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Forced and Spontaneous Imbibition of Surfactant Solution into an Oil-Wet Capillary: The Effects of Surfactant Diffusion Ahead of the Advancing Meniscus Paul S. Hammond and Evren Unsal* Schlumberger Cambridge Research, Cambridge, U.K. CB3 0EL Received October 16, 2009. Revised Manuscript Received February 22, 2010 A previous paper (Hammond, P.; Unsal, E. Langmuir 2009, 25, 12591-12603) reported a simplified model for the flow of a surfactant solution into an oil-wet capillary. Results were computed by neglecting the spreading of surfactant molecules ahead of the moving oil/water meniscus onto the hydrophobic surface. We now present a more thorough version of the theory where such spreading is considered. Both spontaneous and forced imbibitions are studied. As the differential pressure across the capillary increases, a slow increase in the meniscus velocity is observed until the capillary threshold pressure is reached. At this point, the pattern changes and the velocity increases dramatically. The surfactant concentration did not have a significant effect on the speed under differential pressures greater than the capillary threshold. For lower pressures, there is a critical surfactant concentration below which the interface was not able to advance into the capillary even under positive differential pressure.
1. Introduction Surfactants are used to control wettability in certain industrial applications (i.e., oil recovery, printing, and cosmetics) because they can dramatically affect wettability due to their surface activity. The wettability-altering behavior depends on the chemical structure of the surfactant, solid surface properties, and how the surfactant molecules are assembled at the solid-liquid, solid-vapor, and liquid-vapor interfaces.1,2 Spontaneous penetration of water into hydrophobic capillaries is impossible if the contact angle exceeds 90. Wetting of the hydrophobic surface can be improved by the adsorption of surfactant molecules, which are directed with polar groups toward the water phase;3 surfactant molecules are adsorbed on solid surfaces because of favorable interactions between the surfactant molecules and the chemical species on or near the solid surface. Let us consider the situation during the imbibition of a surfactant solution into an oilwet capillary. Surfactant molecules move towards the meniscus in the bulk solution (convective transport). Meanwhile, some molecules are adsorbed on the capillary walls behind the moving meniscus. Consequently, the bulk solution concentration decreases from the inlet in the direction of the meniscus. If the advancing contact line is examined microscopically, then some interesting behavior is revealed. Experiments show that the surfactant molecules are not only adsorbed behind the moving meniscus but also are spread over the solid surface beyond the moving three-phase contact line. This is not unexpected for hydrophilic surfaces, but it is usually neglected in the case of hydrophobic substrates because there is no obvious strong attraction. In fact, this phenomenon is believed to be *To whom correspondence should be addressed. E-mail:
[email protected]. Tel: þ44 1223 325231.
(1) Frank, B.; Garoff, S. Langmuir 1995, 11, 87–93. (2) Qu, D.; Suter, R.; Garoff Langmuir 2002, 18, 1649–1654. (3) Churaev, N. V. Surfactant Solution Behavior in Quartz Capillaries. In Surface and Interfacial Tension: Measurement, Theory, and Applications; Surfactant Science Series; Hartland, S., Ed.; Marcel Dekker: New York, 2004; Vol. 119, pp 313-374. (4) Kumar, N.; Varnasi, K.; Tilton, R.; Garoff, S. Langmuir 2003, 19, 5366– 5373.
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closely associated with the penetration of surfactant solutions into hydrophobic capillaries.4-9 Kumar et al.4 give experimental evidence for the presence of surfactant on the solid surface on the vapor side of an advancing water-vapor interface, and Churaev et al.10 present a model for surfactant imbibition including this feature. Kao et al.11 observe changes at or near the solid surface on the oil side of the advancing interface as a surfactant solution displaces an oil drop from a surface, which is strongly suggestive of the presence of surfactant on the downstream side of the advancing interface in that case too. Eriksson et al.8 used the Wilhemy plate method to study adsorption onto hydrophilic and hydrophobic substrates. For hydrophobic substrates, the dynamic wetting behavior is strongly affected by the surfactant carryover through the advancing threephase contact line, which appears to be the dominant mode of surfactant transport to the solid/liquid interface. Conversely, for hydrophilic substrates, three-phase contact line carryover of surfactant does not appear to be so important, the predominant transport mechanism being bulk diffusion. We present a model for two-phase flow in a single oil-wet capillary tube: wettability-altering surfactant solution is applied to one end of an initially oil-filled capillary, and the subsequent movement of the meniscus is calculated. The model is an extension of the previous work by Hammond and Unsal,12 and the reader is referred to that paper for a more complete discussion of the ideas and assumptions behind the mathematical description. The main addition here is that we show results for cases in which the adsorption of surfactant molecules onto the hydrophobic surface, in front of and behind the moving interface, is significant. This requires a more elaborate (5) Cachile, M.; Cazabat, M. Langmuir 1999, 15, 1515–1521. (6) Starov, V.; Kosvintsev, S. R.; Velarde, M. G. J. Colloid Interface Sci. 2000, 227, 185–190. (7) Starov, V. J. Colloid Interface Sci. 2004, 270, 180–186. (8) Eriksson, J.; Tiberg, F.; Zhmud, B. Langmuir 2001, 17, 7274–7279. (9) Honciuc, A.; Harant, A. W.; Schwartz, D. K. Langmuir 2008, 24, 6562–6566. (10) Churaev, N. V.; Martynov, G. M.; Starov, V. M.; Zorin, Z. M. Colloid Polym. Sci. 1981, 259, 747–752. (11) Kao, R.; Wasan, D.; Nikolov, A.; Edwards, D. Colloids Surf. 1988, 34, 389–398. (12) Hammond, P.; Unsal, E. Langmuir 2009, 25, 12591–12603.
Published on Web 03/12/2010
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Upon rearranging eq 1, we find ΔpðtÞ þ Δpc ¼
8U ðμ X þ μo ðL -XÞÞ a2 w
ð5Þ
which will later be used to find U. The contact angle θ between the surfactant solution and oil at the solid boundary satisfies Young’s equation σ ow ðΓow Þ cos θ ¼ σ os ðΓos Þ - σws ðΓws Þ Figure 1. Problem geometry. Aqueous surfactant solution displaces oil in a circular cylindrical capillary tube. The walls of the capillary are initially oil-wet.
treatment of the processes of surfactant transport and adsorption in the vicinity of the moving contact line.
2. Model Spontaneous and forced displacement of oil by aqueous surfactant solution from a circular cylindrical capillary is considered. The problem geometry is sketched in Figure 1. The capillary radius is a, and the length is L. The meniscus separating oil and water is located at position z = X(t), where t is time and z measures the axial distance along the capillary. Initially, the walls of the capillary are oil-wet, but on contact with surfactant, the wettability changes and the walls become water-wet. We envisage that this occurs by a coatinglike mechanism; surfactant molecules adsorb with tails down on the initially hydrophobic solid surface, exposing headgroups and thus creating a hydrophilic surface. The velocity of the meniscus is U(t), so dX ¼ U dt
ð1Þ
subject to X(0) = X0 (which should be greater than zero in order to avoid early-time velocity singularities). By ignoring flow processes in the immediate vicinity of the meniscus, the velocity of the meniscus is taken to be linked to pressure differences between various positions in the capillary through the Poiseuille flow formulas13 U ¼
a2 ðpL - p - Þ a2 ðp þ - pR Þ ¼ 8μw X 8μo ðL -XÞ
ð2Þ
Here μw and μo are the viscosities of the surfactant solution and oil, respectively, pL is the aqueous solution pressure at the entrance to the capillary, pR is the oil pressure at the exit, p- is the pressure in the aqueous solution immediately adjacent to the meniscus, and pþ is the pressure in the oil adjacent to the meniscus. The difference between the externally applied pressures at the ends of the capillary is taken to be pL - pR ¼ ΔpðtÞ
p þ - p - ¼ Δpc
ð4Þ
(13) Batchelor, G. K. An Introduction to Fluid Dynamics; Cambridge University Press: Cambridge, U.K., 1967.
