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Articles Viscoelasticity of the Onion Phase P. Panizza,*,† D. Roux,† V. Vuillaume,† C.-Y. D. Lu,‡ and M. E. Cates‡,§ Centre de Recherche Paul Pascal, Avenue Dr A. Schweitzer, 33600 Pessac, France, Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, U.K., and Department of Physics & Astronomy, University of Edinburgh, King’s Buildings, Edinburgh EH9 3JZ, U.K. Received May 23, 1995. In Final Form: August 25, 1995X We studied the viscoelasticity of two lyotropic lamellar systems prepared in the so-called “onion phase”. This state consists of monodisperse multilamellar vesicles filling up space. The elastic (G′) modulus is measured as a function of the frequency in the linear regime (small amplitude of the oscillations) for different onion sizes and characteristic distances between membranes. The results are interpreted in terms of the elastic deformation of smectic beads.
Introduction Recently, it has been observed that lyotropic lamellar phases under steady shear flow present different orientations.1 These orientations can be located on a so-called shear diagram.2 Such a diagram represents the steady states the system adopts under shear flow, as a function of the shear rate and the repeat distance d of the smectic order. Three main orientations have been described: At very low shear rate (typically below 1 s-1) the layers are mainly parallel to the flow with defects (dislocations) in the two other directions. At very high shear rates, the orientation of lamellae is similar but the defects in the flow direction have been suppressed. For intermediate shear rates, the lamellar phase organizes itself into multilamellar vesicles which are closed-packed and fill up space. In what follows we will call them onions. Note that cryofracture observations reveal that the structure of these onions is not spherical but polyhedral.3 One characteristic of this intermediate state is that the multilamellar vesicles formed under shear flow all have the same size, R2. This size can be tuned from a few micrometers to a tenth of this1,2 by increasing the shear rate of formation. It has been proposed that this size results from a balance between the viscous stress applied to the onion and the elastic stress.4 This argument leads to the prediction that the size varies as the inverse of the square root of the shear rate in good agreements with experiments.1,2,4 Another feature of this intermediate state is that the onion structure can be quenched by quickly stopping the shearing. Then the onion structure relaxes very slowly over a few days to several months depending particularly upon the lamellar composition. Taking advantage of this last property, we have studied the linear viscoelasticity of the lamellar phase in the onion * To whom correspondence should be addressed. E-mail address
[email protected]. † Centre de Recherche Paul Pascal. ‡ Cavendish Laboratory. § University of Edinburgh. X Abstract published in Advance ACS Abstracts, November 1, 1995. (1) Diat, O. Effets du cisaillement sur les phases lyotropes: phase lamellaire et phase eponge; Ph.D. Thesis Universite de Bordeaux I, 1992. (2) Diat, O.; Roux, O. J. Phys. II 1993, 3, 9. Roux, D.; Diat, O. French patent 9204108. (3) Gulik, T.; Roux, D. To be submitted for publication. (4) Diat, O.; Roux, D.; Nallet, F. J. Phys. II 1993, 3, 1427.
0743-7463/96/2412-0248$12.00/0
state as a function of both the onion size (fixed by the shear rate of preparation) and the distance between membranes (fixed by the lamellar chemical composition). Note that this particular spatial organization can be seen as a concentrated medium of soft (smectic) monodisperse elastic beads. Material We studied two lamellar systems. One was made of SDS, dodecane, pentanol, and water with a SDS/water mass ratio of 1.55. The phase diagram of this system has been widely described in the past.5 The SDS was puchased from Touzart & Matignon Co. (France) and was used without any purification. The composition in mass fraction of the lamellar phase, we studied, consists of 59.67% dodecane,13.245% pentanol, 10.62% SDS, and 16.465% water. This lamellar phase can be seen as constitued of water films surrounded by surfactant molecules and separated with the dodecane. The second system is made of AOT and brine. The brine solution made of 15 g/L of NaCl separates the membranes.6 The AOT, from Sigma Chemical Co., was used as received. Both lamellar phases are stabilized by undulation interactions,7 their intermembrane distance can be changed by dilution continuously from 50 to a few hundreds of angstroms.8 Experimental Section The measurements have been performed on a rheometer (CSL 100 Carrimed). We have used two geometries depending on the system studied. The viscoelatic properties of the SDS/pentanol/ dodecane/water system have been investigated using a couette cell (Mooney type). Such a cell consists of two concentric cylinders with the inner one ending in a cone shape at the bottom in order to ensure a constant shear rate throughout the cell. In our experiment, the radii of the cylinders are 24 and 25 mm. For the AOT/brine system, we used a cone/plane geometry. The cone diameter is 40 mm and its angle is 2°. Both cells are thermostated, and the temperature has been fixed at 21 °C. The experiments have been done as follows. The lamellar phase studied (corresponding to a given d spacing) is loaded into (5) Roux, D.; Bellocq, A. M. Phys. Rev. Lett. 1984, 52, 1895. (6) Skouri, M.; Marignan, J.; Appell, J.; Porte, G. J. Phys II 1991, 9, 1121. (7) Helfrich, W. Z. Naturforsch. 1978, 33a, 305. Roux, D.; Safinya, C. J. Phys. (Paris) 1988, 49, 307. (8) Roux, D.; Bellocq, A. M. In Physics of Amphiphiles; Degiorgio, V., Corti, M., Eds.; North Holland: Amsterdam, 1985.
