Article pubs.acs.org/Langmuir
Dynamic Wetting on a Thin Film of Soluble Polymer: Effects of Nonlinearities in the Sorption Isotherm Julien Dupas,† Emilie Verneuil,*,† Marco Ramaioli,‡ Laurent Forny,‡ Laurence Talini,† and Francois Lequeux† †
PPMD-SIMM, UMR 7615, CNRS, UPMC, ESPCI ParisTech, 10 rue Vauquelin, 75231 Paris, France Nestle Research Center, route du Jorat 57, 1000 Lausanne 26, Switzerland
‡
ABSTRACT: The wetting dynamics of a solvent on a soluble substrate interestingly results from the rates of the solvent transfers into the substrate. When a supported film of a hydrosoluble polymer with thickness e is wet by a spreading droplet of water with instantaneous velocity U, the contact angle is measured to be inversely proportionate to the product of thickness and velocity, eU, over two decades. As for many hydrosoluble polymers, the polymer we used (a polysaccharide) has a strongly nonlinear sorption isotherm ϕ(aw), where ϕ is the volume fraction of water in the polymer and aw is the activity of water. For the first time, this nonlinearity is accounted for in the dynamics of water uptake by the substrate. Indeed, by measuring the water content in the polymer around the droplet ϕ at distances as small as 5 μm, we find that the hydration profile exhibits (i) a strongly distorted shape that results directly from the nonlinearities of the sorption isotherm and (ii) a cutoff length ξ below which the water content in the substrate varies very slowly. The nonlinearities in the sorption isotherm and the hydration at small distances from the line were not accounted for by Tay et al., Sof t Matter 2011, 7, 6953. Here, we develop a comprehensive description of the hydration of the substrate ahead of the contact line that encompasses the two water transfers at stake: (i) the evaporation−condensation process by which water transfers into the substrate through the atmosphere by the condensation of the vapor phase, which is fed by the evaporation from the droplet itself, and (ii) the diffusion of liquid water along the polymer film. We find that the eU rescaling of the contact angle arises from the evaporation−condensation process at small distances. We demonstrate why it is not modified by the second process.
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INTRODUCTION
θ 3 ≈ θe 3 +
The wetting of a soluble material by its solvent is a common situation when water is placed in contact with food powders or with a tablet of detergent. Although ubiquitous, it is nevertheless a poorly understood situation owing to the numerous coupled phenomena involved that result from the diverse mutual transfers of solvent and soluble material. By studying the motion of a water droplet onto supported films of watersoluble polymers, our group has recently identified different spreading regimes2 where parameters such as the film thickness, humidity or contact line speed were shown to tune the wetting dynamics. Let us briefly recall the classical hydrodynamic theory for moving contact lines of velocity U and dynamic contact angle θ on solid substrates3 under partial wetting conditions. The contact line dynamics is set by a balance between the energy dissipated by viscous shear and the energy of the driving capillary force. The former scales as ((3ηl)/θ)U2 where η is the liquid viscosity and l = ln(R/a), with a being the molecular size and R being the size of the drop.4 The driving force is F = γS − γSL − γ cos θ, where the surface tension γi is related to the substrate i = S, substrate/liquid i = SL, and liquid interfaces, respectively. The driving energy scales as γθ2U for small angles. Hence, for a clean wetting solid substrate © 2013 American Chemical Society
η Ul γ
(1)
where θe is the contact angle in the static case for which γS − γSL − γ cos θe = 0. A fictitious increase in the substrate energy ahead of the contact line γS enhances the driving force F. In eq 1, this is accounted for by the term in θe, which depends on the substrate state on the microscopic level. An increase in the substrate wettability leads to a decrease in θe and thus a decrease in θ for a given velocity. Returning to the wetting on soluble materials, our previous work5 shows that a hydrosoluble material, although eventually dissolved in water, may not be wetted by water if it is dry. The increased hydrophobicity when drier was attributed to the preferred orientation of the carbon backbone of the polymer at the air/film interface. However, the hydration of the substrate causes an increase in its wettability.6 Moreover, in another study,1 we show that the hydration of the substrate arises from the solvent transfers from the droplet itself, and the wettability (i.e., the droplet contact angle) is found to depend on the spreading dynamics of the droplet. We listed the different ways Received: June 18, 2013 Revised: September 4, 2013 Published: September 5, 2013 12572
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contact line. For the first time, the model encompasses both water transfers: the transfer of liquid water by diffusion through the polymer along the horizontal direction in addition to the hydration by the evaporation and condensation process accounted for in the previous studies.1,2 Eventually, the measured dependence of θ with eU−1 is proven to be due to the nonlinearities in the sorption isotherm of the polymer and to the hydration of the substrate at small distances (smaller than 100 nm).
