Friction in surfactant layers at solid–liquid interfaces

Colloids and Surfaces A: Physicochem. Eng. Aspects 270–271 (2005) 252–256

Friction in surfactant layers at solid–liquid interfaces Christophe Cheikh, Ger Koper∗ DelftChemTech, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands Available online 2 August 2005

Abstract The flow behavior of solutions of three different surfactants through pores of a laser-etched membrane is presented. Partial non-linear flow-pressure curves are obtained and the cross-over between the two linear regimes is interpreted as a stick–slip transition. Two parameters are extracted. The critical shear stress, needed to initiate slip, describes the transition and the slip length, the interfacial friction. © 2005 Elsevier B.V. All rights reserved. Keywords: Surfactant; Stick–slip transition; Solid–liquid interfaces

1. Introduction Various systematic experiments have shown slip of hydrophobic fluid on hydrophilic surfaces or of water on hydrophobic surfaces [1] as well as with more complex systems such as surfactant solutions [2,3] or polymer melts [4]. Such deviations from the Poiseuille law are usually quantified with an experimental parameter called the “slip length”. Stick–slip transitions of liquid over solid surfaces have been clearly demonstrated more than a decade ago by Yoshizawa et al. [5] using a surface force apparatus. Providing thus the proper conditions, i.e. an appropriate capillary radius [6], such flow behavior is expected to occur under high enough wall shear stresses. To address this shear dependence, we can write the flow through a pore as
 πR4 4λ˜ J= P 1+ (1) 8η R where P = −∂p/∂z is the pressure gradient and where λ˜ is the empirically related to the slip length λ as  σc  λ˜ = λ 1 − θ (σ − σc ) (2) σ

∗

Corresponding author. Tel.: +31 15 278 8218; fax: +31 15 278 4135. E-mail address: [email protected] (G. Koper).

0927-7757/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2005.06.011

with σc the critical shear stress at which the transition is occurring. When one substitutes Eq. (2) into Eq. (1), one finds the equation given previously in Ref. [6]. This Eq. (2) is presenting two major advantages. First, such expression is dealing directly with forces and can thus be directly introduced into the boundary condition for the Navier–Stokes equation. Second, relating the slip length to the interfacial friction coefficient µ = η/λ˜ (see for example [7]) we observe that the friction coefficient is very large below a critical value of the shear stress σc , i.e. the no-slip boundary condition applies. At σc and beyond, the friction coefficient is finite and tends towards a constant. This provides a more physical picture, as is sketched in Fig. 1, than an abrupt transition as suggested by the step-function. As long as the friction below the transition is large enough it will effectively result in an abrupt change in the flow curves. Slip can greatly contribute to mass transport, especially in porous systems, i.e. in the case of small radii, where the deviations from the Poiseuille law are expected to be large. Industrial applications can be found in detergency [8], in food processing [9,10] and more generally in membrane science [11,12]. In this article, we will use the phenomenological equations derived in [13], and recalled in Appendix A, allowing for the consideration of electrokinetic effects. These equations will be applied to the three different surfactant systems that are presenting discontinuities.

C. Cheikh, G. Koper / Colloids and Surfaces A: Physicochem. Eng. Aspects 270–271 (2005) 252–256

Fig. 1. Friction coefficient vs. shear stress. A smooth transition from an almost infinite value, where the stick boundary condition applies, to a finite value is observed. The critical shear stress has been taken equal to 400 Pa for a viscosity of 10−3 Pa s and for a slip length of 10 nm.

2. Experimental Within experimentally accessible pressure gradients, the stick–slip transition is expected to be observed for small diameter capillaries only. In order to obtain a measurable flow many capillaries would have to be put in parallel, which is the case in a laser-etched polymer membrane. Such membranes are commercially available from Millipore (Massachusetts, USA) as Isopore VCTP and have a pore diameter 0.1 ␮m, a thickness of 10 ␮m, a total diameter of 47 mm and a porosity of 4%. This corresponds to a pore number of 8.8 × 109 pores. The pore size distribution is, by electron microscopy, found to be very narrow, with a coefficient of variation of 1%. The membranes are made of poly(carbonate) to give them strength and coated with poly(vinyl pyrrolidone) (PVP) to make the surface hydrophilic. The membranes are guaranteed by the manufacturer to remain up to 2 bar without deformation. Accurate flow rates are imposed by a syringe pump Postnova PN1610 (Salt Lake City, USA) in the pressure range of 0–20 bar. The pressure drop across the membrane is measured

