Connectivity of the Hexagonal, Cubic, and Isotropic Phases of the C 12 EO 6 /H 2 O Lyotropic Mixture Investigated by Tracer Diffusion and X-ray Scattering
Doru Constantin, Patrick Oswald, Marianne Impéror-Clerc, Patrick Davidson, Paul Sotta
aa r X i v : . [ c ond - m a t . s o f t ] A p r Connectivity of the Hexagonal, Cubic, and IsotropicPhases of the C EO /H O Lyotropic MixtureInvestigated by Tracer Diffusion and X-ray Scattering
D. Constantin ∗ and P. Oswald Laboratoire de Physique de l’ENS de Lyon,46 All´ee d’Italie, 69364 Lyon Cedex 07, France
M. Imp´eror-Clerc, P. Davidson, and P. Sotta
Laboratoire de Physique des Solides, Univ. Paris-Sud,Bˆatiment 510, 91405 Orsay Cedex, France
Abstract
The connectivity of the hydrophobic medium in the nonionic binary systemC EO /H O is studied by monitoring the diffusion constants of tracer molecules atthe transition between the hexagonal mesophase and the fluid isotropic phase. Theincrease in the transverse diffusion coefficient on approaching the isotropic phasereveals the proliferation of bridge-like defects connecting the surfactant cylinders.This suggests that the isotropic phase has a highly connected structure. Indeed, wefind similar diffusion coefficients in the isotropic and cubic bicontinuous phases. Thetemperature dependence of the lattice parameter in the hexagonal phase confirmsthe change in connectivity close to the hexagonal–isotropic transition. Finally, anX-ray investigation of the isotropic phase shows that its structure is locally similarto that of the hexagonal phase.
PACS : 61.30.Jf, 64.70.Md, 66.30.Jt, 61.10.E ∗ Author for correspondence. E-mail address : [email protected]; Tel : +0033-4 72 72 83 75; Fax :+0033-4 72 72 80 80 Introduction
The phase diagram of the C EO /H O system was determined more than 15 years ago[1]. Since then, extensive studies have focused on the structure of the three mesophases itexhibits, which are (on increasing surfactant concentration) hexagonal H α , bicontinuouscubic V , and lamellar L α phases. Their characteristics are by now very well established[2]. The situation is less clear concerning the organisation of the isotropic phase thatborders all the above-mentioned mesophases at higher temperature. The lack of long-range order and of optical birefringence prevents the use of common techniques such asX-ray diffraction or optical microscopy. Therefore, the isotropic phase has been mostlystudied by light scattering and NMR, but the interpretation of experimental data is largelymodel-dependent [3, 4, 5, 6, 7].The isotropic phase in C EO /H O and other related systems has also been studiedby elastic and quasi-elastic neutron scattering [8]. It is suggested from the observed fastrelaxation times that the isotropic phase is formed of small, globular micelles, even in thevicinity of the hexagonal phase. However, no direct evidence is presented to support thisassertion.Even at low concentration ( c < EO /H O system [17]. Two transitions were investigated : the H α → I atthe azeotropic point of the hexagonal phase (50.0 % C EO weight concentration) andthe V → I (concentration 59.0 %). The reason why we chose the azeotropic point forthe first transition is that the mixture behaves like a pure compound at this particularconcentration, i.e. the freezing range vanishes, which allows us to approach the transitionvery closely [15].We also measured very precisely the lattice parameter of the hexagonal phase by X-ray scattering and obtained additional qualitative information from the X-ray scatteringof the isotropic phase as a function of temperature (sections 2.2 and 3.2). Results arediscussed in section 4 within a numerical model for the diffusion of the probe moleculeswhich takes into account the lifetime of the defects. Conclusions are drawn in section 5. The surfactant was purchased from Nikko Ltd. and used without further purification. Weused ultrapure water from Fluka. The mixture was carefully homogenized. The samples3ere prepared between two parallel glass plates, with a spacing of 75 µ m as described indetail elsewhere [15]. The hexagonal phase was then oriented by the directional solidifi-cation technique presented in reference [18]. No orientation procedure was needed for thecubic phase, the diffusion tensor being isotropic.We used a hydrophobic fluorescent dye (NBD-dioctylamin) at a concentration of about0 . ∼ o C the transition temperature T h at the azeotropicpoint of the hexagonal phase , but the freezing range remains negligible (about 0.1 o C).Our method of measuring the diffusion constant is a variant of the technique knownas fluorescence recovery after photobleaching (FRAP). The experimental setup was orig-inally designed for the study of thin liquid films [19]. We focused the TEM mode of amultimode Ar + -ion laser (total power 70 mW) on the sample, bleaching a spot about 40 µ m in diameter. The intensity profile of the beam was approximately Gaussian. Typicalbleaching times were of the order of 5 seconds. About 0 . o C below the transition tem-perature the power had to be decreased to around 15 mW to avoid local melting of thesample. The evolution of the fluorescence intensity profile (proportional to the concen-tration of non-bleached molecules c n ( x, y, t ) ) was then monitored for one minute using acooled CCD camera (Hamamatsu C4742). It is however easier to write the equations forthe concentration of bleached molecules, c ( x, y, t ) = c tot − c n ( x, y, t ). We write the diffu-sion equation for the hexagonal phase, with D k and D ⊥ the diffusion constants along andacross the columns, respectively. In the equation for the cubic or isotropic phase, D k and D ⊥ both take the same value, D C or D I respectively. The bleaching is considered uniformthrough the sample, so the concentration obeys a two-dimensional diffusion equation (inthe plane of the sample); the x axis is along the columns : e x k n . ∂c∂t = D k ∂ c∂x + D ⊥ ∂ c∂y + βI c n (1)where the source term βI c n accounts for the bleaching due to the observation light, ofhomogenous intensity I ; its effect is negligible so we will ignore it from now on.The initial concentration profile is : c ( x, y,
0) = Ca a exp − x a − y a ! (2)where a and a are the semi-axes of the initial spot. The time dependence reads :4 ( x, y, t ) = C exp (cid:18) − x a +4 D k t − y a +4 D ⊥ t (cid:19)q ( a + 4 D k t )( a + 4 D ⊥ t ) (3)In the hexagonal phase, where the diffusion is anisotropic, the concentration profile iselliptical. The diffusion coefficients are deduced from the images by fitting a Gaussianfunction whose adjustable parameters are the two coordinates of the center, the amplitude C , the two semi-axes and the angle between the major axis of the ellipse and a given axisin the plane. High resolution X-ray scattering experiments were performed at the H10 experimentalstation at the LURE synchrotron radiation facility in Orsay, France [20]. A wavelength λ = 0 .
155 nm was selected by a two crystals monochromator. Harmonic rejection wasobtained by reflection of the X-ray beam on two Rh-covered mirrors. The beam size was0 . × . The sample was set at the center of a Huber diffractometer equipped witha crystal analyzer (Ge 111) . The scattered X-rays were detected with a Bicron point-detector after reflection on the crystal analyzer. The resolution of the diffractometerfor a 2 θ – scan was 0.002 o for the value of 2 θ and the FWHM of the direct beam wasequal to 0.005 o in 2 θ . For temperature regulation we used a home-made oven and acomputer-driven temperature controller.Samples were contained in flat glass capillaries of thickness 0.1 mm (Vitro Com Inc.,Mountain Lakes, New Jersey, U.S.A.). Capillaries were filled with the hexagonal phaseat room temperature by suction using a vacuum pump, as described in detail in reference[21]. This procedure allows us to obtain very well aligned samples, the surfactant cylindersbeing oriented along the capillary long axis by the flow.In order to investigate the structure of the isotropic phase, X-ray scattering experi-ments were also performed at the Laboratoire de Physique des Solides using a rotatinganode set-up (Cu K α , λ = 0.154 nm). The X-ray beam delivered by the anode is punc-tually focused by two perpendicular curved mirrors coated with a 60 nm nickel layer [21].The mirrors cut the high energy radiation issued from the anode and a 20 µ m nickel foilfilters the K β emission line. The beam size on the sample was 0 . × . . The X-ray5ntensity at the sample level is about 10 photons/s mm . The scattered X-rays weredetected on imaging plates and the sample – detection distance was 30 cm. Exposuretimes were