by which the solvent contained in the droplet can hydrate the substrate during the course of a spreading experiment. Similar analyses have been performed for water droplet spreading on hydrogels.7 Of course, liquid water diffuses from the droplet edge into the substrate, but remarkably, at long distances from the contact line, the solvent evaporation and its absorption by the film is the most efficient transfer. Note that wetting by volatile liquids has been extensively studied in the recent years in the case of receding contact lines:8 suspensions drying on solids, which leads to the so-called coffee-ring effect,9,10 or receding alkane droplets.11 Here, the novelty lies in that we work on the advancing of contact lines of volatile liquids that dissolve their substrate. A theoretical analysis of the sole evaporation/condensation process2 allowed for the derivation of a model describing the extent of solvent uptake of a supported film of thickness e in front of a contact line advancing at a velocity U. It was limited to the simple case of a polymer having a linear dependence of its water content ϕ with the water activity in the vapor aw. The calculated hydration of the substrate was proven to be controlled by the diffusion of water in air: the relative swelling depends on the distance that water molecules will be able to travel at any given time and therefore on how fast the contact line is advancing. Once water molecules condense on the substrate, the hydration of the film across its thickness e is limited by the diffusion of water through the polymer (diffusion coefficient D). Hence, the velocity U and thickness e were shown to be the two key parameters governing the hydration by condensation. Furthermore, we predicted that hydration by condensation leads to the homogeneous water content of the polymer film across its thickness e if e is small enough. At a given distance x from the contact line, this is obtained if the time τ required for the water to diffuse vertically into the polymer over a distance e (τ ≈ e2/D) is shorter than the time x/U for the droplet to move by x. Altogether, the thin-film approximation holds at distances from the contact line larger than xc = ((e2U)/D): note that xc depends on e and U. In a previous study,1 measurements of the advancing contact angle θ of water droplets on soluble polymer films and of the water content at large distances from the droplet were compared to the theoretical prediction.2 Interestingly, θ was found to depend on e and U through the product eU on a series of suspended films of varied thickness. However, the water content measurements at distances ranging from 100 μm to 10 mm fail to support the interpretation for the rescaling of θ with eU, as opposed to what is claimed by Tay et al., for two reasons. First, some of these measurements happen to be outside the thin regime, and the measure of the water content, which is averaged along the thickness of the film, is not relevant in this case. Second, when measurements were made in the thin-film regime, the water content was indeed observed to depend on the contact line velocity U, but we will prove in the following text that the evaporation/condensation process at large distances (100 μm to 10 mm) cannot account for the rescaling of θ with eU. In this article, the measurements are carefully restricted to thin films for which the volume fraction of water does not depend on the vertical coordinate. We present the results of the contact angle measurements, and we find that θ is inversely proportionate to the product eU in a very reliable way: over 2 orders of magnitude in e and in U. This result backs up our previous study. From this, we experimentally observe what happens very close to the contact line at distances ranging from 5 μm to 1 mm. The analysis of the results allows for the derivation of a comprehensive model of the hydration of the substrate ahead of the
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EXPERIMENTAL SECTION
Materials. The polymer used is a maltodextrin, a polysaccharide consisting of D-glucose units, provided by Roquette, France, with molecular mass Mw = 2500 g·mol−1, dextrose equivalent DE = 29, and polydispersity index 4.9 (supplier data). To avoid any degradation, the maltodextrin is carefully stored under vacuum with desiccant. The same batch was used throughout the experiments presented in this study. Maltodextrin is very soluble in water, and the water content of the polymer equilibrates with the ambient humidity. The activity of water in the vapor phase aw is defined as the ratio between the partial pressure of water in air and the saturated vapor pressure. It equals the relative humidity in air, that is, the ratio of the concentration of water in air to the concentration at saturation csat. For maltodextrin, the equilibrium volume fraction of water ϕ is a nonlinear function of the water activity in the polymer, that is, equal to water activity in air aw at equilibrium. Thermogravimetric analysis measurements were conducted to quantify those variations. Figure 1 shows the experimental data that are well described using a Flory equation:12
Figure 1. (Left axis, ■ and ---) Sorption isotherm of the maltodextrin polymer ϕ(aw): data collected by thermogravimetric analysis are fit to a Flory model (eq 2) yielding χ = 0.6. Error bars are too small to appear. (Right axis, −) Exponent k calculated from eq 10 for U = 3 × 10−4 m·s−1, θ = 14, and e = 300 nm. 2
a w = ϕe1 − ϕ + χ * (1 − ϕ)
(2)
