Imbibition Kinetics of Spherical Colloidal Aggregates - Ipcms

week ending 11 JULY 2014

PHYSICAL REVIEW LETTERS

PRL 113, 028301 (2014)

Imbibition Kinetics of Spherical Colloidal Aggregates A. Debacker,1,2 S. Makarchuk,1 D. Lootens,2 and P. Hébraud1 1

IPCMS/CNRS 23 rue du Loess, 67034 Strasbourg, France 2 Sika, Tüffenwies 16, CH-8048 Zürich, Switzerland (Received 6 February 2014; revised manuscript received 1 April 2014; published 7 July 2014) The imbibition kinetics of a millimeter-sized aggregate of 300 nm diameter colloidal particles by a wetting pure solvent is studied. Three successive regimes are observed. First, the imbibition proceeds by compressing the air inside the aggregate. Next, the solvent stops when the pressure of the compressed air is equal to the excess of capillary pressure at the meniscus of the wetting solvent in the porous aggregate. The interface is pinned and the aggregate slowly degases up to the point where the pressure of the entrapped air stops decreasing and is controlled by the capillary pressure. Finally, the imbibition starts again at a constant excess of pressure, smaller than the capillary pressure but larger than the one of the atmosphere. This last stage leads to the complete infiltration of the aggregate. DOI: 10.1103/PhysRevLett.113.028301

PACS numbers: 82.70.Dd, 47.56.+r, 81.05.Rm

The dispersion of a powder in a liquid is a process that appears in everyday life [1] as well as in many industrial processes, from mining [2] to elastomer reinforcement [3] and drug delivery [4]. In its dry state, the powder consists of small particles agglomerated due to attractive surface forces. During the dispersion, the aggregates must be disrupted. This can be achieved through the aid of mechanical forces by mixing the powder with the solvent [5] or it may be induced by the imbibition itself [6]. Chemical dissolution may also play a role, but the very first stage of the dispersion always consists in the imbibition of the aggregate by the dispersing solvent. It is governed by the difference of pressure between the inner part of the aggregate and the outside pressure: on one hand, the capillary pressure drives the infiltration, on the other hand, the charge loss slows down the flow. Such a very simple description of the capillary rise inside a pore has been known for almost a century [7], and it can be used to describe the imbibition of a porous medium whose one side remains in contact with the atmosphere provided gravitation [8] and inertia [9] are negligible. Nevertheless, in the case of powders [10], complex kinetics is observed and is yet to be understood: the rate of infiltration abruptly accelerates after a transient time [11]. This kinetics may be attributed to the inherent complexity of the systems studied up to now (heterogeneities of the pore size and shape [12,13], distribution of molecular weight for polymeric solvents [11], fluctuations of the air-liquid front [14], differences in wetting properties for a mixture of solvents). Nevertheless, the presence of air inside the aggregate has been overlooked: while the liquid penetrates the aggregate, the air must find its way out. These two processes obey different driving forces: the reduction of the interface surface is responsible for the liquid infiltration whereas air goes out through dissolution in the solvent or by pushing the interface out, drainage occurs, leading to the formation 0031-9007=14=113(2)=028301(5)

interfaces rough at length scales larger than the pore scale [15–17]. The goal of this Letter is to study the imbibition kinetics of a colloidal aggregate in its simplest form, where a Newtonian fluid imbibites a spherical aggregate composed of monodisperse spherical colloidal particles that neither dissolve nor disrupt when infiltration occurs. We thus prepare spherical aggregates of 300 nm diameter silica colloidal particles (Nissan Corp.). A 30% volume fraction colloidal suspension of silica particles in water is poured in a spherical mold whose radius ranges between R0 ¼ 1 and R0 ¼ 3.5 mm, connected to a reservoir and centrifuged at 2000 g for 10 mn. After drying at 70 °C during 5 h, the aggregate is unmolded and particles hold together due to attractive surface forces. It can be manipulated and is subsequently dried during 6 h at 100 °C until its weight no longer decreases. The resulting structure is a spherical aggregate composed of nonsintered colloidal particles. The aggregate possesses an elastic modulus of 200 MPa. It consists in a random packing of colloidal spheres whose porosity is ϵ ¼ 0.36 0.01. We will model this system as a continuous medium of permeability κ. The permeability may in general be related to the porosity by the semiempirical Kozeny-Carman relation [18], κ ¼ ðϵ3 =Cð1 − ϵÞ2 Þσ −2 , where C is a numerical constant and σ the specific surface. In the case of a packing of hard spheres, κ ¼ ðϵ3 =C0 ð1 − ϵÞ2 ÞR2c , where Rc is the radius of the colloidal particles and C0 a constant measured to be equal to C0 ¼ 35 [19]. For the aggregates under study here, we have κ ¼ 73 nm2 . The infiltration solvent is chosen so that it wets the silica particles surface, its viscosity allows for an observation of the imbibition kinetics in accessible time scales, and its refractive index matches the index of silica. Moreover, in order to avoid the phenomena associated with different wettabilities, we choose a pure solvent, diisododecyl phtalate (DIDP),

