J . Chem. SOC., Faraday Trans. 1, 1985, 81, 273-283 Structural Aspects of Certain Phase Transformations in Lyotropic Liquid-crystal Systems B Y RAYMOND M. WOOD* AND MALCOLM P. MCDONALD Departments of Physics and Chemistry, Sheffield City Polytechnic, Pond Street, Sheffield S1 1WB Received 20th December, 1983 Aqueous binary systems of potassium oleate and sodium dodecyl sulphate exhibit an hexagonal phase at low surfactant concentrations and appropriate temperatures. However, in the former system the phase is succeeded at higher surfactant concentrations by the so-called rectangular phase, while in the latter system a distorted hexagonal phase follows. X-ray diffraction results for these systems indicate that the rectangular phase is a consequence of axial growth of the hexagonal-phase surfactant cylinders with surfactant content at substantially constant radius.Water in the hexagonal phase occupies positions other than along and between cylinder lengths. Radial growth of cylinders leads to the distorted hexagonal structure. The cylinder growth and transformation behaviours are associated with characteristics of the respective molecular structures. When the composition of a single-phase crystalline substance is changed, the change in the size of the repeat unit of the structure is three dimensional. In contrast, a change in the composition of a single liquid-crystal phase results in an equivalent dimension change which may be one, two or three dimensional, according to the number and chemical form of the constituents. The importance of this latter range of behaviour lies in its relationship to the mechanism of the structural change which takes place when that phase undergoes a transformation and also to the subsequent phase structure.Thus a study of the way in which liquid-crystal phase structures change dimensions with composition is a prerequisite for an understanding of the physical basis of structural transformations in these substances. In this work, we have examined the effect on phase structure and transformation behaviour of two types of anionic amphiphilic molecule with different structures, namely sodium dodecyl sulphate (SDS) and potassium oleate (KO). Fig. 1 gives details of the published data for the liquid-crystal phases in the binary systems SDS + water and KO + water.In the case of the SDS+water system, Husson et a1.l claim that the hexagonal phase is followed at higher amphiphile concentrations by a complex hexagonal phase and finally a lamellar phase. The results of Bergeron2 do not contain any observation of a phase between the hexagonal and lamellar ones. For the KO + water system there is agreement that increasing the amphiphile content results in a set of phases which are hexagonal, rectangular, complex hexagonal and lamellar, in that order,lT although Ekwall et aL4 reported only a rectangular phase in the intermediate region. EXPERIMENTAL Parts of the binary systems SDS +water and KO + water have been examined by polarised light5 and low-angle X-ray diffraction. The SDS (specially pure) was obtained from B.D.H. and was recrystallised from ethanol, while the water was deionised and doubly distilled from alkaline 273274 SDS AND KO LIQUID-CRYSTAL PHASES I H u s s o n ef a l , 348 K 1 1 1 HJsson ei a l., 293 K B a l m b r a etal.,I 293 K I I I I I I I I I 0.2 0.3 0 .4 0 . 5 0.6 0.7 0.8 0.9 1.0 Fig. 1. Relevant phase structures and their compositions in the binary SDS+water and KO+water systems as observed by previous workers. -, Hexagonal; w, rectangular; \, complex hexagonal; /, lamellar; 0, two-phase. surfactant (mass fraction) potassium permanganate solution. Mixtures were obtained by repeatedly centrifuging the weighed constituents through a constriction in a sealed glass tube. Potassium oleate was prepared by dissolving redistilled oleic acid in ethanol and titrating the solution with potassium hydroxide to pH 9.The resulting solution was freeze-dried and the solid so produced was dried over phosphorus pentoxide. Mixtures of KO and water in selected proportions were prepared in the same manner as the SDS + water mixtures. X-Ray exposures were carried out using a Warhus low-angle camera with a specimen-to-film distance of 255 mm and filtered copper Ka radiation. For this purpose, samples were sealed either inside 0.5 mm Lindemann tubes or inside a disc-shaped cell of thickness 1 mm with Mylar windows. Sample temperatures were monitored during exposures (usually 16 h) by means of a thermocouple mounted in the sample heating chamber. Diffraction line positions on films were measured using a Hilger and Watts film measuring device with a precision