Diffusion Effects in the Wetting of a Contaminated Surface

Journal of Colloid and Interface Science 229, 168–173 (2000) doi:10.1006/jcis.2000.6996, available online at http://www.idealibrary.com on

Diffusion Effects in the Wetting of a Contaminated Surface Martin E. R. Shanahan Centre des Mat´eriaux P.M. Fourt, Ecole Nationale Sup´erieure des Mines de Paris, Universit´e d’Evry Val d’Essonne, B.P. 87, 91003 Evry C´edex, France Received January 31, 2000; accepted May 30, 2000

Industrial wetting situations often involve the presence of a third component near an interface, potentially modifying the local effective capillary balance. We here consider a simplified situation in which a drop is placed on a substrate supporting a thin layer of “contamination.” As spreading of the drop to equilibrium occurs, the contaminant diffuses into the drop, modifying the effective substrate/drop interfacial free energy (increasing it, it is assumed). Thus the kinetics of spreading are altered. The essential effect when the final equilibrium contact angle is zero is that of accelerating the process. However, when the final value of contact angle is finite, “overshoot” may occur. The drop spreads beyond its equilibrium state and then retracts to attain its equilibrium state asymptotically at long times. °C 2000 Academic Press Key Words: contaminant; diffusion; pollutant; spreading; wetting.

INTRODUCTION

The wetting and spreading of liquids on solid surfaces constitute important phenomena in various biological and industrial situations. Dynamic aspects invoke not only capillary parameters, in particular surface/interfacial free energies, but also the viscosity of the liquid (1). For the case of wetting (or dewetting) of soft substrates, mechanical properties of the solid can also play a significant role (2). In the majority of wetting problems however, treated both theoretically and experimentally, surface/interfacial free energies (tensions) are treated as given constants for the system in question [an exception is in the recently studied field of reactive wetting (3–6)]. Nevertheless, there exist cases where the otherwise “simple” interface between, say, a solid S and a liquid L, and corresponding to a unique value of interfacial tension, γSL , becomes modified with time, possibly by chemical interaction or swelling of the solid, or, as in the case of interest presented here, by the presence of a third component. For example, in the preparation of fibers to be used in the manufacture of composite materials, it is common practice to apply a sizing material (7), presumably to enhance wetting and/or interfacial bonding in the final product (although the exact composition of such materials is often a closely guarded industrial secret!). In the motor industry, a steel sheet is often covered with a thin layer of oil applied to improve corrosion resistance and lubricate to some extent the stamping process. 0021-9797/00 $35.00

C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.

Adhesive bonding to such steel is often effected without any prior decontamination. The adhesive bonds “through” the oily layer by mechanisms incompletely understood at present (e.g., 8–10), although absorption is certainly involved (11). The purpose of this study is to give an assessment of the situation corresponding to processes similar to the examples given above, while (over)simplifying the problem to make it more tractable. Our basic problem is the following: a drop of liquid is placed on a solid substrate which has previously been covered with a (thin) film of a second liquid, which may diffuse into the first. What spreading behavior may be expected? DIFFUSION AND INTERFACIAL FREE ENERGY

The usual simplifying assumptions will be made about the solid substrate, S: it will be taken to be homogeneous, isotropic, flat, smooth, and rigid. Its inclination with respect to the horizontal will be considered immaterial since we shall assume sufficiently small liquid masses that gravitational effects will be negligible. On the surface of solid S is a very thin, homogeneous (constant thickness) layer of a liquid 2. This liquid may be considered to be a contaminant, or pollutant, at least as far as the wetting behavior is concerned. To fix our ideas, this layer may typically be of the order of a few microns thick. Its surface coverage, M, is then typically ca. 10−3 kg · m−2 . Due to its being extremely thin, we shall assume that the layer is effectively fixed with respect to any potential motion parallel to the surface: any possible sliding or lubrication is neglected. This aspect will be reconsidered later, in the Discussion and Conclusions. A drop, or ribbon (see below), of liquid 1 is placed on the solid surface covered by a thin layer of liquid 2. The liquids are assumed to have quite similar surface properties and are miscible. Their interfacial free energy is low. After initial contact, there will be mutual diffusion, but by taking the viscosity of liquid 2 to be considerably less than that of liquid 1, the diffusion process can be approximated by motion of 2 into the bulk of 1. At time t = 0, the drop will “see” a substrate of surface free energy corresponding to that of liquid 2, and therefore interfacial free energy γ12 pertains, but as time passes, diffusion will occur, resulting in the drop establishing some contact with the solid itself: the system evolves. The effective solid (S)/drop (D) interfacial free energy, as a function of time, t, will be denoted

