Langmuir 2006, 22, 5991-5993
5991
New Method to Determine the Viscoelastic Properties of Admicelles around the Stick-Slip Transition C. Cheikh,† G. J. M. Koper,*,† and T. G. M. van de Ven‡ DelftChemTech, Delft UniVersity of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands, McGill UniVersity, Department of Chemistry, Pulp & Paper Research Centre, 3420 UniVersity Street, Montreal QC H3A 2A7, Canada ReceiVed March 21, 2006. In Final Form: June 2, 2006 We report on a new method by which, for the first time, the viscoelastic properties of an adsorbed surfactant layer on a solid surface are measured. It is based on an analysis of the amplitude and the phase angle of the pressure fluctuations induced by a pulsating flow of a Newtonian surfactant solution through cylindrical pores. This method is subsequently used to determine the viscoelastic properties of an admicelle, formed when flushing surfactant solutions through nanopores, around the stick-slip transition. We find that the admicelle responds elastically for flow strengths below the transition and beyond the viscous. This is in agreement with the hypothesis formulated earlier (Cheikh, C.; Koper, G. J. M. Phys. ReV. Lett. 2003, 91, 156102).
In pores through which a surfactant solution flows, a surfactant layersthe admicellesis formed between the solid surface and the flowing solution. In the case of an aqueous solution and a hydrophilic pore surface, the surfactant molecules form a bilayer with the hydrophilic headgroups oriented toward the solid for the bottom layer and toward the liquid for the top layer. The flowing liquid exerts a shear stress on the top layer, of which the strength is given by
∂V P σ(R) ) η0 |R ) - R ∂r 2
(1)
where R is the pore radius, η0 is the bulk viscosity of the liquid, and V is the velocity field inside the pore. For surfactant concentrations that are not too high, the surfactant solution may be considered to be Newtonian, and hence the velocity gradient is linearly proportional to the pressure gradient, P, which yields the second part of eq 1 where we used the Poiseuille flow expression for the velocity field. Under mild flow conditions, the wall shear stress is small, and the admicelle will respond elastically. For larger pressure gradients, the wall shear stress may overcome a critical value beyond which the bilayer will exhibit viscous flow inside the admicelle where the top layer slips over the bottom layer. For the flow through the pore, this transition implies a suddenly stronger dependence on the pressure gradient as if the pore radius has increased. Such stick-slip transitions have been observed for various systems as reported in refs 1 and 2. The enhancement of flow through membranes and other porous materials is interesting both from a technological point of view, for instance, in the fields of oil recovery3 and membrane emulsification,4 and in the field of electrokinetics as applied in microelectromechanical systems.5 Unfortunately, the simple flow experiments described in refs 2 and 6 and the squeeze flow * Corresponding author. E-mail:
[email protected]. † Delft University of Technology. ‡ McGill University. (1) Zhu, Y.; Granick, S. Langmuir 2002, 18, 10058. (2) Cheikh, C.; Koper, G. J. M. Phys. ReV. Lett. 2003, 91, 156102. (3) Morrow, N. R.; Mason, G. Curr. Opin. Colloid Interface Sci. 2001, 6, 321. (4) Charcosset, C.; Limayem, I.; Fessi, H. J. Chem. Technol. Biotechnol. 2004, 79, 209. (5) Yang, J.; Kwok, D. Y. Langmuir 2003, 19, 1047. (6) Cheikh, C.; Koper, G. J. M. Colloids Surf., A 2005, 270-271, 252.
experiments described in ref 7 are not capable of providing sufficient detail about the phenomenon; therefore, more elaborate experiments are called for. In this letter, we present a new experimental method by which more can be learned about the properties of the admicelle around the stick-slip transition. The essence of the method is to superimpose a small oscillating component on the constant flow through the pore. The resulting pressure variations will exhibit a phase difference with respect to the flow oscillations. Because the fluid itself is viscous in the relevant frequency domain, it can be judged from this phase difference whether the admicelle responds elastically or viscously at the applied constant flow rate. With the constant flow rate, we can select the flow regime of the admicelle before or beyond the stick-slip transition. The picture sketched above implies that the fluid near the pore wall possesses different properties than the bulk fluid. This can be modeled by radius-dependent viscosity η(r) where it is assumed that the pore has simple cylindrical symmetry. It is to be understood that η(r) * η0 close to the wall only. For an incompressible fluid and stationary flow, the Navier-Stokes equation reduces to
1 ∂ ∂V rη(r) + P ) 0 r ∂r ∂r
(
)
(2)
where all body forces, such as gravity, are absorbed in the pressure gradient, P. We keep the no-slip boundary condition (i.e., V(R) ) 0), and because of cylindrical symmetry, we also have ∂rV|r)0 ) 0. Because over the larger part of the pore the viscosity is equal to the bulk viscosity, we subtract the field V0(r) that results from an overall constant viscosity η0 and find for the excess velocity field
Vex(r) )
(
)
1 1 r′ dr′ ∫rR η(r) η0
P 2
(3)
Because of the imposed no-slip boundary condition, the excess fluid velocity at the wall vanishes, and the above expression yields only a finite value for the excess fluid velocity inside the pore when the actual viscosity deviates from the bulk viscosity. Integrating over the total pore cross section yields the flow rate (7) Tang, H. S.; Kayon, D. M. Rheol. Acta 2004, 43, 80.
