.; American Chemical Society: Washington, DC, 1989;pp 204. (9) Anderson, D. M. U.S. Patent Application052,713,1986;EPO Patent Application 88304625; Japanese Patent Application 63-122193. (b) Anderson, D. M. In Proceedings of International Workshop on Geometry and Interfaces, Aussois, France, Sept. 17-22,1990. Suppl. 23 to J . Phys., Colloq. p C7-1. (10) Eriksson, J. C.;,Gillberg, G. Acta Chem. Scand. 1966, 2019. (11) Hyde, S.2. Knstallogr. 1989, 187,165. (12) Anderson, D. M.; Wennerstrdm, H.; Olsson, U. J . Phys. Chem. 1989, 93,4243.
working in the same direction at stabilizing the present cubic phases at low water content; namely, the presence of styrene is acting to reduce the disparity between the low preferred mean curvature (which is generally low in the binary DDAB-water system) and the high mean curvature required by the constraints of volume fractions, both by increasing the preferred mean curvature through penetration and decreasing the curvature required by volume fractions through increasing L. Thus a precise determination of the bilayer width bears directly on a delicate balance which contains information about the underlying physics of this and related systems. In addition this question may also bear on the general question of whether cubic phases made with ionic surfactants can be swollen with oil, where by "swollen" we mean the actual increase of structural dimensions due to separate regions of pure oil; the answer in the case of lamellar phases made with single-tailed ionic surfactants appears to be "n0*.13 The complex phase behavior in this cubic phase region called for the mixing of a very large number of samples (several hundred), with samples of at least 1g which were equilibrated for several months. This was combined with NMR self-diffusion data and with SAXS results in the literature for DDAB cubic phases withother hydrophobes, which were in agreement with our own SAXS taken for at least one sample in each of the five regionsin the present styrene system. Accurate determination of the phase behavior then justified the development of accurate theories for interpretation of the phase behavior, SAXS, and self-diffusiondata, all of which provide further support for our structural assignments. Experimental Procedure Materials. DDAB obtained from Fluka was purified as described elsewhere (see refs 14 and 15). Styrene 99%obtained from Merck contained 20 ppm 4-tert-butylbrezcatechin as inhibitor and was used as received. DVB (divinylbenzene) obtained from Dow was purified from inhibitor by passing it through a column of basic aluminumoxide (obtained from BDH). The water was twice-distilled. DzO (99%)obtained from Norsk Hydro was used in the deuterium NMR measurements to determine the phase boundaries of the lamellar phases, and in some experiments on polymerized samples. Sample Preparation. All samples were prepared by weighing the components in glass ampules or test tubes, which were immediately flame sealed, except for the samples made for polymerization which were flushed with inert nitrogen gas. The samples for the NMR self-diffusion measurement were all prepared and equilibrated into test tubes that fit exactly in 10 mm 0.d. tubes designed for NMR. Due to danger of polymerization of styrene, and particularly DVB, the usual procedure for equilibration (see e.g. ref 5), involving temperatures in the 100 "C range, could not be used. The samples were therefore equilibrated by placing them, after a mild centrifugation for several minutes in a table centrifuge (3000 rpm) at 20 "C, in darkness to avoid initiation of polymerization by light, for at least 1 week. After that the samples were centrifuged back and forth in the test tubes in a table centrifuge several times. Then theywere placed back in darkness at 20 O C for at least 1 month before any measurements were taken. The measurements were taken at 20 O C . Every sample was checked for optical isotropy between crossed polars. All samples used for the phase diagram determination showed no sign of polymerization and were unchanged over a time period of more than 1 year. ~~~~
~~
(13) Jansson, B. PhD Thesis, University of Lund, 1981. (14) Chen, S.J.; Evans, D. F.; Ninham, B. W. J. Phys. Chem. 1984,88, 1631. (15) Ninham, B.W.; Chen, S. J.; Evans, D. F. J. Phys. Chem. 1984,88, 5855.
Cubic Phase Region in DDAB- Water-Styrene styrene
807
4
0
,,/
40
4
c i p! \ \
I
\\c
20
Figure 1. Partial phase diagram for the DDAB-waterstyrene system a t 20 O C , showing two lamellar phases, a liquid isotropic phase area, and five separate cubic phases.
NMR Measurements. Proton NMR measurements were performed on a Bruker operating a t 300 MHz. Deuterium NMR measurements were performed on a Bruker MSL-100, operating at 15.371 MHz. For the determination of self-diffusion coefficients the same spectrometer, operating a t 100.15 MHz, was used with the Fourier-transformpulsed-gradientspin-echo (FT-PGSE) method described by Stilbs.lG The temperature in the NMR probe was 20 i 0.5 "C. The self-diffusionrate of bulk water was measured after every measurement to check the stability of the gradients in the NMR magnet; it showed no perceptible change. SAXS Measurements. For the samples with styrene as oil, the Luzzati-type camera operated by Krister Fontell was employed, as described elsewhere." Polymerized samples were broken into small pieces and loaded into capillaries. Polymerization. Polymerization reactions were initiated by light at 3500 A in a Rayonet light reactor. The temperature was controlled by placing the samples in a jacketed glass vessel connected to a water bath, with a temperature stability of f0.5 OC. The reaction time was a t least 48 h.
Experimental Results Phase Behavior and Physical Characteristics. The phase diagram DDAB-water-styrene shows, as with other hydrophobes,' two lamellar phases, one between 4 and 30 wt % water swollen to 3 wt % by styrene and one between 83 and 89 wt % water swollen to -15 wt % by styrene. The LZphase region is rather small, compared to the L2 phases in phase diagrams with alkanes and alkenes.4~5J4J5J&24 The cubic phase region extends over a huge range, from 9% to 80% water (see Figure 1). In agreement with the four observed two-phase regions within the cubic phase region and with other data presented herein, the cubic phase area was separated into (16) Stilbs, P. Prog. N u l . Magn. Spectrosc. 1987, 19, 1. (17) Fontell, K.; Mandell, L.; Lehtinen, H.; Ekwall, P. Acta Polytech. Scan. 1968, Ch 74 111. (18) Ninham, B. W.; Evans, D. F.; Wei, T. J.Phys. Chem. 1983.84, 5029. (19) Angel, L. R.; Evans, D. F.; Ninham, B. W. J. Phys. Chem. 1983, 87, 538. (20) Hashimuoto, S.; Thomas, J. K.; Evans, D. F.; Ninham, B. W. J. Colloid Interface Sci. 1983, 95, 594. (21) Talmon, Y.;Evans, D. F.; Ninham, B. W. Science 1983,95,594. (22) Brady, J. E.; Evans, D. F.; Kachar, B.; Ninham, B. W. J. Am. Chem. SOC.1984,106,4279. (23) Pickup, S.; Ninham, B. W.; Chen, S. J.; Evans, D. F. J. Phys. Chem. 1986,89,711. (24) Chen, S . J.; Evans, F. D.; Ninham, B. W.; Mitchell, D. J.; Blum, F. D.; Pickup, S. J. Phys. Chem. 1986, 90,482.