of 0.05 mm.Specimens subjected to the diffraction technique were examined visually after exposure for deterioration and/or capsule failure. Any indication of crystal formation or capsule leakage led to rejection of the diffraction pattern. Since three-dimensional crystal structures generate diffracted beams of higher intensity than those of two-dimensionally ordered liquids, the absence of identifiable crystal diffractions was taken to indicate zero or insignificantly small quantities of crystalline SDS. RESULTS In association with Tiddy and coworkers we have published results for the SDS+ water system which show that the hexagonal structure becomes d i ~ t o r t e d . ~ This behaviour is to be compared with that indicated by the X-ray results for the KO + water system in fig.2, which shows that, in agreement with the other workers,'! we have obtained hexagonal, rectangular and complex hexagonal phases. There is only a narrow two-phase region between the hexagonal and rectangular single phases which contrasts with the broad region where both hexagonal and distorted hexagonal phases occurred in the SDS system. Such two-phase regions, being inherent parts of binary phase diagrams, must, by their absence, introduce doubt about the results of Husson et aI.,l which, as shown in fig. 1, contain no two-phase regions. In contrast, the phasesR. M. WOOD AND M. P. MCDONALD 275 1070 H 010 R 3 . 6 I I I I 1 0.30 0.34 0.38 0.42 0.46 0.50 0.54 0.58 0.62 0.66 0.70 KO (mass fraction) Fig. 2. X-Ray diffraction observations of phases in the KO+water binary system at room temperature.H, Hexagonal ; R, rectangular; H, complex hexagonal. and phase boundaries presented by Balmbra et are little different from those of the present work, and Balmbra et al. acknowledged the difficulty of establishing the precise locations of phase boundaries. The identification of the complex hexagonal phase in the work of Balmbra et al. rests on electron microscopy and a single diffraction line, designated as a reflection from (1 120) planes with a spacing of 5.60 nm. We have obtained a diffraction line from planes of spacing 5.70 nm at compositions similar to those of Balmbra et al. and accordingly have designated this a (1 120) reflection from a complex hexagonal structure. The effect of composition and temperature on the edge length of the rhomboid repeat unit of the hexagonal distribution of right circular SDS cylinders in the 'middle' phase of the SDS+water system is shown in fig.3. Plotted points are the means of values of the rhombus parameter calculated from two to four X-ray diffraction lines appearing on several films at each indicated composition and the probable error associated with each mean is presented as an error bar. Of interest are the two points which deviate from the best straight line for the results at 3 13 K. Their probable errors indicate major compositional errors or a real effect as the cause of the deviations. In view of the fact that the points closely follow the pattern of the deuterium n.m.r. results at this temperat~re,~ which was interpreted as indicating a change in the water environment, it is probable that they are similarly involved in an incipient transformation. Apart from the actual values for the size of the rhombus, the two main features of these results are the linear relationship between rhombus size and composition of the sample and the varying effect of temperature on the magnitute of this linear relationship, the slopes of the lines going through a minimum between 303 and 333 K.Fig. 4 shows the variation of the same parameter with composition for the potassium oleate + water system. Presented in fig. 4 are published results for the hexagonal phase at room temperature together with results for four compositions obtained in the present investigation. The data obtained from X-ray diffraction studies by Ekwall276 SDS AND KO LIQUID-CRYSTAL PHASES 5.2 5.0 4 .8 4 * 6 5.0 4 . 8 4 . 6 4 . 4 5 .O 4 . 8 G \ 4 .8 4 . 6 4 . 