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where Γij is the number of moles of surfactant adsorbed per unit area at the interface between materials i and j and σij(Γij) is the interfacial tension of that interface. Indexes i and j can each take the value o denoting oil, w denoting the aqueous solution, or s denoting the solid of the capillary wall (thus σow is the interfacial tension of the oil-aqueous solution meniscus). In the absence of surfactant, ð7Þ
σow ð0Þ cos θ0 ¼ σos ð0Þ - σ ws ð0Þ whence σow ðΓow Þ cos θ - σ ow ð0Þ cos θ0 ¼ σ ws ð0Þ - σ ws ðΓws Þ -½σ os ð0Þ - σos ðΓos Þ
ð8Þ
The adsorption of surfactant onto an interface reduces the interfacial tension, and the surface properties of an ideal monolayer are related to each other by σ ij ðΓij Þ ¼
σ ij ð0Þ þ RTΓm ij
Γij log 1 - m Γij
! ð9Þ
where R is the universal gas constant, T is the absolute temperature, and Γijm is the monolayer capacity. Combining eqs 8 and 9, we find σ ow ðΓow Þ cos θ -σ ow ð0Þ cos θ0 ! ! Γws Γos m m ¼ -RTΓws log 1 - m þ RTΓos log 1 - m Γws Γos
ð10Þ
This shows how the adsorption of surfactant at the various interfaces will change the contact angle. For low-capillary-number flows, that is, slow flow in which viscous stresses at interfaces are much weaker than stresses due to interfacial tension, the menisci are the surfaces of constant curvature. In the present circular cylindrical geometry, this means that the meniscus is the cap of a sphere. If the contact angle between the sphere and the capillary wall is θ, then the pressure difference across the meniscus is Δpc ¼
ð3Þ
In the case of spontaneous imbibition, this pressure difference will be zero. The capillary pressure at any instant, Δpc, is defined as the difference between the oil and water pressures across the meniscus so that
ð6Þ
0
2σow ðΓow Þ cos θ a
2 Γws ¼ @σow ð0Þ cos θ0 - RT Γm ws log 1 - m Γws a þ RTΓm os log
Γos 1- m Γos
!
!! ð11Þ
using eq 10. Hence, if the amounts of surfactant adsorbed on the solid surface on either side of the meniscus are known, then eq 11 DOI: 10.1021/la903924m
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combined with eq 5 gives the speed of advance of the meniscus, which can then be used in eq 1 to update the position of the meniscus. To find the amount of surfactant adsorbed on the solid surface near the moving meniscus, we must consider the processes of surfactant transport through the bulk aqueous phase, along the meniscus, through the moving three-phase contact line onto the solid, and over the surface of the solid. Upstream from the meniscus, surfactant is adsorbed on the walls of the capillary. This effect was not included in the model presented in ref 12. We assume that the amount adsorbed, Γws(z,t), is in instantaneously achieved equilibrium with the cross-sectional averaged surfactant concentration in the adjacent fluid, c(z,t), according to the Langmuir isotherm Γws ðz, tÞ ¼ Γm ws
Kws cðz, tÞ 1 þ Kws cðz, tÞ
ð12Þ
where Γwsm is the monolayer capacity and Kws is a measure of the relative rates of adsorption and desorption from the interface. Inlet bulk surfactant concentrations must be restricted to be less than the cmc (critical micelle concentration) so that the use of the Langmuir isotherms, eqs 12 and 18, is valid. If we write πa2Ξ(z, t) for the total amount of surfactant per unit length of capillary at location z, then πa2 Ξðz, tÞ ¼ πa2 cðz, tÞ þ 2πaΓws ðz, tÞ
ð13Þ
However, Γws(z, t) is related to c(z, t) by eq 12, and by inserting this into 13 we obtain a quadratic equation linking the total and flowing amounts of surfactant. The physically meaningful (i.e., positive) solution of the resulting equation for the flowing concentration is
Ξðz, tÞ Γm 1 - ws 2 2Kws a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 Ξðz, tÞ Γm 1 Ξðz, tÞ - ws þ þ 2 2Kws Kws a cðz, tÞ ¼
ð14Þ
The adsorbed concentration can then be found from eq 12 or 13. When the Peclet number aU/Db is small and for times large compared to a2/Db, surfactant transport within the capillary is described by DΞ Dc D2 c þ U ¼ Db 2 Dt Dz Dz
ð15Þ
where Db is the bulk diffusivity of surfactant within the aqueous phase, U is the average axial velocity of fluid within the capillary, which is of course equal to the speed of the meniscus, and c is related to Ξ by eq 14. In writing eq 15, we have dropped a small term 2Ds∂2Γws/a∂z2 describing the transport of surfactant by surface diffusion. For bulk concentrations of less than 1/Kws, this m 2 term is approximately 2DsKwsΓws ∂ c/a∂z2 from eq 12, so our m approximation requires that Db . 2DsKwsΓws /a. Using the parameter values given later, the right-hand side of this inequality is around 2.5 10-11 m/s whereas Db ≈ 10-9 m2/s, so diffusive transport in the bulk dominates diffusive transport on the solid surface. We shall consider below what happens ahead of the advancing meniscus, where the bulk diffusion path is not available because the surfactant is taken to be insoluble in the oil. See ref 12 for a discussion of the underlying assumptions in this treatment of transport by flow in the bulk. 6208 DOI: 10.1021/la903924m
Equation 15 is subject to the inlet condition cð0, tÞ ¼ cb so that Ξð0, tÞ ¼ cb þ
2Γm Kws cb ws a 1 þ Kws cb
ð16Þ
2Γm Kws cb ws a 1 þ Kws cb
ð17Þ
and to the initial condition cðz, 0Þ ¼ cb so that Ξðz, 0Þ ¼ cb þ
where cb is the bulk concentration of surfactant in the initial aqueous solution. The remaining boundary condition required by second-order eq 15 will be presented 32 below. At the meniscus, the amount of surfactant adsorbed on the oil-water interface is in Langmuir isotherm equilibrium with the concentration of surfactant in the immediately adjacent aqueous phase so that Γow ¼ Γm ow
Kow cðX, tÞ 1 þ Kow cðX, tÞ
ð18Þ
m where Γow is the monolayer capacity and Kow is a measure of the relative rates of adsorption and desorption from the interface. In writing this expression, we are assuming that the surface concentration of surfactant on the meniscus is spatially uniform, consistent with treating the solution concentration as uniform across the capillary radius by virtue of the restrictions leading to eq 15. To complete the system of equations, it is necessary to give relations linking c(X, t), Γow, Γos(X, t), and Γws(X, t), that is, a description of the transfer of surfactant from the oil-water interface to the liquid-solid interfaces through the moving contact line. These are processes occurring on a length scale that is very much shorter than the capillary diameter or other length scales considered up to this point. In effect, we now make an analysis valid on a short, or inner, length scale, with the previously considered, outer, variables playing the part of boundary or matching conditions at infinity for the subsequent inner analysis. (In effect we are using the method of matched asymptotic expansions in chapter 5 of ref 14, although we do not attempt to set up the formal machinery of this approach here). Our treatment is a more elaborate version of that given in ref 12. We consider a system in which surfactant molecules adsorb on top of any pre-existing deposits on the solid surfaces. Thus, the picture is one of wettability modification by coating the solid surface rather than of wettability restoration by cleaning it. Zhmud et al.15 and Tiberg et al.16 assume that the total time over which the transfer of surfactant from the oil-water to the watersolid interface takes place is equal to h/U, where h is the thickness of the surfactant-loaded oil-water interface and U is the speed of the contact line; the configuration is sketched in Figure 2. They further assume that surfactant is transferred at a rate given by the Langmuir kinetic equation (section 2.1.1 of ref 17). That is, the rate of transfer from the oil-water to the water-solid interface is taken to be proportional to the product of the amount of surfactant on the oil-water interface and the number of unoccupied adsorption sites on the water-solid interface, with
(14) Van Dyke, M. Perturbation Methods in Fluid Mechanics, The Parabolic Press: Stanford, CA, 1975. (15) Zhmud, B. V.; Tiberg, F.; Hallstensson., K. J. Colloid Interface Sci. 2000, 228, 263–269. (16) Tiberg, F.; Zhmud, B.; Hallstensson, K.; Von Bahr, M. Phys. Chem. Chem. Phys. 2000, 2, 5189–5196. (17) Krotov V. V.; Rustanov A. I. Physicochemical Hydrodynamics of Capillary Systems; Imperial College Press: London, 1999.