Langmuir, Vol. 8, No. 2, 1992 693 five different cubic phases which we will call C1 through C5. Photographs of samples from three of these four twophase regions are presented in Figure 2, and in each case the meniscus between the two cubic phases is clearly visible. Each of these samples was equilibrated over 1 year. Both phases are optically isotropic. Centrifugation of these two-phase samples produces extremly sharp menisci. After centrifuging, these become slightly diffuse with time and temperature variations, which for instance were unavoidably encountered in the transfer of these samples to the photographic lab. This effect is completely reversible, however, and centrifuging for several minutes always restores the sharp interface. Interestingly these five different cubic phase regions show differences in physical characteristics. Equilibration in C1 and C2 is rather easy compared to C3, C4, and C5, which in same cases took several months, particularly in C4 and close to the phase boundaries. The stiffness of the samples generally increases with decreasing water concentration, from relatively soft in C5 and C4 to the highest stiffness of all in C2. This is expected in general because interbilayer interactions should be strongest at the lowest dilutions. However, with further decrease of water concentration, the cubic phase in C1 again becomes softer, particularly for compositions close to those in the LZphase. Possibly this is due to the rather high styrene contents in C1. In C2 the stiffness increases noticeably with increasing styrene concentration. These observations may or may not indicate important changes in structural characteristics, because very little is presently known about the underlying physics of viscoelasticity in bicontinuous surfactant microstructures, and the observed behavior may depend on secondary properties such as microcrystallite size or grain boundary structure. Indeed, it is rather easy to demonstrate that the viscosity of some lyotropic liquid crystals can vary tremendously depending on the conditions of sample preparation and equilibrati~n.~~ Nevertheless, the variations that we have mentioned in this paragraph are those that were consistently observed, even in cases where other sample preparation protocols were used. This is particularly the case with the softness of C4 in comparison with C3, which is quite dramatic and always observed. The magnitude of this drop in elastic modulus would be difficult to explain solely on the basis of the water dilution effect just discussed and must be correlated with the change in microstructure, whether through direct or indirect effects. C4, with a water content between 56 and 73% and around 5% styrene, has a very interesting temperature dependence. When slowly cooled below 18 "C the cubic phase becomes much softer, and actually melts upon cooling at a water content between 64 and 73 % . Melting is also observed when heating slowly to over 45 "C in C4. This is in contrast to C2 and C3, where over most of these regions, the cubic phases can be heated to over 60 "C. In these heating experiments, at these temperatures the time scale of any polymerization would be on the order of an hour or more, whereas the melting, when it occurred, took only 1min or less. Thus we can be reasonably assured that there was n o t any interference from polymerization during the determination of the melting behavior. This was further corroborated by the fact that the samples returned to cubic after melting and returning to room temperature. C5 is closest to the water corner, with a water content between 74 and 80% and around 2% styrene. It is very (25) Bohlin, L.; Fontell, K. J. Colloid Interface Sci. 1978, 67, 272. (26) Anderson, D. M.; Gruner, S.; Leibler, S. Proc. Nat. Acad. Sci. U.S.A. 1988,85, 5364.
694 Langmuir, Vol. 8, No. 2, 1992
Str6m and Anderson
Figure 2. Photos of three of the four two-phase regions appearing in the cubic phase region showing the interface between C1 and C2 (a, left), C2 and C3 (b, center), and C3 and C4 (c, right). styrene
Figure 3. An extension in the DDAB-styrene direction of the DDAB-wate-tyrene phase diagram, showingthe differentcubic phaes and the multiphase areas above it.
soft and melts with increased temperature as in C4, but in contrast to C4 it does not melt or become softer when cooled. This observation also aids in the determination of the phase boundary between C4 and C5. Clearly the changes in structure are playing an important role, since many of these changes are not gradual with addition of water but rather occur at phase boundaries. Above the cubic phase region (see Figure 3) is a large region with different two- and three-phase regions. There is a three-phase area with one point in C2 in equilibrium with the L2 phase and with almost pure water. At lower styrene concentration, C2 at around 23% water, is in equilibrium with almost pure water. With an increase in the water concentration to around 28% and decrease in the styrene concentrationto around 17% ,the cubic phase C2 is in equilibrium with the water-rich lamellar phase. Between these two-phase areas there is a three-phase area with a point in C2, a water-rich lamellar phase, and almost pure water in equilibrium. If we further decrease the styrene concentration, there is a three-phase region with C2 and C3 and the water-rich lamellar in equilibrium.