168

169

WETTING OF A CONTAMINATED SURFACE

dependence is then F γSD (t) = γSD −

FIG. 1. Schematic of the cross section of a two-dimensional drop (ribbon) of liquid 1 on a solid substrate covered with a thin layer of contamination, liquid 2. Liquid 2 diffuses into liquid 1 and modifies spreading.

¤ kM £ F I γSD − γSD , 1/2 (π Dt)

[3]

where k = C −1 (δt). During spreading, the triple line will gradually gain on fresh solid, but Eq. [3] should represent a reasonable approximation for γSD (t) averaged over the solid /drop interface. EQUATIONS OF SPREADING

γSD (t). The similar surface properties of 1 and 2 ensure a low value for γ12 , whereas the true interfacial free energy between the (clean) solid and 1 may be significant. Thus we assume that γSD (t) increases with time. (Whether, in a given real system, γSD (t) increases or decreases with time depends, among other things, on entropy considerations and is outside the scope of the present treatment. Reactive wetting, for example (3–6), involves increasing γSD (t).) Some diffusion of liquid 2 onto the exposed surface of liquid 1 near the triple line will also take place (corresponding effectively to a Marangoni effect), but given our (simplifying) assumption of liquids 1 and 2 having similar surface properties, this effect will be considered to be of secondary importance in the present context (although not in general). As shown schematically in Fig. 1, with diffusion occurring perpendicularly to the interface, the phenomenon may be considered as one-dimensional. Taking the (constant) diffusion coefficient to be D, the evolution of the concentration of 2 in 1, C(x, t), corresponds to the situation described by Crank for a finite, plane source diffusing into a semi-infinite medium (12), µ 2¶ −x M , exp C(x, t) = (π Dt)1/2 4Dt

[1]

where x is the perpendicular distance from the interface (approximated to be infinitely thin) into the drop, and M is the initial surface coverage of S by 2. Although the drop is of finite size, the quantity of liquid 1 per unit area of solid can reasonably be taken to be sufficiently great compared to that of 2 for the situation to be adequately described by Eq. [1]. Equation [1] presents a singularity at t = 0, but the problem may be avoided by considering the system after some small “settling” time, δt. We are interested by the concentration of 2 in 1 at the interface, C(0, t), thus: C(t) = C(0, t) =

M ; (π Dt)1/2

t ≥ δt.

Up until now, we have assumed the presence of a drop of liquid 1 without further specification. Although an axisymmetric drop could be modeled, it was thought more judicious to use a two-dimensional geometry, corresponding for example to a ribbon of adhesive being applied to a solid. This leads to (slight) simplification without changing the basic physics of the problem and, as the example suggests, could be of some practical use. From the classic description of wetting (1), the capillary force, Fm , causing spreading (per unit length of triple line) is given by Fm (t) = γS − γSL − γ cos θ (t) = γ (cos θo − cos θ (t)), [4] where γS and γ are respectively solid and liquid surface free energies, and γSL their common interfacial free energy. Angles θo and θ (t) correspond to the equilibrium value and the value at time t. Equation [4] amounts to an unbalanced Young equation where, if θ (t) is equal to θo , clearly the force Fm (t) becomes zero and spreading stops. In the present context, γSL is replaced F − by γSD (t), which is given by Eq. [3]. We define β = k M[γSD I 1/2 γSD ]/(π D) and, combining Eqs. [3] and [4], obtain Fm (t) = γ (cos θo − cos θ (t)) + βt −1/2 .