We found that a Flory parameter χ = 0.6 nicely describes the experiments in the [0.05, 0.7] range of ϕ, in agreement with results found in the literature.13,14 The mutual diffusion coefficient of water in maltodextrin D is a function of the water content in the polymer. Herein, all of the presented experiments have been performed at a humidity of 0.75, which is above the glass transition that the polymer undergoes in water content at room temperature. At the large water activities we have here, we found experimentally that the mutual diffusion coefficient varies only slightly and can be set to the value of the polymer self-diffusion coefficient: D = 10−10 m2·s−1. Finally, we have verified that maltodextrin is not tensioactive by measuring the surface tension of aqueous solutions of maltodextrin at mass fractions in polymer ranging from 1 to 20 wt % using the pendant drop method. Within experimental error, we find the same surface 12573
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tension as with pure water. This shows that the surface energy γ of the liquid is not affected by a change in polymer content at low concentrations of polymer. This is different from the increase in surface energy γS expected when solid maltodextrin is hydrated with water, where much lower water contents are discussed. For the preparation of the films, first, aqueous solutions of polymer are prepared at concentrations ranging between 1 and 40 wt %. The polymer films are then obtained by spin-coating or dip-coating the solutions onto solid substrates. As solid substrates, we use bare silicon wafers (ACM France), 2 in. in diameter, that were modified in some specific cases to tune their surface properties. In the general case, the wafers are first oxidized using a H2O plasma treatment and immediately coated with the polymer solution. Films with thicknesses ranging between 100 nm and 1 μm were obtained by spin-coating at a speed of 4000 rpm and by adjusting the solution concentration. Thicker films were obtained by dip-coating the plasma-treated wafers into a solution at 25 wt % (for e = 3 μm) or 40 wt % (for e = 8 μm). The samples are then dried on a hot plate at 70 ± 10. The length of this drying step is adjusted so as to avoid the onset of cracks in the substrate: overnight for e = 3 μm and 2 h for e ≈ 8 μm. The thickness is measured by ellipsometry or interferometric profilometry. The films are homogeneous across the central area of the wafer, greater than ≈4 cm, and their roughness is measured by AFM to be subnanometric. The reflection of light at the silicon surface allows for the precise measurement at all times of the dynamic contact angle and of the film thickness, as detailed below. Procedures. The polymer-coated substrates are placed in a tailormade transparent chamber (Figure 2a). The chamber is tightly closed at
Figure 3. (a) Time-variation of the contact angle θ (left axis) and of the velocity U (right axis) during the spreading of a droplet. e = 300 nm. The droplet slows down as it spreads whereas its contact angle decreases to a few degrees. (b) Contact angle versus velocity of the contact line for the same experiment. Determination of the Film Thickness and Water Content. Figure 2b shows a typical color image of a water droplet spreading on a maltodextrin-supported film. Different hues can be seen in the substrate that are due to changes in thickness. Two alternative interpretations of the thickness increase were initially considered: (i) the film swells with water or (ii) water penetrates between the film and the silicon substrate. In the latter case, the polymer would be peeled off of the substrate, and the moving contact line would actually connect the peeled polymer film, the silicon wafer, and water. Therefore, the dynamic contact angle should be affected by a change in the surface energy of the underlying substrate. To check this hypothesis, we use a modified silicon wafer obtained by spin-coating a 250-nm-thick layer of polystyrene (PS) in toluene. The hydrophobicity of the PS layer is further reduced by exposure to H2O plasma for different times. On these plasma-treated PS layers, maltodextrin solutions are spin-coated as usual. Performing wetting experiments on those films, we clearly find that the different substrates do not have any effect on the contact angle. We conclude here that the polymer film is not peeled off of the substrate and the change in thickness can be attributed to the swelling of the polymer by water. The variations in hue result from the interference between the light reflected by the film and the light reflected by the silicon wafer. By collecting the intensity of the red, green, and blue sensors of the color camera and comparing the theoretical and experimental hues, the film thickness e can be measured. The analysis method was described in an earlier work.1 It applies to films of thickness larger than 50 nm, below which no more interference is obtained in visible light, and, in practice, smaller than 600 nm, with this limit being set by the loss of contrast. We define x as the distance to the contact line so that x = 0 corresponds to the edge of the droplet at all times. From the film thickness map e(x), we compute water content profiles ϕ(x) ahead the contact line that are used to discuss the hydration mechanisms. More precisely, our measurement assumes that the volumes of water and polymer in the swollen polymer film can be added. This hypothesis was validated experimentally by comparing the measurements by thermogravimetry of the volume fraction at a prescribed humidity (Figure 1) and of the thickness of swollen films measured by our technique. The method provides the average value of the water volume fraction across the film thickness. A typical curve is plotted in the Figure 4 inset. As stated in the following discussion, the thickness and velocity ranges are such that the thin film condition holds, and the water content is homogeneous in the vertical direction.