Looking at C3 we see that at around 33% water C3 is equilibrium with the water-rich lamellar and at around 36 % water there is an equilibriumwith C5. Between these two-phase areas there is a three-phase area with a point in C3, the water-rich lamellar, and C5 in equilibrium. At higher water concentrationthere is a equilibrium between C3, C4, and C5-three cubic phases in equilibrium! As one can see in Figure 3 the three-phase areas above the cubic area are very narrow and almost parallel to the phase boundaries of the one-phase regions. This makes it very difficultto determine the boundaries of these three-phase regions exactly. Self-Diffusion. We have measured the water and surfactant self-diffusion rates, DWand Ds, as functions of water concentration at four different ratios of DDAB to styrene. The NMR self-diffusion technique has the advantage of measuring directly on large samples prepared and equilibrated in the same tubes as used in the experiment. SAXS and specific heat measurements, the techniques used until now for studies of progressions in cubic phases, have the disadvantage of sampling only a small portion removed from a larger sample-which may in fact be multiphase-rather than sampling the whole tube precisely as mixed, sealed, and equilibrated. The results for water are presented in Figure 4a. The solid lines present in Figure 4a are not a least-squares fit nor an aid to the eye but rather the results of a theory which is discussed in detail in a later section. The water self-diffusionrate Dw as a functionof water volume fraction @W shows five regions of continuous, very nearly linear dependence. (Throughout we use the letter f for weight fractionsand @ for volume fractions.) These five different parts show five different slopes and are separated by four discontinuitiesthat reflect the microstructure dependence of the water obstruction factor. The size of the discontinuities in water self-diffusion at the phase boundaries, on the order of 10-20%, is much larger than the experimental error, which is less than the size of the symbols
Langmuir, Vol. 8,No. 2,1992 695 a 0.60
OSO 0.40
0.30
+
-
+
t
1
@@J @@
I
i
0.10 o.20 0.00 1 0
I 20
40
80
60
Vol % Water
0.60 0.50
lb 1
8
0
XO O
4
0.00' 0
'
'
'
' 20
'
'
' " ' ' 40 60 Vol % Water '
'
'
'
' 80
'
'
Figure 4. (a)Water self-diffusionas function of volume percent
water. Densitiesusedare0.985,1.o00,and0.906g/cm3forDDAB, water, and styrene, respectively, and Do = 1.9 X lo+' m2/s. different symbolsrepresent the different one-phaseareas C1 (A), C2 (a),C3 ( O ) ,C4 (O),and C5 (+). The small marks inside the different symbols represent different ratios DDAB to styrene (s/o): s/o = 2.96 ( ), s/o = 3.35 (w), s/o = 4.25 (O), and s/o = 5.45 (+). The solid curvesrepresent the theoreticalcalculated results for theself-diffusionwithintheG,D,andPstructures. (b)Resulta of water diffusion in polymerized samples (*). The circles correspond to the values of the water self-diffusion presented above in part a.
-
2e N
2.50
1
2.00
I
0.50
1
@@@
0 0
0.00 L
0
@
+1
' 20
40
60
80
Vol % Water
Figure 5. Surfactant self-diffusion as a function of volume percent water. Symbols are as in Figure 4.
in Figure 4a. The linear parts of Figure 4a show that the diffusion coefficient for water is essentially independent of the ratio DDAB/styrene, as long as it is in the same one-phase region, and depends only on the volume fraction of water. The surfactant self-diffusionrates measured are plotted versus water volume fraction in Figure 5 for completeness. We will not attempt a quantitative analysis of the surfactant self-diffusion behavior, for two main reasons. First of all, in contrast to the case of water self-diffusion, the surfactant self-diffusion is seen to be strongly affected by
the surfactant-to-styrene ratio, as expected on first principles. Second and even more important, the surfactant self-diffusion rate is in general expected to be sensitive to variations in mean area per head group (and other geometric measures), in a manner that has not been analyzed, to our knowledge. Qualitatively, the general increase of the surfactant self-diffusionrate with increasing water (and decreasing mean curvature) is as expected for reversed cubic phases from theoretical computations.27 We return in the next section to a detailed analysis of the water self-diffusion data. For now we merely point out that the discontinuities across the phase boundaries are much too large to be consistent with an interconnectedrod description of these cubic phases but are closer to the discontinuities that are expected based on a parallel surface or constant-mean-curvature surface description of the water/surfactant dividing surface. SAXS. Our analysis of the space groups and corresponding microstructures in regions C1through C4 is based on SAXS data in the literature and corroborated for at least one sample in each region by our own SAXS data. The literature data that we rely on come from three sources: the data of Barois et aL2with cyclohexane as hydrophobe, the data of the Toprakcioglu group with octane,3 and one SAXS scan from one of the author's t h e ~ i s .The ~ working hypothesis is that if the same space group and approximately the same lattice parameter (i.e., within a few percent) are observed for two samples with different hydrophobes but otherwise comparable compositions, then the microstructures and minimal surfaces are the same in the two; in the case of the P and D minimal surfaces, the same space groups and lattice parameters are in fact observed with at least three different hydrophobes. Unpublished data of Krister Fontell also support our assignments. The theoretical interpretation given in this subsection and the following subsection leave little doubt as to the validity of our assignments. The issue of the microstructure in region C5 is discussed briefly at the end of the next section. Table I summarizes our SAXS results, with styrene (or DVB)as hydrophobe. In each of regions C1, C2, and C 3 we have shown results for two samples, one with monomeric styrene and one with DVB which was polymerized before SAXS. In each of these regions, the polymerized and unpolymerized cases exhibited the same space group and approximately the same lattice parameter (these cannot be directly compared because of the small differences in composition). Peak intensities and widths, although not analyzed quantitatively, were qualitatively similar in the two samples. In region C4, the low melting point of the cubic phase has made SAXS analysis of unpolymerized samples so far impossible. However, after polymerization this is no longer a problem and Table I shows the results for a polymerized sample in region C4. The space group (Im3m) and lattice parameter (about 270 A) for this polymerized sample agree well with those for a monomeric cubic phase, with decane as hydrophobe, at a comparable comp~sition.~ Accuracies in the lattice parameter in the case of the C4 cubic phase are considerably lower than those in (2143, and are on the order of f12 % . This is due to the large lattice parameter (and the low order (110) allowed reflection), meaning that the reflections were observed at very low angles. This implies an even higher error on the area per head group deduced, since this also depends on the model used for the polar-apolar interfabe. In particular, since the deduced area per head group is increasing (27) Anderson,D. M.; Wennerstrom, H. J.Phys. Chem. 1990,94,8683.