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whose refractive index is 1.48 and viscosity η ¼ 0.1 Pa × s. It wets silica surface, has a surface tension with air equal to γ ¼ 20 mN × m−1 , and displays a wetting angle with silica of θ ∼ 20°. In a typical experiment, the aggregate is deposited at the bottom of a square cell on top of a tetrapod that allows for its complete surrounding by the solvent. The air pressure inside the cell is controlled between 100 mbar and 1 bar. The solvent is poured onto the aggregate and the pressure is released to 1 bar. This defines the initial time of the experiment. The imbibition kinetics is observed by shadow imaging the aggregate illuminated with a parallel beam of white light. The dry core of the aggregate strongly diffuses light and appears dark whereas the infiltrated shell is almost transparent. Special care is taken to control the sharpness of the image: the sample is scanned along the optical axis up to the point where the border between the wet and the dry regions is not thicker than a pixel. The image is then acquired with an 8 bit CCD camera (Baumer HXC13) and is thresholded, allowing for the computation of the area of the dry region. This setup allows us to visualize the air interface inside the porous medium with a resolution of 10 μm, larger than the particle diameter. In particular, we do not resolve the front position down to the pore scale. We observe that the air-liquid interface is isotropic and the spherical symmetry of the system is conserved until the end of the experiment [Figs. 1(a)–1(c)]. The evolution of the area of the dry core as a function of time is given in Fig. 1 for aggregates of four different radii, R0 ¼ 1, 1.5, 2, and 2.5 mm. At the mesoscale, imbibition shows spherical symmetry and the current j of the solvent is purely radial. Darcy’s law gives j ¼ −ðκ=ηÞðdP=drÞ. Taking into account the difference in pressure due to charge loss, the mass conservation law leads to the difference of pressure Pin − P0 between the inner dry core, of radius R, and the outside of the aggregate: ηϵ dR R2 Pin − P0 ¼ Pc þ R− ; κ dt R0

ð1Þ

where Pc is the capillary pressure due to the front meniscus between adjacent colloidal particles. This equation relates the velocity of the infiltration to the difference in pressure between the inner gas and the outer atmosphere. It has been used to describe the imbibition of a silica powder by polydimethylsiloxane [10] when the air inside the aggregate is in equilibrium with the outside atmosphere, Pin ¼ P0 . This model is called the Washburn model [7,10]. Nevertheless, the actual kinetics is more complex. Three regimes are indeed observed (Fig. 1). In the first one, the volume of the wet shell rapidly increases. In the second, it saturates after a few minutes and no more infiltration is observed for a duration ranging from 1000 s up to 10000 s, depending on the diameter of the aggregate. In the last, infiltration resumes and proceeds up


FIG. 1. Top: Image of the aggregate during the infiltration by DIDP during the first regime (a), at the plateau (b), and at the end of the last infiltration regime (c). Bottom: Evolution of the square of the noninfiltrated core radius as a function of time for aggregates of radius 1 mm (black squares), 1.5 mm (dark gray squares), 2 mm (medium gray squares), and 2.5 mm (light gray squares). Inset: Imbibition curves for 2.5 mm radius aggregates when the initial inner gas pressure is equal to 1 bar (light gray squares), 400 mbar (medium gray) and 200 mbar (dark gray).