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Figure 2. Sketch of the model for the transfer of surfactant from the meniscus to the solid surface across the moving contact line region, showing the possibility of adsorbed surfactant present on the solid surface on the oil side of the advancing meniscus.
rate constant kþ. The rate of backward transfer (i.e., desorption from the water-solid interface either back to the oil-water interface or into the adjacent bulk solution) is taken to be simply proportional to the amount of material adsorbed on the water-solid interface, with rate constant k-. Thus, at a fixed point on the solid surface the time rate of change of the amount of adsorbed surfactant, Γ, due to exchanges with the oil-water interface is ! dΓ Γ þ ¼ k Γow 1 - m - k - Γ dt Γws
ð19Þ
It seems reasonable to limit the adsorbed amount, on the aqueous side at least, to be less than the value that would be in equilibrium with the concentration in the bulk aqueous solution immediately adjacent to the contact line, c(X, t). This can be achieved by setting the value of k- such that the net rate of transfer vanishes when the surface concentration attains the appropriate equilibrium value. If we assume that this equilibrium value is given by the Langmuir isotherm (eq 12), then by inserting eq 12 into eq 19 and setting the result equal to zero, we find k- ¼
kþ Γow m Γws Kws cðX, tÞ
ð20Þ
dΓ ¼ kþ Γow dt
ð21Þ
On that part of the solid surface exposed to the aqueous solution, surfactant can exchange with the bulk, and we model this through ! dΓ 1 þ kws cðX, tÞ þþ ¼ hk cðX, tÞ 1 - Γ m dt Γws kws cðX, tÞ
ð22Þ
Here, kþþ is a rate constant for surface-bulk exchange, analogous to that for surface-meniscus exchange; the overall expression is modified only to take account of the different dimensions of the adjacent concentration variables in the two cases. The surfactant can diffuse over the solid surface, with surface diffusivity Ds. This carries with it the possibility of surfactant being present on the solid surface downstream (i.e., on the oil side) of the advancing oil-water interface. In a frame of reference moving with the three-phase contact line, assuming Langmuir 2010, 26(9), 6206–6221
ð23Þ where y = z - X(t), so y = 0 corresponds to the water side of the finite thickness surfactant-loaded oil-water interface and y = h corresponds to the oil side. For the purposes of evaluating the surface adsorptions in eq 11, we take Γws(X, t) = Γ(y = 0) and Γos(X, t) = Γ(y = hþ). Several inner-region length scales can be discerned in eq 23: h, the meniscus thickness; Ds/U, the length scale over which surface diffusion and advection m þþ /k hcb, the length scale for inner-region balance; and UΓws adsorption to reach saturation. Using the parameter values given later, these can all be shown to be small compared to the capillary diameter a. At great distances from the meniscus, we have boundary condition Γ f 0 as y f ¥, corresponding to no surfactant on the solid surface far downstream from the advancing meniscus m eq Kwsc(X, t)/(1 þ Kwsc(X, t)) = Γws as y f -¥, and Γ f Γws corresponding to the adsorbed concentration in equilibrium with the bulk solution far behind the advancing meniscus. At y = 0, Γ and its y derivative are continuous (i.e., the adsorbed surfactant concentration and the diffusive flux along the surface are continuous there). The conditions at y = h require some further comment. Following Churaev et al.,10 we assume that surfactant molecules adsorbed on the oil-solid interface are in a higherenergy, or less favorable, state than those on the water-solid interface, and the chemical potential of adsorbed surfactant, μis, is μis ¼ μQis þ RT log Γ
It follows that ! 1 þ Kws cðX, tÞ 1-Γ m Γws Kws cðX, tÞ
the surfactant distribution to be in a quasi-steady state, we therefore have ! d dΓ -UΓ -Ds dy dy 8 ! > þþ 1 þ Kws cðX, tÞ > > k hcðX, tÞ 1 - Γ m in y > Γws K ws cðX, tÞ > > > > < ! þ 1 þ Kws cðX, tÞ ¼ k Γow 1 - Γ m in 0 < y < h > > Γws Kws cðX, tÞ > > > > > > > 0 in y>h :
ð24Þ
Q Q We set μos > μws so that at equal chemical potential the adsorbed concentration on the water-solid interface is greater than that on the oil-solid interface. At y = h, we demand that the chemical potential of surfactant be continuous, representing the local equilibrium between adsorbed surfactant at the edge of the transfer zone and that at the start of the region in contact with oil. As a result, the surfactant concentration must be discontinuous at y = h, with ! μQos -μQws þ ð25Þ Γðy ¼ h Þ ¼ Γðy ¼ h Þ exp RT
We also require that the total flux of surfactant along the surface is continuous at y = h so that
dΓ -UΓ -Ds dy
y ¼h -
¼
dΓ -UΓ -Ds dy
y ¼hþ
ð26Þ
By virtue of eq and the conditions at y f ¥, the total flux at y = hþ is in fact zero. Because Γ is not continuous at y = h, dΓ/dy is not continuous there either. DOI: 10.1021/la903924m
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Equation is a second-order linear differential equation and can be solved analytically by standard means. The details of the calculation are given in the Appendix. From eqs A12, 12, and 18, we find that Γws ðX, tÞ ¼ Γm ws 0 B B B1 @
Kws cðX, tÞ 1 þ Kws cðX, tÞ
1
C λ - λþ C C λ þ Ds λ þ h λ - Ds λ - h A e þ ðν - - λ þ Þ 1 þ e ðλ - - ν - Þ 1 þ U U
ð27Þ and from eq A11, we find that Γos ðX, tÞ ¼ Γm os
Γm Kws cðX, tÞ Q Q ws e - ð μos - μws Þ=RT m Γos 1 þ Kws cðX, tÞ
1 λ - Ds λ - h λ þ Ds λ þ h e þ ðλ - - ν - Þ e ðν - - λ þ Þ C B C B U U C B @ λ þ Ds λ þ h λ - Ds λ - h A e þ ðν - -λ þ Þ 1 þ e ðλ - -ν - Þ 1 þ U U 0
ð28Þ where 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Uh @ 1 þ Kws cðX, tÞ Kow 4Ds kþ Γm ow A λ( h ¼ 1( 1þ 2Ds 1 þ Kow cðX, tÞ Kws U 2 Γm ws ð29Þ and 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Uh @ 4Ds hkþþ A ν -h ¼ -1 þ 1 þ ð1 þ Kws cðX, tÞÞ 2Ds Kws U 2 Γm ws
ð30Þ
These expressions give the adsorbed amounts on the watersolid and oil-solid surfaces immediately adjacent to the meniscus in terms of the bulk concentration of surfactant there and the speed of advance of the meniscus. When inserted into eq 11, they permit the contact angle and capillary pressure to be computed. The final piece of the model is a link between the flux of surfactant onto the meniscus from the adjacent bulk solution and the fluxes of surfactant away from the three-phase contact line along the solid surface. In the frame of reference moving with the meniscus, the flux of surfactant onto the meniscus and into the inner region from the bulk solution is -πa2Db(∂c/∂z)(X, t). In the same frame of reference, the flux of surfactant out from the inner region is -2πaUΓeq ws by virtue of the conditions on Γ as y f -¥. From eq and the conditions at y f ¥, there is no net flux of surfactant out into the region y > h. Because the surfactant mass is conserved, the difference between these two fluxes is equal to the rate of accumulation of surfactant on the meniscus and in the inner region. The total amount of surfactant stored on the meniscus is AΓow, where A = 2πa2(1 - sin θ)/(cos θ)2 is the area of the meniscus. R The amount of surfactant stored in the inner region is 2πa -Hh Γ(y) dy where -H represents a value for y where the inner and outer regions join, and we assume that this is much smaller than the amount stored on the meniscus because the characteristic length scale of the inner region is much smaller than the diameter of the capillary. The mass balance for surfactant on 6210 DOI: 10.1021/la903924m
the meniscus and inner region is thus Rh dðAΓow þ 2πa -H ΓðyÞ dyÞ Dc ¼ -πa2 Db ðX, tÞ - 2πaUΓeq ws dt Dz ð31Þ Now, because the adsorbed concentration on the meniscus is in equilibrium with the adjacent bulk solution, the time derivam Kow(1 þ tive term on the left-hand side of eq 31 is πa2(Γow Kowc(X, t))2)∂c(X, t)/∂t, which has an order of magnitude of m Kowc(X, t)/τ if we write the characteristic timescale πa2Γow for variations of the adsorbed amount on the meniscus as τ. The flux of surfactant along the solid surface on the right-hand m side is 2πaUΓws Kwsc(X, t)/(1 þ Kowc(X, t)), so the ratio of the magnitudes of the time derivative to the surface convective flux m m term is approximately (Γow Kow/Γws Kws)(a/2Uτ). For the typical parameters given below, the first term in parentheses is 0(1) and the second term will be small if the distance traveled in a characteristic time, Uτ, is large compared to the capillary diameter a. We might expect the adsorbed amount on the meniscus to vary with the total distance the meniscus has penetrated into the capillary, X(t), so that τ = X/U, then a/2Uτ will be small provided X(t) . τa, that is, at large meniscus penetrations or large times. However, the order of magnitude of the concentration flux term on the right-hand side of eq 31 is πa2Dbcb/X if we estimate the concentration gradient to be cb/X, which is appropriate at long times as we shall see from the solutions presented below. The ratio of the magnitudes of the time derivative to the diffusive flux term m is thus Γow KowXc(X, t)/τDbcb. Now, if we again take τ = X/U, m this becomes Γow KowUc(X, t)/Dbcb, which becomes small if either the velocity U or the bulk concentration near the interface c(X, t) decays to zero. We shall assume that this is the case for now, and will see later from the computed solutions that this is indeed so. As a result, we may neglect the time derivative term in eq 31 at long times or large meniscus penetrations. This equation then becomes a balance between the bulk diffusive flux of surfactant onto the interface and surface fluxes in and out of the three-phase contact line region. Equations A2 and A10, when inserted into eq 31 with the time derivative neglected, then allow us to specify the remaining boundary condition for eq 15: Dc 2U m Kws cðX, tÞ ðX, tÞ ¼ Γ Dz aDb ws 1 þ Kws cðX, tÞ
ð32Þ
Inserting eq 11 into eq 5 gives an equation for U 0
aΔpðtÞ @ Γws ðX, tÞ þ σ ow ð0Þ cos θ0 -RTΓm ws log 1 Γm 2 ws !! Γos ðX, tÞ þ RTΓm os log 1 Γm os ¼
4U ðμw X þ μo ðL -XÞÞ a
!