4 0.30 0.40 0.50 0.60 Fig. 3. Dependence of unit-cell dimensions on composition for the hexagonal phase in the SDS+water binary system at (a) 303, (b) 313, (c) 323 and (d) 333 K. SDS (mass fraction) et aI.,* Balmbra et aL3 and the present investigation are in good agreement, while the electron microscopy values of Eins6 follow a similar trend with composition, although slightly displaced to smaller rhombus dimensions. A significant difference between the hexagonal phases in the two systems, as shown by fig. 3 and 4, is the manner in which their basic dimension varies with composition, i.e. linear for SDS compared with a more complicated relationship for KO.It has been pointed out7 that the spacing, d, of the (1010) Bragg reflection for the hexagonal phase will be related to the mass fraction, c, of surfactant present by a negative half-power relationship (i.e. dlolo a c - O . ~ ) if the surfactant cylinder radius is invariant with composition. The relationship between surfactant cylinder dimensions and the dimensions of the rhomboid prism of the hexagonal phase is more fully given by (1) 7 t ~ ~ L ~ [ ( c - ~ - 1) v;' + 11 = L, a2 sin 60" when the partial specific volume of water is taken as unity. In eqn (l), L, and L, are the lengths of the surfactant cylinder and rhomboid prism, respectively, c is the mass fraction of surfactant which has partial specific volume v, and a is the rhombus edge length.When I,, and L, are assumed equal and v, is assumed to have the same value as that for water, the expression reduces to a GC c - O . ~ . Deviatiom from this relationship have been suggested7 to indicate a cylinder radius changing with surfactant concentration.R. M. WOOD AND M. P. MCDONALD 277 9 .o 8 . 6 8 . 2 7.8 7.4 7.0 6.6 2 \ ct 6.2 5 . 8 5 . 4 5.0 I I I I I I 0.20 0.30 0.40 0.50 Fig. 4. Dependence of unit-cell dimension on composition of hexagonal phase in the KO + water binary system at room temperature. 0, Balmbra et A, Ekwall et a1.;4 +, Eins;6 a, this work. KO (mass fraction) SDS +WATER The clearly linear variation of rhombus dimension with composition (fig. 3) indicates a marked deviation from the relationship a K c - O - ~ with the ensuing implication of a major dependence of cylinder radius on SDS content, The variation displayed in fig.3 will be expressed in the form A = A,-k(c-c,) (2) where A is the rhombus area at any SDS mass fraction c and A , is the rhombus area at an arbitrary mass fraction c, within the hexagonal-phase region. The gradient of the graph of area against mass fraction is k. An assumption of equality of specific volumes of surfactant and water leads to the equality of mass and volume fractions of surfactant (and of water). Since the water surrounding a cylinder will have approximately the same length as the cylinder itself, the fraction of rhombus area occupied by surfactant cylinder and by water will be the same as their volume fractions and hence their mass fractions, i.e.A , = CAR r2 = [(A, + kc,) c - kc2]/n. where A , = cylinder area = nr2. Eqn (1) now takes the form (3)278 1.90 1.85- 1.80- E 1.75 2 -a .- 2 1.70 4 -0 1.65 1.60 1.55 5. - - - - - SDS AND KO LIQUID-CRYSTAL PHASES 1.50 L I I I I 0.30 0 .GO 0 S O 0.60 0.70 SDS (mass fraction) Fig. 5. Plot of hexagonal-phase cylinder radius dependence on composition in SDS +water binary system derived from X-ray diffraction measurements of unit-cell dimensions. Arrows indicate minimum SDS contents at which the distorted hexagonal phase was observed. V, 333; a, 323; A, 313; 0, 303 K. It can be shown that on introducing the partial specific volumes of surfactant and water as v, and vH, respectively, eqn (3) becomes Using the Husson et al.l value for v,, namely 0.92 cm3 g-l for SDS and a mass fraction of 0.50, r is 2.2% smaller than the approximate value given by eqn (3), which permits the specific volume difference to be ignored. The data contained in fig.3 for the hexagonal phase of the SDS+water system at four temperatures have been analysed using eqn (3). Four resulting curves showing the variation of cylinder radius with composition appear in fig. 5 and each curve is marked to indicate the lowest composition at which the additional X-ray spacing of distorted hexagonal phase were observed in the present investigation. KO + WATER The non-linear relationship between size and composition shown by the data of fig. 4 for the hexagonal phase is well described for the Ekwall’s results4 by a oc c--O.~~, which is similar to the relationship for invariant cylinder radius, namely a cc c - O .~ . Of the data available, these results were selected for analysis because they most closely agreed with the measurements obtained in the present investigation. However, as will be apparent, it is departure from the above negative half-power relationship which is the vital factor in defining the growth behaviour of hexagonal-phase cylinders. The above dependence is in distinct contrast with that of the SDS hexagonal phase and implies a relatively small change of surfactant cylinder radius with composition.R. M. WOOD AND M. P. MCDONALD 279 However, if the water contribution to the rhombus area remains the same irrespective of whether a cylinder radius changes or not (i.e. the partial specific volume of the water is unaffected by the mixing), it would be necessary for