typically 10 hours in the liquid isotropic phase. The results for the hexagonal-isotropic transition are presented in figures 2 and 3. Farbelow the transition, both D k and D ⊥ in the hexagonal phase follow Arrhenius laws, withactivation energies of 0.35 eV and 0.75 eV respectively. Starting at T ∼ T h − o C, thebehaviour of D ⊥ changes. Its value increases rapidly, departing from the activation lawand reaches 4 . − m / s at the transition temperature T h = 38 . o C. The behaviourof D I (in the isotropic phase) also fits with an Arrhenius law, giving an activation energyof 0.65 eV.Close to the liquid crystal transition temperature we attribute the increase in D ⊥ to the proliferation of defects connecting the cylinders and providing passage for thetracer molecules. The next section presents a detailed and quantitative discussion of thisphenomenon.In order to confirm this conclusion we measured the diffusion coefficient at the cubic-isotropic phase transition. This type of experiment has already been performed by Mon-duzzi and coworkers [22] for the self-diffusion coefficient of surfactant in the ionic systemCPyCl/NaSal/D O. Using PFG-NMR, they found little or no difference between the dif-fusion coefficients in the cubic and isotropic phases.Figure 4 shows our experimental results. The transition temperature is T c = 38 . o C.Within experimental accuracy, no discontinuity was observed at the transition. Since thediffusion rate is closely related to the local structure of the phase, we can then concludethat, at least at concentrations of about 60 %, the connectivity of the isotropic phase issimilar to that of the bicontinuous cubic one.6 .2 X-Ray Scattering
A well oriented sample of the hexagonal phase (50 w% of surfactant) contained in a flatglass capillary of thickness 100 µ m was mounted with a (10) hexagonal Bragg peak inreflection condition [21]. We performed 2 θ –scans of this peak at different temperatures.One such scan is shown in figure 5. The value of the hexagonal lattice parameter a at agiven temperature is obtained from the position 2 θ max of the maximum of the diffractionpeak : a = 2 √ d = 2 √ λ θ max ) = λ √ θ max ) (4)Thanks to the high resolution of the diffractometer, we were able to detect very smallchanges (down to 0.002 o ) of the value of 2 θ max versus temperature. The results are plottedin figure 6. Far below the transition to the isotropic liquid phase, a increases linearly withtemperature : a = a lin ( T ) = (5 . − . T h − T ))nm (5)The value of the thermal expansion coefficient is 1 a d a d T = 3 . − K − . Its positive signaccounts for the observation of the zig-zag structure on cooling the hexagonal phase [23].Near the phase transition, one observes (figure 6) a deviation from the linear temperaturedependence, with an additional increase in a , ∆ a = a − a lin . This effect is observed in therange ( T h − T ) < o C, in good agreement with birefringence and NMR observations [15]and the diffusion results.We performed X-ray scattering measurements on the isotropic phase at the samesurfactant concentration (figure 9). For comparison, we also plotted the first two Braggpeaks of the hexagonal phase. It is possible to notice that there is practically no shift ofthe maximum scattering vector q upon crossing the transition temperature (compare thecurves for 37 o C and 40 o C). This indicates that there is little change in the local structureof the system. Moreover, the fact that the peak in the isotropic phase is narrow and thata wide shoulder is detectable (corresponding to the second Bragg peak, in position √ d that canbe estimated as : 7 ∼ a q ∆ q ∼ a (6)where ∆ q is the width of the peak (in the isotropic phase) from which we subtract theexperimental resolution, taken as roughly equal to the width of the Bragg peak in thehexagonal phase [24]. The positional order is maintained up to fourth-neighbours. Thisshould not surprise us, since we are investigating a very concentrated system. In this section, we try to relate the pretransitional evolution of D ⊥ to the appearanceof structural defects. Indeed, NMR and birefringence data [15] show that, in the sametemperature range ( T h − o C up to T h ), the hexagonal order weakens when approachingthe transition. This feature can be explained by considering that an increasing fractionof