253

by a Bronckhorst (Ruurlo, The Netherlands) pressure transmitter P-506C-FAC-22V connected to a digital computer. The accuracy of this device is ±0.5%. Pressure readings are averaged over 100 values (measured each 5 s) and showed deviations less than 1%. Pressure gradients are expressed in terms of measured pressure over actual membrane thickness. Three surfactant solutions have been used. Crystalline sodium dodecyl sulphate (SDS), commercially available from Fisher Scientific, and crystalline cethyltrimethylammonium bromide (CTMA), commercially available from Acros Organics, were dissolved in doubly distilled water. Solutions of a nonionic surfactant, Triton X-100 (TX100, p(5,5-dimethylhexyl)phenoxy(polyethoxy)ethanol, pure from Serva, have been prepared by dilution. Concentrations are given in terms of the critical micelle concentration (CMC). The CMC values were determined by measuring the change of the interfacial tension with respect to the concentration. They are found to be equal to 2.3, 0.23 and 0.11 g/l, for SDS, CTMA and TX100, respectively. The viscosities were measured using three capillaries of different radius. For a given concentration, the viscosity readings do not vary significantly with capillary radius, i.e. remain within experimental error. Further, the flow of the surfactant solutions in membranes is varying linearly with respect to the pressure for smaller and bigger pore diameters than 0.1 ␮m, i.e. 0.05 and 0.2 ␮m, respectively. We conclude that the solutions are Newtonian (see also [14] for TX100). These data are summarized in Table 1 for the solutions presenting nonlinearities in the flow-pressure curves. In a typical experiment, the membrane is first flushed with doubly distilled water at high flow rates. Then, a surfactant solution is flushed through the membrane stepping from lowpressure values to high values and back again. Fouling of the membrane is largely avoided by recycling surfactant solutions.

3. Theoretical The results are fit to the following phenomenological model, for stick and slip boundary conditions [13], respectively, of which the details are reproduced in Appendix A:

Table 1 Relative viscosity, zeta potential and slip lengths for SDS, CTMA and TX100 at various concentrations Concentration (CMC)

Relative viscosity

ζ (mV)

SDS

5 7 10

1.057 1.088 1.151

−15.4 ± 0.8 −18.3 ± 0.9 −22.3 ± 1.2

CTMA

2 5 10

0.994 0.999 1.01

12.2 ± 0.6 19.0 ± 1.0 26.1 ± 1.3

TX100

5

1.01

Surfactant

Model A λ (nm)

Model B µ (bar/m)

λv (nm)

µ (bar/m)

7.3 ± 3.7 9.8 ± 3.8 6.5 ± 2.8

1.3 ± 0.7 1.0 ± 0.4 1.6 ± 0.7

0.29 ± 0.03 0.29 ± 0.03 0.29 ± 0.03

31 ± 3 31 ± 3 31 ± 3

6.7 ± 3.4 15.9 ± 1.8 37.3 ± 2.1

1.3 ± 0.7 0.6 ± 0.08 0.2 ± 0.02

0.40 ± 0.04 0.35 ± 0.03 0.33 ± 0.04

23 ± 3 26 ± 3 28 ± 3

14.7 ± 5.2

0.6 ± 0.2

The error in relative viscosity is less than 1%. Since the surfactant TX100 is nonionic, no zeta potential or model B values are given.

C. Cheikh, G. Koper / Colloids and Surfaces A: Physicochem. Eng. Aspects 270–271 (2005) 252–256

254



R

J = 2π 0



R

I = 2π 0

and J s = 2π



rv(r) dr = LVV,0 P + LVI,0 E, ri(r) dr = LIV,0 P + LII,0 E

R

0

I s = 2π



0

R

(3)

rv(r) dr = LVV P + LVI E + KV , ri(r) dr = LIV P + LII E + KI

(4)

here J and I are the flow rate and the electrical current, respectively, P and E, the pressure gradient and the electrical field, and R is the pore radius. The derivation of the coefficients Lkk = Nkk , where N is the pore number of the membrane, is given in [13]; their expressions are recalled in Appendix A. In the absence of electrical current one finds from the above equations an expression for the electric potential across the membrane. The flow rate reads with stick boundary conditions:   L2VI,0 J = LVV,0 − P = a1 P (5) LII,0 where a1 is the experimental slope. For slipping flow:   
L2VI kI = a2 P + b P + kV − J = LVV − LII LII

(6)

where a2 is the experimental slope and b the flow rate extrapolated to the origin.