Figure 2. (a) Experimental setup. (b) Top view of a spreading drop onto a 300-nm-thick film of maltodextrin. (c) Side view showing the determination of the dynamic contact angle θ. e = 550 nm. Scale bars represent 1 mm. the top by a removable lid with vaseline. The lower level of the chamber is a drawer that contains a saturated solution of NaCl that sets the humidity at 75% inside the box. A sensor monitors both the humidity and temperature at all times. Prior to any experiment, the coated wafer is allowed to sit in the chamber until complete equilibration of the substrate with the atmosphere. The experiment consists of depositing a 3 μL droplet on the coated substrate with a micropipet and monitoring its spreading from the side and from the top with two cameras. Side illumination is provided by a white LED screen (Phlox, France). The two cameras are synchronized, and acquisition is made at a varied frame rate ranging between 1 and 30 Hz depending on the spreading speed. The lateral views are acquired with a black and white camera (Sony XCD-SX90). They show clear views of the droplet and its mirror image in the wafer, making the determination of the dynamic contact angle θ of the droplet accurate (the error in the measurement of θ was estimated to 0.5° at worst), as shown in Figure 3a. The top view (Figure 2b) is acquired with a color camera (Sony XCD-SX90CR) placed at an angle of α =15° with respect to the vertical direction. The images show that the droplet shape remains circular during the whole spreading, with a radius R used to compute the contact line velocity U = Ṙ (Figure 3a). The spontaneous spreading of the sessile droplet yields velocities ranging from 6 × 10−6 to 6 × 10−4 m/s. The top images also exhibit the Newton hues that enable the thickness calculation, as detailed in the next section.
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RESULTS In the first experiment, we measured the contact angle θ and the contact line velocity U during the course of the spreading of a droplet on a polymer film of thickness e = 300 nm. The result is 12574
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distances smaller than ξ, the activity is very close to saturation (aw ≈ 98%).
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DISCUSSION First, we check that our experimental conditions lead to a uniform water concentration across the film in the vertical y direction. As detailed in the Introduction, in the case of hydration by condensation only, this is true for distances from the contact line x larger than xc ≈ Ue2/D. For U = 10−4 m·s−1, D ≈ 10−10m2· s−1, and e = 300 nm, this condition reduces to x > 100 nm. Therefore, in the range of distances x probed experimentally, hydration is homogeneous across the film thickness and the problem reduces to a 1D problem along coordinate x. We now turn to a comprehensive analysis of the water transfers at play, as schematically depicted in Figure 5. In the framework of
Figure 4. Water activity difference to saturation 1 − aw vs distance to the contact line x at different contact line velocities U during the spreading of a water droplet onto a polymer film of thickness e = 300 nm. Calculated from eq 2 with χ = 0.6. Error bars were omitted for clarity. (Inset) Water volume fraction ϕ vs x at U = 3 × 10−4 m/s. At distances smaller than ξ, ϕ varies very little. Solid line: Fit to eq 9 with k = 0.5. Dashed line: k = 0.1.