10.1021/la0607555 CCC: $33.50 © 2006 American Chemical Society Published on Web 06/13/2006
5992 Langmuir, Vol. 22, No. 14, 2006
Letters
that can be expressed in the form
Q)
4
Vˆ (r) ) -
πR 4λ 1+ P 8η0 R
(
)
(4)
where the slip length can now be expressed in terms of the excess viscosity as
λ)
(
η
∫0Rr3 η(r)0 - 1
1 R3
)
dr
(5)
The important message of eq 5 is that small deviations in the viscosity from bulk behavior will give rise to a finite slip length. Less-viscous material near the pore wall (i.e., η(r ≈ R) < η0) will render flow enhancement, and more-viscous material, η(r ≈ R) > η0, a flow decrease at the same pressure gradient. In other words, the apparent radius of the pore depends on the viscous behavior of the fluid near the boundary: smaller for more than bulk viscosity and larger for less than bulk viscosity. The advantage of eq 5 is that we can use the Navier slip boundary condition to account for deviations from uniform bulk viscosity near the wall, of which the details are not known. The slip length as such is an excess quantity analogous to the equation for the surface tension of a fluid-fluid interface in terms of surface excess densities, the Gibbs adsorption equation.8 Therefore, the slip length λ may in general depend on the wall shear stress and on frequency. The expression obtained above is also essentially the basis for the analysis by Tuinier et al. for the case of polymer depletion near a wall.9 To obtain more detailed information on the viscoelastic properties of the fluid near the pore wall, we will apply a pulsating pressure gradient as
P ) P0 + pˆ eiωt
(6)
with pˆ being the amplitude of the (negative) pressure gradient oscillations at frequency ω that are superimposed on the constant (negative) pressure gradient P0. We substitute this pulsating pressure gradient in the Stokes flow equation and use as boundary conditions ∂rV|r)0 ) 0 and V(R) ) λ∂rV|R. As discussed above, the latter boundary condition is effectively the Navier slip condition and accounts for any deviations from the bulk viscosity near the pore wall. The Stokes equation is linear in the pressure gradient; therefore, the resulting velocity field is the superposition of a constant flow field
V0(r) ) -
P0 2 [r - R2 - 2λ0R] 4η0
(7)
and an oscillating field
Vˆ (r) ) -
[
I0(kr) pˆ 1iωF I0(kr) - λωkI1(kr)
]
(8)
with In being the modified Bessel function of nth order and of the first kind and where k2 ) iωF/η0 with F being the fluid mass density. In the limit of low frequencies such that kR , 1, we may expand the Bessel functions up to second order, and we find (8) Stokes, R. J.; Evans, D. F. Fundamentals of Interfacial Engineering; WileyVCH: New York, 1997. (9) Tuinier, R.; Taniguchi, T. J. Phys.: Condens. Matter 2005, 17, L4.
