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Evaporation of Liquids and Solutions in Confined Geometry F. Cle´ment and J. Leng* School of Physics, The University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, Scotland Received February 20, 2004. In Final Form: May 25, 2004 We describe a drying setup where a liquid droplet is confined between two circular glass plates and allowed to evaporate. Optical imaging of the shrinking droplet provides the temporal evolution of the solvent volume. For a pure liquid, we measure the diffusion coefficient of the evaporating gas with a good accuracy. For a solution, the slowing down of evaporation by progessive concentration of the solute yields a direct measurement of the solvent activity.

Evaporation of a liquid droplet on a substrate is an intricate phenomenon as it couples effects such as evaporation, hydrodynamics, wetting, etc.1 When the liquid contains a solute, drying becomes even more complicated due to the hardly avoidable contact line pinning.2 Understanding drying is thus challenging from a fundamental point of view, relevant for related phenomena such as shrinkage, fracture, etc., and important for applications.3 Drying as such is also a major route for out-of-equilibrium studies, e.g., concentration quench.4 But studying it is difficult: boundary conditions are hard to control and observation is often awkward. Here, we present an experimental proceduresthe drying in confined geometrysthat bypasses most of those difficulties. In our experiments, we squash a small droplet of a liquid between two circular glass plates and allow it to dry. Evaporation proceeds by gas diffusion from the edge of the droplet toward the edge of the cell: the confinement naturally casts well-defined boundary conditions. Besides, the quasitwo-dimensional (2D) arrangement facilitates observation. Evaporation in confined geometry proceeds at room temperature (23 ( 1 °C) in a cell simply consisting of two circular glass plates (Figure 1A,B). A few microliters of a liquid are deposited onto the bottom plate and squeezed with the upper plate. The gap between the plates (∼100 µm) is controlled by thin linear spacers laid out to match a radial geometry. The cell is illuminated with parallel white light, and we image the droplet during its shrinking process with simple optics. During a typical evaporation kinetics (∼4 h), several thousand images can be framegrabbed on a PC (at a rate up to 3 frames/s), as sometimes required for demanding signals (e.g., activity measurement). A typical image features a circular dark ring on a light gray background (Figure 1B,C1): it is the meniscus whose specific shape has refracted the light out of the field of view. An intensity profile across the meniscus shows a Heaviside-like function. The high contrast eases up the identification of droplet boundaries with the use of image-processing software. Here we measure two * To whom correspondence may be addressed. E-mail: [email protected]. (1) Cachile, M.; Be´nichou, O.; Cazabat, A.-M. Evaporating droplets of completely wetting liquids. Langmuir 2002, 18, 7985. (2) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Capillary flow as the cause of ring stains from dried liquid drops. Nature 1997, 389, 827. (3) Keey, R. B. Introduction to industrial drying operations; Pergamon Press: Oxford, 1978. (4) Haw, M. D.; Gillie, M.; Poon, W. C. K. Effects of phase behavior on the drying of colloidal suspensions. Langmuir 2002, 18, 1626.

bounds for the droplet area A(t), one that includes the dark meniscus (Figure 1C2) while the other excludes it (Figure 1C3). A close estimate of the solvent volume can thus be calculated. The accuracy of shape detection depends on the magnification, pixel size, and the intensity dynamic range. With our setup, we can resolve (10 µm, hence a good accuracy [O(2%)] on the area of a millimetric droplet. An experiment provides A(t) with good spatial and temporal resolutions (Figure 2a). We derive now a simple model for it. We first assume the geometry is cylindrical and ignore the cell thickness. The gas that escapes from the droplet meniscus is driven outward from the cell by diffusion; this empties the droplet at a rate deduced by the law of mass conservation. Let us assume the gas number density is nvs at the edge of the drop (saturated vapor) and nv at the edge of the cell (density at partial vapor pressure in air, for simplicity we assume nv ) 0). In a steady state, the density profile of the gas inside the cell follows n(r) ) nvs ln(r/Rs)/ln(R0/Rs) with r, R0, and Rs the radial coordinate, the time-dependent droplet radius, and the substrate radius, respectively.5 It yields a gas flux, j(r) ) -Dnvs/r ln(R0/Rs), which empties the droplet at a rate δV/δt ) -j(R0)vL dS, with D the gas diffusion coefficient, vL the molecular volume of the liquid phase, and dS ≈ 2πR0h, the gas/liquid exchange surface (we neglect the exact shape of the meniscus as the latter is likely to be filled with saturated vapor and use only the gap height h, along with V ≈ hA). We consider the case where the liquid may contain a nonvolatile solute. D is independent of the solute, but the saturated vapor pressure depends on the solute concentration through the solvent activity a [a(c) ) Pvs(c)/Pvs(0) ) nvs(c)/nvs(0), where Pvs(c) and Pvs(0) are the vapor pressures when there is a solute at concentration c and without a solute, respectively,6 and assuming a perfect gas]. Introducing the relative surface R(t) ) A(t)/As, the reduced diffusion coefficient D ˜ ) DnvsvL, the drying time τe ) As/4πD ˜ , and the reduced time τ ) t/τe, we rewrite the equation of mass conservation to obtain

