Langmuir 1999, 15, 7495-7503
7495
Concentration Fluctuations in Surfactant Cubic Phases: Theory, Rheology, and Light Scattering J. L. Jones†,‡ and T. C. B. McLeish*,§ University of Leeds, Department of Physics and Astronomy and IRC in Polymer Science and Technology, Leeds LS2 9JT, U.K., and University of Durham, Department of Chemistry, Polymer IRC, Durham DH1 3LE. U.K. Received January 26, 1999. In Final Form: May 27, 1999 Ternary surfactant cubic phases exhibit a sharp melting transition. We investigate the effect of concentration fluctuations in this system which are a possible driving mechanism for the phase transition. We consider local variations in the surfactant concentration and impose the conditions of a fixed topology, volume, and surface area of surfactant. The energy is derived from changes in curvature and is given by the Helfrich Hamiltonian in contrast to a naive expectation from dilation invariance. We find that the constraints give rise to an energy change which contributes to the bulk modulus and gives the expected (∇δφ(r))2 term. The decay of the fluctuations is also examined, and a relaxation time that contains both diffusive and soft modes is obtained. We compare estimated times with experimental values which we have measured by performing dynamic light scattering on cubic phases of didodecyl-dimethylammoniumbromide/ toluene/H2O. In the range of wave vectors studied we observe only the diffusive modes; these times are predicted by the theory. In addition we consider the consequences of fluctuations to rheological measurements. We present rheological results that clearly show the melting transition.
Introduction During the past few years self-assembled phases of aqueous, nonaqueous, and surfactant components have been the focus of much interest.1-3 They are able to form a myriad of complex ordered phases, and consequently, they display remarkable physical characteristics. There have been only a few studies of the dynamic properties of cubic surfactant phases, concentrating principally on the rheological response.4-6 Recently, however, attention has turned to the local dynamics of the surfactant sheet which may be probed via dynamic scattering techniques. Clerc et al.6 have shown that offreciprocal-lattice dynamic neutron scattering exhibits an inelastic decay with an effective cooperative diffusion constant (so that the relaxation times τ(q) ∼ q-2)) on a gyroid phase. The authors speculate that thermal fluctuations of the ordered membrane are responsible. Extending the fluctuation measurements (via light scattering) to longer length scales is therefore highly desirable and may permit such local structural probes to be correlated with bulk measurements such as rheological response. Indeed this approach is now well established for polymeric complex fluids. In this article we present both experimental dynamic light scattering and theoretical results on the effect of thermal fluctuations in the surfactant (Schwartz) cubic phase and its melting transition. The particular system that we have studied here is composed of three components, water/toluene/didodecyl† Current address: Unilever Research Laboratories, Port Sunlight, Winal, U.K. ‡ University of Leeds and University of Durham. § University of Leeds.
(1) Radiman, S.; Tobracioglu C.; Faruqi A. R. J. Phys. (Paris) 1990, 51, 1501. (2) Bruinsma, R. J. Phys. II 1992, 2, 425. (3) Toprakcioglu, C. In Theoretical Challenges in the Dynamics of Complex Fluids; McLeish, T. C. B., Ed.; Kluwer: Dordrecht, 1997. (4) Radiman S.; Toprakcioglu C.; McLeish T. C. B. Langmuir 1994, 10, 61. (5) Jones, J. L.; McLeish, T. C. B. Langmuir 1995, 11, 785. (6) Clerc, M.; Hendrikx, Y.; Farago, B. J. Phys. II (France) 1997, 7, 1205.
Figure 1. (a) The Scharwz P minimal surface that is formed by the surfactant bilayer. (b) The unit cell, where W is the aqueous phase and the toluene resides between the two parts of the bilayer.