plotted in Figure 3b. Measurements of the contact angle are recorded when the droplet has relaxed to a spherical cap, within 100 ms after deposition. The droplet slows down as it spreads, and the contact angle decreases to 3°. Referring to eq 1, we can verify that this decrease in contact angle is not due to the viscous dissipation term on the right-hand side of eq 1. Indeed, for a constant viscosity of η = 0.87 mPa·s, the decrease would be Δθ ≈ ηΔUl/γθ3 ≈ 0.01° for ΔU = 4 × 10−4 m·s−1. By far, the equation does not account for the measured decrease in θ with U. Besides, by accounting for the increase in viscosity induced by the dissolution of the polymer into the droplet over time, we can estimate the resulting increase in the dynamic contact angle for a given contact line speed.3,15 The dependence of θ with η scales as η1/3 (eq 1), that is, a power law with a small exponent. With θe = 10° and U = 10−4 m·s−1, an increase of 10° in θ would require an increase in polymer content of around 70% in the droplet. This is much higher than the available amount of polymer in the thin films here. Therefore, the viscosity effects will be neglected in our discussion. Eventually, the change in wettability is likely to be due to the water transfers in the film that tune the term θe in eq 1. These transfers depend on the relative values of the velocity at which the contact line moves and the flux of water hydrating the substrate. For smaller velocities, the water flux hydrates the substrate for a longer time, so the hydration is greater, yielding a smaller contact angle. We also measured the water volume fraction ϕ as a function of the distance to the contact line x as the droplet spreads. An example is shown in the Figure 4 inset for an instantaneous velocity of U = 3 × 10−4 m·s−1. At millimetric distances from the drop, the volume fraction is set by the humidity prescribed in ∞ the box (a∞ w = 0.75, ϕ = 0.26). The water fraction increases as the distance x decreases down to a distance denoted as ξ where the water fraction jumps to a value of ϕξ. The ϕξ values barely depend on the velocity U whereas ξ decreases with U. Closer to the contact line, the water fraction varies only slightly. From the water fraction measurements, we extract the activity of water within the polymer aw using the Flory equation (eq 2). For contact line velocities spanning 0.1 to 0.4 mm/s, the results are reported in Figure 4 where we plotted 1 − aw as a function of the distance x to the contact line. Close to the contact line, at
Figure 5. Schematic drawing of the water transfers into a thin polymer film during the spreading of a droplet of solvent with instantaneous velocity U. The horizontal (respectively vertical) distance to the contact line of the droplet is denoted as x (respectively y). The evaporation flux Jvap is defined by eq 18, the condensing flux jcond is defined by eq 5, and the convective transfer jU is defined by eq 7. Along the x axis, L is the macroscopic distance over which the water content of the solid is no longer changed by the droplet, and ξ is the experimental coordinate of a jump in the water content of the polymer. (Inset) Close-up on the contact line evidencing the diffusion process from the droplet into the film that extends over a distance κ.
the droplet, the general balance equation for the mass of water in the polymer film is Jvap ∂ϕ ∂ϕ ∂ ⎛ ∂ϕ ⎞ = + ρ ⎜D ⎟ + ρ U ρ (3) ∂t ∂x ⎝ ∂x ⎠ ∂x e where Jvap is the mass of water per unit area diffusing through the vapor at the surface of the film (y = 0 in Figure 5), D is the mutual diffusion coefficient of water in the polymer, and ρ is the water density. As derived in the Appendix, this flux is a functional form depending on the function aw(x). It reads Λ
Jvap (x) = − csatDv
∫−Λ
∂a w dx′ ∂x x − x′
(4)
It is the Hilbert transform of (∂aw)/(∂x), and as such, the evaporation flux is a nonlocal function of the water activity. Here, Dv is the diffusion coefficient of water in air, and csat is the saturation concentration of water in air. Indeed, the two possible mass transfers of water into the film are (i) water evaporation from the droplet and condensation onto the polymer film, for which the flux jcond is derived from the flux through the vapor phase across the air/polymer interface through jcond = Jvap/e, and 12575
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⎛ x ⎞k 1 − a w (x) = (1 − a w∞)⎜ ⎟ ⎝L⎠
(ii) water diffusion within the polymer along the horizontal x direction from the contact line, corresponding to the second term in eq 3. A convective term arises in eq 3 because this equation is derived in the moving frame of the spreading droplet of velocity U. The corresponding time variations of the water mass fractions in eq 3 scale as jcond
(1 − a w ) 1 1 = Jvap ≈ csatDv e e x
jdiff = ρD jU = ρU
δϕ x2
δϕ x
where k=
(5)
∂ϕ + Jvap [a w (ϕ)] = 0 ∂x
⎛ ∂a 1 ⎞ 1 arctan⎜ w ⎟ π−θ ⎝ ∂ϕ Pe v ⎠
(10)