4. Results Experimental flow, such as for CTMA presented in Fig. 2, are exhibiting discontinuities interpreted as stick–slip transition, i.e. the sudden increase in the flow rate is explained as the onset of slip. For higher concentration of TX100, i.e. 10 CMC, a very strong non-Newtonian behavior is observed. In that case, the slip length is very difficult to measure since no linear region is observed. From the results of Fig. 2, values for the critical shear stress, Fig. 3, and for the slip length are extracted using a model A which does not involve electrokinetics [6] or using a more detailed model B involving streaming potentials [13]. In the first model, the flow rate is expressed with the Eqs. (5) and (6) by formally setting the coefficients LIV and LVI equal to 0. The slip length is then simply found by expressing the ratio of the experimental slopes: a2 λ =1+4 a1 R

Fig. 2. Flow rate of surfactant solutions through a nanoporous membrane vs. applied pressure gradient for some concentrations of CTMA: 2× CMC (a), 5× CMC (b) and 10× CMC (c), where the critical micellar concentration CMC = 0.23 g/l.

can subsequently be determined. The expression 5 is then rewritten with the experimental slope a1 as  a1  2 lVV,0 − (8) lII,0 = lVI,0 N where the two coefficients lVI,0 and lII,0 both contain the zeta potential ζ. Results are presented in Table 1.

(7)

In the model B, one needs first to determine the zeta potential. Knowing the membrane properties, the flow in a single pore

Fig. 3. The shear stress at which the stick–slip transition is occurring varies linearly with surfactant concentration.

C. Cheikh, G. Koper / Colloids and Surfaces A: Physicochem. Eng. Aspects 270–271 (2005) 252–256

Using the Debye–H¨uckel approximation, the zeta potential has been calculated solving the Eq. (8) by neglecting the micelle and the free surfactant contributions to the conductivity. The results are verifying the initial assumption of low potentials. Furthermore, these values are found to be in a good agreement with the literature (see for example Ref. [15]). The error in zeta potential is estimated from the accuracy of the experimental slopes and the viscosity. A zeta potential for the solution of the non-ionic surfactant TX100 could not be determined since its conductivity is very close to 0, almost that of pure water. The slip length is subsequently found using the slope a2 , i.e. in the slip regime. From Eq. (6): a2 L2 = LVV − VI N LII

(9)

By expanding the coefficients, we get a quadratic expression in λ, see Appendix A. The error is roughly estimated to be 5% using the model A and to 10% using the model B. In the model B, the electrostatic contributions are accounted for separately and, as a result, the obtained slip length values become smaller, as clearly observed in Table 1.

5. Discussion As suggested in Fig. 1, the slip length is linked to the friction at the solid–liquid interface and is thus describing the energy dissipation within the interface. We decompose the slip length into several components that describe a specific interfacial mechanism in analogy with other surface properties such as surface tensions. Here, the simplest model would be a sum of an electrokinetic term λel : λ = λv + λel

(10)

where λv is representing a viscous dissipation within the interface and λel the electrostatic friction. In the model A, one indeed measures the sum λ, which corresponds to an interfacial region including the electric double layer. With the model B, we are in fact subtracting the electrokinetic contribution and we measure λv . Such result is found to be in agreement with Lauga [16] in the sense that apparent slip lengths are decreasing when electrokinetic effects are considered, albeit that our analysis shows the existence of slip also in the absence of electrokinetics. Our results show a very strong influence on the electrokinetic effect with ionic surfactant systems. For these systems, λ ≈ λel . Both the electrostatic friction and the zeta potential are dependent on the surfactant concentration which is to be expected because both vary with the surfactant adsorption on the walls which in turn depends on the bulk surfactant concentration. To conclude, the flow behavior of three surfactant systems in nanopores is presenting discontinuities interpreted as stick–slip transitions. The critical shear stress characterizing the transition is found to be concentration dependent. The slip

255

regime is described by a slip length that can be decomposed into the sum of an electrostatic contribution and a viscous one. The electrokinetic contribution is experimentally shown to vary linearly with the zeta potential.

Acknowledgements The authors thank Dick Bedeaux and Theo van de Ven for illuminating discussions.