Note that, in Eq. [5], θo corresponds to the final equilibrium value of contact angle of the drop of liquid 1 (surface tension = γ ) on the denuded solid substrate, after total absorption of liquid 2. With the assumption of infinite dilution of 2 in 1, θo is simply the equilibrium contact angle of liquid 1 on the substrate with no “interference” in the wetting behavior from the thin film. ˙ corresponding to viscous flow With a dissipative term, T S, of the Poiseuille-type (parabolic) and reasonably small contact angle (1), we have energy dissipation during spreading (per unit length of triple line and per unit time) of the form

[2]

The dependence of γSD (t) on C(t) will be simplified and taken I as the initial value of interfacial free to be linear. We define γSD F as the final, or equilibrium, value energy, i.e., γSD (δt), and γSD when all of liquid 2 has diffused away from the interface, i.e., γSD (t → ∞). (Again, infinite dilution is assumed at t = ∞ due to the vastly different relative amounts of 1 and 2.) A plausible

[5]

T S˙ ≈

3η` θ (t)

µ

dr dt

¶2 ,

[6]

where η is the viscosity of liquid 1 (assumed unmodified by diffusion of liquid 2) and ` is a number (approximately constant) representing the logarithm of the ratio of a macroscopic distance (ca. the drop contact radius, or ribbon half-width here, r ) and a

170

MARTIN E. R. SHANAHAN

microscopic cut-off. The dynamic energy balance takes the form Fm

dr ˙ ≈ T S. dt

[7]

We assume the contact angle to be small (cos θ ≈ 1 − θ 2 /2) and using the condition of constant cross-sectional area, A, for the liquid ribbon (A ≈ 2r 2 θ/3), Eqs. [5] to [7] lead to dr 9A3 γ = dt 16η`

·

¸ 1 1 Aβ − 2 4 + , 6 r r ro 2η`r 2 t 1/2

[8]

where ro is the equilibrium value for the half-width of the ribbon of liquid 1, corresponding to contact angle θo . If complete spreading ensues, clearly θo → 0 and ro → ∞ [neglecting any effects of long-range forces (1)] and we have k2 k1 dr = 6 + 2 1/2 ; dt r r t

t ≥ δt,

Case of Zero Final Equilibrium Contact Angle We consider Eq. [9]: it would not seem to possess any analytical solution. Let us nevertheless look at the overall type of behavior. If there is no diffusion effect, or if the second term on the right-hand side of Eq. [9] is small, initially, compared to the first term (k2 relatively small, initial value of r small, and non-negligible value of “settling” time), then the behavior will be dominated by the term in k1 . We readily find a solution, r 7 − rI7 ≈ 7k1 t = k˜ 1 t,

[15]

where rI is the initial value of r . Thus we have a scaling law of the form r ∼ t 1/7 . However, assuming that diffusion occurs, k2 6= 0, the spreading rate will be somewhat higher. We shall allow for this in Eq. [15] by modifying the power law, to give an approximate expression, 7−ζ

r 7−ζ − rI

[9]

≈ k˜ 1 t,

[16]

where ζ is a small, positive constant. Substitution of expression [16] into Eq. [9] leads to

where k1 =

9A3 γ 16η`

[10]

and £ F ¤ I Ak M γSD − γSD . k2 = 2η`(π D)1/2

[11]

For the (probably more common) case of θo and ro finite, we define y = r/ro and obtain dy = k3 dt

µ

1 1 − 2 y6 y

¶

k4 + 2 1/2 ; y t

t ≥ δt,

[12]

where k3 =

γ θo3 6η`ro

[13]

and £ F ¤ I − γSD θo k M γSD . k4 = 3η`ro (π D)1/2

[14]

In the next section we shall consider trends and approximate solutions corresponding to Eqs. [9] and [12]. WETTING BEHAVIOR

We shall first briefly consider the case of θo = 0, for completeness, although the behavior of a system with finite θo is intrinsically more interesting. The latter will be covered in more detail.

1 dr ≈ 11/2 dt r

(

) 1/2 k1 k2 k˜ 1 r ζ /2 +£ ¤1/2 , r 1/2 1 − (rI /r )(7−ζ )

[17]

and thus it can be seen that, after a transition period, the term in k2 will become dominant. From Eq. [9] we now obtain a scaling relation of the type r ∼ t 1/6 .