© 1996 American Chemical Society
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Figure 1. G′ versus frequency measured for two different couette gaps: 0.5 and 1 mm (respectively black and white circles). G′′ versus frequency measured for two different couette gaps: 0.5 and 1 mm (respectively black and white triangles). The wall surface is made of Plexiglas. The system consists of 20% (mass fraction) AOT diluted in a solution of 15 g/L of NaCl. A preparation shear rate of 2 s-1 has been applied for 2 h. its measurement cell. In order to prepare the onion phase, the material is sheared at a constant shear rate (for which the onion phase is stable) for typically an hour after a constant stress is reached. We have observed that this amount of time is needed in order to have good reproducibility of the data. The shear rate is then stopped, and viscoelastic properties are measured. We performed creep and oscillatory experiments. The amplitude of the applied stress for the viscoelastic measurements is reduced until the linear response regime is apparently reached (the observed modulus or compliance no longer changes with amplitude). To prevent the structure (onions) from being too much affected by the viscoelastic measurements, we very often apply a steady shear rate for a few minutes in between two series of measurements. Previous experiments on colloidal suspensions and emulsions have revealed the importance of wall slip.9-11 If wall slip is present, one expects the measurements to be dependent on the gap. (Indeed, the wall slip contribution to the bulk elasticity should become smaller as the gap increases.9) We have checked that our measurements are independent of the gaps of the cell (0.5 mm and 1 mm), we use (Figure 1). We do then not need to account for wall slip in our measurements.
Results Creep Measurements. A creep measurement J(t) (ratio of the observed deformation at time t over the applied stress) reveals the viscoelastic behavior of the onion phase. When submitted to a step stress , the onion phase shows a very rapid elastic response like a solid, but flows like a liquid at long time (Figure 2). This viscoelastic behavior can be qualitatively well understood, if one assumes that the onion state consists of monodomains of size ξ > R separated by grain boundaries (Figure 3). For length scales shorter than ξ, the system can be seen as a perfect elastic solid (i.e. without any defects). However, for length scales larger than ξ, there is no correlations between the orientation of monodomains: the system appears disordered, and then can be seen as a viscous liquid. The compliance curves J(t) corresponding to a creep experiment (Figure 3) can be well fitted by a phenomenological model consisting of a Maxwell and a Voigt element in series12 (9) Princen, H. M. J. Colloid Interface Sci. 1986, 112, 112. (10) Yoshimura, A.; Prud’homme, A. K.; Princen, H. M.; Kiss, A. D. J. Rheol. 1987, 31, 699. (11) Mason, T. Elasticity of compressed emulsions; Ph.D. Thesis; Princeton University, 1995. Mason, T.; Bibette, J.; Weitz, D. A. Phys. Rev. Lett., in press. (12) See for instance, Ferry, J. D. In Viscoelastic properties of polymers; John Wiley & Sons: New York, 1961; p 60 and p 61.
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Figure 2. Creep curve for an onion phase of characteristic length R ) 2.3 µm. The onion phase is made of SDS, pentanol, water, and dodecane. Its composition has been given in the material section. The applied stress is 0.03 N m-2. The stress is applied for 300 s and then turned off. The fit gives for the Maxwell and Voigt elements respectively 6 and 5.4 N m-2 for the elastic moduli and 708 and 146 Pa s for the viscosities. A preparation shear rate of 10 s-1 has been applied for 2 h.