First, the macroscopic cutoff length L in eq 9 is the distance from the droplet where the water activity in air is unchanged by the 16 evaporating drop and equals a∞ w . As previously observed, it is on the order of the radius of the droplet. Second, the dependence of the water uptake with velocity U and film thickness e is embedded in exponent k. This exponent also depends on the sorption isotherm aw(ϕ): here, we emphasize that k depends on the slope (∂aw/∂ϕ), but eqs 9 and 10 are valid only in the range where this slope varies slowly. In Figure 1, we plot the variations of k given by eq 10 as a function of the water activity for a typical velocity U = 10−4 m·s−1. For aw ranging between 0 and 0.9, k ≈ 0.5. Indeed, k is constant when (∂aw/∂ϕ) is large (at small ϕ), owing to the arctan functional dependence. Remarkably, in the range of water activity between 0.9 and 1, exponent k abruptly decreases from 0.5 to 0. This yields two distinct regimes: at smaller ϕ, the exponent is a constant and close to 0.5, and at larger water contents, the variations of the water content with the distance become very weak, with exponent k rapidly decreasing to 0. This second regime is limited to the distance x ranging between κ and ξ. These theoretical results were compared to our experimental data. In the intermediate distance between ξ and the longdistance limit, all of the data 1 − aw were fit to a power law in x/L with exponent 0.5 and L as a fitting parameter with very good agreement. It is shown in Figure 4 for U = 3 × 10−4 m·s−1. We find that the macroscopic cutoff distance L is of millimeter size, as expected, and is slightly larger for later times in the course of the experiment (corresponding to smaller velocities). The convection of the atmosphere above the film can be compared to the diffusion of water in air using the Peclet number in air: Peair = (LU)/(Dv). We find Peair = 10−2: the water transfer in air is purely diffusive; therefore, L is set by the time for water to diffuse along x. The dependence is indeed weak. For x < ξ, the water content ϕ is found to vary very little with x. A power law (eq 9) with k = 0.1 describes the data fairly well. This order of magnitude of k can be compared to our prediction by eq 10. Using ϕξ ≈ 0.6, we find from Figure 4 that it corresponds to an exponent k ≈ 0.3, in fairly good agreement with our experiments. If our linear model proves to be able to describe the hydration behavior before and after the jump at x = ξ, then it cannot account for this jump, where both the slope (∂aw/∂ϕ) and ϕ vary greatly. Nevertheless, as shown by eq 8 and whatever the nonlinear relation between aw and ϕ, the water content depends on the product eU. We conclude that, at distances ranging between ξ and L, the water absorbed by the polymer film is set by the evaporation flux from the advancing droplet and the condensation from the vapor. It is independent of both U and e because k is a constant, and the variations of L with U are small. The nonlinearities of ϕ with aw account for the observed jump in the water content for x ≈ ξ. In the micrometer-sized range of distances, the water content of the film is a power law in x with an exponent k that rescales in 1/Pev and therefore in eU.
(6)
(7)
First, we compare the transfers in the air versus polymer phases. Roughly, the transfer coefficient, that is, the product of the concentration of water by the diffusion coefficient, is larger by 1 order of magnitude in the vapor (csatDvap) than in the polymer (ρD). Therefore, at large distances from the droplet, the diffusive process jdiff does not play a role. Looking into further details, we find that the amounts of water transferred through air (eq 5) and through the polymer (eq 6) into the film exhibit a dependence on ex and x2, respectively, showing that their relative importance depends on the distance from the contact line x. Therefore, in the following section, we perform an analysis of the characteristic lengths at stake. Within the polymer phase, a typical length appears when comparing jdiff and jU. This length δ = D/U is the distance above which diffusion in the polymer can be neglected. For a typical U = 10−4 m·s−1, δ = 1 μm and is below our observation range. Besides, at distances less than δ, the two dominant terms are jdiff and jcond. Their relative importance depends on the distance to the contact line. This yields a typical length κ = e(D/Dv)(ρ/csat) over which transfers through the vapor phase dominate. For a typical U = 10−4 m·s−1, with Dv = 2.6 × 10−5 m2·s−1, csat = 23 × 10−3 kg·m−3, and ρ = 103 kg·m−3, we find κ = 30 nm. First, κ is smaller than the film thickness e = 300 nm. Second, κ is small compared to the range of distances where we measure the water content. Therefore, diffusion effects are negligible at the distances from the contact line that we probe, and the water fraction profiles we measure (Figure 4 inset) result from the water transferred by evaporation and condensation and from the movement of the contact line only. The coupling between these two processes has been widely analyzed in our previous paper.2 It is convenient to introduce the Peclet number comparing these two transfers jU and jcond: Pev = (ρeU/csatDv). In the range of velocities explored, we find Pev ≈ 10−1, which is smaller than 1 as expected. In the following section, we focus on this condensation/ convection regime. For x ≫ κ and under stationary conditions, eq 3 reduces to eUρ
(9)
(8)
Equation 8 shows that the volume fraction of water depends on the product eU, even if aw is a nonlinear function of ϕ. Solutions of this equation have been theoretically calculated2 for the case where the water activity aw is proportional to the volume fraction ϕ; that is, (∂aw/∂ϕ) is a constant. We can therefore extend this result to cases where the slope of the sorption isotherm varies slowly with x. Note that in our case this approximation is valid only at low water volume fractions, as seen in Figure 1, or when ϕ barely varies in x. The solution is 12576
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It is therefore expected not to modify the dependence on the product eU of the amount of water transferred by the evaporation/condensation process.