ln R

dR )a dτ

(1)

This result, which provides the (geometrical) instantaneous drying rate, is crude as it simplifies the shape of the (5) Crank, J. The mathematics of diffusion; Clarendon Press: Oxford, 1979. (6) Levine, I. N. Physical Chemistry; McGraw-Hill: New York, 1995.

10.1021/la0495534 CCC: $27.50 © 2004 American Chemical Society Published on Web 06/29/2004

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Figure 1. Schematic setup and analysis. (A) Setup: the sample cell consists of two circular glass plates (we used three diameters: 87, 117, and 148 mm of a 3 mm thick ordinary float glass) between which a liquid droplet is squeezed; the gap is controlled by thin linear spacers (pieces of plastic shim, thickness from ≈50 to 200 µm, length 5 mm, width 2 mm). Parallel white light is passed through the sample and (after deflection with a first-front miror M) the objective (two identical achromats of focal length L1 ) L2 ) 25 cm and aperture 6 cm) creates the image of the drop onto a standard CCD chip. Images are frame-grabbed on a PC at a frequency up to 3 frames/s. Spatial calibration is achieved with accurate micrometers. (B) Close-up of the confined drop showing the evaporation mechanism (gas diffusion) and notations. (C) Typical snapshots of evaporation (decaline evaporating onto a 87 mm diameter substrate, images taken ≈20 min apart). An image is analyzed by a two-step procedure: a first intensity threshold yields the outer contour of the drop (including the meniscus) and related area (out); within this limit, the same procedure gives the inner coutour and area of the droplet (in).

Figure 2. (a) Relative area of a decaline droplet R ) A/As against time (cis/trans mixture). (As is the area of the substrate. The ≈200 data points have been sampled out for clarity. The line corresponds to a fit using eq 2, on a time range excluding the initial transient state (left of the dashed line), which yields τe and therefore D ˜ .) (b) Rescaled evaporation curves for several organic liquids. (c) Scaling of the drying time τe versus substrate area As for decaline (mixture of cis/trans), yielding a more accurate determination of D ˜ ) (5.41 ( 0.02) × 10-11 m2 s-1.

drop, assumes a steady state, no ambient vapor (nv ) 0), and neglects the nonideality of the gas. Yet, interesting results follow immediately. First, the time scale is essentially fixed by the geometry (τe ∼ As for a given gas) as a natural outcome of the confinement. It represents a convenient means to tune evaporation kinetics. This is especially relevant for pure solvents (a ) 1) where integration of eq 1 with initial condition R(τ ) 0) ) R0 yields

R ln R - R - R0 ln R0 + R0 ) τ

(2)