696 Langmuir, Vol. 8, No. 2, 1992
Strom and Anderson
Table I. SAXS on Both Polymerized and UnDolvmerized SamJes from Four of the Observed Five One-Phase RBBiona. region
c1 c2
c3 c4
hydrophobic component styrene
polyDVB styrene
composition (w/o/s) 15.87119.81164.32
26.18/16.93156.89 27.71/15.96/56133
symmetry Za3d
Pn3m
polyDW3 styrene
46.70/8.70144.60 50.1117.23142.66
Zm3m
polyDW3
64.5814.99130.43
Zm3m
lattice parameter, A 74.8
56.7 60.1
A, A2 60.5
63.6 63.6
97.4 100.17
70.9 67.8
270
61.2
(h2+ kZ + 12) m 6 8 14 16 2 3 4 6 2 4 6
intensities 8
m W
W 8 8
W
m s
m W
2
s
6 8 12
W
m W
a The composition of each sample is given in weight percent. The calculation of a0 is described in the text. Peak intensities are given as s = strong, m = medium, and w = weak. Errors in the case of C4 are considerably higher than in the other cases due to the large lattice.
with increasingwater content, as expected, in regions C l C3,we strongly suspect that the actual area per head group is higher than the value recorded in Table I. A comparison between the cubic phase region presented here and that with cyclohexane studied by Radlinska et al.7 (specific heat) and Barois et (SAXS) and with octane studied by Radimanet aL3(SAXS)shows that their two single-phase regions compare very well with our C2 and C3. For these two regions they reported one with primitive, space group Pn3m, at lower water content and one with body centered, space group Im3m, cubic symmetry. Our own SAXS results (Table I) agree with the assingment of C2 as Pn3m, with the model proposed by Barois et al. based on the Schwarz D minimal surface providing an excellent interpretation of the structural dimensions in these samples, as does our SAXS analysis of the sample in C3 agree with the designation of the minimal surface as the Schwarz P minimal surface. We have also reported a P structure for a polymerized cubic phaset8with methyl methacrylate as the hydrophobe, at a composition which would put it near the center of C3; namely, 55.0% water, 35.0% DDAB,and 10.0% methyl methacrylate. (Note the misprint in that publication, that the water and DDAB concentrations were reversed.) It should be noted that a mistake in the original calculation of the fit in the data of Barois, given in Figure 5b of ref 2, was corrected in a later publication,28and the corrected fit is quite good. At about 56% water, in addition to the observation of two-phase samples there are obvious signatures of a structural phase transition at this water content. There is the aforementioned strong decrease in viscosity in C4 in comparison with C3,which is immediately evident, and which has been measured by rheometry to be an order of magnitude decrease in elastic modulus [C.Toprakcioglu and S . Radiman, personal communication]. This is in agreement with the proposal by Fontell’ of a structural change at around 60% water, with decane as oil in his case. For the structure in C4, with water content between 57 and 7 2 % , we propose the Neovius surface C(P)29(C(P) stands for the “complement of P”, as designated by Schoen30) as the minimal surface that describes the structure (see Figure 6 ) . Although the symmetry group (28) Hyde, S. In Proceedings of International Workshop on Geometry and Interfaces, Aussois, France, Sept. 17-22,1990. Suppl. .~23 to J . Phys. Colloq. p (3-209. (29) Neovius, E. Bestimmung Zweier Speciellen Periodischen Minimalflachen; J. C. Frenckell and Son: Helsingfors, 1883.
of C ( P )is Im3m as in C3, the lattice parameters in C4 are much higher than the values extrapolated from C3. The lattice parameter at 65 % water is 270 A, as measured by SAXS (see Table I). For the case with decane as oil: a similar value of 300 A has been reported; however, the conjecture of the I-WP as the structure of that phase in ref 4 was incorrect. In the next section we present precise analyses that give strong evidence in favor of our designation of C ( P )for the structure in C4. For the moment however, we proceed with an approximate analysis that will give three results. First, it will show that despite the Im3m space group, the structure cannot possibly by the P structure; this of course agrees with the phase behavior and the sharp drop in viscosity, both of which indicate a change in structure. Second, it will recreate the analysis that led us to investigatethe C ( P )as a structural possibility, and that virtually rules out every other minimal surface which is presently known. And third, it will show that the assumption of a C ( P ) structure leads to structural dimensions that agree with those in the other cubic phases and the lamellar phase^.^ This approximate analysis begins with an exact formula for the parallel surface models
Here @B is the volume fraction occupied by the bilayer, including the hydrophobe, Sb is the dimensionless surface area of one unit cell of the minimal base surface, L is onehalf the thickness of the bilayer, which is assumed to be uniform over the structure (thus, the parallel surface model), and a is the lattice parameter. From this formula we see that the bilayer volume fraction determines the ratio Lla, given the dimensionless surface area of the minimal surface; or alternatively one can estimate the surface area given an estimate of Lla from experiment. This formula, with L = 13.9 A and {Sb,XE)= (2.345107 ..., -4) for the P surface, yields lattice parameters that are within a few percent of the values measured by Barois et al. Extrapolating yields an expected lattice parameter of 186 A at 65% water. The ratio of the measured lattice parameter 270 A to this value extrapolated for the P minimal surface structure, 270&186A = 1.45, is of course very much higher than errors that could be encountered in the extrapolation. The bilayer half-width L would have (30) Schoen, A. Infinite Periodic Minimal Surfaces Without Selfintersections;NASATechnical InformationServiceDocument N70-29782, Springfield, VA.
Cubic Phase Region in DDAB- Water-Styrene
Langmuir, Vol. 8, No. 2, 1992 697
I
Figure 6. Computer graphic of the Neovius minimal surface C(P) (from ref. 4). The space group is #229 (Im3m). The surface divides space into two congruent subvolumes.