the cylinder to contract in radius as the surfactant mass concentration increases in order to obtain an increase in the power of c from -0.5 to -0.44.Since the volume fraction of surfactant increases with its mass fraction, cylinder contraction is most unlikely and the cylinder radius probably remains essentially constant. The variation of rhombus edge length, a, with mass fraction, c, of KO may be described by the general equation a = mc-p (4) and eqn (I), on simplification by taking v, M 1, becomes L,/LR = (ca2/nr2) sin60" so that L,/LR = (m2 sin 6Oo/nr2) F 2 P . ( 5 ) Applying eqn ( 5 ) to the ideal situation of invariant cylinder radius and L, = LR leads to r = m(sin6O0/n)i = 0.525m.Hence eqn (4) can be interpreted as presenting the size of the invariant cylinder radius by the term m and the variation of the ratio LJL, by the index p . Ekwall's data appearing in fig. 4 can best be descnbed by the equation a = 4.176 c-0.443 obtained from a least-squares fit to the data in logarithmic form. This expression then provides a value of 2.19 nm for the surfactant cylinder radius. DISCUSSION The results presented here for the systems SDS+water and KO+water are of special interest for their indications of contrasting dependence of phase structure on composition. Although both systems contain a hexagonal phase followed at higher surfactant content by a non-hexagonal phase, these phases in the two systems have their own particular characteristics which appear to be determined by the lengths and structures of the hydrocarbon chains in the different molecules.Of striking contrast is the dependence of the hexagonal phase rhombus area for the two systems on surfactant content. In the case of the SDS+water system this dependence is a manifestation of a continuously increasing surfactant cylinder radius, as shown in fig. 5 where this parameter is plotted against composition for four temperatures. Note that at 313, 323 and 333 K the onset of a two-phase region was observed at almost identical values of the cylinder radius. It is possible that the two-phase region for the 303 K results would have been observed at lower SDS concentration had specimens containing between 0.50 and 0.55 mass fraction of SDS been examined.As it is, there is clear indication in fig. 5 that the hexagonal phase shows signs of changing its structure when the surfactant cylinder reaches a radius of ca. 1.77 nm. This value compares well with the length of an SDS molecule, 1.76 nm to the centre of the sulphur atom, obtained when the hydrocarbon chain is in the fully extended trans configuration. It is suggested that the hexagonal-phase structure in the SDS +water system is changed with increased surfactant content by the additional amphiphile molecules taking up positions radially within existing layers of the280 SDS AND KO LIQUID-CRYSTAL PHASES cylinders and so increasing their molecular content. Such a process would require increased cylinder circumference to maintain equilibrium separation of the polar head groups of the molecules and hence increased cylinder radius.A concomitant of this behaviour would be the ‘straightening’ of the SDS hydrocarbon chains to ensure that the available cylinder cross-section remained filled. If the additional SDS molecules were to adopt positions which resulted in longer cylinders, a different relationship between cylinder radius and rhombus dimensions would result from the requirement to maintain approximately equal lengths of cylinder and surrounding water. Fig. 3 and 5 reveal that, as pointed out earlier, temperature effects on the SDS hexagonal- phase dimensions are irregular. The results of fig. 3 show that up to a mass fraction of SDS of cu. 0.55 and a maximum temperature of 323 K, the rhombus dimensions decreased with temperature.This observation is in agreement with that of Luzzati et aZ.8 for all the systems which they reported on and for which the explanation was that the decrease in dimension follows from a contraction of cylinder radius because of the increased effect of higher temperatures on hydrocarbon-chain motion. The present results (fig. 5) show that for the lower SDS concentrations this explanation is probably correct but at higher concentrations the cylinder radius increases with temperature up to ca. 328 K and then decreases again. This behaviour is consistent with the restricted opportunity for chain flexing so leading to thermal expansion, which must occur with the proposed ‘ straighter’ chain. The chain-motion effect will then dominate at the higher temperatures.When an SDS hydrocarbon chain approaches its fully extended length, an increase in amphiphile