Pore-Scale Analysis of Evaporation and Condensation Dynamics in

Aug 2, 2010 - The basic wedge-film solution was scaled up to represent 3D transient evaporation and condensation processes within an assembly of rough...
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Pore-Scale Analysis of Evaporation and Condensation Dynamics in Porous Media Ebrahim Shahraeeni*,†,‡ and Dani Or*,† †

Soil and Terrestrial Environmental Physics (STEP) and ‡Institute of Fluid Dynamics, Department of Mechanical and Process Engineering, Swiss Federal Institute of Technology (ETHZ), Zurich Received April 21, 2010. Revised Manuscript Received June 30, 2010

We developed analytical models for surface- and capillary-assisted condensation and evaporation dynamics considering phase-change behavior in idealized wedge-shaped pores and in adsorbed liquid films as building blocks for condensation and evaporation in granular media. Phase-change rates are important for dust mobilization and deposition, vapor transport through partially saturated media, and for residence times of emitted combustion particulates. The basic wedge-film solution was scaled up to represent 3D transient evaporation and condensation processes within an assembly of rough spherical particles. Model comparisons with experimental data for evaporation from a single meniscus and condensation rates into sand samples show consistent agreement for a range of media and ambient conditions.

Part I: Introduction Vapor condensation or drying of porous media in response to variations in ambient conditions is of interest in many research fields such as food science, meteorology, hydrology, pharmacy, and civil engineering. The wide range of scales ranging from nano to macro scale, and the wide range of applications share a common fundamental process of evaporation or condensation from water menisci or films anchored on solid surfaces. The process can be expressed as the diffusive growth or shrinkage of liquid-vapor interfaces in confined geometry where interfacial forces and ambient conditions control phase-change rates. Vapor condensation is controlled by capillary and adsorptive forces resulting in a moving free boundary (interface) known as a Stefan problem1 in the context of mass diffusion or heat transfer. Under certain conditions, analytical scaling solutions based on similarity variables such as length/sqrt(time) are possible;2,3 however, the solution of most Stefan problems require numerical methods.4 In this study we have used a similarity solution to describe evaporation-condensation in a wedge-shaped pore that is subsequently used as a building block for modeling capillary-assisted vapor behavior in a granular bed. Figure 1 illustrates different scales used for this purpose. Vapor density gradients induce vapor transport through capillaryassisted condensation-evaporation and film adsorptiondesorption sequences that affect mass flux and accumulation in the porous medium. We harness microscale vapor-transport dynamics to deduce macroscale mass and energy transport mechanisms towards better quantification of vapor-transport enhancement processes. *Corresponding authors. (E.S.) E-mail: [email protected]. Tel: þ41446336113. (D.O.) E-mail: [email protected]. Tel: þ41446336015.

(1) Crank, J. Free and Moving Boundary Problems; Oxford University Press: New York, 1987; p 438. (2) Harris, S. Microscopic theory for the diffusive evolution of an isoconcentration surface. Phys. Rev. A 1990, 42, 3504-3504. (3) Harris, S. One-dimensional Brownian motion with an accumulating boundary: asymptotic results. Phys. Rev. A 1989, 40, 387-387. (4) Krapivsky, P. L. Growth of a single drop formed by diffusion and adsorption of monomers on a two-dimensional substrate. Phys. Rev. E 1993, 47, 1199-1199. (5) Philip, J. R. Kinetics of capillary condensation in wedge-shaped pores. J. Chem. Phys. 1964, 41, 911-916.

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Philip5 studied the kinetics of capillary condensation as an unsteady diffusion problem with a moving boundary. He also developed a “unitary approach” for capillary condensation and adsorption and applied it to rough surfaces and porous media to establish equilibrium interface configurations represented as a surface of constant partial specific Gibbs free energy or chemical potential μ composed of an adsorptive component (F) and a capillary component (C).6,7 Philip’s unitary approach is similar to the augmented Young-Laplace (AYL) equation,8,9 which has been simplified for practical applications in porous media to the Shifted Young-Laplace equation (SYL) by Tuller et al.10 considering adsorptive and capillary individual contributions to the shaping of liquid-vapor interfaces. In this study, we investigate the transient behavior of capillary condensation and film adsorption and extend the results to assess the dynamics of mass transport through an assemblage of sand particles due to gradients in saturation (or vapor density). Following a brief literature review, we focus in Part II on a pore-scale mathematical model for a quasi-static approximation of the spatial and temporal distributions of liquid accumulation (or loss) and vapor density for a pore and a population of rough grains. The experimental setup is presented in Part III where the measured data are compared with upscaled modeling results. For validation purposes, we also examined the results of modeling versus experimental data for an evaporation experiment on the pore scale. And finally in Part IV, the dynamics of capillary condensation and film adsorption are investigated using the analytical model to assess the roles of different parameters. (6) Philip, J. R. Unitary approach to capillary condensation and adsorption. J. Chem. Phys. 1977, 66, 5069-5075. (7) Philip, J. R. Adsorption and geometry: the boundary layer approximation. J. Chem. Phys. 1977, 67, 1732-1741. (8) Blunt, M.; Zhou, D.; Fenwick, D. Three-phase flow and gravity drainage in porous media. Transp. Porous Media 1995, 20, 77-103. (9) Novy, R. A.; Toledo, P. G.; Davis, H. T.; Scriven, L. E. Capillary dispersion in porous media at low wetting phase saturations. Chem. Eng. Sci. 1989, 44, 1785-1797. (10) Tuller, M.; Or, D.; Dudley, L. M. Adsorption and capillary condensation in porous media: liquid retention and interfacial configurations in angular pores. Water Resour. Res. 1999, 35, 1949-1964.