dimethylammonium bromide (DDAB). The surfactant molecules naturally prefer to sit at the interface between oil-filled and water-filled regions, but depending on the relative concentration of the three components, many composite structures can be realized. One of the most interesting families of surfaces comprises those with an infinitely periodic surface of cubic symmetry, which divides space into two equivalent but disconnected parts. These cubic phases themselves have several possible symmetries: one of the simplest and more common is the Schwarz P-surface. This is the phase identified for this system1 in which the aqueous component forms a bicontinuous phase and the surfactant (DDAB) bilayer forms a minimal surface with the symmetry group Pn3m, as shown in Figure 1. The equilibrium structure emerges from energy considerations based on the premise that all the energy of the system is derived from the curvature energy of the surfactant layer. The free energy can be written in terms of the Helfrich Hamiltonian7 which is
10.1021/la990079p CCC: $18.00 © 1999 American Chemical Society Published on Web 10/01/1999
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Langmuir, Vol. 15, No. 22, 1999
H)
κ 2
∫ (R11 + R12) d2s + 2κj ∫ R11R2d2s 2
Jones and McLeish
(1)
where R1 and R2 are the two local principal radii of curvature on the surface, parametrized by s. In eq 1, the first term is the contribution from the bending energy of the surface and the second is the GaussBonnet term, which depends on the connectivity of the system. Provided that the topology remains fixed, the second term is constant; in our work we assume that this is the case and subsequently that the total number of cells in the crystal structure is fixed, even though they may be deformed. X-ray and neutron scattering studies1,4,6 have already revealed the periodicity of the structures and have shown that, although the cell size depends on the precise volume fraction of water, it is generally of the order of 10 nm. These results have already indicated the existence of a melting transition: as the temperature is raised above a certain value, the Bragg peaks defining the ordered state disappear. Another property of this phase is its ‘ringing′ behavior when a small sample is tapped. The melting exhibits interesting rheology: on increasing temperature, the material melts into one with much lower modulus and the ringing disappears. The mechanism by which the order/disorder transition occurs is unknown, and there is still speculation as to the order of the transition. It is anticipated that the work presented here goes some way toward understanding this process, and the theoretical work examines the role of concentration fluctuations as a mechanism driving the phase transition. In section 1 we examine one candidate for a divergent degree of freedom as the system approaches the transition. We suggest that thermally induced fluctuations occur in the lattice parameter. The possibility that these diverge is suggested by the scale invariance of the Helfrich Hamiltonian; indeed Bruinsma suggests that higher order terms in the Hamiltonian are needed to stabilize thermal fluctuations.2 However, we will find that such terms in the local free energy density of higher order in the curvature are not required to stabilize the cubic phase against such composition fluctuations. We present a theoretical study from which we predict the dynamic scattering function and the relaxation times for the decay of the fluctuations. This theory involves the treatment of concentration fluctuations in the surfactant component which leads to a contribution to the bulk energy. A weak first-order melting transition (involving small discontinuities in mechanical susceptibility and osmotic pressure) to a phase of higher symmetry is more likely. In section 2 we describe some rheological results. From an experimental viewpoint, the rheological data presented here confirm the results of Radiman et al.4 and Jones and McLeish,4,5 in that the melting transition can be observed from a sharp decrease in the two components of the complex modulus. Section 3 presents the dynamic lightscattering results and an analysis of these measurements in terms of our theoretical predictions. We show that bulk relaxations exist and obtain the wave vector dependence of the relaxation times.
falls short of predicting the actual melting transition, but studies the role of concentration fluctuations in the modification of the free energy. The quantity of interest is the local surfactant concentration φ(r). Naturally, we have to define this concentration on a length scale greater than one cell length because this component occupies the space in the walls of the cell and over the length scale of one cell, the concentration varies hugely from 100% in one wall, to zero in the aqueous phase in the center, then 100% again in the opposite wall composed of surfactant material. In this work we consider concentration variation wavelengths of the order of many cell units. We calculate the effect of fluctuations δφ(r) from the mean value φ0(r), where φ(r) ) φ0(r) + δφ(r). Within our system the number of cells is constant because we do not allow the topology of our network to alter. Under this constraint the variations in the local density of the surfactant manifests itself in the form of local regions of small cells where there is a higher than average surfactant concentration and regions of large cells where the volume fraction of the surfactant phase is lower than the mean value. Evidently, to achieve these gradients in concentration, material must flow within the surfactant wall from one region to another and the length scale of the concentration variations will determine how quickly the fluctuations may relax. Before looking at the dynamics we must address the issue of how they affect the free energy. We turn our attention to the Hamiltonian and look at the role played by concentration variations. 1.1. Calculation of the Free Energy. We already know that the mechanical properties of the system are determined principally by the curvature energy of the surfactant layer and that any deviations away from the minimal curvature surface shape will increase the energy. Consideration of the terms that appear in the free energy expansion leads us to the following form for the lowest order contributions