surfactant molecules belongs to the defects. These defects may consist either in afragmentation of the columns, which become spherically capped cylinders, or in bridgesconnecting neighboring columns ( figure 1), as discussed in the Introduction (see alsoreference [15]). A possible structure for the latter type of defects has been proposed interms of Karcher surfaces connecting columns three by three [25, 26]. It is clear that connections between cylinders can provide a passage for the tracer molecules,thus leading to the observed increase in D ⊥ . The presence of defects of the first type(capped cylinders) can be ruled out, since it would bring about a decrease in D k , whichdoes not appear in our measurements. We will assume in the following that the onlydefects present are of the ‘bridge’ type (figure 1 – b). This conclusion is coherent with thefact that the curvature of the aggregates diminishes with increasing temperature (due tothe decreasing hydration of the nonionic polar groups [27, 28]) which clearly favours themerging of the cylinders over their breaking up.Let us now estimate the density of defects n def (number of defects per unit volume).In order to do so, we must quantify their role in transverse diffusion by relating n def to D def = D ⊥ − D norm (where D norm is the “normal” behaviour of D ⊥ , extrapolated from low8emperature). We shall denote by a ≃ L be the average distance between connections along a cylinder. Theadimensional parameter x = a/L provides a quantitative measure of the defect density,since n def = √ a x [29]. We shall see in the following that the diffusion contribution ofthe defects also depends on their lifetime, τ . Throughout the discussion, we will consideronly the type of defects depicted in figure 1 – b, connecting three neighbouring cylinders.In the case where the lifetime of the defects is very short compared to the characteristicdiffusion time of the molecules, D def can be evaluated analytically. The molecule has aconstant probability x to encounter a defect. Once a defect is reached, the molecule caneither cross it, or continue to move along the cylinder, with equal probability, so that itwill spend (on average) x/ − x/ z the axis of the cylinders and by ρ the position vector in theplane orthogonal to z , the presence probability for a particle starting a random walk atthe origin is (time is given in units of t , the elementary step) : P ( ρ , z, t ) = C ( t ) exp − z a (1 − x/ t ! exp − ρ (4 / a ( x/ t ! (7)where C ( t ) is a normalisation factor depending only on the time; the numerical factors2 and 4/3 are the ones for random walks respectively in 1D and in 2D on a hexagonalnetwork. Keeping in mind that we are interested not in the density along ρ , but in itsprojection on the fixed plane of the sample (which gives an additional factor 1 / √ r = D def /D k as being : r = 23 √ x/ − x/ z . Defects connecting three nearest-neighbour cylinders are introducedin order to allow transverse diffusion. When a defect is encountered, the particle has aprobability of 1/2 to remain on the same cylinder and equal probabilities 1/4 to jump onone or the other of the two connected neighbours.9ne elementary RW step defines the unit time in the simulation. Defects have alifetime τ which means that, every τ elementary steps, all the defects are erased andreplaced again at random.Each RW has 10 steps and the probability distribution is averaged over 10 RWs.The statistics of particle positions gives us the diffusion coefficients both the parallel(along the z axis) and transversal diffusion coefficients. Their ratio r = D def /D k is plottedagainst x = a/L (figure 8) for different lifetimes τ .For τ = 1, the defect configuration should be changed at every RW step and the sim-ulation would require too much time. However, since the defect positions are completelydecorrelated from one step to the other, the solution adopted was to give each particle aconstant probability x of encountering a defect. We see that the dependence r ( x ) is verywell described by equation (8) (solid line in figure 8).It is clear from figure 8 that increasing the stability of the defects lessens their efficiencyfor tracer diffusion or, in other words, more defects are needed to obtain a given value of r when τ becomes larger : if for τ = 1 a density x ≃ .