to be over 21 A if the minimal surface were the P surface, and this is clearly ruled out. In general, a tell-tale signature of the C(P) structure is an Im3m lattice with lattice parameters about 1.5times those in (or better, extrapolated from) a nearby P cubic phase. We proceed to the second argument, using the experimental data to estimate the surface area of the minimal surface,thus providing an important clue as to its identity. If we first assume that the bilayer half-width L is very nearly the same, and thus approximately 14 A,in C4 as in C1, C2, and C 3 (the value in the oil-free lamellar phase at high surfactant content is 13A4),then taking the lattice parameter to be a = 270 A at 65 w t % (or 64.3 vol %) water, so @B = 0.357, and solving eq 1provides an estimate
for the normalized area per unit cell &,;in this calculation the dependence on the Euler characteristic XE is small because L / a is small (~0.052)and this is cubed, so we simply take XE = 4 as in the P structure for the moment. This calculation yields S b = 3.46 as an estimate of the area per normalized unit cell of the minimal surface. At present there are very few minimal surfaces known which have this low a surface area. Furthermore, it can be well argued that the triply periodic minimal surfaces which do have very low surface area have already been discovered; for example,all of the minimal surfaceswhich can be generated by surface patches spanning space quadrilaterals are known (cf. ref 4). A survey of the minimal surfaces of cubic symmetry known to have low values of S b reveals
Strom and Anderson
698 Langmuir, Vol. 8, No. 2, 1992 that the Neovius surface is the only one that is close to this value, namely s b = 3.51048 ...,which is within 2 % of the estimate 3.46, and indeed the C(P) surface describes an Im3m structure with two congruent subspaces (i.e., it is a balance surface).3l Proceeding then with the third argument, we compute the actual bilayer half-width L and the area per head group A from the experimental data with the help of eq 1 and the formula for the dimensionless area per unit cell over the parallel surface
s, = s, + 2TxE(Lla)2
(2)
where XE = -16 for a unit cell of the C(P) structure. At @B = 0.357 we solve eq 1 (iteratively) for Lla, giving a value of 0.052, and eq 2 yields SL= 3.239. We then deduce that L = 0.052.270 A = 14.0 A. Then, again using the observed lattice parameter of 270 8,to convert to dimensional area per unit cell and then per unit volume, one obtains 61.2 A2 per surfactant molecule, which compares well with typical values for DDAB. The computation assuming the P minimal surface yields 47.0 A2, which is much too low at this high water content. Both of these dimensions assuming the C(P) compare well with those from samples in C1, C2, and C3, and with those reported elsewhere in the literature for DDAB liquid crystals (e.g. refs 1-4). Thus we have seen that the only way to explain the structural dimensions-within the range of cubic minimal surfaces known-is with the C(P) structure, and the space group of the C(P) bilayer structure is consistent with the SAXS indexing. Furthermore it is very unlikely, we believe, that another minimal surface of Im3m symmetry will be found that also matches the dimensionless area required by these data. In addition, the following very approximate analysis of the peak relative intensities, using as a model of C(P) structure, provides further support for the C(P) designation. One of the authors (D.M.A.) has developed an extension of the form factor calculation derived in ref 32 that handles parallel surface models. The method is exact if the Gaussian curvature over the minimal base surface is known; however, the authors do not know this function for the C(P) minimal surface, and so the results for the limiting case @B -,0 will be compared with the experimental data. The lowest order intensities using this model, including multiplicities and normalized to I(110) = 100, are I(200) = 1.0, I(211) = 4.4, I(220) = 15.7,I(310) = 0.0, I(222) = 1.8, and I(321) = 0.6. This is qualitatively consistent with our observation of peaks, in order of decreasingintensity, as (110), (220),(211), and (222). Both the theoretical and experimental intensity orderings are qualitatively different for the P surface models. Much more support for the C(P) designation is given in the next section. Polymerization. We have also made a large number of samples with divinylbenzene (DVB) instead of styrene (vinylbenzene)for the purpose of polymerizing the samples and subsequently characterizing them. The fixation of lyotropic cubic phases via polymerization produces structured polymers with a high degree of order and uniformity, of interest as materials in the separations, catalysis, controlled release, composites, and biomaterials area^.^,^ DVB was used instead of styrene because of its higher concentration of vinyl groups (important because of the additional steric hindrances due to the presence of surfactant), because it has better initiation characteristics, (31)Fischer, W.;Koch, E. Z.Kristllogr. 1987,179,31. (32)Anderson, D.M.;Thomas, E.L.Macromolecules 1988,21,3221.
and because it leads to a highly cross-linked polymer. Not suprisingly, we found the phase behavior with DVB to be nearly identical to that with styrene, even quantitatively. Polymerization was performed in each of the five cubic phases except for region C5. The polymerization was carried out by photoinitiation using 350-nm lamps as described elsewhere.* The conversion was followed by proton NMR. As expected, the NMR spectra in all polymerizations showed a broadening of the DVB/polyDVB signal and a sharp reduction in the intensity of the vinyl proton signal. Conversion was less then 100% as indicated by the presence of a small vinyl proton signal, but this was not quantifiable. To determine the conversion more quantitatively, a known amount of the polymerized cubic phase was placed in pure methanol, which is a solvent for the surfactant, water, and the monomeric and oligomeric DVB but not for polyDVB. The methanol was replaced with fresh methanol every 12 h until there was no detectable amount of surfactant as checked easily by titration with silver nitrate on the bromide counterion. In all cases, the final weight of the remaining polyDVB was close to 70 % of the originalweight of the monomeric DVB. The 30% DVB lost was either monomeric or low-MW oligomers. Also as expected, the self-diffusion rate of the surfactant (see Figure 5) was reduced with the polymerization of the DVB, providing further corroboration that much or most of the DVB (or styrene) in the cubic phase is dispersed among the surfactant tails rather than in a separate layer between the tails in the middle of the bilayer. A change in visual appearance upon polymerization was observed, but other methods of characterization, sensitive to a range of length scales, indicate no significant structural changes with polymerization. Thus, in regino C2, polymerized samples were translucent, and moving to either higher or lower water contents (where viscosities decrease as compared to region C2), opacity increased. Nevertheless, in the previous subsection we presented SAXS results on polymerized samples that are indistinguishable from those of unpolymerized samples, showing that the lattice order is maintained throughout polymerization. Furthermore, deuterium NMR showed no anisotropy in the form of quadrupolar splittings. Perhaps most significantly, the water self-diffusion rates are exactly identical to those for unpolymerized samples (see Figure 4b), and since the pulsed-gradient NMR self-diffusion technique measures diffusion over macroscopic distances, this would seem to indicate that there is no disturbance of structure even on longer length scales. It is highly unlikely that the absence of any measurable changes in water self-diffusionbehavior is due to a fortuitous cancellation of two or more effects, because it is observed over a very large range of water contents. Thus the translucency remains unexplained in the light of considerable evidence that the short- and long-range structure in the cubic phase is maintained throughout polymerization. Most likely there is a rearrangement of the surfactant, within the ordered polymeric matrix, that leads to birefringence and multiple scattering; the eye is very sensitive to slight differences in refractive index. There is little doubt that the driving force for the surfactant rearrangement is the incompatibility of the surfactant hydrocarbon chains with the polyDVB (hydrocarbon is a solvent for DVB but a nonsolvent for polyDVB). In related experiments in this group with other polymerizable cubic phases, we have shown unequivocally that loss of transparency is due to rearrangement of the surfactant and that removal of the surfactant is sufficient to yield a clear
Langmuir, Vol. 8, No. 2, 1992 699
Cubic Phase Region in DDAB-Water-Styrene polymer with triply periodic ordering. Thus, when acrylamide is added to the aqueous phase of the cubic phase at 60 "C in the dihexadecyldimethylammonium bromidewater-decane system, and the acrylamide polymerized, the result is a clear material-until the sample is brought down below 50 "C, at which point it quickly becomes completely opaque. Then, raising the temperature back above 50 "C brings the sample back to complete clarity, and in fact the phenomenon is completely reversible. After removal of the surfactant, the material is clear even at room temperature, and SAXS has shown that cubic ordering has been obtained.33 Similarly, the cubic phase in the lecithin-water-styrene system becomes opaque after polymerization, but removing the lecithin by bathing in toluene restores transparency. The experiments referred to above have also demonstrated that the polymer-surfactant compatibility is much more important in determining the degree of clarity of the final material than is any volume change upon polymerization. Thus, for example, while isoprene and styrene have comparable density changes upon polymerization-at least in bulk-the polymerization of isoprene in the DDAB-styrene-water cubic phase yields a substantially clear polymer, much clearer than in the case of styrene or DVB. This is clearly due to the compatibility of polyisoprene and the hydrocarbon tails of DDAB (polyisoprene is soluble in hydrocarbon at room temperature). One must also consider that while the volume change on the polymerization of DVB is on the order of 15-20% in bulk, the fact that the styrene comprises only 3-20% of the cubic phase means that the overall volume change is very small (and the linear change, e.g., in lattice parameter, is smaller still). The polymerization of a monoglyceridewater cubic phase, with an estimated change in volume on the order of 5 % ,has yielded perfectly clear (and isotropic) polymers, which show cubic ordering in SAXS.gb Many similar observations have led us to believe that the density change on polymerization is not a key factor in determining clarity in polymerized cubic phases but rather that polymer-surfactant compatibility is the main determinant. The primary change in solubility properties upon polymerization is due to the loss of entropy associated with the change from monomeric to polymeric DVB, and since this is a relatively small free energy difference the driving force for rearrangement is necessarily fairly small. This would explain the apparent low degree of structural disruption.