Figure 3. Schematic representation of an onion phase. ξ is the characteristic length of monodomains. Each monodomain is made of monodisperse closed-packed onions of size R. Within an onion, the lamellar structure is kept.
(Figure 4). The elastic moduli of the Maxwell and the Voigt elements are of the same order of magnitude and are quite reproducible (20%). These moduli are expected to be similar because the onions fill space and grain boundary motion must induce deformation of the onions themselves. The Maxwell viscosity, however, is not reproducible: it depends on the sample history. This is quite understandable, since we believe this viscosity to be directly related to the flowing of the material and therefore to grain boundaries that are probably irreproducible. The Voigt viscosity, which is related to dissipation in a monodomain (i.e. in or around a single onion), is more reproducible. The Voigt characteristic time is typically on the order of a few seconds. The characteristic time associated with the Maxwell element (i.e. grain boundaries) is larger, from a few tens to hundreds of seconds. In summary, the onion phase appears to behave as a viscous solid at short scales and as a glassy state at larger scales. Oscillation Measurements. We have measured the elastic (G′) and the viscous (G′′) moduli of the onion phase as a function of frequency to 2 decades (0.1 to 10 Hz). Note that below about 0.1 Hz, the measurements becomes irreproducible, since this frequency region is sensitive to the viscosity of grain boundaries. The existence of a linear response regime in this frequency range is not certain. A
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a
b
Figure 4. Schematic representation of the so-called Voigt 1 model. Such a “model” is a phemenological superposition of springs and dashpots. It consists of a Maxwell and a Voigt element in series.
Figure 6. (a) Elastic modulus G′(f) versus frequency measured from oscillations (black circles) and computed from the Laplace transform of the Voigt 1 model that best fits Figure 2 (plain line). This system is prepared as in Figure 2. (b) Loss modulus G′′(f) versus frequency measured from oscillations (white circles) and computed from the Laplace transform of the Voigt 1 model that best fits Figure 2 (plain line). The system is prepared as in Figure 2. Figure 5. Elastic G′ (black circles) and loss G′′ (white circles) moduli versus oscillation frequency. The system is made of SDS, pentanol, water, and dodecane with the same mass fraction as in Figure 2. The onion size R is 2.3 µm. G′ and G′′ are respectively expressed in N m-2.
typical plot for G′ and G′′ versus f(Hz) is given in Figure 5. These curves are not unusual for a viscoelastic medium. For this range of frequencies, the system is more elastic than viscous (i.e. G′ > G′′). Note that the value of this plateau for G′(f) corresponds to the inverse of the instantaneous compliance in a creep experiment in agreement with the previously proposed model. In the linear regime, G′(f) and G′′(f) are indeed related to creep compliance J(t), through a Laplace transform. For the same system, Figure 6 compares G′(f) measured with oscillations to G′(f) derived from the Laplace transform of the creep model12 that best fits the compliance curve J(t) as given in Figure 2
G′(f) + iG′′(f) )
[
]
Gm + i2πfηm 1 + i2πfηmGm GV + i2πfηV
-1
(1)
Note that the agreement is relatively good between the two curves for G′(f) (Figure 6a). However the predicted decrease in G′′(f) at frequencies above the Voigt relaxation rate is not observed (Figure 6b). This suggests the presence of other relaxation modes not reflected in the
Figure 7. Evolution of G0 for the same system as Figure 2 prepared at different bead sizes. The preparation shear rates, applied for 2 h, are 7, 10, 15, and 25 s-1.