concentration should result in little or no increase in cylinder radius by further chain ‘ straightening’. However, an increase in cylinder circumference can be obtained if the circular cross-section becomes elliptical so that the central core of a cylinder remains filled by the hydrocarbon groups. It is therefore tempting to speculate here that the continued increase in cylinder radius with amphiphile concentration of the distorted hexagonal phase is accompanied by a continued increase in eccentricity of the cylinder cross-section until eventually the ellipses are so eccentric as to be indistinguishable from the well known bilayers of the lamellar phase. Hendrikx and Charvolin, examining the SDS + decanol+ water ~ystem,~ obtained a phase which they identified as having a two-dimensional centred rectangular structure.They claim that the molecular aggregates at the lattice points are ‘ribbon shaped’ (i.e. cross-section similar to that of a lath) and incorrectly interpret their qualitative X-ray diffraction intensities as supporting evidence. As Oster and Riley have indicatedlO and Gale and Wood have demonstrated,ll the relative intensities of X-ray beams scattered by structures comprising cylinders at points of a two- dimensional net vary significantly as the ratio of cylinder-centre separation to cylinder diameter is changed. Taking up the concept of ribbon-shaped aggregates, Chidichimo et uZ.l2 studied part of the potassium [2H,]palmitate + potassium laurate + water system using deuterium n.m.r. While divergence from right-circular cylinders could be inferred from their results, it is clear that the maximum divergence was limited to an approximate ellipse of axial ratio 2: 1.Thus there is little basis for proposing cross-sections of great eccentricity. As has already been pointed out, the new phase appearing in the SDS+water system on the higher amphiphile concentration side of the hexagonal phase contains cylinders distributed on a two-dimensional lattice having a parallelogram as the repeat unit. Fig. 6 shows the variation of parallelogram acute angle with amphiphile concentration. These results may be interpreted as the outcome of a conflict between one set of forces attempting to retain the close-packed structure of a rhomboid arrangement of circles and another set of similar magnitude striving for the more openR.M. WOOD AND M. P. MCDONALD 28 1 1 I I I I 1 0.50 0.54 0.58 0.62 0.66 0.70 SDS (mass fraction) Fig. 6. Effect of composition on parallelogram acute angle in the distorted hexagonal phase structure of SDS + water binary system at 3 I3 K. t B 1711 SDS (mass fraction) Fig. 7. Effect of composition on the area of the repeat unit of the two-dimensional hexagonal and distorted hexagonal structures at 313 K. (-), Hexagonal phase; A, distorted hexagonal structure. 0.40 0.45 0.50 0.55 0.60 0.65 packing of a rectangle to fit the reduced symmetry possessed by an ellipse. In consequence, distorted hexagonal packing of the cylinders would be expected to form within a two-phase region extending over a relatively wide composition range until true hexagonal packing is unfavourable.Evidence in support of this proposal appears in fig. 7, which is taken from our previous paper and shows the two-phase region extending over ca. 0.1 mass fraction of SDS. As further evidence the n.m.r. data of the previous paper5 can be cited since, as might be expected from the above reasoning, this data displayed a continuous variation from hexagonal to distorted hexagonal structure, i.e. the transformation is not first order. In the KO + water system the cylinder radius, 2.19 nm, remains invariant with KO concentration (as calculated earlier for hexagonal phase) and agrees well with the length of a KO molecular, 2.25 nm, implying that for this molecule the hydrocarbon chain has a restricted range of conformations. The only factors which could be282 SDS AND KO LIQUID-CRYSTAL PHASES responsible for the lower flexibility compared with the apparently highly flexible state of the SDS molecule are the stronger intermolecular forces of alignment for the much longer oleyl chain and the nature of the head-group. A constant cylinder radius necessitates an increase in the relative total length of cylinder with increasing surfactant content since the monomer surfactant content of the water component will not increase.Eqn (1) states that for constant cylinder radius the rhombus area is related to the mass fraction of surfactant uia the partial specific volume of the surfactant and the relative total lengths of the cylinders and their surrounding water.However, the value of 0.96 for the partial specific