-
Theoretical Interpretation The progression with increasing water of G (Ia3d) D (Pn3m) P (Im3m) C(P) (Im3m) observed in this cubic phase area can be readily and even quantitatively derived using two approaches given in this section. The first, and main approach, which actually yields quantitative interpretations of the phase boundaries, is a free energy minimization calculation in which there are two contributions to the free energy: chain deformation energy and a head group area term. It is worth pointing out that the Hamiltonian which has been very successfullyused in RPA free energy calculation^^^ of block copolymer phase behavior comprises a chain deformation term and a x interaction term, which is obviously very analogous. This treatment will simultaneously yield lattice parameter values that are in excellent agreement with those observed. The second treatment will be a mean curvature calculation and is simpler in that the surfactant chain length and
-
-
(33) Anderson, D. M.; Strom,P. Proceedings of International Workshop on Ordering in Supermolecular Fluids, Nov. 24-26. Phys. A. 1989,176,
151.
head group area are boiled down to one parameter, the mean curvature at the polar-apolar interface. Thus it yields less information but has the advantage that the sequence of cubic phases can be deduced almost immediately, without explicit free energy calculations, by virtue of the one-parameter characterization of the candidate microstructures. In our opinion this is one of the most important features of using curvature energies rather than actual head group interaction and chain deformation energies, and a recent reformulation of the curvature free energy3Iwill be used that decomposes the curvature free energy in such a way that the dominance of this mean curvature term is manifested. We emphasize that the first approach, using stretch lengths and head group areas directly, is the main approach of this work, and the second apparoach using curvatures is given only for completeness and comparison. We contend that curvature energy approaches are inherently less accurate, and much less intimately tied to experimentally measurable quantities, than approaches based on quantities such as areas and lengths which do not require (second order) differentiation. Free Energy Calculation. We now perform a free energy calculation that will yield lattice parameter predictions and surfactant film free energies for the four candidate cubic microstructures G, D, P, and C(P). The results of this section will make it obvious how the results would come out for other cubic phases structures, characterized by triply periodic minimal surfaces with certain Euler characteristics and surface areas, that could be envisioned. We will not attempt to include noncubic structures in the competition and will limit our analysis to the relative surfactant film free energies for these four cubic structures along a particular path through the middle of the observed cubic phase region. It should first be pointed out that because of the volume fraction constraint, for a given structural type at a known composition the only degree of freedom available to the system is the lattice parameter, and therefore the area per head group and chain stretch length are linked by a relation and cannot vary independently. Thus the free energy in question could in principle be written as a single term, incorporating this relation. However, there are two reasons for not doing so. First, the relation depends on the structure; that is, it is difficult for the G and the D, etc. (more specifically, the constants (Sb,XE) in the relation depend on the structure). And second, the relation is defined only implicitly, by equations which must be solved iteratively, and so the relation cannot be written in closed form. These are simply practical reasons and should not be considered profound in any sense. We have assumed that L is the same for every surfactant molecule, equivalent to our use of the parallel surface equations (1 and 2). Implicit in our use of a bilayer description is the convention that the quantities referred to, in particular the preferred values A0 and LO,refer to the system in the locally bilayer configuration. One can well imagine that interdigitation and such effects could make these values different than in the locally monolayer situation. The (dimensionless) bilayer free energy density associated with the parallel surface model interface will be written as (3) where L and LOare the actual and "preferred" bilayer half-widths and similarly for the head group areas A and Ao. One could of course insert an adjustable factor that would give different relative weights to the two terms;
700 Langmuir, Vol. 8, No. 2, 1992
however, we will not need to do so in the calculationsherein. This is avoided in order to minimize the number of adjustable parameters and reflects the result that the free energy competition is not particularly sensitive to the relative weighting of the two terms; this is really related to the interdependence of A and L discussed in the paragraph above. The implied constant needed to produce a quantity with units of energy is irrelevant in the present free energy competition. Concerning more detailed formulations of this free energy, we maintain that one important advantage of this approach is that the constants for stretch length and area per head group are in principle more closely connected to experimental measurable such as trans-gauche transition energies and electrostatic potentials than more phenomenological constants such as curvature rigidities. The presence of the third component (styrene) gives rise to an extra degree of freedom with regard to local packing, and we will treat two cases that represent the two extremes in this respect. These cases are as follows: (i) We assume that the styrene (and the hydrating water molecules) can redistribute so as to keep the area per head group uniform even in the presence of nonuniform Gaussian curvature over the minimal surface; this calls for variations in the local styrene concentration in the bilayer of on the order of *5 % ,from average values on the order of 20%. ( i i ) We a s s u m e t h a t t h e s t y r e n e c a n n o t redistribute-that is, one sees a uniform distribution as one moves laterally over the bilayer. (Obviously the local styrene concentration will always be a strong function of the distance moving along a normal away from the bilayer minimal midsurface; here