ð33Þ
where Γws(X, t) and Γos(X, t) are given by eqs 27 and 28. Only positive roots can be meaningful because we have assumed the movement of surfactant from the oil-water to the liquid-solid interfaces. The terms within the parentheses on the left-hand side of this expression are equal to σow(Γow) cosθ. This completes the description of the basic mathematical model. Langmuir 2010, 26(9), 6206–6221
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Figure 3. Γws(X, t), Γos(X, t), and the left- and right-hand sides of eq 33 as functions of U for Δp = 0, μw = μo = 10-3 Pa s, c(X, t) = 0.01 mol/m3,
Q Q Ds = 5 10-14 m2/s, and μos - μws = RT. The values of the other parameters are given in the main text. Note that from the bottom track m that there is a unique root near U = 1 10-5 m/s. In the upper two tracks, the dotted red lines show the values of Γws and Γm os respectively, -(μosQ - μwsQ)/RT eq and the dashed green lines show the values of Γws and Γeq e . ws
The previous version of this model12 in which there is no surface diffusion of surfactant and no adsorption of surfactant from the bulk directly onto the water-solid interface away from the meniscus can be recovered by setting ν_ = 0 and λ_ = kþΓow/ m U and taking λþ f -¥. Γws The present model is assembled from a collection of simplified representations for individual processes. Each of these brings with it some restrictions on the admissible values of parameters for it to be valid, and we summarize these here. The treatment of the bulk hydrodynamics requires that the length of the capillary significantly exceeds its diameter so that contributions to pressure drops from processes near the meniscus are relatively small. The capillary number μU/σow must be small for the meniscus not to be significantly deformed from a static shape. The timescales of all variations need to be large compared with a2/Db in order for concentrations to be approximately uniform across the capillary diameter so that a cross-sectional-averaged description of surfactant transport in the bulk is appropriate. Slow variations are also necessary for the assumption of instantaneous attainment of adsorption equilibrium between surfaces and the immediately adjacent bulk solution. The Peclet number aU/Db must be small so that the molecular diffusivity, rather that the Taylor shearLangmuir 2010, 26(9), 6206–6221
enhanced diffusivity, appears in the bulk surfactant transport equation. The neglect of the time derivative in the surfactant mass conservation equation at the meniscus, eq 31, is essentially a latetime approximation, valid once any starting transients have died away and once velocities (or near meniscus concentrations) have decayed to low values. The bulk concentration of surfactant must everywhere be less that the cmc so that the Langmuir isotherm is appropriate. The surface equations of state used here follow from the Gibbs equation given the Langmuir isotherm, provided that the bulk solutions are sufficiently dilute that activities and concentrations may be equated; this again requires concentrations be below the cmc. The main mathematical assumption in the treatment of the inner region is that the inner-region dynamics may be treated as quasi-steady because changes in the outer region occur on a very much slower time scale than relaxation processes in the inner region; this is reasonable because inner-region length scales are so much smaller than outer-region length scales and the timescale for mass transfer by diffusion varies with the square of the characteristic length. There are of course many modeling assumptions made about the physical and physical-chemical processes acting in the inner region, for example, those relating to the mode of transfer of surfactant between the meniscus and solid surfaces and to the DOI: 10.1021/la903924m
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Figure 4. Γws(X, t), Γos(X, t), and the left- and right-hand sides of eq 33 as functions of U for c(X, t) = 0.0001 mol/m3, Ds = 5 10-14 m2/s,
Q and μQ os - μws= RT. The values of the other parameters are given in the main text. Note that there is no root for U.
partitioning of surfactant between the oil-solid and water-solid interfaces. The validity of these can be assessed only from appropriate experimentation or molecular-level simulation. Taken together, we can state that the present model is valid for slow flow at long time, with the inlet concentrations below the cmc. It was shown in ref 12 that a similar solution exists when the applied differential pressure is not too large, in which the velocity decays to zero as time increases. By analogy, we can say that the conditions for the validity of the present model will become ever better satisfied as time increases. When applied differential pressures are large compared to the capillary entry pressure -2σow(0) cos θ0/a, we see in Figure 10 that the surfactant concentration no longer affects the speed of the meniscus; the central requirement then is simply that the capillary number be small. We conclude this section by plotting the left- and right-hand sides of eq 33 to illustrate that either one, two, or zero roots for U can be found, depending on the values of the parameters. We take Δp(t) = 0, a = 0.0001 m, L = 0.01 m, μw = μo = 10-3 Pa s, σow(0) = 30 10-3 N/m, cos θ0 = -0.1, Kws = Kos = 1.3 Q Q m m - μws = RT, Γws = Γos = 104 m3/mol, RT = 8.314 300 J, μos -6 2 -9 þ þ2 (1.96 10 /RT) mol/m , h = 4 10 m, and hk = hk = (18) Lin, S. Y.; Tsay, R. Y.; Lin, L. W.; Chen, S. I. Langmuir 1996, 12, 6530–6536.
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10-5 m/s. (See ref 18 for measurements of dynamic surface tension and inferred rate parameters for subsurface-interface exchange; their average value for a parameter equivalent to hkþ2 is not inconsistent with the value used here.) Taking oil and water viscosities to be equal causes the dependence on X to disappear from the right-hand side of eq 33, and then from eqs 27 and 28, Γws(X, t) and Γos(X, t) depend only on the bulk concentration adjacent to the meniscus c(X, t), the velocity U, and the various fixed parameters. Figures 3 and 4 show cases with surface diffusivity Ds = 5 10-14 m2/s,9 whereas in Figure 5 the diffusivity value is reduced to Ds = 10-17 m2/s so that Uh/Ds = 0(1) for typical spontaneous imbibition velocities. In Figure 3, c(X, t) = 0.01 mol/m3 (about 1/10 of the cmc value reported for C12E10 by Kumar et al.4), and there is a single positive root to eq 33 for U. In Figure 4, the bulk concentration of surfactant is lower, c(X, t) = 0.0001 mol/m3, and there is now no root for U. These observations demonstrate the existence of a concentration threshold, below which spontaneous imbibition is not possible according to the present model. In Figure 5 at the same value of c(X, t) but a much smaller value of Ds, there are two roots for U. For the faster root the adsorbed concentration on the oil-solid interface is small whereas for the slower root the adsorbed concentrations are comparable with the maximum possible values on both oil-solid and water-solid surfaces. On the basis of a Langmuir 2010, 26(9), 6206–6221
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Figure 5. Γws(X, t), Γos(X, t), and the left- and right-hand sides of eq 33 as functions of U for c(X, t) = 0.0001 mol/m3, Ds = 10-17m2/s, and
Q μQ os - μws= RT. The values of the other parameters are given in the main text. Note that with the low value of surface diffusivity there is now a double root for U.