[18]

Thus, even if spreading commences at a rate of the form r ∼ t 1/7 , provided k2 6= 0, this will evolve quite rapidly to r ∼ t 1/6 . If k2 is sufficiently large and the “settling” time, δt, small, the second term on the right-hand side of Eq. [9] is dominant immediately and behavior follows relation [18] straight away. Thus the conclusion here is that the presence of a diffusion term leads to a power law dependence of r which follows t 1/6 rather than t 1/7 . The overall behavior is shown schematically for both k2 positive finite and k2 = 0 in Fig. 2. Note that this analysis will become invalid at very long times (r → ∞, θ → 0) since the drop thickness will become comparable to that of liquid 2 and Eq. [2] will no longer be valid. Also long-range forces may have important effects. Case of Finite Final Equilibrium Contact Angle In the case of finite equilibrium contact angle, θo , we shall consider the behavior in three regimes; as before, with Eq. [9], Eq. [12] apparently has no analytical solution. Since y −2 ¿ y −6 initially, Eq. [12] behaves as Eq. [9] in the first regime. Neglecting any potential short initial period of spreading in t 1/7 (see previous section), we may consider the first regime to obey a scaling law of the form r ∼ t 1/6 (cf. relation [18]). However,

171

WETTING OF A CONTAMINATED SURFACE

We pose y = (1 − ε), whence Eq. [12] may be expressed as ε −k4 dε + ≈ 1/2 . dt τ t

FIG. 2. Schematic representation of spreading behaviour: logarithm of r , ribbon half-width, vs logarithm of t, time, for zero equilibrium contact angle. Solid line is that obtained with diffusion of liquid 2 into drop of liquid 1; broken line is that in the absence of diffusion.

this regime is restricted and valid only for k4 y 4 /(k3 t 1/2 ) À 1, implying a large value of β (high surface coverage, M, and /or low value of the coefficient of diffusion, D). This first regime corresponds to a time range, say, of δt ≤ t ≤ t1 . For the second regime, corresponding to the time span t1 ≤ t ≤ t2 , the problem is somewhat complex, but qualitatively, we see that the presence of a term in k4 , adding to that in k3 , leads to an increased spreading rate. The triple line thus arrives in the vicinity of y = 1 somewhat quicker when the diffusion term is present. A reasonable solution of Eq. [12] in this regime is probably best obtained by numerical methods. The third regime corresponds to t ≥ t2 when y is not very different from unity. Since the acceptability of approximate solutions is always subjective, no absolute figure for the start of this regime may be put on the value of y, but something like 0.8 to 0.9 might be considered reasonable. In the absence of the diffusion term, this regime corresponds simply to an exponential decay from y < 1 toward y = 1 with time constant τ = (4k3 )−1 . The behavior is of considerably more interest when k4 is finite and positive.

[19]

At the start of this third regime, with y approaching unity from below, we expect a negative value of dε/dt, corresponding to a positive value of dy/dt. When ε = 0 (for the first time), at t = t3 , −1/2 dy/dt = k4 t3 . The liquid ribbon is still spreading despite the fact that the asymptotic spreading limit for the case of k4 = 0, i.e., y = 1, has already been reached! (See Fig. 3.) Nevertheless, the spreading rate is decreasing, and at t = t4 , corresponding to dε/dt = dy/dt = 0, we obtain maximum −1/2 spreading extent at y = 1 + k4 τ t4 . The ribbon has “overshot” its equilibrium position of y = 1, to an extent determined by the ratio of the coefficients in Eq. [12], viz., k4 /k3 , and also the value of t4 , which depends on the previous history (including the initial half-width of the liquid ribbon at t = δt). Henceforth, the ribbon will start to recede and y will pass through an inflection point at t = t5 , corresponding to y = 1 + −1/2 (1 + 0.5τ t5−1 ). At large values of time, t, the normalik4 τ t5 zed half-width, y, will tend to a value of unity with an asymptotic decay. Both ε and t −1/2 tend to zero at t → ∞, confirming a zero value of dε/dt in Eq. [19]: this corresponds to total diffusion of liquid 2 into liquid 1 and, with the infinite dilution hypothF , we obtain y = 1 and esis and an interfacial free energy of γSD θ = θo . In fact, we may obtain a solution for Eq. [19], which itself may be expressed differently as Z ε ≈ −k4 exp(−t/τ ) t −1/2 exp(t/τ ) dt. [20] The integral in Eq. [20] can be evaluated to give a series solution: Z

r −1/2 exp(t/τ ) dt = 2t 1/2 exp(t/τ )

µ ¶ ∞ X (−1)n 22n n! t n n=0

(2n + 1)!