we refer instead to the lateral variation, moving along a direction tangent to the surface.) To quantify this, we consider dividing the bilayer region into frustrum-shaped subvolumes, each with volume u which is the bilayer volume fraction divided by the number density of surfactant molecules; u is thus the volume of surfactant and styrene per surfactant molecule. The bounding surface of each frustrum is ruled by normals to the minimal surface, extending a distance L to the parallel surface, and the top and bottom comprise patches of the parallel and minimal surfaces. If K is the Gaussian curvature on the minimal surface (actually an average over the patch), then in case ii the area A ( r ) of the patch on the parallel surface identified with this surfactant molecule is A = (u/L)[(l+ KL2)/(1+ KL2/3)l This area per head group thus varies over the bilayer in accordance with the variation of Gaussian curvature over the minimal surface, as given by the fraction in this equation. By contrast, in case i the area A on the parallel surface for each frustrum is the same,i.e., not a function of position, and thus u is given by u = A L [ ( 1+ KL2)/(1+ KL2/3)]-l. This variation of volume per surfactant with position is presumed to be effected by avariation in volume of styrene in one frustrum as opposed to another. For a P structure at 50 vol % water, for example, umar/uave = 1.11. In case ii, where A and therefore the second term in eq 3 varies over the minimal surface, it is necessary to perform the integration over the minimal surface by expanding the fraction in powers of KL2 and using the results tabulated in ref 26 for the moments of K. Since the G and P minimal surfaces are associates of D, these moments for D are easily extended to moments for G and P by simple rescaling. We expanded the fraction to fourth order in
Strom and Anderson KL2. Case ii could be not be treated for the C(P) structure since the higher moments of K were not known. The mixing entropy associated with the styrene favors the uniform distribution of case ii, whereas the enthalpy associated with the head group interactions would favor constant area per head group and thus a redistribution of styrene as in case i. The actual distribution of styrene would in general be expected to lie between these two extremes. The preferred bilayer half-width LOwill be taken to be constant throughout the region of the phase diagram treated, not varying with composition (at fixed temperature, of course). In particular, we will take LO= 13.90 A for all compositions along the path examined. For less penetrating hydrophobes, such as medium and longer chain length alkanes, LOcould be expected to increase with increasing hydrophobe:DDAB ratio (and could in fact be longer than the fully extended DDAB chain length), and indeed the SAXS data for octane3 indicate that this dependence is nonnegligible, as shown below. The preferred area per surfactant molecule A0 measured at the head group surface is well-known to increase with increasing hydration. We will take the simplest possible approach and write it as a linear function of the water content. It will also be shown that the observed sequence of structures is predicted by the theory even when A0 is constant with water content, but the sequence occurs over a much narrower concentration range than that observed. The computation is very straightforward, and was performed on a MacIntosh SE30 using MatLab software. For each water content fw between 10 and 80 7% ,a styrene weight fraction was chosen that was close to the center line of the cubic region. The function which was used to do this conveniently was fstr = 0.03 exp(2.33(1 - fwP5). The composition was converted to volume fractions (the densities of DDAB and styrene being 0.9855 and 0.906, respectively), and @B is the sume of the DDAB and styrene volume fractions. Equation 1was solved iteratively for Lla, and this was inserted into eq 2 to give the dimensionless area SL.In case i, the free energy in eq 3 was then minimized with respect to the lattice parameter a , using L = (L/a)aand A = 2S~/(aps);here ps is the surfactant number concentration (MW = 462.65). In case ii, it was found natural to introduce the quantity x = v/(A&) and to minimize with respect to x , which is tantamount to minimizing with respect to L. Formally there are three independent adjustable parameters; however, particularly in case ii there are for all intents and purposes only two, in the free energy modeling. This is because in the final formulas, the parameters LO and Aoj do not appear alone, but rather as the product. Here Aoj = 1, 2) refers to the two parameters in the expression A0 = AOlfW + A02U - f w )
(4)
Thus the two adjustable parameters LOA01 and LOA02 dominate in the formulas for the free energies. For the lattice parameter calculations, the value of LO itself becomes important once again. Results. To begin with, Figure 7a gives the resulting free energies for case i with A0 = 60 A2,constant with composition (i.e., A01 = Ao2). With increasing water content we see that the lowest free energies occur for the G, then D, then P, and then C(P), the order which is observed experimentally. With fixed Ao, however, the range of water concentrations over which the sequence occurs is much smaller than that observed experimentally.
Langmuir, Vol. 8, No. 2, 1992 701
Cubic Phase Region in DDAB- WaterStyrene
c
1.0.
I
I
I
I
I
I
0.8 -
\
f Om2I
I
1
oI2
0.4 0.6 0.8 Water weight fraction Figure 7. Free energy as a function of water weight fraction: (a, case i) with A. constant; (b, case ii) with A0 varying linearly with water content, with (-) G structure, (- -) D structure, (. * .) P structure, and (- -) C(P) structure; (c, case ii) with A0 varying linearly with water content (C(P)not included). See O'
text.
In particular, according to this theory, with A0 constant the water concentration range of the D and P structures would be on the order of only a few percent, and for the G and C(P)structures, except near the minimum in free energy, one would expect that other noncubic structures would win out. Then taking A0 to increase (linearly)with water content spreads out the theoretical phase diagram over the range of water contents, to where very good agreement with experimental data is obtained in case i with A01 = 70.5 A2 and A02 = 47.8 A2; that is, the preferred area per head group increases linearly from A0 = 47.8 to 70.5 A2 going from the limit of zero water to the limit of 100% water.