limited number of numerical experiments, it appears only to be possible to obtain multiple roots when Ds is so small that the thickness of the transfer region, h, and the length scale for diffusion on the oil side of the interface, Ds/U, are comparable. If we assume that at the threshold concentration the velocity U is zero and that the bulk surfactant concentration adjacent to the meniscus is equal to the inlet bulk concentration, then we can equate the left-hand side of eq 33 to zero using the (finite) U f 0 values for the adsorptions on the solid surface given in the Appendix to obtain a relationship between the threshold concentration cbmin and the applied differential pressure Δp(t): 2σow ð0Þ cos θ0 ΔpðtÞ ¼ a 2 ! Kws cmin 2RT 4 m b þ Γws log 1 a 1 þ Kws cmin b -
Γm os
Γm -ðμQos - μQws Þ=RT Kws cmin b log 1 - ws e Γm 1 þ K cmin ws os b
!# ð34Þ
Equations 11 and 16 imply corresponding values for σow(Γow), which must be positive, and for cos θ, which must be less than 1 in Langmuir 2010, 26(9), 6206–6221
magnitude; these requirements set a lower limit for the values of Δp(t) for which a meaningful solution to eq 34 exists. The resulting relationship between Δp(t) and cbmin is plotted in Figure 6 using the parameters of Figure 3. Imbibition is possible above and to the right of the solid line. The overall model is valid only for inlet bulk concentrations below the cmc, which thus sets an upper limit to cbmin or rather a limit on how large and negative we can allow Δp(t) to become without compromising the validity of the model. Equation 34 has a simple physical interpretation: when U = 0, the capillary pressure jump across the meniscus (right-hand side of eq 34) must exactly balance the applied differential pressure (left-hand side). This requires that the contact angle takes a particular value, which in turn requires a finite amount of surfactant adsorption so as to provide the necessary changes in solid-liquid surface free energies, as represented through the logarithmic terms containing Langmuir isotherms on the right-hand side of eq 34. Clearly, a finite bulk concentration is necessary if finite changes in the surface free energies are to be achieved. Put another way, as the bulk concentration decreases, so the attainable change in contact angle decreases and eventually a point is reached where there is insufficient surfactant available to change wettability to the point that imbibition can occur. The concentration appearing in eq 34 is that adjacent to the DOI: 10.1021/la903924m
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negative, and presumably eventually reaches the cmc value, at which point the model is no longer valid. The characteristic velocity in this case is a little harder to define because the dynamics is such that meniscus velocities continuously decay in time and the meniscus penetration is proportional to t1/2 (as in the similar solution found in ref 12). In the present model, eq A17 eq )(kþh) ≈ shows that a characteristic velocity scale is (Γow/Γws þ (k h) (at least when Ds is not too large). This can be interpreted as the meniscus speed below which the adsorbed amount at the advancing meniscus, Γ(y = 0), takes its maximum, fully saturated value and above which the adsorbed amount is less than its saturated value. (This corresponds to the region of rapid transition in the bottom tracks of Figures 3-5.) When conditions are such that velocities have decayed below this value, we expect a solution structure analogous to the similar solution in ref 12 to set in. By analogy with the scalings found there, we expect this to occur for times large compared to (Db/(kþh)2) and for meniscus penetrations large compared to (Db/kþh). This places the concept of long time onto a more secure footing.
3. Results
Figure 6. Threshold concentration computed using eq 34. The dashed vertical line marks the capillary threshold pressure -2σow(0)cos θ0/a for forced imbibition into the original oil-wet capillary. With the present parameter values, the capillary threshold pressure for a fully water-wet capillary is -600 Pa, which is far off to the left on this graph; here surfactant adsorption on the oil-water interface lowers the interfacial tension and so reduces the magnitude of the threshold pressure.
meniscus, but for vanishingly small values of the meniscus speed this will be close to the inlet concentration because the rate of creation of “new” surfaces on which a surfactant is adsorbed is very small in very slow flow so only a tiny bulk concentration gradient is needed to drive the required small flux of surfactant into the meniscus. The model in ref 12 does not show a concentration threshold effect because it treats adsorption on the solid surface differently; transfer of surfactant onto the solid surface there occurs only from the oil-water meniscus, is treated as irreversible (k- = 0), and is limited only by the maximum coverage Γwsm rather than through the constraint of being in equilibrium with the adjacent bulk concentration. As a result, high surface adsorption can occur at low velocities, even for low bulk concentration, because of the long time then available for transfer and the lack of any mechanism to limit adsorbed amounts. In the present model, the amount adsorbed on the solid in such cases is limited by the Langmuir isotherm 12 and the corresponding nonzero value of k-. The model predicts that if Δp(t) is sufficiently large and positive then imbibition will occur independently of the surfactant concentration. The characteristic meniscus velocity in this case is the Washburn value ! 0 2σ ð0Þ cos θ ow a2 Δp þ a 8μo L (when μo = μw) and the meniscus penetration is proportional to t. Once Δp(t) decreases below the entry threshold capillary pressure of the initial oil-wet capillary, ΔP0c ¼ -
2σ ow ð0Þ cos θ0 a
a finite surfactant concentration is necessary for imbibition to be possible. This concentration increases as Δp(t) becomes more 6214 DOI: 10.1021/la903924m
We now present the results of numerical simulation of spontaneous and forced imbibition of a wettability-altering surfactant solution into an oil-wet capillary. As in ref 12, the governing equations in 0 e z e X(t) are transformed onto a fixed domain 0 e ξ e 1 and then time stepped numerically. In the mathematical model described above, surfactant adsorption onto the solid surface takes place in three places: (1) from bulk solution onto the capillary walls behind the meniscus, (2) from the meniscus onto the three-phase contact line, and (3) from the meniscus onto the hydrophobic surface in front of the three-phase contact line. The surfactant solution is applied to one end of the oil-filled capillary, and the other end is left open to a large reservoir of oil at atmospheric pressure. Once the solution comes into contact with the capillary inlet, a meniscus forms and advances through the capillary. Surfactant molecules move toward the meniscus in the bulk solution (convective transport). Meanwhile, some molecules are adsorbed on the capillary walls behind the meniscus. Consequently, the bulk solution concentration decreases. Once at the meniscus, molecules move along the meniscus and transfer onto the solid surface. Some molecules, however, are assumed to spread beyond onto the hydrophobic surface. Adsorption always acts here to reduce the surface energy, and through Young’s equation, its effect on the contact angle is described by eq 10. We see that the adsorption of surfactant on the water-solid interface makes a positive contribution to σow (Γow) cos θ (i.e., acts in such a way as to make the contact angle smaller and the system to appear more water-wet) whereas adsorption on the oil-solid interface provides a negative contribution (i.e., acts to make the contact angle larger and the system to appear more oil-wet). Provided the adsorption on the water-solid interface is suitably greater than that on the oil-solid interface, when both effects are combined the result will be a more water-wet system (i.e., a contact angle of less than 90). Thus, in the present model adsorption onto the oil-solid interface hinders imbibition whereas adsorption onto the water-solid interface aids it. Surfactant transport and wettability alteration have been studied under both spontaneous and forced conditions. A pressure difference Δp(t)=Pd was imposed between the ends of the capillary, and simulations were performed at a series of differential pressures, starting from negative values. Parameter values are listed in Table 1. We will refer to this case as the reference case to which the other results will be compared. The capillary threshold pressure (ΔPc0) is calculated to be 60 Pa. Langmuir 2010, 26(9), 6206–6221