H)
∫B(δφ(r))
2
+ λ(∇2δφ(r))2d3r
(2)
Here we calculate the effect of thermal fluctuations that may become important as the transition is approached. This work
The term(∇δφ(r))2 is absent because first-order changes in the concentration gradient ∇δφ(r) do not cause first-order changes in the curvature, but the lowest order correction to the curvature energy is second order. The next contribution therefore comes from the second derivative in the gradient which is the second term above. The coarse-grained parameter λ is directly related to the mean curvature parameter κ as seen by comparing eqs 1 and 2. Dimensional analysis gives κa4 ≈ λa3 (where a is the lattice spacing of the cubic phase) so that λ ≈ κa. Additionally we can consider the Lindemann condition at the melting transition (by which estimates of melting temperatures Tm are made by equating mean thermal displacements of local structures with the lattice parameter of the crystal.8 At the transition the magnitude of the second term in eq 2 may be estimated by using a for all spatial length scales to be λa〈δφ(r)2〉 ) kT/2 which for then gives λ ) kTma. We now discuss the contribution from concentration fluctuations to the compositional bulk modulus B. We suppose that thermal fluctuations cause changes in the cell sizes so that we have a distribution of sizes, and the topology of the system is preserved. Bruinsma2 assumes that cell size fluctuations may occur without a change in cell shape, i.e., the crystal consists of Schwarz P-cells of different sizes. The interesting corollary of this is that no free energy is expended in changing the size of the cell provided that the shape remains on the minimal surface. Therefore, according to this model higher order terms in the Helfrich Hamiltonian are needed to prevent the instability inherent in the lowest order modes. However, because we have to preserve the total volume of material in the sample and the total surface area of surfactant in the walls, it is impossible for the cells to maintain the shape of a minimal surface. There are strong arguments for justifying that the area of the wall material is a constant, because the modulus of dilation for the wall is much higher than the bending energy associated with deforming the surface away from minimal.
(7) Helfrich, W. Z. Naturforsher 1973, 28c, 693. Deuling, H. J.; Helfrich W. J. J. Phys. (Paris) 1976, 37, 1335.
(8) Zemansky, M. W.; Dittman, R. H. Heat and Thermodynamics; McGraw-Hill: New York, 1981.
1. Theoretical Treatment-Concentration Fluctuations
Fluctuations in Surfactant Cubic Phases
Langmuir, Vol. 15, No. 22, 1999 7497 in δr(x) is the amplitude of the perturbation which varies sinusoidally across the crystal and is q-dependent. At this stage we introduce the most important parameter in this model: a factor that takes into account the shape changes which we will label K(x). The x-dependence signifies the possible slow spatial gradients in the cell shapes over many cells. We are able to write the conservation of area in the following way,
A0 ) Figure 2. Schematic representation of the sinusoidal variation in cell size a(x). In essence, we argue that the constraints, of total number of cells, total volume of the system, and the total area, force the system to deform away from the minimal surface and that this is the origin of the extra contribution to the Hamiltonian from the concentration fluctuations.9 To see simply how the argument works, consider first a crude partitioning of the system into n1 small cells of size a1 ) a0(1 δ) and n2 large cells of size a2 ) a0(1 + δ), where n ) n1 + n2. The conservation of volume gives the result that n1 ) n/(2(1 + δ)) and similarly n2 ) n/(2(1 - δ)). Using these expressions and writing the total area as A ) n1a12 + n2a22, we find that expanding this gives A ) A0(1 - δ2) and the area cannot be conserved, but is reduced. To preserve the number of cells, volume, and area we have to introduce a shape change and we write the area in the form A ) K-n1a12 + K+n2a22, where K- and K+ account for the change needed to recover an area of A0 in the smaller and larger cells, respectively. First-order departures of K- and K+ from unity imply a cost to the Helfrich Hamiltonian. A full treatment of this problem requires a calculation of the new shape of the surfactant surface subject to thermal fluctuations. One approach to this problem would be to map out the positions of all the points on the surface, which would entail extensive mathematical analysis. It is inappropriate to use this for our purposes because we only require the general form of terms in the free energy and particularly their dependence on wavenumber q. In our case it suffices to examine changes in curvature by means of two scalar fields, representing a local variation in cell size and an accompanying variation in the mean curvature. We evaluate the contribution to the free energy from the changes in curvature in the following way. We assume that we can write the variation in the cell size as a sinusoidal variation over many cells and that a(r) is the cell dimension then
a(r) ) a0 + δ1 + δ2 sin(q.r)
(3)
as shown in Figure 2. At this stage we make the distinction between the position of a cell, r(x) and its number, x, in the crystal. The analysis is easier conceptually if we work in terms of x and just count the total number of cells. In all that follows we consider the cell size a(x) and the average change to the radiusof-curvature δr(x) of its surface to be functions of the cell number x rather than distance, and we assume that the variations are the same in all three coordinate directions. δr(x) is the change in the local radius of curvature which accompanies the perturbation in the cell size a(x). The amplitude of the cell size fluctuations is δ2 and is small compared with a0. The wavelength of the fluctuations is 2π/q. The offset δ1 is necessary to preserve volume for a sinusoidal variation in the cell size as we will see below. The form of δr(x) is chosen to be a sinusoidal function of position and is written as
δr(x) ) δ3 + δ4 sin(qa0x)
(9) Semenov, A. University of Leeds, U.K. Personal communication, 1997.