to the complete imbibition. When the size of the aggregate increases, the overall infiltration slows down, but the same general behavior is observed. The plateau value of the fraction of the area that is not yet infiltrated does not depend on the size of the aggregate when it is prepared at atmospheric pressure. Conversely, when the aggregate is prepared under lower pressure, the amount of trapped air is smaller and the radius of the dry core at the plateau decreases down to a value corresponding to an excess of pressure in the inner bubble equal to the capillary pressure (Fig. 1, inset). This kinetics cannot be rendered by the Washburn model, whose result is a complete infiltration of the aggregate in a time of the order of τW ¼ ηR20 ϵ=6Pc κ ≈ 130 s for the smallest aggregate (Fig. 2, dashed lines). This model assumes that the air inside the aggregate finds its way out and is in equilibrium with the atmospheric pressure. The saturated molecular concentration of air in DIDP has not been measured, but the molar solubility of oxygen and nitrogen in organic solvent at atmospheric pressure is of the order of 10−3 mol=mol [20]. The air may thus dissolve in the solvent according to Ostwald kinetics. The dissolution time of the dry bubble inside the solvent is R2 P

P

0 Lap 0 [21] τdiss ¼ cs DRTðP ≈ 107 s, where cs is the molar Lap −P0 Þ

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FIG. 2. Evolution of the square of the radius of the noninfiltrated core as a function of the logarithm of time, during the first regime, for aggregates radii equal to 1, 1.5, 2, and 2.5 mm (same color legend as Fig. 1). Dashed lines are fits according to the Washburn model, assuming that the pressure of air inside the aggregate is equal to the atmospheric pressure. The continuous lines are fits in which the air is assumed to be trapped in the aggregate, according to Eq. (2). The two fit parameters, κ (filled circle) and Pc (filled square), are plotted in the inset as a function of the aggregate radius.

concentration of gas in the solvent, P0 the atmospheric pressure, PLap the pressure inside the bubble of radius R0 , and D the molecular diffusion coefficient of DIDP. This dissolution time is several orders of magnitude longer than the observed kinetics, and air dissolution does not play a role in the infiltration kinetics. The air remains trapped inside the dry core, so that its pressure increases during the infiltration process. Assuming a Boyle-Mariotte law for the evolution of the air pressure as a function of the volume, Pin ¼ P0 R30 =R3 , Eq. (1) leads to the following kinetics equation: 3 R0 ϵηR20 dR R R2 ; − P0 3 − 1 ¼ Pc þ κ dt R20 R30 R

ð2Þ

where the first term on the right-hand side is the capillary force, and the last one the charge loss term. This equation can be solved analytically: as time goes to zero, the slopes of the infiltrated radius as a function of time according to Washburn and to the trapped air model are identical. Then, the kinetics of the trapped air model is slower, and at longer times, the infiltration stops when the excess of pressure inside the dry core equals the capillary pressure of the airliquid interface. The radius Rp at which the infiltration stops is given by Rp =R0 ¼ ðPc =P0 þ 1Þ−1=3. The first infiltration regime is well described by this model (Fig. 2, continuous lines) with two fit parameters: the permeability κ and the capillary pressure Pc .


These parameters do not depend on the size of the aggregate (Fig. 2, inset). The permeability deduced from the fits is κ ¼ 78 3 nm2 , in good agreement with the Kozeny-Carman value (73 nm2 ). The capillary pressure value, Pc ¼ ð4.7 0.1Þ × 105 Pa, is the value of the imbibition pressure. For total wetting, a semiempirical relationship for random packing of spheres is given by Pc ¼ γð2=Rin − 1.6=Rp Þ [22], where Rin is the radius of the sphere tangent to the four particles that define a pore space. Taking into account solvent partial wettability, the imbibition radius is smaller by an amount calculated by Gladkikh and Bryant [23]: one finds Pc ≈ 4.5 × 105 Pa, in good agreement with the measured value. At the end of this first regime, the system is in a nonstable state: an increase of the pressure inside the dry core pushes the nonwetting and less viscous gas into the wetting and more viscous liquid: drainage occurs. The capillary number may be evaluated to be Ca ¼ ηDIDP v=ϵγ ≈ 5 × 10−5 , where v=ϵ is the local velocity. Ca is much smaller than 1: the viscous forces are very small in comparison with the capillary forces. The ratio of the viscosities of the pushing fluid (the air) and the solvent is ηair =ηDIDP ¼ 2 × 10−4 . In this regime, the interface of the drainage is not smooth, but exhibits fingering driven by capillarity [17,24], leading to the formation of a connected pathway between the dry inner core and the outer surface of the aggregate. The formation of this pathway has been observed in granular media [25], where it is associated with local fracturing of the medium. Nevertheless, in our case, the attractive forces between the particles is much larger than the capillary forces (Supplemental Material, Sec. 4 [26]), and local rearrangement of the particles induced by capillary forces is not expected to occur. The heterogeneity of the pore size is another mechanism for the instability: the excess of Laplace pressure in the smallest pores destabilizes the largest ones [13]. During the plateau regime, an isochoric transformation occurs during which the pressure inside the dry core decreases down to a pressure value that we will determine by analyzing the kinetics of the last regime, to find 4.2 × 105 Pa (Fig. 4, inset). The plateau duration T p increases as the square of the radius of the trapped gas bubble Rp (Fig. 3). This relation holds both when the radius of the dry core changes due to a change of the aggregate radius or due to a change of the initial air pressure. Conversely, when the aggregate is prepared at low pressure, the shell thickness increases whereas the plateau duration decreases (Fig. 3, inset). The characteristic duration of the plateau is not related to the time needed for the air to diffuse from the air-liquid interface to the outside of the aggregate, but to the surface of the inner air droplet. This may be quantified by considering the proportionality constant, q ¼ 4πR2p ϵ=T p ¼ 8 × 10−10 m2 × s−1 . The last regime leads to the complete infiltration of the aggregate. If the air inside the dry core was in equilibrium with the outside atmosphere, the front dynamics would be