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3.1. Effect of Adsorption onto a Solid Surface Behind the Moving Meniscus. Figure 7 shows a spontaneous imbibition case where surfactant adsorption onto the solid surface from the bulk solution is neglected behind, and beyond, the moving interface. Surfactant is transferred only at the meniscus and from the bulk to the water-solid surface in the inner region immediately behind it. While the meniscus is moving from the inlet toward the outlet of the capillary, its position is plotted against time in Figure 7a, and we Table 1. Model Variables for the Reference Case variable
value
L (tube length) a (tube diameter) h (meniscus thickness) μo (oil viscosity) μw (water viscosity) Db (bulk diffusivity) Ds (surface diffusivity) cb (initial surfactant concentration) Kow (adsorption/desorption capacity) Kws (adsorption/desorption capacity) σow(0) (initial surface tension) cos θ0 (initial contact angle) R T μosQ - μwsQ Γwsm Γosm Γosm hkþ
1 cm 0.1 mm 4 10-9 m 1 10-3 Pa s 1 10-3 Pa s 1 10-9 m2 s-1 5 10-14 m2 s-1 1 10-2 mol m-3 1.3 104 m3 mol-1 1.3 104 m3 mol-1 30 10-3 N/m -0.1 8.314 J K-1 mol-1 300 K RT J mol-1 1.96 10-6 mol m-2 1.96 10-6 mol m-2 1.96 10-6 mol m-2 1 10-5 m s-1
see that as time increases, the meniscus moves more slowly. In Figure 7b, the bulk concentration profiles at different times are shown. At the inlet (z = 0), the surfactant concentration is the highest and is equal to the initial surfactant concentration, cb. As the meniscus advances, more solution is fed into the capillary from the inlet, and this maintains the concentration at the inlet. There is no surfactant adsorption directly from the bulk solution onto the solid surface in the main part of the capillary; however, in the inner region close to the meniscus, there is transfer onto the solid surface from the meniscus itself and also from the bulk solution immediately behind it. This causes a depletion of the concentration in the solution adjacent to the meniscus, but because of the nonlinearity of the Langmuir isotherm, the adsorbed concentration on the meniscus does not decrease very much initially. As time progresses, further depletion occurs because diffusion is not fast enough to resupply the bulk concentration completely and the concentration at the oil/water meniscus continues to decrease and eventually becomes almost zero there. The bulk concentration by this time is on the linear part of the Langmuir isotherm, so the adsorbed concentration on the meniscus also falls. Surfactant is deposited onto the solid surface only as the meniscus passes through; nothing further is added later. Figure 7c plots the amount of surfactant deposited by the meniscus as it advances. The dotted line shows the amount adsorbed at each point as the interface of thickness h passes over axial position z (i.e., at each z, we plot Γ(y = 0) evaluated at time t such that X(t) = z). The same data is plotted versus time in Figure 7d. The adsorbed amount is almost constant in time at a value of ∼9 10-7 mol/m2
Figure 7. (a-d, Top to Bottom) Spontaneous imbibition with no additional adsorption of surfactant from the bulk solution onto the solid surface or transfer of surfactant ahead of the moving interface. (a) Meniscus position vs time. (b) Surfactant concentration in bulk solution along the capillary at equispaced times between t = 0 and 8 104 s. (The leftmost line corresponds to the earliest time, and the rightmost, to the latest.) (c) Amount of surfactant adsorbed on the solid surface through inner-region processes. At each z, we plot the inner-region solution values evaluated at the time when the inner region is at that point (i.e., when t is such that X(t) = z). The dotted line shows inner-region Γ(y = 0) values, and the dashed line shows inner region Γ(y f -¥) values. (d) The same inner-region variable values plotted vs t (with a log scale used to expand the behavior at early times). Langmuir 2010, 26(9), 6206–6221
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Figure 8. (a-d, Top to Bottom) Spontaneous imbibition, with adsorption on a solid surface behind the interface and none beyond.
(a) Meniscus position vs time. (b) Surfactant concentration in the bulk solution along the capillary at equispaced times between t = 0 and 105 s. (c) Amount of surfactant adsorbed on the solid surface at different times, Γws(z, t) (solid lines); the dashed line (---) shows region quantity Γ(y f -¥) (far behind the meniscus); the dotted line ( 3 3 3 ) shows inner-region quantity Γ(y = 0) (at the meniscus); crosses () show region quantity Γ(y = hþ) (just beyond the meniscus). (d) The same inner-region variable values plotted versus t (with a log scale used to expand the behavior at early times).
(dotted line). This is the quantity that sets the surface energy in Young’s equation and thus the contact angle. As was explained in ref 12, weak bulk diffusion implies that the contact angle must be close to 90, so in this case there is a single, nearly constant in time, value of the adsorbed surfactant required to achieve this nearly constant contact angle. The dashed line shows the total adsorbed amount at the outer limit of the inner region (i.e., Γ(y f -¥)) again evaluated at time t such that X(t) = z. Because we have temporarily turned off transfers between the bulk of the solid surface in the outer region, the amount of surfactant deposited by inner-region processes as the inner region passes over a particular location gives the amount of adsorbed surfactant to be found at that location at all later times. When the meniscus has not advanced very far into the capillary (e.g. X(t) < 0.0005 m), we see that the adsorbed amount is significantly larger than at later times. This is because the bulk concentration adjacent to the meniscus has not yet fallen because of depletion, so the Langmuir equation between the bulk concentration of the adsorbed amount gives a large adsorbed amount. A simple explanation of the decay of the meniscus velocity, noted above in relation to Figure 7a, is as follows: For spontaneous imbibition into an initially oil-wet capillary to occur, sufficient surfactant must be adsorbed onto the solid surface to change the contact angle to a water-wet value. The necessary surfactant is transported by diffusion relative to the bulk flow of the aqueous phase, and once the concentration within the bulk fluid near the meniscus has dropped to a value much smaller than cb, the flux of surfactant onto the meniscus and hence through onto the solid surface is on the order of πa2cbDb/X(t). As the 6216 DOI: 10.1021/la903924m
meniscus moves forward, the rate of creation of a “new” watersolid interface is 2πaU(t), on which finite surfactant adsorption of m magnitude Γws is required to effect the necessary change in the contact angle. The required flux of surfactant is therefore on the m order of 2πaU(t)Γws . Equating this to the incoming diffusive flux m from the bulk leads to 2U(t)X(t) = d(X2)/dt ≈ acbDb/Γws , from m 1/2 which we deduce X(t) ≈ (acbDbt/Γws) and U(t) ≈ (acbDb/ m 1/2 4Γws t) . As claimed, the velocity decays as time increases. (A detailed calculation showing a similar solution having this form can be found in ref 12.) This order of magnitude calculation is appropriate for late times because when estimating the diffusive flux in the bulk we have assumed that the concentration near the moving meniscus is near zero. At early times, this is not the case. (See the top track of Figure 10). Also, we have tacitly assumed that there is no process acting to limit surfactant transfer rates between the meniscus and solid, thus causing the flux of surfactant leaving the meniscus for the solid surface to be different from that onto the meniscus from the bulk. At early times, this is not so; there is a maximum rate at which surfactant can leave the m meniscus, on the order of 2πahkþΓow according to eq 21. If the adsorbed amount on the meniscus is already near the maximum value, then the diffusive flux of surfactant onto the meniscus is m limited to be less than O(2πahkþΓow ) and as a result the bulk concentration adjacent to the meniscus cannot depart much from its inlet value because if it did then it would drive an excessive diffusive flux of surfactant onto the meniscus. A similar analysis is made when there is surfactant adsorption directly onto the capillary walls in the main part of the tube from Langmuir 2010, 26(9), 6206–6221
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Figure 9. (a-d, Top to Bottom) Spontaneous imbibition, with adsorption both in front of the moving meniscus and on a solid surface behind the meniscus. (a) Meniscus position vs time. (b) Surfactant concentration in the bulk solution along the capillary at equispaced times between t = 0 and 105 s. (c) Amount of surfactant adsorbed on the solid surface at different times Γws(z, t) (solid lines); the dashed line (---) shows Γ (y f -¥) (far behind the meniscus); the dotted line ( 3 3 3 ) shows Γ (y = 0) (at the meniscus); crosses () show Γ(y = hþ) (just beyond the meniscus). (d) The same inner-region variable values plotted vs t (where a log scale is used to expand the behavior at early times).