2
(5)
We write the volume as below where f is a geometrical factor that depends on the exact lattice type,
V0 )
∫ fa(x) dx 3
(6)
The form of K(x) requires considerable care but we will find a consistent choice within our perturbation theory to be
( ( ))
K(x) ) K0 1 + c
2
δr(x) a(x)
(7)
where c is a geometrical factor of order unity. The quadratic dependence on the cell-size perturbation arises from the minimalsurface property of the equilibrium surface. Combining the equations above immediately gives us the following relations between the amplitudes. The volume constraint gives
δ1 ≈
-δ22 2a0
(8)
and the area gives an equation that is used during the minimization of the energy. This constraint equation is
g(δ2, δ3, δ4) ) δ42 + 2δ32 - δ22 ≈ 0
(9)
1.2. Minimization of the Energy. Here we solve for the energy change subject to the constraints. As discussed earlier, the energy is all derived from the curvature and is written down mathematically in terms of the Helfrich Hamiltonian. Shape changes away from the minimal surface give a positive contribution to the free energy, and the nonzero part of the Hamiltonian that depends directly on the local curvature changes is
∆H ) 2κ
∫( a(x) ) d s ) 2κ∫( a(x) ) δr(x)
2
2
δr(x)
2
dx
(10)
We assume a separation of time scales such that a(x) is a slow variable (because it couples to the local surfactant concentration that is conserved) and δr(x) is a fast variable (because it couples to the local shape change of the surface which does not require long-range convection of the surfactant). To calculate the free energy cost of a fluctuation in a(x) we first impose a sinusoidal concentration fluctuation of amplitude δ2 (see eq 3), then find the shape change given by δr(x), which minimizes the free energy. The minimization requires the solution to the following two equations together with eq 9 (constraint equation) with a Lagrange multiplier R:
∂∆H ∂g +R )0 ∂δ3 ∂δ3
∂∆H ∂g +R )0 ∂δ4 ∂δ4
(11)
Two simultaneous linear equations emerge which are
(4)
where δ3 represents a change in the shape (i.e., the curvature), which is the same for all values of q and for all cells. As for the offset δ1, we will find δ3 is necessary to satisfy volume and area constraints when the perturbation is present. The second term
∫ K(x)a(x) dx
(P + 2cR)δ3 + Qδ4 ) 0
(12)
Qδ3 + (R + cR)δ4 ) 0
(13)
from P, Q, and R, from eqs 12 and 13, are given below with their values to the lowest order in the perturbation amplitudes.
P ) 2κ
∫ (a
0
κN dx ≈2 2 + δ1 + δ2 sin(qax))2 a0
(14)
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Langmuir, Vol. 15, No. 22, 1999 Q ) 2κ
∫ (a
R)κ
sin(qax)dx 2
0
+ δ1 + δ2 sin(qax))
Jones and McLeish ≈
sin2(qax)dx
∫ (a
2 0 + δ1 + δ2 sin(qax))
-2κNδ2 a30 κN a20
≈
(15)
(16)
This procedure gives the solutions for the Lagrange multiplier R and the curvature perturbation parameters δ3 and δ4 as
R)
-κN x2κNδ3 ( a20c a30c
δ3 ) (
δ2
x2c
δ4 ) (
δ2
xc
(17a)
r(x) )
∫ (a
(1 ( x2 sin(qax)) δr ) x2c
(17b)
∆E δr 2 1 ≈κ V a a3
( )
∆E )
2ca20
∫
(
(1 + x2 sin(qax))2
1-
δ22 2a20
+
δ2 a20
)
dx
2
K(x)a (x)b a3(x)
2
a-1(x)b
(21)
which to first order in δ2 is
(
(26)
An interesting point is that this expression contains a constant mean concentration that must be subtracted from the energy expression to compare directly with the term in the Hamiltonian. This is associated with the constant terms required in the perturbations functions for cell size and radius of curvature. Viewed alternatively, the constraints on the system mean that pure harmonic functions are not eigenmodes of composition fluctuations. Each finite q mode of composition must couple to a q ) 0 mode from the conservation of total area and topology. The composite bulk modulus picks up a contribution from this “mode-coupling” at all q. Once this is done we obtain an expression for the energy per unit volume B
B)
κ a0K20b2c
(27)