creep fit. By Kramers-Kronig, such modes must also give a rise in G′ over the same frequency range which is indeed visible in Figure 6a. Let us now examine the evolution as a function of R, the bead size. As previously described, we can prepare the same system (same d-spacing) at different shear rates corresponding to different sizes of the beads. Figure 7 represents the values of the high-frequency (plateau) elastic modulus G0 for the solution made of SDS, pentanol,
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Figure 10. Zones of high deformation in a polyhedral droplet. Note that the exterior shape is “forgotten” only very close to the center (within λ of it). Figure 8. G0 versus 1/R for the solution made of SDS, pentanol, dodecane, and water (Figure 2). The best fit gives G0 ) 3.84 + 8.11/R with G0 in N m-2 and R in µm. For this dodecane h )1/2/R. concentration, this corresponds to G0 ) 3.84 + 0.23(KB
Figure 9. (a) G0 versus 1/R for a solution of 20% in mass of AOT in a brine solution of 15 g/L. The best fit gives G0 ) 53.63 + 40.446/R with G0 in N m-2 and R in µm. For this concentration in membrane, this corresponds to G0 ) 53.63 N m-2 + 0.57(KB h )1/2/R. (b) Shear rates of preparation and corresponding onion sizes for the system of Figure 9a. Each preparation shear rate is applied for 3 h.
dodecane, and water (the intermembrane distance is 120 Å) prepared at different sizes, R. Note that from (1) G0 ) Gm. When the size decreases, G0 increases. Indeed, G0 varies linearly with the inverse of the bead size (Figure 8): G0 ) R + β/R . This variation with the bead size is also checked for the system made of AOT in a solution of NaCl (15 g/L) (see Figure 9). However, the total change in G0 is not large so that other fits (such as G0 ) R′ + β′R) cannot be ruled out. From dimensional analysis and what is observed in other systems, like compressed emulsions and incompressible foams,13,14 this R dependence of the (13) Princen, H. M. J. Colloid Sci. 1983, 91, 160.
elastic modulus suggests that β is an effective surface tension associated with deformation of the onions. Theory For spherical onions, it is known15 that σ ≈ (KB h )1/2 plays the role of an effective surface tension in determining the cost of small deformations. Here K and B h are respectively the curvature and the compression modulus of the lamellar phase.16 For polyhedral shapes, the deformation energy is stored somewhat differently: it is concentred near certain planes of stacked edges of successively nested, faceted droplets (Figure 10). In fact, the deformation is localized within a width λ of these planes, where λ ) (K/ B h )1/2 is the penetration depth.16 Nonetheless, a straighforward estimate can be made as follows: the volume of deformed material per droplet is of order λR2, and the local curvature throughout this region is λ-1. Hence the stored energy per unit volume is of order K/λ2 (R2λ/R3) ≈ σ/R. The area of planes has a fractional increment of order γ2 when a strain is applied; hence dE ) σγ2/R. This shows that for small strains γ about a symmetric polyhedron, the deformation energy is still of order (σ/R)γ2. (Here, “symmetric” means that strains γ,-γ cause equivalent changes in the droplet shape.) On this basis, for a perfectly ordered array of polyhedra, one would predict β ≈ σ, consistent with the experiment, but R ) 0. We now argue that R is connected with the disordered arrangement of the onions on a scale ξ > R. Because of this disorder (arising from polydispersity and other effects), any individual onion is subject to asymmetric mechanical forces from its neighbors: the pre-existing quenched strain, of characteristic magnitude γ0 (say) interacts nontrivially with an applied deformation, giving a modulus contribution. To estimate this, note first that even for a symmetric onion, an affine deformation γ would lead to a compression of the layers in some places and a dilatation in others. This would result in a high modulus h ). However, the layers within each onion (G0 of order B can rearrange themselves,17 sacrificing bending energy to minimize the compressional cost: this restores the mean layer spacing to its original value almost everywhere (apart from the regions of size λ). The result is a reduction in G0 by a factor λ/R , 1. This mechanism exploits the (14) Khan, S. A.; Amstrong, R. C. J. Non-Newtonian Fluid Mech. 1986, 22, 1. (15) Van der Linden, E.; Droge, J. H. Physica A 1993, 193, 439. (16) de Gennes, P. G. In Physics of Liquid crystals; Clarendon Press: Oxford, 1976. (17) The rearrangement from an affinely distorted sphere to one of uniform layer spacing can, for infinitesimal distortions, take place at constant concentration. Linear relaxation is therefore likely to proceed very rapidly by second sound (see ref 16). However, when background strains are present, the situation is more complicated and there could be some coupling to the baroclynic mode.