Appendix A The phenomenological coefficients for stick boundary conditions read: πR4 εR2 , LVI,0 = LIV,0 = − ζ(1 − G), 8η 4η  R  R
2 ε2 dφ = 2π rσ(r) dr + r dr (11) 8πη 0 dr 0

LVV,0 = LII,0

with ζ = φ(R) the zeta potential, η the fluid viscosity and ε its dielectric permittivity, and with the conductivity σ is given by

 zj eφ(r) σ(r) = en0 zj νj µj exp − (12) kT j

where the sum runs over all ion types in the fluid with zj , νj and µj the valency, the stoichiometric coefficient, and the mobility of ion type j, respectively. Furthermore, n0 is the ion concentration outside the pore, e the electron charge, k the Boltzmann’s constant, and T is the temperature. Following [17], the function G is defined as  R 2 rφ(r) dr (13) G= 2 R ζ 0 For low surface potentials, ζ
25 mV, the Debye–H¨uckel approximation applies in which the exponential terms in the L-coefficients are expanded up to the first order and rewrite as

LII,0

πR4 , 8η

εR2 ζ(1 − GDH ), 4η
  eζ 2 (14) = πR en0 zj νj µj 1 − zj GDH kT

LVV,0 =

LVI,0 = LIV,0 = −

j

A simple expression for the potential φ remains [18]: φ(r) = ζ

I0 (κr) I0 (κR)

(15)

where I0 is the zeroth order modified Bessel function of the first kind and κ is the inverse Debye screening length. The

C. Cheikh, G. Koper / Colloids and Surfaces A: Physicochem. Eng. Aspects 270–271 (2005) 252–256

256

function G is then given by 2 I1 (κR) GDH = κR I0 (κR)

References (16)

with I1 the first order modified Bessel function of the first kind. In the case of slip the coefficients read: LVV λ = 1+4 , LVV,0 R LII = LII,0 − λR KI = −λ

LVI LIV ζ = , =1−λ LVI,0 LIV,0 ζ(1 − G) ε2  2 (ζ ) , 8πη

KV = −λ

εR  ζ σc 2η

πR2 σc , η (17)

where we use the shorthand ζ  = ∂φ/∂r|r=R . The critical surface shear rate is given by σc . In the Debye–H¨uckel approximation the set of coefficients becomes: LVV λ = 1+4 , LVV,0 R LVI LIV κ2 GDH = = 1 − λR , LVI,0 LIV,0 2(1 − GDH )
 ε2 κ2 I1 (κR) 2 2 LII = LII,0 − λR ζ , 8πη I0 (κR) KV = −λ

πR2 σc , η

KI = −λ

εR  ζ σc 2η

(18)

[1] N.V. Churaev, V.D. Sobolev, A.N. Somov, J. Coll. Int. Sci. 97 (1984) 574. [2] Y. Zhu, S. Grannick, Phys. Rev. Lett. 87 (2001) 096105. [3] Y. Zhu, S. Grannick, Langmuir 18 (2002) 1058. [4] H.A. Barnes, J. Non-Newtonian Fluid Mech. 56 (1995) 221. [5] H. Yoshizawa, Y.L. Chen, J. Israelachvili, J. Phys. Chem. 97 (1993) 4128. [6] C. Cheikh, G.J.M. Koper, Phys. Rev. Lett. 91 (2003) 156102. [7] P.G. de Gennes, Langmuir 18 (2002) 3413. [8] A.M.D.E. Timmerman, Ph.D. Thesis TU Delft, Understanding detergency, 2002. [9] F.X. Pierre, I. Souchon, M. Marin, J. Memb. Sci. 187 (2001) 239. [10] H.S. Alkhatim, M.I. Alcaina, E. Soriano, M.I. Iborra, J. Lora, J. Arnal, Desalination 119 (1998) 177. [11] K. Scott, Handbook of Industrial Membranes, Elsevier Advanced Technology, Oxford, 1995. [12] K. Scott, R. Hughes, Industrial Membrane Separation Technology, Blackie, London, 1996. [13] C. Cheikh, G.J.M. Koper, Physica A, in press. [14] R. Pal, Coll. Surf. 64 (1992) 207. [15] A.E. Childress, M. Elimelech, Environ. Sci. Technol. 34 (2000) 3710. [16] E. Lauga, Langmuir 20 (2004) 8924. [17] S. Levine, J.R. Marriott, G. Neale, N. Epstein, J. Coll. Int. Sci. 52 (1975) 136. [18] C.L. Rice, R. Whitehead, J. Phys. Chem. 69 (1965) 4017.