This may be rewritten in the more convenient form
( ( ))
φ(x) )
δ22 E ) κ 5(δφ + δφ)2 V a0
(20)
where b is the thickness of the surfactant wall. Substituting for K(x) we obtain
δr(x) a(x)
(25)
(19)
As indicated above, we find that the coefficient of the secondorder term in the free energy arises from the mean curvature constant κ, so that no more than the Helfrich Hamiltonian is required to stabilize the fluctuations. This contrasts with Bruinsma’s suggestion that higher order terms are required for bulk stability. Our conclusions are different because we have assumed the additional constraint of conserved global surface area of the surfactant. This constraint produces a second-order stabilizing term from the Helfrich Hamiltonian alone. 1.3. Change of Variables to the Local Volume Fraction. So far the development of the theory has been achieved in terms of the parameter a(x), (cell size) as a function of cell number. We now want to rewrite this in terms of the local volume fraction of surfactant material φ(x) because it is this that gives rise to the scattering by determining the average index of refraction. We define this in the following way by dividing the volume of surfactant per cell (Ka2b) by the cell volume (a3):
φ(x) ) K0 1 +
1 a3c
To derive a bulk modulus B, this expression has to be related to the one for the energy per unit volume in terms of the concentration fluctuations, which is B/(2(δφ(r))2). To do this we rewrite the wavelike perturbations of the lattice parameter in eq 25 in terms of the perturbations in local density via eq 22. The result is:
which to lowest order in δ2 gives
φ(x) )
)
2
(18)
sin(qax)
∆E ≈ κNδ22/ca20
where we write the size of all length scales and derivatives of magnitude a. Substituting the form of δr (from eq 17b) gives the following expression for the energy per unit volume:
(
2
(23)
(24)
δ2(1 + x2 sin(qr)) ∆E )κ V a
The change in the free energy is
κδ22
+ δ1 + δ2 sin(qax))dx
However, the difference between this and the zero-order relation r(x) ) a0x only contributes to higher order terms in the perturbation theory, so we may consistently use δφ(r) ) -bK0δ2/ a02 sin(qr). 1.4. Calculation of B. We now possess all the tools needed to relate the curvature energy within our cell-size perturbation to the coarse-grained form of the free energy identified in eq 2 by finding an expression for B (we argue for a value for λ in section 1.6). The energy per unit volume from the integral of eq 18 scales as
Consequently, the local radius of curvature carries the associated sinusoidal perturbation and background offset
δ2
0
)
bK0 δ2 1 - sin(qax) a0 a0
and bK0δ2 δφ(x) ) - 2 sin(qax) (22) a0 To relate the energy to the scattering function we need to work in terms of the variable δφ(r), where r is the actual position of the surfactant. Strictly speaking converting between x and r requires the change of variables through the equation relating them, which is
B)
κ φ02a30c
(28)
This simple expression confirms our hypothesis that the mean curvature energy κ does contribute to a bulk modulus in terms of surfactant concentration. It will therefore control the dynamics of concentration fluctuations in general, rather than higher order terms in the curvature. 1.5 Dynamics. We now turn our attention to the time evolution of the fluctuations. We assume that the decay obeys the simplest type of dissipative equation of motion for a conserved order parameter. In this picture spatial gradients in the chemical potential µq drive currents in surfactant concentration, j against a molecular drag coefficient ζ. The dynamic equation is derived from the conserved order parameter δφ under the driving Hamiltonian of eq 2.10 As usual11,12 we take Fourier transforms of fields and fluxes to give (10) Onuki, A.; Kawasaki, K. Physica A 1982, 11, 607. (11) Fredrickson, G. H.; Larson, R. G. J. Chem. Phys. 1987, 86, 1553.
Fluctuations in Surfactant Cubic Phases ζ
∂φq ) -iqjq ∂t
Langmuir, Vol. 15, No. 22, 1999 7499 (29)
where jq ) -iqµq and µq ) (δH/δφq) H is given by eq 2 with the subsequently determined values of B and λ. This model gives a relaxation time that contains both diffusive terms in q2 and ‘soft modes’ varying as q6. The relaxation time is
τq )
ζ q2(B + λq4)
(30)
exp(-t/τq) (B + λq4)
(31)
This simple analysis shows that two different regimes of q-scaling of the relaxation time are anticipated. For large wavelengths the bulk modulus B dominates the relaxation time; for small wavelengths it is the gradient energy λ. The two regimes are separated by a “crossover” wavenumber q* ) (B/λ)1/4. We need to examine the ranges in which we would expect each type of behavior. This is discussed in the following section. 1.6 Consequences for Light Scattering and Rheology. We must obtain the orders of magnitude for the important parameters B and λ. We have an average cell size of approximately 10-8 m and if we take κ to be 100 kT (so that thermal fluctuations in quantities are at the 10% level) and φ ) 0.5, we obtain a value for B of 2 × 106 J/m2. Similarly, we can approximate λ with the result that at the melting transition a Lindemann criterion gives λ2π/q〈δφ(r)2〉 ) kT/2, then λ ) kTa ≈ 5 × 10-29 J m. From the considerations above we estimate the value of q for which the q2 term dominates the relaxation time. We find the crossover to be approximately 2π/a0; and for all q values greater than this value, Bq2 is the dominant term. Essentially, this means we do not expect to be able to see the q6 dependence in the lightscattering experiments to within the accuracy of the approximations above, because it lies beyond the range of wave vectors in which the continuum approximation is valid. Given the magnitude of the controlling term in the free energy we can estimate the relaxation times expected assuming that the hydrodynamic back-flow associated with the cell-size perturbation is the dominant source of dissipation.4 We use ζ ) η/a2 and η ) 10-3 Pa (the viscosity of water). If we introduce n, the number of cells in a wavelength of the scattering perturbation, then we can write q ) 2π/na0 and use n as a convenient way of describing wavelength. In this notation, our estimates lead to relaxation times of τ ≈ 10-9 n2s. So perturbations of a characteristic size corresponding to 100 cells (∼ 1 µm) have relaxation times of the order of 10-5 s-1. In some complex fluids near transitions, such as block copolymers at an order-disorder transition, there is a contribution to the stress in addition to that which arises from the elasticity of the surface curvature. This arises from deformations of the composition fluctuations. In principle a similar term from fluctuations in the surfactant concentration should give a contribution to the shear modulus in the case of the cubic phases. In fact this extra component of the stress is too small relative to the total stress to show up in the rheological data as is shown below. The contribution to the loss modulus via the extra contribution to the viscosity may be estimated from the Kubo relation12
ηflux ≈
∑G τ
q q
ηfluc ≈
(32a)
q
where Gq is the effective contribution to the dissipative shear modulus from modes in the range d3q (which we assume to be entropy dominated so that the modulus picks up of the order of (12) Chaiken, P. M.; Lubensky T. C. Principles of Condensed Matter Physics; Cambridge University Press: New York, 1995.
kTζ 2π
∫ dq (Bq
2
1 + λq6)
(32b)
This integral is dominated by the range of wave vectors up to the crossover value of q, q* ) (B/λ)1/4. An estimate of the fluctuation contribution to the viscosity is
ηfluc ≈
with a dynamic scattering function of the form
S(q,t) ≈
kT per degree of freedom, giving Gq ∼ kTd(q3)), and τq their relaxation time. Thus
kT (q*)3τ(q*) ≈ 5 × 10-2 Pas 6π
(33)
which, although much larger than that of the background solvent (water), would be much too small to be seen in simple viscometric rheology against the background of the very large effective viscosities generated by deformation and relaxation of the viscoelastic cubic phase itself (of the order of 106 Pa or greater). Consequently, we expect that both the transition itself and the rheological response of surfactant cubic phases are dominated by the variables describing a change of symmetry, rather than large fluctuations associated with critical transition. However, the principle contributing factor to the small value of the viscosity is the short time scale for relaxations at the crossover wave vector. The characteristic modulus itself, kTq*3, is between 103 and 104 Pa compared with the measured modulus of the cubic phases away from their melting transitions, which is of order 105 Pa. So a significant dynamic slowing down at the transition (which we cannot rule out in a very weakly first-order system) might give a measurable rheological response. The signature would be a small rise in G′(ω) at temperatures just below the melting transition.
2. Rheology We have studied the rheology of the cubic phase. We have noted above that we do not expect to see fluctuation contributions at the transition. However, rheology gives a very sharply defined value for the melting temperature itself and is sensitive to much slower dynamic processes that give rise to hysteresis. So far measurements have revealed that the cubic phase has a nonlinear response at considerably lower strain than might be expected.1,4 The two components of the complex modulus both depart from linear behavior at about 1% strain for a frequency of 1 rad/s. In addition, in the linear regime, the gels follow a typically Maxwell element behavior, displaying a peak in the frequency spectrum of the loss modulus. Correspondingly, above the critical strain, where the nonlinearities become evident, the frequency dependence of the loss modulus is that of a liquid. A typical time scale for the bulk relaxation of these materials is 1 s. All measurements were done on samples of weight composition ratio DDAB/H2O/toluene equal to 60.25/24.08/ 15.67 or DDAB/D2O/toluene equal to 58.73/25.99/15.27 using a cone and plate geometry on a Rheometrics RDA II rheometer. The samples were prepared by dissolving the DDAB in the D2O/toluene mixture. This requires heating in an oil bath to about 100 °C while shaking in a well-sealed glass tube. The cubic structure may form immediately or the sample may need several weeks to equilibrate to this morphology depending on how well mixed it is during heating. The initial rheology performed on these samples was a series of strain sweeps at different frequencies followed by frequency sweeps, all at room temperature. The strain and frequency ranges were between 0.1 and 100% and ω ) 0.1-102 rad/s, respectively. These results, shown in Figure 3, confirm previous measurements by Radiman et al.1 that the gel exhibits a near-Maxwellian relaxation spectrum.
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Langmuir, Vol. 15, No. 22, 1999
Jones and McLeish
Figure 3. (a) Strain sweep at frequency ω ) 5 rad/s-1 and T ) 22 °C. The storage and loss modulus are shown to fall off steeply for strains above 1%. (b) Frequency sweep for a strain of 0.1% at T ) 22 °C. The loss modulus shows a broad relaxation peak.
Figure 4 shows a temperature ramp for the two moduli. The total time taken for the cycle is about 45 min and the striking features are: first, the melting transition is very sharp; second, the material completely recovers its modulus after melting. Moreover, it is able to do this over a very short time scale. Figure 5 traces the behavior of the loss modulus through the transition. A further significant feature is the appearance of a premelting peak in the elastic modulus (Figure 4), particularly noticeable on melting, although weakly present on freezing. Although an order of magnitude smaller than the discontinuity in G′ on melting, it is well outside the noise of the experi-
mental measurements, which are another factor of 10 smaller. This effect is consistent with a contribution from fluctuations of the composition mode via dynamic slowing down as discussed above. The conjecture is supported further by an initial increase in G"(ω) at low temperatures on heating (Figure 5). Another appealing feature of the melting transition is seen in the frequency spectra for the two moduli which were measured exactly on the melting temperature. There is evidence1 that the storage and loss moduli both scale in an identical way with frequency which would give key signatures of a percolation-type melting behavior.4
Fluctuations in Surfactant Cubic Phases
Langmuir, Vol. 15, No. 22, 1999 7501
Figure 4. This figure shows the variation of the moduli with temperature. A clear melting transition appears at about 55 °C, and the moduli recover their full values when cooled to room temperature.
Figure 5. This figure shows the frequency spectrum of the loss modulus for a series of different temperatures, 22 °C, 55 °C, 60 °C, and 65 °C. The dramatic reduction in the modulus above the melting transition is clearly shown. In addition, the measurements at the transition temperature (55 °C) give indications of power law scaling with frequency.
3. Dynamic Light Scattering Dynamic light scattering was performed using a HeNe laser (λ ) 633 nm). The measurements were made for a series of different q vectors and two different temperatures. The autocorrelation function was measured at temperatures of 45 °C and 50 °C, which was just below
the melting transition of 55 °C for these gels. The results were then analyzed using the iterative numerical scheme for inverse Laplace Transforms “CONTIN”13 to extract the distribution function of relaxation times G(τ). This is shown for several different angles in Figure 6, and the (13) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 229.
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Jones and McLeish
Figure 6. Shown are the distribution functions, G(τ), for the relaxation times, τ, for a series of wave vectors q defined by different scattering angles θ. The four graphs are for (a) θ ) 30°; (b) θ ) 50°; (c) θ ) 90°; (d) θ ) 140°.
mean relaxation time is extracted from G(τ). In addition we note that the distribution function is narrow, confirming single-exponential behavior. The plots of τ versus 1/q2 for the two temperatures are shown in Figure 7 and are straight lines even at the highest q values, showing that the q6 term is not seen in this range. These experimental results confirm the theoretical prediction for the form of the q-dependence, furthermore, our calculations for the magnitude of the time scales are correct to an order of magnitude. The relaxation times vary from about 3 to 300 µs. In addition they do provide evidence for some dynamic slowing down as the transition is approached; the relaxation times are shifted to longer times as the temperature is altered from 45 °C to 50 °C, even though the molecular dynamics are faster. However it is not possible to access the very low q-range which would correspond to the low frequencies (∼1 s-1) observed rheologically. At the higher end of the DLS q-range the fluctuation dynamics are just 2 orders of magnitude in spatial scale larger than the lower range of the neutron-spin-echo work of Clerc et al.6 These authors also found a dispersion relation of the form q-2Deff ) τ(q) and found values of Deff of 4 × 10-11 m2 s-1. In the light-scattering range of our work the corresponding value is approximately 2 × 10-10 m2 s-1. Given that the chemistry and the symmetry of the two systems is different, the “coincidence” is remarkable.
It strongly suggests that we are seeing different spatial frequency ranges of the same mode. In this case, our data would support the conclusion of Clerc et al. that the fluctuations of the membrane are responsible for the dynamic scattering. Moreover, we can suggest that membrane fluctuations at short length scale uniformly extend to the lattice-size and shape fluctuations responsible for light scattering at longer length scales. 4. Conclusions We have examined the effect of concentration fluctuations of wavelengths equal to several cell lengths in the surfactant cubic phase. This work has shown that fluctuations in this system, which has the constraints of conservation of the total number of cells (topology), the total surfactant area, and the total volume give rise to changes in the curvature of the membrane that are first order in an expansion in terms of variation in lattice spacing. This leads to a contribution to the osmotic bulk modulus which may be observed in dynamic light scattering. Considering the energy in terms of the Helfrich Hamiltonian we find that the compositional bulk modulus term is directly proportional to the bending modulus κ in the Helfrich Hamiltonian and is in fact
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the possible presence of fluctuation modes experiencing dynamic slowing down at the transition. A very sharp melting transition is observed and upon cooling the sample recovers its original structure very readily. There is some indication that both the storage and loss moduli scale identically with frequency when at the transition temperature but further work is needed to confirm this. This scaling behavior does give very relevant information as to the order of the transition, which if it is first order is very weak. One picture of the melting mechanism is of fluctuations in the membrane curvature becoming so large that local topological changes can occur. Cylindrical sections of the surface may develop necks so that the “bonds” between centers of cubic symmetry may be thought of as “on” or “off” as some neck completely (creating topological defects in the surface). This would lead to a percolation theory for the transition and would be second order in character.4 Our experimental values for the transition suggest scaling at a lower frequency dependence than that expected from percolation. However, the second-order character is supported by the lack of hysteresis in the temperature ramp experiments and an apparent extra contribution to the elastic response at the transition. In summary, the picture for the melting transition is still incomplete at this stage. The decay of these dilational modes has been studied and a dynamic structure factor and the corresponding relaxation time have been calculated. These are of the form given by eqs 35 and 36.
τq )
ζ q2(B + λq4)
S(q,t) ≈
Figure 7. Relaxation time τ(q) versus 1/q2 at (a) T ) 45 °C and (b) T ) 50 °C. The plots indicate the absence of any ‘soft′ modes for this range of q.
B)
4κ φ0a20c
(34)
where c is an order-one constant depending on the shape of the surfactant membrane and relating the perturbations of curvature and area. The contribution that this makes to the elastic moduli is too small to be observable by rheological methods away from transitions. However, rheology on these systems has given some interesting results pertaining to the melting transition and indicates
exp(-t/τq) (B + λq4)
(35)
(36)
It can be seen that B determines the q2 relaxations and that there are soft modes of the form λq6. We give estimates for the sizes of all the quantities above, and the theoretical predictions indicate that the q2 relaxations are the only ones that will be observable for this particular system. We measuremed the dynamic light scattering on these gels and have observed a correlation function with the features above. The relaxation times agree with those predicted both qualitatively (order-of-magnitude estimates) and in the form of the wave vector dependence, and with measurements by others on smaller length scales by dynamic neutron scattering. They provide additional evidence of dynamic slowing down near the melting transition. Acknowledgment. We are very grateful to A. Semenov, D.R. Read, S. Ramaswamy, and D. M. A. Buzza for their insight and helpful comments about this work. We would also like to thank R. W. Richards for the use of light-scattering facilities at Durham University. LA990079P