τ

.

[21] The solution of Eq. [19] is thus found, and therefore the (approximate) solution of Eq. [12] in the regime of most interest, is finally given by y = 1 − K exp(−t/τ ) + 2k4 t 1/2

µ ¶ ∞ X (−1)n 22n n! t n n=0

(2n + 1)!

τ

, [22]

where K is a positive constant depending on k3 , k4 , and the initial conditions of the drop deposition (initial value of y). The overall behavior for the case of finite θo is shown schematically in Fig. 3. DISCUSSION AND CONCLUSIONS FIG. 3. Schematic representation of (normalized) ribbon half-width, y, vs time, t, for finite equilibrium contact angle. Solid line represents behavior with diffusion and broken line that without.

There are many examples of wetting phenomena in which a third (or possibly even fourth or more) component may intervene

172

MARTIN E. R. SHANAHAN

and modify surface or interfacial free energies and thus modify kinetic behavior. Two industrial cases which came to mind were the sizing of fibers destined for inclusion in composite materials and the application of oil to a steel sheet, later to be adhesively bonded without prior removal of the oil. The mechanisms involved when a third component is present may be complex, but we have modeled the behaviour for a very simple case. A thin layer of a second liquid present on the substrate may diffuse within a sessile drop (or ribbon) of liquid deposited on the solid surface and modify the effective interfacial free energy between the substrate and the drop, in a time-dependent manner, leading to modification of the wetting behavior. We have modeled diffusion as Fickian and developed two equations, viz. [9] and [12], describing the spreading rate (in two dimensions) for, respectively, the case where the final equilibrium contact angle is zero, and the situation when the final angle is finite, θo . For the former, we find that spreading behavior is speeded up by the presence of a supplementary diffusion term adding to the already existing capillary imbalance. Although the intricate behavior may be quite complex, given the form of the differential equation governing wetting, the essential conclusion is that the more rapid spreading with diffusion follows a scaling law of t 1/6 , rather than t 1/7 as in the classic case with constant interfacial free energy. It should nevertheless be pointed out that this spreading tendency cannot continue ad infinitum, since for very large values of ribbon half-width, r , drop thickness will become comparable to the thickness of the initial layer of liquid 2, invalidating our basic assumptions concerning diffusion and (possibly) invoking modification due to long-range forces [“pancake” formation (1)]. However, when θo is finite, the behavior becomes intrinsically more interesting. A short first regime exists in which we expect spreading as t 1/6 . The second phase is complicated mathematically, but we see how, again, the spreading rate is increased compared to the classic scenario. More curious is the third regime, when the ribbon of liquid is approaching its equilibrium contact width. We have “overshoot” of the equilibrium value with a return leading to asymptotic arrival at the equilibrium value. This behavior is somewhat akin to drop shrinking, previously discussed in the context of reactive wetting (6). (Although not treated here, the similarity of the situation described with that of reactive wetting, when γSD increases with time, may lead to spontaneous motion of the drop as a whole (5).) Such phenomena of change of direction of wetting are not unknown in the context of liquids spreading on liquids. For example, if benzene is added to a water surface, initial spreading is rapid, but following mutual saturation, the benzene retracts to a lens (13). The water surface tension is reduced and corresponds to a Gibbs monolayer for a saturated solution of benzene in water. Although the situation is different in the present case, a certain analogy is clear. A technological note may be added at this point. Adhesive ribbons, when applied to substrates, often go through a heating cycle. Initially, temperature increase helps spreading by reduc-