Figure 7b shows the free energies calculated with this expression for Ao. Also shown are the phase boundaries in the experimental system, and the agreement between theoretical and observed phase boundaries, along this path, is very good. The wide range of water concentrationsover which these first four cubic microstructures are spread is therefore, according to the results of the theory, simply a consequence of the increasein the preferred area per surfactant molecule with increasing water. If there were no increase (or a much milder increase), then the theory predicts that the D and P structures would exist over very narrow concentration ranges of water. It is difficult to say what should happen to the G and C(P) ranges without taking other noncubic microstructures into the free energy competition, but one would expect that the range of stability of these cubic structures would also be diminished. It is important to point out that any attempt to explain the wide range of the cubic phase as a result of a variation of Lo with styrene content will face formidable difficulties. For example, as noted above it is the product A& that dominates in the final free energy equations, and since an increase in A0 with increasing water content was needed to match the phase behavior when LOwas fixed, such an alternative explanation based on variation in LO would require that LOincrease with increasing water-that is, LO would have to increase with decreasing styrene content, since styrene contents are much lower at higher water contents. This is obviously counterintuitive. The fits of the phase behavior and lattice parameters shown in this and section were done without prejudice toward fixed LO, yet the results lead us to believe that it is the variation in A0 with water content that accounts for the wide range of existence of each of the cubic phase structures D and P, and almost certainly for G and C(P) as well. In case i, the calculated free energy does indeed vanish at one water content, for each structure. At this point there is no frustration, because A = A0 and L = LO identically. Thinking of these as two constraints, and the lattice parameter and water content as variables, clearly there must be one water content where both are satisfied. We return to this point later in the next subsection. By contrast, in case ii the energy is always positive because A in eq 3 varies with position. Figure 7c shows the results with AOI= 69.0 A2and Aoz = 45.0 A2. The free energies for all three structures climb at low water volume fraction, where the diminsionless length Lla becomes large and wL21becomes comparable to unity; this in turn implies a strong variation in A. We note that at these low water fractions, the styrene fraction is large, so that even a relatively minor redistribution of styrene can relieve some of the stress associated with the variation in A. The sequence of phases under case ii, given by Figure 7c, is G D P as in case i, but there has been a slight increase in the free energy of the D relative to G and P which greatly reduces the region of theoretical stability for D. This is a resilient result it that it persists over a wide range of the adjustable parameters. At present we are not able to explain why the range of stability of the D structure is larger experimentally than theoretically; nonetheless the theoretical boundaries in case i are quite close to those determined experimentally. The remainder of t h discussion in this section will refer to case i. Figure 8 shows, for completeness, the ratio of a~ to UA, where a~ is the lattice parameter at which the condition L = LOwould be satisified, and likewise A = Ao at a = QA, in case i. Thus, at low water content for each structure, a L / a ~< 1,then comes a water content of no frustration
--
Strom and Anderson
702 Langmuir, Vol. 8, No. 2,1992 1.051
I
I
I
I
I
160
i
I
a
6 0 0.3 0.35
,
I
I I
0.4
0.45 0.5 0.55 Water weight fraction
0.6
0.65
Figure 8. a L / a ~as a function of water weight fraction,where a~ and UA are the lattice parameters at which the conditions L = LO and A = Ao, respectively, would be satisfied.
(which would be a curve in the three-component diagram), and then for high water contents, adaA > 1. The strong slope of the phase boundaries with respect to the lines of constant water content can also be deduced from the theory. When the styrene content is increased in the theory, the theoretical phase boundaries are indeed calculated to shift to lower water content, as observed experimentally (and conversely to higher water content with lower styrene). The effect is perhaps even too large in the theory, predicting a slope that is even more than that observed. However, we believe this is to be due to the redistribution of styrene such that, e.g., at the highest styrene contents for a given water concentration a thin pure styrene layer forms in the middle of the bilayer, which in this theory would call for a small dependence of LOon 'PBt,.
The theory also yields accurate values for the lattice parameters. The most accurate SAXS data available for a DDAB cubic phase are those of Barois et al. with cyclohexane as hydrophobe and those of Toprakcioglu3 with octane. Figure 9a shows a comparison of the lattice parameters of Barois et al. with those calculated from the theory using LO= 13.9 A and Aol = 70.5 A2, the same values used for the styrene free energies, but in this case it was found necessary to increase the value of A02 to 59.0 A2. We hesitate to conclude from this that cyclohexane gives rise to a higher preferred head group area than styrene a t lower water contents; however, with styrene the cubic phase extends to much lower low water contents than with cyclohexane, to where very strong reversed curvatures are called for, and so it would seem consistent that head group areas at low water contents are lower in the styrene case. Using these lattice parameters one can compute the areaaveraged mean curvature using the formula
(H)= ( h ) / a= 2*x,(L/a)/(S,a)
(5)
which is also an exact formula. The values of (H) using the theoretical and experimental lattice parameters for the Barois et al. system are shown in Figure 9b. The jump in (H) observed (and predicted by the theory) at the D P transition will be acentral issue in the next subsection. Figure 10 shows the fit for the octane data of Radiman et al.? where in this case due to the much less penetrating hydrophobe, LOwas taken to be 12(1 + 2amme) A, thus giving a range of LObetween about 12.7 and 15.6 A. This is the only case in which LOwas allowed to vary with composition. Curvature Calculation. The second theoretical treatment of the G (Ia3d) D (Pn3m) P (Im3m) C(P)
-
-
-
-
e
I
I
0.010
1
Q
@m
I
L.
0.005
I
1
1
0.3 0.35 0.4 0.45 0.5 0.55 Water weight fraction
,
I
0.6
0.65
Figure 9. (a) Lattice parameter and (b) mean curvature as a function of water weight fraction for the system DDAB-water-
cyclohexane. The symbol (+) is for SAXS data with primitive cubic and body-centered cubic symmetriesat low and high water weight fraction, respectively, taken from and Barois et al.2 The is the theoretical data for the D and P structures at symbol (0) low and high water weight fraction, respectively. (Im3m) progression is based on a very simple mean curvature argument. The expansion of curvature free energy G, that underlies our approach is that recently proposed in ref 34, which incorporates group theoretic properties of G,(cI,cz), where c1 and c2 are the principal curvatures measured at the polar-apolar interface. The expansion furthermore decomposes the curvature energies in such a way that mean curvature effects are isolated in a single term, resurrecting the primary incentive for introducting the concept of mean curvature in the first place-namely, to reduce the description of the interface from two parameters (A and L)to one (H).By contrast, in many mean and Gaussian curvature free energy formulations, the primary contribution, coming from the mean curvature a t the polar-apolar interface, enters as a difference between mean and Gaussian terms of comparable magnitude. The expansion for G, in ref 34, to second order, is
G, = a J J ( H - cOl2dA + bJJ(cl - c2I2dA
(6)
As argued in ref 34, we can take b