25 is needed in order to reach thevalue measured just before the transition to the isotropic phase r = D def /D k = 0 . τ = 10 one already has x ≃ . r ( x ) also changes its character with increasing τ : for τ = 1, r ∼ x (to leading order), while for frozen-in defects ( τ = 10 ), r ∼ x . This is because, in orderto jump from a cylinder to the next one at a distance a , the molecule has to diffuse along z for a distance L : t jump ∼ a D def ∼ L D k = ⇒ r = D def D k ∼ a L = x (9)Since the defects are at equilibrium, their density n def is given by a Boltzmann factor n def = n exp (cid:18) − E def k B T (cid:19) (10)where E def is the energy of the defect as compared to the perfectly ordered structure.The fact that the number of defects increases abruptly close to the transition tempera-ture means that their energy decreases (we will neglect the increase in thermal energy : k B T ≃ k B T h in the vicinity of the transition). The simplest assumption is that of a linearbehaviour : 10 def = α ( T h − T ) + E (11)The birefringence and NMR measurements [15] give (by fitting n def ( T )) a value α =225 k B ; our fit for D def ( T ) (see figure 3) gives α = 180 k B . We can thus consider that D def ∼ n def , which supports our hypothesis that the defects have very short lifetimes. Itis then justified to use (8) for evaluating the mean distance between connections at thetransition temperature L : L ≃ a ≃
25 nm, in fairly good agreement with results ofreference [15], which gives L ≃
17 nm. An important consequence is that the densityof defects in the hexagonal phase is large at the transition; the picture of the hexagonalphase as being formed of infinitely long parallel columns is no longer accurate in theseconditions.We can thus infer that the isotropic phase above the hexagonal mesophase has a highlyconnected structure.
The additional increase in lattice parameter ∆ a (figure 6) can also be explained by theappearance of connections close to the transition. In this paragraph, we derive a verysimple relation between ∆ a and the fraction f of surfactant molecules involved in theconnections.The X-ray peak provides a measure for the parameter of the hexagonal lattice formedby the surfactant cylinders. If a fraction f of molecules is involved in transversal junctionsbetween cylinders, a fraction (1 − f ) of molecules are still inside the cylinders. Massconservation then demands that in a given volume, the total number of cylinders is dividedby 1 / (1 − f ) and the lattice parameter is multiplied by a factor 1 √ − f (because thecylinders form a 2D hexagonal array). The value of f as a function of temperature isgiven by the relation : a ( T ) = a lin ( T ) √ − f = ⇒ f = 1 − " a lin ( T ) a ( T ) (12)where the value a lin ( T ) is extrapolated from the low-temperature behaviour in the regionnear the transition (solid line in 6). From this relation we obtain f ( T ), which fits wellwith an exponential law, as plotted in figure 7.11he fact that we find an exponential law for f ( T ) for 0 < T h − T < o C is inagreement with the tracer diffusion results and the NMR and birefringence experiments[15]. However, we obtain f ( T h ) = 1 . α ≃ k B T , about half of that given by thebirefringence experiment). This is not very surprising, since it is reasonable to assumethat the connections induce an elastic deformation of the hexagonal lattice, which willalso affect the value of the average lattice parameter a as measured by X-ray scattering.If such a contribution tends to reduce locally the value of a (the cylinders getting closertogether), then the simple relation (12) underestimates the value of f . The investigation of the C EO /H O system by means of diffusion coefficients measure-ments and X-ray scattering allowed us to obtain a clearer image of its isotropic phase forhigh surfactant concentration (50–60 %). The data was mainly obtained by studying thepretransitional effects that appear in the hexagonal and bicontinuous cubic mesophasesclose to the transition towards the isotropic phase.The increase of the diffusion coefficient across the cylinders ( D ⊥ ) in the hexagonalphase for a hydrophobic fluorescent dye proves that very mobile defects, consisting inconnections between the cylinders, appear close to the transition.This information is corroborated by the anomalous increase of the lattice parameter a in the same temperature range. The X-ray scattering of the isotropic phase shows awell-defined and fairly narrow peak corresponding to the first Bragg peak in hexagonalphase.No detectable jump in diffusion coefficient occurred at thetransition between the cubicand isotropic phases, showing that the surfactant aggregates in the two phases are verysimilar.We can therefore conclude that, in the concentration range that we investigated, theisotropic phase of the C EO /H O system is probably composed of very long surfactantcylinders locally preserving the hexagonal order (even though long-range order is lost),forming a highly connected and rapidly fluctuating structure.12 cknowledgements.