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Figure 11. log-log plot of R versus membrane concentration for a solution of AOT in brine (15 g/L). For each membrane concentration, we applied a preparation shear rate of 2 s-1, for 3 h. The best fit shows that R varies as Φ2.93.
key fact that (by symmetry) the fractional area change of each shell in the onion is of order γ2 and may be neglected. For an asymmetric (prestrained) onion, in contrast, the fractional area change of each shell is of order γ0γ, with a coefficient whose sign is the same for all shells in a given onion, but varies randomly from one onion to the next (see ref 18 for a discussion of similar effects in disordered foams). This means that, even after various local relaxation processes,17 the layer spacing will depart at order γ from its ideal value (which has changed in response to the new surface area). Accordingly, a compression cost of order B h γ02γ2 is incurred and cannot be relaxed by bending. This corresponds to a modulus contribution R ≈ B h γ02 . A more careful calculation, based on a 2-D array of hexagonal “onions” gives R ) B h γ02 with 19 prefactor unity. Qualitatively we expect an R-like term in the Voigt modulus GV as well as in G0, since local shape changes and grain boundaries motion both couple to the quenched strains: GV and GM should remain of similar magnitude. Our prediction for R is in reasonable accord with the data, requiring quenched strains of order 3% in the SDS h system (B h ≈ 5000 N m-2) and 9% in the AOT system (B ≈ 7000 N m-2), Figures 8 and 9. Morever, if we assume that γ0 does not depend strongly on d, we obtain at once the scaling R ≈ B h ≈ Φ3, as found experimentally (Figure 11). However, we have as yet no interpretation of the large difference in γ0 between the SDS and the AOT systems, which seems to indicate an intrinsic difference in the type of disorder. (This could be linked with different residual polydispersity, or other factorsswe do not know.) (18) Buzza, D. M. A.; Lu, C.-Y. L.; Cates, M. E. J. Phys. II 1995, 5, 37. (19) Lu, C.-Y. L. Ph.D. Thesis, University of Cambridge, in preparation.
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It is interesting, finally, to consider the origin of the (reproducible) Voigt viscosity ηV. this comes from the dissipation associated with local onion deformation within a monodomain. There are two dissipative contributions; flow of fluid between adjacent onions and flows within a single onion. The first would contribute a viscosity ηV ≈ ηsR/d, where ηs refers to the solvent; there is a geometrical enhancement factor R/d because shear gradients are concentrated within the thin solvent film that separates an onion from its neighbors. (This corresponds to rubbing one onion past another.) The second term, arising from the shape changes within a single onion, gives ηV ≈ ηeff, an effective (average) viscosity of the material within a smectic droplet. (This is hard to estimate; see ref 20 .) For the system of Figure 1 (R/d) ) 200, one has ηs(R/d) ) 0.2 Pa s compared with ηv ) 146 P as as measured (Figure 2). This suggests an intraonion contribution at least of order ηs(R/d)2,which suggests a coupling to a baroclynic type relaxation mode,17 in which Poiseuille flow occurs between successive layers of the onion. It was noted already above that the loss modulus G′′(f) is higher than that predicted from the Maxwell/Voigt model. This too may suggest a strong viscous contribution from internal modes with characteristic frequencies in the range probed by the linear viscoelastic measurements. Conclusion Onion phases behave in creep as viscoelastic solids. Their small-strain viscoelastic response involves two mechanisms. The first mechanism is responsible for liquidlike behavior at long time and has a very slow characteristic frequency, typically 0.001-0.01 Hz. This appears to correspond to slow relaxation of grain boundaries or other long range organization. The details of its viscosity are irreproducible and we have not tried to explain them. The second mechanism has a characteristic frequency of typically 0.01-0.1 Hz and is probably connected with the local response of individual onions; its viscosity is much larger than that predicted from the “rubbing together” of two adjacent droplets but could arise from intraonion relaxations. The plateau modulus G0 can also be rationalized theoretically as the sum of two terms. The first is associated with the effective surface tension σ ≈ (KB h )1/2 and varies inversely with the onion size; the second is a size-independent contribution B h γ02 where γ0 is a characteristic scale of quenched strains, arising from the random arrangement of the onion domains. Acknowledgment. We thank P. Boltenhagen, H. Gayvallet, F. Lequeux, S. Milner, and J. F. Palierne for discussions and P. Sierro for drawing Figure 3. The work of C.-Y. D. Lu is supported by the Oppenheimer Fund and the Colloid Technology Programme. LA9504016 (20) Lu, C.-Y. L.; Cates, M. E. J. Chem. Phys. 1994, 101, 5219.