the bulk solution (Figure 8). Figure 8a shows the position of the meniscus at different times. The meniscus velocity is now smaller than in the previous case (compare Figures 7a and 8a). A similar, nearly linear bulk concentration profile is again seen: At the inlet (z = 0), the surfactant concentration is equal to the initial surfactant concentration, cb (Figure 8b). Toward the oil/water meniscus, the concentration decreases, and at the meniscus, it is much less than cb. Over most of the distance between the inlet and the meniscus, the surface concentration of adsorbed surfactant is close to saturation; see the solid lines in Figure 8c. Adsorbed concentrations are below the maximum possible, and significant transfer of surfactant from the bulk to the capillary wall only takes place, in a small region close to the meniscus. From the Langmuir isotherm (eq 12), the bulk concentration at which the adsorbed amount reaches the arbitrarily large value of 90% of the maximum possible value occurs when Kwsc(z,t) = 9 (i.e., for the current parameter values, c(z, t) = 6.9 10-4 mol/m3). As can be seen in Figure 8b, this occurs close to the moving meniscus and is consistent with the large gradient in Γws(z, t) close to the meniscus seen in the third track. The meniscus moves slowly because some of the incoming flux of surfactant is consumed in surface adsorption before it reaches the meniscus. Hence, only a fraction of the incoming flux is available to adsorb at the moving contact line, alter surface adsorption, and hence influence contact angles. With solid lines, Figure 8c shows the amount of surfactant adsorbed onto the capillary from the meniscus and bulk solution, Γws(z, t). Over much of the capillary, the surface concentration is near saturation, and there is a decrease close to the meniscus in Langmuir 2010, 26(9), 6206–6221
response to the reduced bulk concentration there. The broken lines in Figure 8c show various surface-adsorbed amounts in the inner region. At each ordinate z, we plot the inner region quantities obtained when the meniscus is at that point (i.e., at time t such that z = X(t)). Three different inner-region quantities are plotted: (1) at the meniscus Γ (y = 0), (2) far behind the meniscus Γ (y f -¥), and (3) just ahead of the meniscus Γ (y = hþ): (1)
(2)
At the meniscus: There is adsorption from the meniscus itself only while it is passing through a given location. The amount deposited by the meniscus is a function of the meniscus speed. The more slowly it advances, the more time there will be for surfactant adsorption. The meniscus speed in this case study is changing very slowly (Figure 8a), and we see an almost constant amount of surfactant deposited at the meniscus during the process. In spontaneous imbibition, the amount deposited is that required to create a contact angle of ∼90 for the reasons explained in ref 12. Far behind the meniscus: At long times, once the meniscus has passed we see that there is little additional adsorption in the inner region from the bulk solution. This is because equilibrium with the local bulk concentration has been approximately reached at the meniscus; there is no scope for significant extra adsorption in the inner region. The situation is different at early times. The meniscus is moving comparatively fast, and there is not sufficient time for an amount of DOI: 10.1021/la903924m
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Figure 10. Plot of meniscus properties of the moving interface during spontaneous imbibition.
surfactant in equilibrium with the local bulk to be deposited at the meniscus. Further adsorption occurs, at a finite rate governed by kþþ, until equilibrium with the local bulk concentration is reached. As a result, the inner-region concentration far from the meniscus is larger than that at the meniscus. Because the surfactant concentration in the bulk near the meniscus has not yet fallen to its long-time value, the adsorbed amount is larger than at late times. (3) Beyond the meniscus: There is no adsorption beQ Q - μws yond the meniscus in this case because μos has been set equal to a large, positive value so as to suppress it. The amount of surfactant deposited by the meniscus is plotted versus time in Figure 8d. 3.2. Effect of Adsorption in Front of the Moving Meniscus. Figure 9 shows a case where there is adsorption in front of and behind the moving three-phase contact line. Compared to the previous case, the imbibition process is fractionally slower, as can be seen by comparing the points of maximum advance of the concentration profile in Figures 8b and 9b. For spontaneous imbibition into an initially oil-wet capillary to occur, sufficient surfactant must be adsorbed onto the solid surface to change the contact angle to a water-wet value. If there is adsorption beyond the meniscus as well as behind it, then more surfactant molecules will be required to create the changes in surface energy and contact angle necessary for imbibition to start. The molecules are transported at a finite rate via diffusion. Consequently, it will take longer for the required additional molecules to diffuse, and hence the motion will be slower. 6218 DOI: 10.1021/la903924m
Figure 11. Meniscus velocity at t = 1 s vs differential pressure for different surfactant concentrations. When cb = 0.0001, no solution can be found for Pd < 5 Pa.
The concentration and Γws(z, t) profiles are similar to those in Figure 8. The main difference is that because of adsorption beyond the moving meniscus the surfactant concentration in this zone is not zero (Figure 9c, crossed line). Γws(z, t) is also plotted versus time in Figure 9d, and the behavior at early times can be seen in detail. Beyond the meniscus, Γws(z, t) increases slightly and tends toward a constant value. Figure 10 is a plot of different properties at the moving meniscus. The normalized surfactant concentration at the meniscus decreases gradually from 1 to near zero at late times. This, together with the decay of meniscus velocity, justifies the assertion Langmuir 2010, 26(9), 6206–6221
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Figure 12. Meniscus velocity at t = 1 s vs surfactant concentration under different applied differential pressures. The velocity is timedependent, and we choose to compare velocities at the same time. The reason that 1 s is used is because at high pressure the meniscus moves very fast and reaches the other end of the capillary very quickly.
made in section 2 that the time derivative term in eq 31 could be neglected at late times. The number of surfactant molecules on the meniscus (Γow) is also decreasing with time because more and more molecules are adsorbed on the solid surface (Γws, Γos). This results in an increase in the oil-water surface tension. It is assumed that once imbibition starts some surfactant molecules transfer beyond the meniscus and some are adsorbed behind the meniscus from the bulk solution. There is always a minimum number of surfactant molecules on the solid surface both beyond (Γos) and behind the meniscus (Γws). This number is equal to the number that changes the contact angle from oil-wet to water-wet. The number adsorbed from the meniscus onto the solid surface is proportional to the time that the surface is in contact with the moving meniscus. It is determined by the meniscus thickness, meniscus velocity, and surfactant concentration at the meniscus. The thicker the meniscus, the more adsorption that will take place because it will take longer for the meniscus to pass across a given position on the capillary wall. It is similar to the increasing concentration; more surfactant molecules are likely to be adsorbed if the concentration is high. If the meniscus is moving quickly, then the contact time will be less. As a result, fewer molecules will be adsorbed onto the surface. At late times, the velocity U decreases as the imbibition slows down for the same reason explained in relation to Figure 7c. 3.3. Further Results. The behavior of the meniscus has also been studied under different applied differential pressures, when there is adsorption onto the solid surfaces both behind and in front of the moving meniscus. Surfactant molecules are adsorbed at the capillary walls and alter the wettability of the capillary, lowering the contact angle. Adsorption on the meniscus reduces the surface tension. Consequently, even under negative applied pressure the meniscus is able to imbibe within the capillary, provided the inlet surfactant concentration is large enough (Figure 11). However, the process is extremely slow, and the velocity values are tiny. As the pressure increases, a slow increase in the front velocity is observed until the capillary entry threshold pressure ΔPc0 is reached. At this point, the pattern changes and the velocity increases dramatically. The surfactant concentration does not have a significant effect on the speed under applied pressures greater than the capillary threshold; this is because when there is adsorption from the bulk Langmuir 2010, 26(9), 6206–6221
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onto the capillary walls the surfactant concentration front lags a finite distance behind the meniscus so there is no significant surfactant concentration at the meniscus or diffusive flux of surfactant onto it and hence no effect of surfactant on the contact angle or oil-water interfacial tension. For lower surfactant concentrations (i.e., cb = 0.0001), however, the interface was not able to advance even under positive differential pressure until a surfactant-concentration-dependent threshold pressure was exceeded. Sufficient surfactant must be adsorbed onto the solid surface to change the contact angle to a water-wet value, otherwise imbibition cannot start. When there is adsorption beyond the meniscus as well as behind it, even more surfactant is required, and with very low concentrations, there will never be enough surfactant molecules accumulated on the meniscus to start the imbibition. The meniscus velocity at t = 1 s is plotted in Figure 11. Because the velocity is time-dependent, we need to choose a reference time at which to make comparisons. This time should be large enough that starting transient effects have died away; 1 s is used because at high pressure the meniscus moves very fast and reaches the other end of the capillary very quickly so longer times are not possible. Similar analysis has been made to observe the velocity versus surfactant concentration under different applied differential pressures (Figure 12). Under Pd values greater than the capillary threshold pressure (e.g., 70 Pa), the concentration had no effect on the speed because the surfactant does not reach the meniscus and so cannot effect the surface wettability or interfacial tension. For lower Pd values, we again see a threshold concentration; as the pressure decreases, the surfactant concentration needs to be higher for the meniscus to advance. The physical reason for the appearance of a threshold was explained in the paragraph following eq 34.