We now turn to distances even closer to the contact line, where the diffusion along the x direction from the liquid droplet is expected to become relevant. We cannot probe the water content in the polymer at submicrometric distances from the contact line. However, we can test the rescaling in eU on the values of θ: the contact angle is believed to be set by the amount of water in the polymeric substrate in the vicinity of the contact line. To that end, in a second series of experiments, we measured the contact line velocity U and the contact angle of droplets spreading onto polymer films of varied thicknesses e. An example is shown in Figure 3b for e = 300 nm. For a given value of the contact angle, the data (e and U) are collected so as to build a series of iso-θ curves in Figure 6. Remarkably, all the iso-θ curves were fit to a
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CONCLUSIONS In this study, we investigate what happens very close to the contact line of a water droplet spreading on a hydrosoluble substrate. We focus on thin films for which, at large distances from the contact line, the solvent fraction in the soluble substrate is homogeneous across the thickness. We show that the substrate wettability, measured through the droplet dynamic contact angle θ, is directly tuned by the solvent content ϕ close to the contact line. At distances smaller than 100 μm and for the first time, we find that the nonlinearities of the sorption isotherm, the direct diffusion of the liquid into the polymer, and condensation of the vapor phase fed by evaporation from the droplet itself strongly influence the hydration of the substrate. By measuring ϕ ahead of the droplet, we find segments where ϕ is a power law of the distance to the contact line, with a jump, and exponents that are directly related to the nonlinearities of the sorption isotherm of the polymer ϕ(aw). The power-law behavior is a consequence of the solvent flux arriving by the evaporation/condensation process. This transfer imposes a rescaling of ϕ with the product eU where e is the thickness of the soluble substrate and U is the velocity of the spreading droplet in the range of distances from the contact line between 1 and 100 μm. Finally, contact angle measurements show that θ also rescales as eU. From this, we conclude that, first, the dynamic contact angle is set by the solvent content at the contact line and, second, that the solvent transfer by the diffusion of liquid solvent through the polymer substrate increases the solvent content by an amount that does not depend on e or on U; therefore, the dependence of ϕ on e and U is entirely accounted for by the derivation of the flux of solvent by the evaporation/condensation process. Altogether, the mechanisms at stake in the wetting dynamics of a droplet of solvent on a thin supported film of soluble polymer are now well understood. The next steps include the study of thicker films where a gradient of solvent content arises in the vertical direction, up to the limit where the soluble substrate behaves as an infinitely thick material. Another aspect yet to be studied is the effect of the molecular mass of the polymer: a longer polymer will cause an increase of viscosity in the wedge of the droplet that will slow down the spreading process. Also, the onset of a gel phase in the vicinity of the contact line at larger polymer masses is likely to modify the spreading dynamic in a complex way. Along another line, this study could be extended to soluble materials with very different sorption behavior: as an example, crystalline materials such as sugar or lactose are likely to exhibit different wetting dynamics owing to the specific behavior of crystals in contact with water vapor. These aspects are to be developed in coming studies.
Figure 6. Symbols: Iso-contact angle θ curves for thin films of maltodextrin of varied thickness e during the spontaneous spreading of water droplets of instantaneous velocity U. Data are fit to a power law with exponent −1 (lines).