pˆ 2 [r - R2 - 2λωR] 4η0
(9)
For the oscillating part of the flux, we therefore find
Q ˆ )
{
}
4 πR4 1 + (λ′ω + iλ′′ω) pˆ 8η0 R
(10)
where the complex slip length λω has been split into a real part λ′ω that represents the viscous part of the wall viscoelastic behavior and an imaginary part λ′′ω that represents the elastic behavior. The phase difference φ between the oscillations in flow and pressure is found as the ratio of the imaginary part and the real part of eq 10
tan φ )
4λ′′ω/R 1 + (4λ′ω/R)
(11)
and the relative permeability is
Π)
|1 + (4λω/R)| Q ˆ /pˆ ) Q0/P0 1 + (4λ0/R)
(12)
These parameters can be readily determined experimentally from flow experiments. The above two equations can subsequently be used to calculate the viscous and elastic contributions to the complex slip length at low frequencies. At higher frequencies, inertial effects will contribute that are presently neglected by using the Stokes equation for creeping flow; Reynolds numbers for the flow rates used are sufficiently small to justify this. The experiment is performed in the same way as described in refs 2 and 6. Sodium dodecyl sulfate (SDS) solutions at 5 times the cmc are forced through a laser-etched membrane having straight pores of 50 nm radius. Because of the laser-etching procedure, the membrane has a narrow pore size distribution. Rather than varying the pressure gradient, these experiments are performed with a Postnova PN1610 syringe pump controlled by a computer that records the imposed flow rates and the resulting pressure drop across the membrane. The periods of the flow rate oscillations are chosen to be 60 and 90 s. The amplitude of the oscillations is about 10% of the mean flow rate. In practice, a constant flow rate is first applied until a stationary pressure drop is reached. Then, a sinusoidal oscillation of a given frequency is added. Interestingly, the average pressure across the membrane drops to a slightly lower value when the periodic perturbation is superimposed onto the flow. In Figure 1, an example of the observed pressure variations is given. These slight pressure drops are observed only for average flow rates below the stick-slip transition and vanish beyond the transition. Such effects are observed to be flow enhancements to bulk viscoelastic fluids through pipes (e.g., ref 10). However, in our case, the SDS bulk solution is Newtonian (e.g., ref 11), hence the origin of this effect is to be found in the interfacial region. In fact, these pressure drops indicate slight nonlinearities in the membrane response. Nevertheless, we shall use the small-perturbation approach derived above (cf. eqs 11 and 12) to analyze our results. The pressure variations obtained, as in Figure 1, can be compared with the applied flow rate. The time shift between the two signals divided by the period is used to calculate the phase angle difference. No significant differences are observed for the (10) Mena, B.; Manero, O.; Binding, D. M. J. Non-Newtonian Fluid Mech. 1979, 5, 427. (11) Rodriguez, C.; Acharya, D. P.; Hattori, K.; Sakai, T.; Kunieda, H. Langmuir 2003, 19, 8692.
Letters
Langmuir, Vol. 22, No. 14, 2006 5993
Figure 3. Viscous (dashed line) and elastic part (continuous line) of the complex slip length as obtained from the phase angle and relative membrane permeability as a function of the average surfactant solution flow rate through a nanoporous membrane. The elastic response slowly turns into viscous behavior for larger flow strengths. Figure 1. Pressure and flow rate variation traces for surfactant solution flow through a membrane below the stick-slip transition. Note the pressure drop in the average pressure upon the start of the flow rate oscillation. The dotted line is a guide to the eyes to show the phase angle difference.
membrane permeability to oscillatory flow is higher than for the stationary flow. This is in agreement with the flow enhancement reported above (Figure 1), which is characteristic of viscoelastic fluids. At higher flow rates (i.e., for higher shear stress), the relative membrane permeability tends toward unity. The membrane permeability for the interfacial flow is thus equal to that for the stationary flow. The phase angle and relative permeability data of Figure 2 can subsequently be analyzed to yield the real and imaginary parts of the slip length using eqs 11 and 12. The results are depicted in Figure 3. Apparently, the elastic structure of the interface disappears for higher surface stress, λ′′ω f 0. The viscous part of the interfacial response at the same time becomes equal to the friction for the stationary flow (i.e., λ′ω f λ0).
Conclusions Figure 2. Phase angle and relative membrane permeability as functions of average surfactant solution flow rate through a nanoporous membrane. With increasing flow rate, the elastic response decays toward a completely viscous response beyond the stick-slip transition.
response for the two periods of 60 and 90 s. The results in the rest of this letter are given for the period of 90 s only. In Figure 2 (top), the phase angle difference of the surfactant solution tends toward that of water. It shows that the interfacial region is losing its elastic properties gradually and is becoming more and more viscous with increasing wall shear stress. A constant phase difference, φ0, of a few degrees is obtained at all flow rates through the same membrane. The results in Figure 2 are adjusted for that. This constant phase angle difference is probably due to the combined effect of the delay time of the pressure transducer and of an additional source of elasticity in the setup or the membrane material itself. In Figure 2 (bottom), the results are analyzed in terms of the membrane permeability; see eq 12. At low shear stress, the
We have demonstrated that the viscoelastic properties of an admicelle in a cylindrical pore can be determined experimentally. For this purpose, a pulsating flow was applied through a membrane, and the resulting pressure gradient oscillations were recorded. We have presented a general method to analyze the measured phase angle and permeability to yield information on the viscoelastic properties of the region near the interface in terms of excess quantities that for zero frequency can be related to the slip length. From experiments on SDS solution flow through a nanoporous membrane, we find that before the stick-slip transition the admicelle responds predominantly elastically to the external stress whereas beyond the transition viscous behavior is observed. This confirms our earlier suggestion regarding the origin of the stickslip transition for surfactant flow through cylindrical pores. Acknowledgment. These investigations were supported with financial aid from The Netherlands Technology Foundation (STW). We thank D. Bedeaux for stimulating discussions. LA0607555