The limit R0 ) 1, R f 0, leads to τ ) 1, i.e., τe is the time needed to empty an initially full cell under steady evaporation. Although eqs 1 and 2 are strictly equivalent, the latter is easier to use as it involves fitting only (Figure 2a) (as opposed to numerical differentiation for eq 1, see also Figure 3b,c). To make a proper use of eq 2, we must also account for an initial transient state during which the gas density gradient settles in, which lasts ∼As/D at the beginning of the experiment [O(10 min)] and during which eq 2 does not hold. This regime is excluded from

data analysis (Figure 2a), which reduces to fitting the time t with a two-parameter expression: t ) τe(R ln R R + b) and provides τe. We tested this approach with organic liquids for which the assumption nv ≈ 0 is well verified. For all the solvents we studied, eq 2 gives an excellent agreement with experimental results (regression coefficients ≈ 1 - O(10-4) on ∼200 data points) whatever the substrate size and the gap height (50, 100, and 200 µm; in this small gap limit, no effect of “air” convection can be detected, this may however be a severe limitation of the method when working with too thick gaps for which gas diffusion is not the only evaporation mechanism). Table 1 gives D ˜ measured for several gases along with the experimental error [O(2%)]. The accuracy of the method is best demonstrated by the good rescaling of all evaporation curves onto a master curvesowing the transformation t f τ* ) t/τe + 1 + R0 ln R0 - R0 and A f R ) A/Asswhatever the solvent, the height of the gap, and the size of the substrate (Figure 2b). Importantly, we also checked the scaling, τe ∼ As is valid (Figure 2c). This clearly shows that for a given liquid, evaporation is tunable by simply

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Figure 3. (a) Activity coefficient of water for NaCl/water mixtures obtained from analysis of evaporation curves (b and c). Each set of symbols represents a single drying experiment started at the salt concentration given in the legend. The solid line corresponds to reference values.9 (b) Raw data R against time for pure water and a brine solution. (c) Drying rate obtained by numerical differentiation and smoothing. Table 1. Reduced Diffusion Coefficient of Several Organic Gases Measured by Evaporation in Confined Geometrya organic liquid

D ˜ (m2 s-1)

toluene heptane cis/trans-decaline cis/trans-decaline* cis-decaline

(1.38 ( 0.02) × 10-9 (2.82 ( 0.04) × 10-9 (5.45 ( 0.04) ×10-11 (5.41 ( 0.02) × 10-11 (4.25 ( 0.03) × 10-11

a All results are based on a single experiment except the one marked with an asterisk for which we studied the effect of substrate size (Figure 2c) and obtained a better accuracy.

changing the size of the substrate and that the method itself yields a simple yet efficient experimental control on a drying kinetics. When the liquid contains a nonvolatile solute, its activity may decrease with the solute concentration and one observes a slowing down of evaporation. The latter increases the concentration, which in turn modifies continuously the activity and thereby the evaporation rate, eq 1.7 To test this approach, we studied mixtures of salt and water. NaCl was dissolved in water at a given concentration, and a droplet of the mixture was allowed to dry. We measured A(t) with a very good time resolution (thousands of data points) to be able to calculate the time derivative of R with reasonable statistics (smoothing, by grouping and averaging out bunches of ≈20 data points, was necessary to limit the numerical noise). Evaporation slowing down is obvious when working with concentrated mixtures (Figure 3b): while the drying rate of pure water is constant, that of brine continuously decreases upon evaporation. The ratio of the two rates provides a measurement of the solvent activity (eq 1 and Figure 3b,c). In fact, this drying equation must be modified for water to account for vapor in air (nv * 0). To do this, we measured and used a mean value for the relative air humidity (≈25%8) for all runs while the latter is temperature sensitive and should be measured from run to run. With (7) Note that the concentration dependence of the activity is hidden in eq 1 and only shows up as a time dependence. Equivalence is however straightforward as c(t) ) V(0)/V(t) ≈ R(0)/R(t) (with a necessary correction accounting for the meniscus to obtain the exact concentration). (8) Lide, D. R. CRC handbook of chemistry and physics; CRC Press: Boca Raton, FL, 2001.