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FIG. 3. Evolution of the duration of the plateau regime as a function of the radius of the air bubble inside the aggregate (filled circle). The average values of experiments corresponding to the same aggregate size are also plotted (open circle) with error bars equal to the difference between minimal (or maximal) and average measured values. Values obtained for 2.5 and 1 mm radius aggregates when the initial pressure is smaller than 1 bar (100, 200, and 400 mbar) are also plotted (filled triangle). Lines are power-law fits T plateau ¼ 4πϵR2p =q, from which the proportionality q is computed. Inset: Plateau duration as a function of the shell thickness for an aggregate of 2.5 mm radius prepared at pressures 1 bar, 400 mbar, and 200 mbar.


described by Eq. (1) with Pin − P0 ¼ 0. All parameters being known, the kinetics may be computed (Fig. 4, dashed lines) and are found to be much faster than the observed one. We thus assume that the pressure of the air trapped inside the inner region is at higher pressure Plast and perform fits of the measured kinetics with Eq. (1), Pin ¼ Plast being the sole fit parameter (Fig. 4, continuous lines). This constitutes a precise measurement of the pressure inside the dry core as the model predicts a strong dependence on the inner pressure (see Fig. 3 of the Supplemental Material [26]). We find Plast − P0 ¼ ð3.2 0.2Þ × 105 Pa, independent of aggregate size (Fig. 4, inset). During this stage, the air leaks out of the aggregate and we observe the formation of bubbles at the outer surface of the aggregate. In a typical experiment, bubbles exit from a unique or from a few (three to the most) points of the aggregate’s surface. This suggests that once a path between the inner bubble and the outer surface has been formed by drainage, it is the preferred one for the subsequent air leakage. Remarkably, the leakage occurs at a constant difference of pressure, smaller than the capillary pressure measured in the first imbibition regime. Our experiments do not allow us to determine the microscopic origin of this difference. One possible mechanism could be the existence of capillary bridges along the path between the dry core and the outer surface, acting as a valve that yields at a difference of pressure equal to Plast − P0 . The imbibition kinetics of a colloidal aggregate thus results from two processes that occur simultaneously: the advancement of the liquid and the exit of the gas. These two phenomena lead to complex kinetics during which the system undergoes a nonstable state. Before this state, the inner gas is trapped, whereas after destabilization of the airsolvent interface, the air leaks at a pressure higher than the atmospheric pressure. The mechanisms responsible for these dynamics are yet to be described and understood at the pore scale. This work is supported by ANR Grant No. AgregEx ANR-13-BS09-0002-02.

FIG. 4. Evolution of the square of the radius of the noninfiltrated core as a function of the logarithm of time, during the last regime, for aggregates radii equal to 1, 1.5, 2, and 2.5 mm (same color legend as Fig. 1). Dashed lines are fits according to the Washburn model, assuming that the pressure of air inside the aggregate is equal to the atmospheric pressure. The continuous lines are fits in which the air is assumed to be at constant pressure in the aggregate Pin Eq. (2). The only fit parameter, Pin ¼ Plast is plotted as a function of the aggregate radius in the inset.

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