4. Conclusions 4
Kumar et al. and Eriksson et al.8 have shown experimentally that, in a hydrophobic capillary, surfactant molecules are not only adsorbed behind the moving meniscus but also are spread over the solid surface beyond the moving three-phase contact line onto the bare hydrophobic surface. Here, a theory of forced and spontaneous flow of surfactant solution into an initially hydrophobic capillary has been presented, taking account of this phenomenon. Computations show that the meniscus advances more slowly when such adsorption is present compared to when it is not. The behavior of the meniscus has also been studied under different applied differential pressures. The aqueous solution was able to imbibe even under negative pressure as a result of the wettability-altering effects of the surfactant; without wettability alteration, no penetration of the aqueous phase into the hydrophobic capillary could occur. However, the surfactant-induced imbibition process is extremely slow, and the velocity values are tiny. As the pressure increases, a slow increase at the front velocity is observed until the entry threshold pressure of the hydrophobic capillary is reached. At this point, the pattern changes and the velocity increases dramatically. The surfactant concentration does not have a significant effect on the velocity under applied pressures greater than the entry threshold because surfactant molecules cannot catch up with the moving meniscus as a result of adsorption on the capillary walls and the relativley small value of Db. In a very porous medium, however, this may not be the case because the surfactant molecules might have additional paths in the vicinity of an advancing meniscus available to them. The model makes certain predictions that should be testable experimentally. For lower surfactant concentrations, it is predicted that the interface is not able to advance under weak positive DOI: 10.1021/la903924m
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differential pressure and there is a threshold pressure for different surfactant concentrations that must be exceeded if imbibition is to occur. Equivalently, for a given applied pressure below the entry threshold capillary pressure of the original oil-water capillary, there is a threshold bulk concentration that must be exceeded before imbibition can occur. Furthermore, it was argued in ref 12 that in spontaneous imbibition the contact angle adjusts to a value very close to 90; the same will be true here, and this could be sought experimentally. It would be of great interest to quantify the number and distribution of adsorbed surfactant on the walls of the capillary, ahead of and behind the meniscus, and to probe the mechanisms of surfactant transfer in the vicinity of the moving contact line. This would shed light on the most speculative aspect of the modeling reported here. Acknowledgment. We thank the referees for their comments, which have helped us to improve this article greatly.
Analogous expressions hold for ν(, with kþΓow replaced with hkþþc(X, t). The continuity of Γ and its derivative at y = 0 require
the constant solution being unacceptable at þ¥. Here, Γ(y = hþ) is a constant, which will be determined in due course. For y < 0, we have ðA2Þ Γ ¼ Γeq ws ð1 þ R expðν yÞ þ β expðν þ yÞÞ
ðA3Þ
and R and β are constants to be determined. The growing vþ solution is unacceptable because y f -¥, so β = 0. In 0 < y < h, we have Γ ¼ Γeq ws ð1 þ A expðλ yÞ þ B expðλ þ yÞÞ where λ(
0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 U @ 4Ds kþ Γow ¼ 1 ( 1 þ eq 2 A 2Ds Γws U
ðA4Þ
R ¼ -
λ(
8 U > > > > < Ds þ
k Γow > > > > : Γeq ws U
as Ds kþ Γow =U 2 Γeq ws f 0
6220 DOI: 10.1021/la903924m
λ- - νR λ- - λþ
ðA9Þ
λ - λþ λ þ Ds λ h λ - Ds λ h ðλ - - ν - Þ 1 þ e þ þ ðν - - λ þ Þ 1 þ e U U
ðA10Þ Then, using eqs A4, A5, A8, A9, A10, and 25 we obtain Q
Q
-ðμos - μws Þ=RT Γðy ¼ hþ Þ ¼ Γeq ws e
0
1 λ - Ds λ - h λ þ Ds λ þ h e þ ðλ - - ν - Þ e ðν - - λ þ Þ B C B C U U C B @ λ þ Ds λ þ h λ - Ds λ - h A e þ ðν - - λ þ Þ 1 þ e ðλ - - ν - Þ 1 þ U U ðA11Þ and
1
0 B B B1@
C λ - λþ C C λ þ Ds λ þ h λ - Ds λ - h A e þðν - - λ þ Þ 1þ e ðλ - - ν - Þ 1þ U U
ðA12Þ eq When DskþΓow/U2Γws is small (because Ds is small), these expressions simplify to þ
Γow =Γeq ws U
ðA13Þ
þ
Γow =Γeq ws U
ðA14Þ
R -e -hk
ðA5Þ and
and A and B are also constants to be determined. We note for future reference that sffiffiffiffiffiffiffiffiffiffiffiffiffi kþ Γow ðA6Þ as Uh=Ds f 0 λ( Ds Γeq ws and
B ¼
Γðy ¼ 0Þ ¼ Γeq ws
0
ν(
ðA8Þ
The remaining constant, R, is determined by setting the total flux equal to zero at y = h according to eqs 26 and 23 and the boundary conditions at y f ¥. We find
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 U @ 4Ds hkþþ cðX, tÞ A ¼ 1( 1þ 2 2Ds Γeq ws U
ν- - λþ R λ- - λþ
and
Appendix We detail here the solution of the differential equation for the quantity of surfactant adsorbed on the solid surfaces. For y > h, outside the source region, the basic solutions are either constant or proportional to exp(-Uy/Ds). For y > h, we therefore have Uðy -hÞ þ ðA1Þ Γ ¼ Γðy ¼ h Þ exp Ds
A ¼
A -e -hk and
! Ds kþ Γow hkþþ cðX, tÞ -hkþ Γow =Γeqws U e B - 2 eq 1 kþ Γow U Γws
ðA15Þ
from which we find ðA7Þ Γðy ¼ hþ Þ Γeq ws
Ds kþ Γow -ðμQos -μQws Þ=RT e U 2 Γeq ws
ðA16Þ
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and þ
-hk Γðy ¼ 0Þ Γeq ws ð1 -e
Γow =Γeq ws U
Þ
ðA17Þ
These recover the results of Hammond and Unsal12 if we set Q Q = μws (corresponding to removing any distinction between μos
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 2 Γeq ws R -2 Ds kþ Γow
adsorption on the oil-solid and water-solid interfaces) and eq m = Γws (corresponding to the neglect of further adsorption Γws on the walls of the capillary upstream from the meniscus). When Uh/Ds is small (corresponding to the surface diffusion length scale, Ds/U, being larger than the thickness of the transfer zone, h), we have
1 ! ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þþ þþ eq hk cðX, tÞ hk cðX, tÞ þ 2 kþ h2 Γow =Ds Γeq ws e e - k h Γow =Ds Γws 1þ 1þ þ k Γow k Γow
ðA18Þ
and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 2 Γeq ws A Ds kþ Γow
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hkþþ cðX, tÞ kþ Γow ! ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p þþ eq hk cðX, tÞ hkþþ cðX, tÞ -pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 kþ h2 Γow =Ds Γeq ws e e k h Γow =Ds Γws - 11þ þ þ k Γow k Γow
ðA19Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 2 Γeq ws B Ds kþ Γow
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hkþþ cðX, tÞ 1kþ Γow ! ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þþ þþ eq eq hk cðX, tÞ hk cðX, tÞ þ 2 þ 2 e k h Γow =Ds Γws - 1 e - k h Γow =Ds Γws 1þ þ þ k Γow k Γow
ðA20Þ
1þ
and
The quantities R, A, and B all go to zero as Uh/Ds f 0, so Q
Q
-ðμos - μws Þ=RT Γðy ¼ hþ Þ Γeq ws e
ðA21Þ
Γðy ¼ 0Þ Γeq ws
ðA22Þ
and
Hence, the adsorbed concentrations are finite in the limit as U f 0. The surfactant concentration on the oil-solid interface just downstream from the advancing meniscus, Γ(y = hþ), is seen from eq A16 to be nonzero but small when the surface diffusivity of the surfactant is small. Furthermore, as U increases, Γ (y = hþ) decreases because less time is then available for surfactant transfer from the oil-water meniscus to the solid surface and so less surfactant is deposited on the solid at higher speeds, leading to a
Langmuir 2010, 26(9), 6206–6221
general decrease in adsorption levels. Therefore, for the study of forced flows for which U is significantly greater than the low value obtained in spontaneous unforced imbibition, it therefore seems reasonable to neglect the presence of surfactant downstream of the advancing meniscus as was done in ref 12. But it is clear from eq 10, which for small values of the interface adsorptions may be linearized to give σow (Γow) cos θ - σow(0) cos θ0 ≈ RT (Γws - Γos), that nonzero values of Γos act to reduce the effect of Γws in generating a positive capillary pressure driving imbibition (i.e., a positive value of σow(Γow) cos θ) and if large enough can preclude this altogether. This must have an effect on spontaneous imbibition, as has indeed been demonstrated in Churaev et al.,10 where a slightly different model predicts a finite bulk surfactant concentration below which no imbibition occurs, in contrast to the Tiberg et al.16 model that predicts spontaneous imbibition for all bulk concentrations.12
DOI: 10.1021/la903924m
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