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this correction, we measured the activity of several brine solutions, see Figure 3a: the curves follow satisfyingly reference values (solid line, refs 8 and 9) despite a systematic difference, which may be due to the too rough vapor-in-air correction. It remains however that even with poorly controlled conditions, it is possible to measure continuously the solvent activity by simple observation of an evaporating droplet. To be observable, the effect must be at least of order 10% which restricts the method to strongly interacting species. Much neater results are expected when working with a controlled atmosphere (controllable nv) and with hydrophobic substrates (contact angle >90°) for which pinning of the contact line should not occur. During our experiments, we observed no pinning of the contact line except at cristallization, with a “ramified” crystal often growing inward from the meniscus and thus blocking the line. This leaves open the possibility of salt concentration gradients within the drop even before crystallization. We argue that due to the slow drying rate, as compared to that of a sessile droplet for instance, salt diffusion is efficient enough to equilibrate concentrations at any time. That is, we assume a quasi-equilibrium situation which allows the measurement of an equilibrium featuresthe activitysand explains the correct superimposition of experimental results (Figure 3a). Additionally and as mentioned later in the text, this quasi-equilibrium may fail for other systems.11 We now turn to the comparison with other methods and to the potentiality the present setup offers. Two methods are commonly used to study evaporation of pure solvents: evaporation from a thin capillary tube and evaporation of a sessile droplet.10 The former is the 1D equivalent to the present case with a liquid evaporating from a column under steady gradient of gas density. The advantage of the 2D method is 2-fold: it is accurate due to automation and neat detection and quick. For comparison, the evaporation of 4 mm diameter droplet of toluene squeezed in a As ≈ 8 × 104 mm2 cell takes about 3.5 h to resolve with up to thousands of data points. In a typical capillary tube (10 cm high, cross section 0.25 cm2, half filled10), the level of the liquid would have changed by about 3 mm during the same time. Assuming that the reading of the level is accurate at ≈0.1 mm, this gives about 30 “good” data points to operate the analysis. The statistical difference is enormous and the 2D configuration seems ideal for systematic studies on D ˜ . The method can even be simplified by setting the evaporation cell onto the stage of a microscale. Reading the mass loss versus time would provide the same results. Here again, automation is crucial to make possible the measurement of solvent activity by differentiation of experimental results. As compared to the evaporation of a sessile droplet, the 2D evaporation offers a complementary approach: because the droplet is in contact with a large amount of material (glass plates), it seems reasonable to assume isothermal evaporation, thus excluding difficulties due to self-cooling.1 Additionally, and importantly, the 2D evaporation bypasses issues related to contact angle. The latter does not influence the evaporation kineticssbut for the establishment of the steady statesand any liquid can be evaporated onto virtually any material. This may help determine the transport properties of gases when they are required to (9) Robinson, R. A.; Stokes, R. H. Electrolyte solutions; Butterworth: London, 1965. (10) Yildirim Erbril, H.; Avci, Y. Simultaneous determination of toluene diffusion coefficient in air from thin tube evaporation and sessible drop evaporation on a solid surface. Langmuir 2002, 18, 5113.

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characterize, for instance, the contact angle dynamics of an evaporating sessile droplet.1 We, however, believe that the real potential of this method holds in that it combines the geometrically controlled solvent removal of a droplet to a good observation of the latter. The setup naturally selects a drying time τe ∼ As/D, and changing the substrate size simply modifies the global time scaling. It turns out to be extremely convenient for out-of-equilibrium studies of complex systems. We recently started to study the (confined) drying of hard-sphere colloidal and we recover much of the common phenomenology:2 (self-) pinning of the contact line, capillary driven “jamming” of particles at the edge of the drop, fractures once the solution is nearly dry, etc. Unlike the case of a sessile droplet, the flow of particles is not determined by a combination of contact angle and droplet size2 but by the geometry. By tuning both the evaporation rate and the size and density of particles, we

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have shown that the compacity of the deposit at the outer edge of the drop is a function of the Peclet number only, here defined by the ratio of the Brownian time over the drying time.11 For small particles, the deposit reaches a very high compacity (≈0.74) while it drops down to ≈0.62 for the biggest particles. This illustrates well how drying in confined geometry can be used to finely control an outof-equilibrium system. Acknowledgment. We thank K. Kroy, M. E. Cates, W. C. K. Poon, S. U. Egelhaaf, M. D. Haw, and A. Aradian for helpful dicussions and EPSRC for funding (Grant GR/R42733/01). LA0495534 (11) Cle´ment, F.; Leng, J. Drying of hard-sphere colloidal suspensions in confined geometry. Unpublished data.