ing adhesive viscosity, yet at a later stage, the polymer gels at the onset of cross-linking. The above description of “overshoot” may possibly be exploited in the case of adhesive application to contaminated surfaces. If a judicious choice of heating rate(s) is made, the adhesive may possibly be made to gel at maximum coverage. Our treatment is simplified. In the case of zero equilibrium contact angle, we neglect details of the fact that the spreading coefficient must eventually tend toward zero and that a “pancake” should finally form due to long-range forces linked to co-operation between the substrate/liquid and liquid/air interfaces (1). Likewise, such effects may be present initially “under” the drop since the substrate/liquid 2 and liquid 2/ liquid 1 interfaces are assumed close to each other. No allowance has been made for any possible lubrication effects. Let us briefly consider potential shear in liquid 2. The spreading force, Fm , (Eq. [4] et seq.) is transmitted to the liquid 1/ liquid 2 interface, leading to an average stress Fm /r . Assuming this leads to shear of the thin film of liquid 2, we may take the overall drop spreading rate, dr/dt, to be equal to the sum (v + v˜ ) where v is drop spreading rate with respect to the upper surface of liquid 2, and v˜ is the (average) surface displacement rate of liquid 2 due to shear. With no slip at the interface liquid 1/ liquid 2 and simple Poiseuille flow in the film, we have, in order of magnitude ηr ˜ v˜ 3η`v ∼ , θ h

[23]

where η˜ is the viscosity of liquid 2 and h the (average) thickness of the film. Using constant cross-sectional area (A ≈ 2r 2 θ/3), we obtain 2η`hr v˜ ∼ . v η˜ A

[24]

Provided v˜ /v ¿ 1, our initial assumption of lubrication effects being negligible is valid. As a not unreasonable example, let us take η ≈ 3η, ˜ ` ≈ 10, h ≈ 2 × 10−6 m, r ≈ 3 × 10−3 m, and −6 A ≈ 4 × 10 m2 , from which we obtain v˜ /v ≈ 0.09, and so our assumption is acceptable in this case. Note, however, that the acceptability deteriorates with increasing r , although decreasing h due to diffusion will counteract this to some extent. A simple model of Fickian diffusion is used. With real systems, the diffusion coefficient may well not be constant and may depend on the local concentration of the diffusing substance. This would complicate the mathematics. Viscosity of liquid 1 could be altered following diffusion (and, of course, by any temperature change, as in the example given above), as could surface tension. Of course, if the effective interfacial free energy, γSD , were to decrease following diffusion (see Diffusion and Interfacial Free Energy section), the scenario would be different. Notwithstanding these objections and the simplifying assumptions used in this analysis, it is felt that the above

WETTING OF A CONTAMINATED SURFACE

development gives some insight into possible wetting phenomena occurring in the presence of a third, or “contaminating,” component. REFERENCES 1. de Gennes, P. G., Rev. Mod. Phys. 57, 827 (1985). 2. Carr´e, A., and Shanahan, M. E. R., J. Colloid Interface Sci. 191, 141 (1997). 3. Bain, C. D., Burnett-Hall, G. D., and Montgomerie, R. R., Nature 372, 414 (1994). 4. Domingues dos Santos, F., and Ondar¸cuhu, T., Phys. Rev. Lett. 75, 2972 (1995). 5. Brochard-Wyart, F., and de Gennes, P. G., C.R. Acad. Sci. Paris 321(IIb), 285 (1995).

173

6. Shanahan, M. E. R., and de Gennes, P. G., C.R. Acad. Sci. Paris 324(IIb), 261 (1997). 7. Khazanov, V. E., Kolesov, Yu. I., and Trofimov, N. N., in “Fibre Science and Technology” (V. I. Kostikov, Ed.), p. 147, Chapman and Hall, London, 1995. 8. Debski, M., Shanahan, M. E. R., and Schultz, J., Int. J. Adhesion Adhesives 6, 145 (1986). 9. Commer¸con, P., and Wightman, J. P., J. Adhesion 22, 13 (1987). 10. Ogawa, T., and Hongo, M., J. Adhesion Sci. Technol. 11, 1197 (1997). 11. Greiveldinger, M., Ph.D. thesis, Ecole Nationale Sup´erieure des Mines de Paris, France (to appear). 12. Crank, J., “The Mathematics of Diffusion,” p. 11, Clarendon Press, Oxford, 1956. 13. Adamson, A. W., “Physical Chemistry of Surfaces, 4th Ed.,” p. 106, Wiley, New York, 1982.