M. Gailhanou (LURE, beamline H10) is warmly thanked fortaking part in the synchrotron X-ray scattering experiments. We acknowledge fruitfuldiscussions with Robert Ho lyst. D. C. gratefully acknowledges financial support from theresearch group ‘Liquid crystals in confined geometries’ during his stay in Orsay.
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World Scientific, 1998.[3] Nilsson, P. G.; Wennerstr¨om, H.; Lindmann, B.
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EPJB , , , 93.[22] Monduzzi, M.; Olsson, U.; S¨oderman, O. Langmuir , , , 2914.[23] The ‘zig-zag’ texture is the result of an undulation instability that develops upondilating the system (Oswald, P. et al. J. Phys II , , , 1).[24] In the hexagonal phase, the Bragg peak should be a Dirac δ function (infinite lattice).Its width is only given by the experimental resolution. We must therefore subtractthis width in order to assess the order in the isotropic phase. The remaining ∆ q givesthen a direct measure of local order in the system : if d is the distance over whichhexagonal order is preserved, ∆ q ∼ d − , as q ∼ a − ; hence, the relation 6.[25] Clerc, M.; Levelut, A. M.; Sadoc, J. F. J. Phys. II , , , 1263.[26] Karcher, H. Manuscr. Math. , , , 291.[27] Israelachvili, J. N. Intermolecular and Surface Forces , 2nd ed.;Academic Press: London, 1992.[28] Puvvada, S.; Blanckstein, D. J.
J. Chem. Phys. , , , 3710.[29] To obtain this formula, one can use the fact that each defect connects three cylinders,so the number of individual connections is three times the number of defects, i. e. in a volume V there are N c = 3 n def V connections; in the same volume, the totalcylinder length is given by L tot = V ( √ / a , since ( √ / a is the size of the elementarycell in the hexagonal phase. The mean distance between connections is therefore L = L tot /N c = √ a n − . 14 IGURES
Figure 1:
Possible structures for defects in the hexagonal phase : a) - capped cylinders; b) -bridges. D ( - m / s ) T ((cid:176)C)D || D D I T Figure 2:
Diffusion coefficients in the hexagonal and isotropic phases : D k , D ⊥ and D I . D ( - m / s ) T ((cid:176)C) D norm D norm + D def Figure 3: D ⊥ in the hexagonal phase. Error bars are obtained by an average over three differentmeasures. The straight line represents the extrapolation of low-temperature behaviour. Thesolid curve is an exponential fit (see text). D ( - m / s ) T ((cid:176)C)Cubic IsotropicTransition temperature
Figure 4:
Diffusion coefficients in the cubic and isotropic phases, D C and D I . Error bars areobtained by an average over three different measures. Intensity (a. u.)
2q (°)
Figure 5:
Profile of the (10) Bragg peak in the hexagonal phase (limited by the experimentalresolution). .925.905.885.865.845.82 a ( n m ) -30 -25 -20 -15 -10 -5 T - T h ((cid:176)C) Figure 6:
Lattice parameter a vs. temperature in the hexagonal phase. The line is the extrap-olation of low-temperature behaviour. -3 f -6 -5 -4 -3 -2 -1 0 T - T h ((cid:176)C) Figure 7:
The fraction f of molecules present in the defects as a function of temperature. Lineis an exponential fit (see text). r (cid:13) x(cid:13) lifetime (cid:13)(RW steps)(cid:13) 1(cid:13) 10(cid:13) 100(cid:13) 1000(cid:13) 10000(cid:13)0.072(cid:13) Figure 8: r = D def /D k plotted against x = a/L for different defect lifetimes τ . I ( a . u . ) - i s o t r op i c ph ase q ((cid:176)) I ( a . u . ) - h exa gon a l ph ase
60 (cid:176)C 50 (cid:176)C 40 (cid:176)C 37 (cid:176)C (hexagonal)
Figure 9: