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Role of Permeation in the Linear Viscoelastic Response of Concentrated Emulsions P. He´braud,*,† F. Lequeux,† and J.-F. Palierne‡ E.S.P.C.I., 10 rue Vauquelin, 75005 Paris, and Ecole Normale Supe´ rieure, 46 alle´ e d’Italie, 69007 Lyon, France Received July 31, 2000 We have studied the viscoelastic reponse of a concentrated emulsion. We have detected a slow relaxation in uniaxal flow that does not exist in simple shear flow. We demonstrate that the relaxation originates in the permeation of the water through the network of oil droplets, using a two-fluid model. In addition, we have measured directly the high-frequency behavior up to 10 kHz. It exhibits an anomalous relaxation behavior, in accordance with previous light scattering observations.
1. Introduction Emulsions are mixtures of two immiscible fluids, for instance, oil and water. One of them organizes itself into droplets that may be stabilized by a surfactant in such a way that the emulsion may be stable for weeks or even months. In this paper, we are interested in monodisperse emulsions (droplet size ) 0.6 µm), whose preparation is detailed below. A striking property of these systems is that they exhibit a jamming transition when the concentration of the droplets, φ, reaches the random close packing concentration, φc ≈ 63%, the maximum concentration of randomly packed hard spheres.4,12,13 At lower concentrations, they behave as viscous fluids, whose viscocity increases with increasing concentration of droplets. At higher concentrations, they essentially behave as elastic solids; their elastic modulus is greater than the loss modulus, and they have a yield stress.2,7,9 * To whom correspondence should be addressed. E-mail:
[email protected]. † E.S.P.C.I. ‡ Ecole Normale Supe ´ rieure. (1) Buzza, D. M. A.; Lu, C. Y. D.; Cates, M. E. Linear Rheology of Incompressible Foams. J. Phys. II 1995, 5, 37-52. (2) Durian, D. J. Foam Mechanics at the Bubble Scale. Phys. Rev. Lett. 1995, 75(26), 4780-4783. (3) Kawasaki, K.; Onuki, A. Dynamics and Rheology of Diblock Copolymers Quenched into Microphase-Separated States. Phys. Rev. A: At., Opt. Phys. 1990, 42(6), 3664-3666. (4) Kraynik, A. M. Foam Flows. Annu. Rev. Fluid Mech. 1988, 20, 325-357. (5) Lacasse, M. D.; Grest, G. S.; Levine, D.; Mason, T. G.; Weitz, D. A. Model for the Elasticity of Compressed Emulsions. Phys. Rev. Lett. 1996, 76(18), 3448-3451. (6) Liu, A.; Ramaswamy, S.; Mason, T. G.; Gang, H.; Weitz, D. A. Anomalous Viscous Loss in Emulsions. Phys. Rev. Lett. 1996, 76(16), 3017-3020. (7) Mason, T.; Bibette, J.; Weitz, D. Yielding and Flow of Monodisperse Emulsions. J. Colloid Interface Sci. 1996, 179, 439-448. (8) Mason, T. G.; Bibette, J. Emulsification in Viscoelastic Media. Phys. Rev. Lett. 1996, 77(16), 3481-3484. (9) Mason, T. G.; Bibette, J.; Weitz, D. A. Elasticity of Compressed Emulsions. Phys. Rev. Lett. 1995, 75(10), 2051-2054. (10) Mason, T. G.; Lacasse, M.-D.; Grest, G. S.; Levine, D.; Bibette, J.; Weitz, D. A. Osmotic Pressure and Viscoelastic Shear Moduli of Concentrated Emulsions. Phys. Rev. E 1997, 56, 3150-3166. (11) Poulin, P.; Nallet, F.; Bibette, J.; Cabane, B. Evidence for Newton Black Films between Adhesive Emulsion Droplets. Phys. Rev. Lett. 1996, 77(15), 3248-3251. (12) Princen, H. M. Rheology of Foams and Highly Concentrated Emulsions: (i) Elastic Properties and Yield Stress of a Cylindrical Model System. J. Colloid Interface Sci. 1983, 91(1), 160-175. (13) Princen, H. M.; Kiss, A. D. Rheology of Foams and Concentrated Emulsions: (iv) An Experimental Study of the Shear Viscosity and Yield Stress of Concentrated Emulsions. J. Colloid Interface Sci. 1989, 128(1), 176-187.
At these higher concentrations, they exhibit intriguing rheological properties. First, their elastic regime seems to be extremely small; the maximum amplitude of a linear elastic deformation is smaller than a few percent. The loss modulus is extremely difficult to measure in the linear regime, because it is about 2 orders of magnitude smaller than the elastic modulus.7 Moreover, it seems to be quite independent of the frequency, between 10-2 Hz and a few Hz. In particular, it does not seem to decrease toward zero at low frequencies, as would be expected from general arguments. Thus, one gets a broad time distribution, especially for long times, whose origins remain unclear.1 This paper presents experiments performed in the linear regime with a uniaxial piezorheometer, between 10-1 and 104 Hz, and a two-fluid model which accounts for the lowfrequency behavior. More precisely, it is argued that the low-frequency relaxation observed in uniaxal compression but not in simple shear may originate in the permeation of the continuous phase through the droplets. We have developed a two-fluid model for the flow of these systems, taking into account the permeation of water. On the other hand, our mechanical measurements for the high frequencies confirm those already obtained by diffusing wave spectroscopy (DWS),14 which exhibit a scaling in the square root of the frequency.6 2. Preparation of the Emulsions The emulsions we are interested in are composed of droplets of hexadecane in water stabilized by sodium dodecyl sulfate (SDS). Each droplet’s diameter is 0.6 µm, and its polydispersity is around 10%. They are prepared according to Bibette’s method.8 High concentrations of SDS aredissolved in water, such that we obtain an elastic phase that exhibits a yield stress, σc. We then shear this phase while progressively adding hexadecane. We obtain droplets of oil whose Laplace pressure, ΠLap, equals the applied stress so that their radii, F, obey
ΠLap ) σc F
(1)
The mean radius of the droplets is thus accurately defined, with a polydispersity on the order of 10%. The continuous (14) Weitz, D. A.; Pine, D. J. In DiffusingsWave Spectroscopy, Dynamic Light Scattering: The Methods and Some Applications; Brown W., Ed., Oxford University Press: New York, 1993; pp 652-720.
10.1021/la001091g CCC: $19.00 © 2000 American Chemical Society Published on Web 10/04/2000
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phase is then washed and replaced by a solution of water and SDS just above the cmc, at 40 mM. 3. Piezorheometer The piezorheometer consists of two piezoelectric devices parallel with each other. The sample is located between them. Its thickness is h ) 100 µm, and its radius is R ) 5 mm. It is periodically compressed by one of the piezoelectric devices , with an amplitude of a few angstroms, whereas the other measures by means of a lock-in amplifier the pressure exerted by the sample. The frequency of the excitation ranges between 10-1 Hz and 10 kHz. Let us first explain how we deduce the viscoelastic spectrum from the pressure signal. We exploit two properties of the cell: (i) it is axisymmetric, leading to the assumption that the flow is also axisymmetric, and (ii) its radial dimension is larger than its azimuthal dimension by a factor of 102; therefore, as in the Hele-Shaw cell, the derivatives along the radial component compared to the derivatives along the azimuthal axis will be neglected
∂r , ∂z
(2)
Consider u(r), the field of displacement inside the sample with the cylindrical coordinates (r,θ,z), and u*(r), meaning that it has been subject to a Fourier transform. It should be noted that, in general, the * symbol represents the complex Fourier transform of the quantity without an asterisk, at frequency ω. For symmetry reasons (eq 1), u does not depend on θ, and uθ(r) ) 0 everywhere inside the cell. Moreover, the incompressibility condition leads to
∂ru/r + ∂zu/z ) 0
(3)
Thus, uz ∼ urh/R, and the azimutal displacement, uz, compared to ur may be neglected. Therefore, the global equation of mass conservation simply balances the thickness variation of the cell with the radial flux of the system and may be written
∂r(2πr
∫
h / u (z) 0 r
Figure 1. Schematic drawing of the piezorheometer.
Finally, from eq 6, we get the equation for the complex pressure:
-
h3 ∂ (r∂ P*(r)) + rδh* ) 0 12G* r r
(9)
It is then easy to deduce the total force, Π*, exerted by the sample on the captor of radius R:
Π* )
3π ∂h* G*R4 3 2 h
(10)
Finally, the complex modulus, G*(ω), is the ratio of the pressure and the amplitude of the motion, up to a geometrical factor:
G* )
2 h3 Π* 3π R4 δh*
(11)
This expression shows clearly the advantage of a compression flow over a pure shear experiment. Indeed, imagine that the same cell is used to create a shear flow. Again noting δh as the amplitude of the displacement of the excitator piezoelectric device parallel to itself, the stress, σ, exerted by this flow over the other piezoelectric device would be on the order of
δh h
σ∼G
(12)
whereas, in the case of a compression flow, it is 2
dz + πr δh*) ) 0
(4)
where δh* is the amplitude of the motion of the piezoelectric device as a function of the frequency. On the other hand, consider the equation of elasticity:
-∇P* + G*∆u* ) 0
(5)
where P is the complex pressure field and G*(ω) is the complex elastic modulus. Since we are in the frame of the Hele-Shaw approximation, the pressure is only a function of r. Thus, eq 5 becomes a scalar equation:
-∂rP* + G*‚∂z22u/r ) 0
(6)
Thus, the radial component of the displacement field is quadratic in z, and we may write
u/r (r,z) ) f*(r) z(h - z)
(7)
where f is some function of r to be determined. Then, eq 4 leads to
h3 ∂r(rf*(r)) + rδh* ) 0 6
(8)
σ∼G
δh R h h
()
2
(13)
The detection of the stress is thus amplified by a factor of (R/h)2 ) 104 when compared to a simple shear device, while the strain is amplified by a factor of R/h. This allows us to measure the modulus with strain amplitudes ∼10-4. 4. Results and Discussion Figure 2 shows results for emulsions at three different concentrations. It is seen that the elastic modulus is nearly constant between 0.1 Hz and 10 kHz, regardless of the concentration. For the lowest concentration, it increases a bit because the loss modulus increases. The loss modulus exhibits two features: At high frequency,
G′′(ω) ∼ xω
(14)
showing that the system has a very broad distribution of short relaxation times. This behavior has never been previously observed on such systems by pure mechanical experiments, because such high frequencies are not achievable with standard rheometers; however, it has been observed from multiple light scattering that the loss modulus scales as xω.6 On the basis of the similar behavior of lamellar block copolymers,3 the authors
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He´ braud et al.
Using a classical rheometer, we have checked that, for a simple shear experiment, this time does not appear, and it has not been reported in previous measurements.7 Simple shear and extensional shear give different results, even if mechanical experiments with classical rheometers at these low amplitudes are not very precise. The characteristic time of this relaxation is longer than 10 s. Thus, it is expected that this relaxation is linked to the nature of the flow. Unlike shear flow, compression flow creates a gradient of pressure. Actually, the concentrated emulsion is elastic because the oil droplets behave like elastic objects creating an elastic network. This network can be compressed if the water is allowed to get out of the sample. Here, since there is a pressure gradient, we expect the water to permeate through the network constituted by the oil droplets. Therefore, we may expect that the gradient of pressure induces a flow of the continuous phase through the droplets of oil. A two-fluid model may be written, assuming that water permeates through an elastic network composed of oil droplets. The elasticity of the oil droplet’s network originates into the interfacial tension of the droplets and corresponds to the elastic modulus measured at low frequency. We call µ* the elastic modulus of the oil droplet’s network. We can calculate the deformation field and the stress following the method of section 3. As before, the derivatives along the radial direction will be neglected compared to the derivatives along z. The calculation below thus follows the subsequent guidelines: (1) the displacement field of the elastic matrix is radial and may be written as the product of an r-function with a quadratic function of z; (2) the displacement fields of the elastic matrix and of water are linked by Darcy’s law and by the equilibrium of the water and the elastic matrix; (3) the equation of conservation of mass allows us to completely determine the displacement fields; (4) integration of the z component of the stress over the surface of the captor leads to the total exerted pressure for a given deformation. Note that um is the displacement field of the elastic matrix which is actually a network of oil droplets stabilized by interfacial tension, and uω is the displacement field of water. The only relevant component of um is the radial one, u m, and, according to section 3, it is a quadratic form of the r z-coordinate: Figure 2. Elastic modulus (G′) and loss modulus (G′′) for concentrated emulsions at 65% (O), 78% (4), and 89% (b). Continuous lines represent data obtained by diffusion wave spectroscopy by A. Liu et al.6 for c ) 82% (bold line) and c ) 67% (thin line). Dashed lines are fits obtained with eq 25, the diameter of the pores being the only free parameter.
assumed the existence of weak plans randomly oriented and whose lips slide freely one to the other. The relaxation times of these plans depend on their orientation compared to the direction of the shear, thus leading to a wide distribution of relaxation times. They finally showed that the loss modulus should scale as the square root of the frequency. We confirm here, by direct mechanical measurements, the scaling of G′′(ω) as xω. Quantitatively, our results are in good agreement with Liu et al. Indeed, at high frequency (104 Hz) at 65%, the elastic modulus of our emulsion is identical to the elastic modulus of Liu’s emulsion concentrated at 67% (300 Pa). The value of the loss modulus we measure (500 Pa) agrees with the value derived from diffusing wave spectroscopy (=600 Pa). On the other hand, the low frequency behavior exhibits a surprisingly long relaxation time.
* u/m r ) f (r) z(h - z)
(15)
where f is some function of r to be determined. On the other hand, the displacement field of the water relative to the elastic matrix obeys Darcy’s relation: /
∂t(u/r w - u/m r ) ) κ∂rPH
(16)
where PH is the hydrostatic pressure of water and κ is a permeation coefficient, which is on the order d2/ηw, with ηw being the viscosity of water and d the diameter of the pores. The water and the elastic medium are at equilibrium and note that σm is the stress tensor of the elastic medium. The total force exerted on the elastic medium and on water is zero /
∇PH + ∇‚σ*m ) 0
(17)
neglecting once again the derivatives along r and θ compared to the derivative along z. The radial component
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of this equation simply becomes /
∂rPH ) -∂zσ/rzm ) 2µ*f*(r)
(18)
where the second equality holds from elasticity laws. Returning to Darcy’s law, the radial displacement of water can be expressed as a function of f(r). At frequency ω:2
(
u/w r (ω) ) z(h - z) +
)
2κµ* * f (r) iω
(19)
So, the mass conservation: * /w ∫0h(φu/m r + (1 - φ)ur )2πr dz) + 2πrδh ) 0
∂ r(
(20)
becomes
((
∂r
) )
2µ*κ h3 + (1 - φ) h rf*(r) + rδh* ) 0 6 iω
(21)
This leads to the following expression of f(r):
f*(r) ) -
δh*
(
2
(r + ct)
)
2µ*κ h3 + (1 - φ) h 6 iω
(22)
where ct is an integration constant. Moreover, if R is the radius of the cell, the boundary condition can be written:
δh PH(R) ) k h
(23)
up to the atmospheric pressure and where k is the compression modulus of the emulsion. The integration of eq 18 then leads to the hydrostatic pressure:
PH ) -
µ*δh* 4µ*κ h3 + (1 - φ) h 3 iω
(R2 - r2) + k
δh* (24) h
By integrating over the surface of the captor, the total force exerted by the fluid can be obtained. Using eq 11, we get the “apparent” elastic modulus:
G*(ω) )
µ* *
12µ κ 1 + (1 - φ) iωh2
+k
(Rh)
2
Figure 3. Elastic modulus G′ (b) and loss modulus G′′ (4) of an adhesive emulsion. The concentration of the added salt was cNaCl ) 1.2 M. The oil concentration was 47%.
is the Darcy coefficient, depending on the viscosity of the solvent and the size of the pores. To vary the diameter of the pores, a gel solution was made of the droplets of oil in water at lower concentration. Salt (NaCl) was added to water in order to screen electrostatic interactions between droplets, so that the interactions are dominated by the depletion forces, as shown by Poulin et al.11 The interaction between the droplets is now attractive, and an elastic system at concentrations of oil lower than random close packing may be obtained. The gel network of droplets is formed by aggregation, and the size of the pores is very polydisperse. Some large pores exist between the aggregates and some much smaller ones between the droplets themselves. This leads to a much broader relaxation than that predicted by our simple two-fluid model. Figure 3 shows elastic and loss moduli of a system with cNaCl ) 1.2 M and a concentration in oil of 47%. The long time relaxation is shifted toward higher frequencies, as predicted, but the relaxation is no longer Maxwellian because of the polydispersity of the pore’s diameter. The measured characteristic time is 0.1 s, leading to a mean diameter of the pores on the order of the diameter of the droplets.
(25)
The second term is negligible because the elastic and the compression moduli are of the same order in concentrated emulsions.10 We thus see that the permeation flow induces an additional dissipation on the order of (1 - φ)12µ2d2/ h2ηwω, where d is the diameter of the pores and h the extension of the pressure gradient. Hence, while the elastic modulus scales as φ - φc,5,7 the additional dissipation, because of permeation, scales as (φ - φc)2. The fitting curves are shown in Figure 2b; the only free parameter is the diameter of the pores. Even though the long relaxation times lie on the border of our experimental window of measure, we may deduce from these fits that the mean diameter of the pores is 0.06 µm for φ ) 89% and 0.08 µm for φ ) 78%. The fit is much less accurate for φ ) 65% because of the lack of data at low frequencies but leads to a mean pore diameter of 0.2 µm. The main hypothesis of this two-fluid model is that the continuous phase is supposed to permeate through the droplets. Thus, the main control parameter of our model
5. Conclusion For the first time, compression measurements on concentrated emulsions have been performed at frequencies ranging from 10-1 to 104 Hz, with strain amplitudes smaller than 10-4. At high frequencies, we have recovered the results of D. W. S. from Liu et al.,6 with a slight difference for the amplitude of G′′, thus validating our experiment. On the other hand, at low frequencies, compression experiments give results different from those of simple shear experiments. We have attributed this phenomena to permeation flow. We have shown, with the help of a two-fluid model, that permeation flow can account for the whole low-frequency mechanical behavior of the concentrated emulsions in an uniaxal flow geometry. Actually permeation contributes to additional dissipation with a typical time on the order of τ ) 10 s. Hence, we have evidenced the effect of permeation on uniaxal strain measurements. This appears to be an original method for evidencing and measuring porosity in a concentrated suspension. LA001091G