line with an exponent close to −1 showing how the contact angle θ scales as 1/eU over 2 orders of magnitude in U and e. Hence, we observe that the water fraction ϕ in the vicinity of the contact line must rescale with the product eU. This result is consistent with the condensation/convection process observed far from the contact line and must therefore be reconciled to the penetration of water by diffusion (eq 6) from the droplet and within submicrometric distances from the contact line (i.e., up to x ≈ κ). As stated earlier, the typical length for the diffusion of water in the horizontal direction is κ, which is smaller than the thickness e: κ ≈ 0.1e under our experimental conditions. First, note that κ is smaller than δ ≈ 1 μm. By definition of δ, at distances smaller than δ in the polymer phase, convection is negligible compared to the diffusive term. Similarly, as seen earlier from the value of the Peclet number in air, Peair, transfers by convection are negligible in the vapor phase. Hence, the water transfers are purely diffusive in the range of distances x < κ and do not depend on U. Second, in the vertical direction, the penetration of water by the diffusion of liquid into the polymer is also κ, which is smaller than e. Therefore, the volume fraction of water in the polymer is distributed as a bidimensional field, as schematically depicted in Figure 5. It is therefore independent of the polymer thickness e. Nevertheless, diffusion through the polymer and through air depends on the boundary condition at distances larger than κ, that is, on the concentration field over the intermediate distance κ < x < ξ that does depend (and depends only) on the product eU, through exponent k, as stated earlier. As a consequence, the diffusive process from the liquid droplet transfers an extra quantity of water into the film in the (very) vicinity of the contact line, which does not depend on e or on U.
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APPENDIX: CONDENSATION FLUX In this appendix, we use a 2D description of the evaporation/ condensation problem to derive an equation for the condensation flux ahead the contact line as a function of the water activity of the film. We consider a flat droplet of radius R moving at velocity U and evaporating in the air with a coefficient Dv. The increase in water concentration in air, with reference to the initial humidity, is c(x, y) where x (respectively y) is the 12577
dx.doi.org/10.1021/la402157d | Langmuir 2013, 29, 12572−12578
Langmuir
Article
Notes
horizontal (respectively vertical) direction. Under stationary conditions, the diffusion equation reads Δc = 0. To solve the problem, the boundary condition in the vertical direction is set by introducing a length Λ such that c(x, Λ) = 0. If Λ is larger than every other length in the problem, its choice does not play a role. We set Λ = R. The boundary conditions are c(x, Λ) = 0 and c(x, 0) = aw(x)csat (csat is the maximal concentration of water in air). In this 2D problem, c(x, y) is an odd function of x: we search for a Fourier development of cosine terms with coefficients cq multiplied by a function gq(y) of y only: c(x , y) =
∫0
The authors declare no competing financial interest.
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∞
cq cos(qx) gq(y) dq
(11)
gq(y) is of an exponential type. The long-distance boundary condition (y = Λ) sets gq(y) = sh(q(Λ − y)), and the one at y = 0 yields
∫0
∞
cq cos(qx) sh(q Λ) dq = a w (x)csat
(12)
The odd function aw(x) can also be developed as a cosine Fourier transform with coefficients aq, yielding cq aq = sh(q Λ) csat (13) We now focus on the mass flux of vapor per unit area that transfers between air and polymer Jvap. It is set by the gradient of water content at y = 0. It reads Jvap (x) = −Dv
∂c ∂y
= csatDv y=0
∫0
∞
aqq cos(qx)
ch(q Λ) dq sh(q Λ) (14)
The length Λ is large compared to the distances along x in which we are interested, and qΛ is therefore large compared to 1. Besides, aq can be written as a function of the odd function aw, yielding ∞
Jvap (x) = csatDv
∫0 ∫0
∞
a w (x′) cos(qx′) cos(qx)q dx′ dq (15)
Thus, the condensation flux can be calculated if aw(x) is known. Alternatively, using the definition of aq, we find aq =
∫0
∞
a w (x′) cos(qx′) dx′ = −
∫0
∞
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∂a w sin(qx′) dx′ ∂x′ q
NOTE ADDED AFTER ASAP PUBLICATION This paper was published on the Web on September 23, 2013. Citation information was added to the Abstract and the corrected version was reposted on September 27, 2013.
(16)
Equation 15 gives the expression of the evaporation flux of water from the droplet that condenses on the substrate. It depends on the sorption isotherm via the term (∂aw/∂ϕ): ∞
Jvap (x) = −csatDv
∫0 ∫0
∞
REFERENCES
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∂a w sin(qx′) cos(qx) dx′ dq ∂x′ (17)
Λ
Jvap ≈ − csatDv
∫−Λ
∂a w dx′ ∂x x − x′
(18)
The integral is a Cauchy principal value. Altogether, Jvap is the Hilbert transform of (∂a/∂x) as detailed by Sneddon17 on page 236. Equation 18 holds for x ≪ Λ.
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AUTHOR INFORMATION
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