volume of KO1 when used in that equation causes the index of c to move further away by a small amount from the empirical value of - 0.88 obtained from the data of Ekwall et aL4 than if the specific volume had been assumed to equal that of water. Adopting this last assumption, eqn (1) becomes equivalent to stating that the mass and volume fractions of surfactant are equal. This equality could be obtained for the data of Ekwall et al. plotted in fig. 4, which on conversion gives the relationship between cylinder rhombus areas as A , = c ~ . ~ ~ A ~ by relating the total length of the cylinders and their surrounding water by L, = c0.l2LR.For the approximate composition range of hexagonal phase in the potassium oleate +water system, 0.2-0.6 mass fraction, the ratio L,/LR changes from 0.82 to 0.94. Over this composition range the total length of surfactant cylinders increases with surfactant content relative to the length of the surrounding water until it approaches the water length at ca. 0.6 mass fraction of surfactant. This behaviour is consistent with the need for a constant-radius cylinder to grow in length with increased amphiphile content of the system. Note that for a constant cylinder radius and equal length of cylinder and surrounding water, the results of Ekwall et al. describe a situation where at each surfactant concentration there is more water present between cylinders than even equality of mass and volume fractions permits.Thus, greater surfactant contents require water to come from somewhere other than between the cylinders to maintain the equality of length of the cylinder and the surrounding water, constituting a most unlikely situation. The earlier argument that the flexibility of the hydrocarbon chains in SDS molecules and the resulting deformable amphiphile cylinders are responsible for a relatively wide two-phase region appears relevant in the present context of the transformation from a hexagonal to a rectangular phase in the KO+water system. Here the cylinders of constant radius containing apparently much less flexible molecules could be expected to exhibit a different type of transformation behaviour. This expectation is supported by the datssf fig.1 and 2, which show a relatively narrow two-phase region. In both systems, once the non-hexagonal phase has been established as the only phase present, the size of the two-dimensional lattice repeat unit decreases with increasing surfactant content because of loss of water from between the cylinders. Finally, the proposal of chain flexibility as an important factor in determining the mechanism of transformation from the hexagonal phase received some support from the work of Rendall et aZ.l3 These workers determined, by penetration experiments,14 the general types of intermediate phases occurring in sodium and potassium soaps and in sodium alkyl sulphates. Short hydrocarbon chains, i.e. C , to Cl0, appear to be associated with the so-called cubic phase; for intermediate chain lengths (Clo to C14) the phase is described as I,, which in some instances has been identified as the deformed hexagonal phase, and longer-chain systems exhibit a different phase, designated I,.The C,, sodium sulphate system did not contain an intermediate phase at all. Unfortunately, X-ray diffraction was not employed to ascertain the structures of these phases.R. M. WOOD AND M. P. MCDONALD 283 CONCLUSIONS The hexagonal phase in SDS + water becomes a deformed hexagonal phase at higher amphiphile contents. The hexagonal phase in KO +water becomes a rectangular phase at higher amphiphile contents. In the SDS system the cylinders of the hexagonal phase grow by increasing their cross-sectional area as the amphiphile content is increased.In the KO system the cylinders of the hexagonal phase remain substantially constant in cross-sectional area and grow with increased amphiphile concentration in an axial direction until the cylinders and their containing rhomboids are of the same length. The hexagonal phase in SDS +water appears to become a distorted hexagonal phase when the hydrocarbon chains of the molecules reach a fully extended configuration. The behaviour of the hexagonal phase in the two systems is determined by the length and flexibility of the molecule hydrocarbon chains, SDS molecules appearing to be much more flexible than KO molecules. F. Husson, H. Mustacchi and V. Luzzati, Acta Crystallogr., 1960, 13, 668. J. Bergeron, 1st Congr. Mondial Detergence et Pro& Tensio-Actif. (Paris, 1954), vol. 1, p. 24. R. R. Balmbra, D. A. B. Bucknall and J. S. Clunie, Mol. Cryst. Liq. 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