Published on Web 08/02/2010

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Figure 1. Hierarchy of process and modeling scales (a) The primary building block, a wedge-shaped pore; (b) cross section of two grains, where water evaporates or condenses on a grain surface and in the contact between grains; (c) two spherical sand grains, where roughness is modeled as a series of cone-shaped pits on their surfaces, the dynamics of mass changes in the cones are considered to be a 3D evolution of the interface in the building block as illustrated in part e and the space between the grains is considered to be the body of revolution formed by the building block around the center line axis of the grains as depicted in part f; (d) a population of rough spheres representing a sand sample exposed to known temperature and ambient relative humidity.

Part II: Adsorptive and Capillary EvaporationCondensation: Theoretical Considerations A 2D wedge-shaped pore with an apex angle equal to 2R illustrated in Figure 2a is considered to be the basic geometry for the problem. Subscript “*” is designated for the quantities on the interface, and subscript “0” indicates ambient saturation values. At time t = 0, the initially dry pore is exposed to ambient air with a vapor density F* of a liquid with surface tension σ and vapor diffusivity D and of initial vapor density FsatS0. defined on the basis of relative vapor pressure S = Pv/Psat (S should be distinguished from liquid saturation in porous media often representing the percent of liquid-filled void fraction). The equilibrium liquidvapor interface profile in such geometry for a known ambient chemical potential is described by the AYL (or SYL) equation.6-10 The evolution of interface position during phase-change dynamics relies on the simple SYL formulation (schematically depicted in Figure 2b). Tuller et al.10 provide an extensive discussion regarding SYL implementation. The formation of a meniscus in thermodynamic equilibrium with its vapor phase is governed by the minimization of interfacial energy reflecting contributions of capillary condensation and adsorptive terms. The Concus-Finn condition11 specifies the critical apex semiangle Rc = π/2 - Θ (Θ is the contact angle) that defines whether a meniscus would fill the wedge because of capillary condensation or simply form liquid bridges to minimize the interfacial energy. Considering for simplicity a zero contact angle Θ, for acute or obtuse apex angles we expect both capillary condensation and adsorptive terms whereas for reflex angles we consider only an adsorptive term. In the following section, we first solve for film adsorption and capillary condensation dynamics and then combine them into a dynamic SYL model. II.1. Dynamics of Film Adsorption. The thickness of a planar adsorbed liquid film in equilibrium with S vapor saturation (11) Concus, P.; Finn, R. On the behavior of a capillary surface in a wedge. Proc. Natl. Acad. Sci. U.S.A. 1969, 63, 292-299.

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is given by12,13 x3 ¼

Asvl 6πFRT ln S

ð1Þ

where x is the film thickness (m), Asvl is the Hamaker constant for solid-vapor interactions through the intervening liquid (J), R is the gas constant (J/kg 3 K), T is the temperature (K), and F is the density of the liquid (kg/m3), which was assumed to be constant independent of the controlling parameters. For an initially dry surface introduced to S0, we iteratively calculate the mass deposition rate using Fick’s second law until the interface reaches equilibrium (S* = S0). We thus need to solve the diffusion growth equation in the geometry of Figure 2c for a constant S*: ∂S ∂2 S ¼ D 2 in x g x ðtÞ ∂t ∂x initial condition : S ¼ S0 at t ¼ 0 boundary condition 1 : S ¼ S on x ¼ x Z t  ∂S boundary condition 2 : ðF - Fsat S0 Þx ¼ Fsat D dt ∂x x ¼ x 0 ð2Þ Equation 2 assumes that diffusion extends between x = x* and x f ¥ over which the vapor saturation changes from S = S* for a known S* value to S = S0 ambient vapor saturation. This is the physical interpretation of the partial differential equation subjected to the initial and first boundary conditions. The second boundary condition expresses the equality of mass deposition with the integration of the mass change rate. Introducing a similarity variable η = x/(4Dt)1/2 enables the transformation of the partial differential equation (eq 2) into an ordinary differential (12) Iwamatsu, M.; Horii, K. Capillary condensation and adhesion of two wetter surfaces. J. Colloid Interface Sci. 1996, 182, 400-406. (13) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. A review of surface forces. J. Dispers. Sci. Technol. 1988, 9, 319-319.

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Figure 2. Schematic of the key phase-change processes in an idealized wedge-shaped pore. (a) Capillary condensation and adsorption of water in a wedge-shaped pore, where the thick curve is the liquid-vapor interface calculated from the AYL equation and the thin line is the interface according to the Kelvin equation. (b) Partitioning of deposited water as described by the capillary condensation term (C) and the adsorptive term (F) based on the SYL approach. (c) Parameters used in the formulation of the capillary condensation problem in a pore with wedge semiangle R, where A(t) is the domain of the solution for eq 7, which is time-dependent. F(t), E(t), and κ(t) are time-dependent functions due to the moving boundary B(t). (d) Parameters used in the formulation of the adsorption problem; the film thickness ν(t) is changing with time while the length of the wedge is assumed to be constant.

equation as2 2

dS dS þ 2η ¼ 0 in ηgη/ dη2 ∂η initial condition : S ¼ S0 as η f ¥ ð3Þ boundary condition 1 : S ¼ S/ for η ¼ " η/ #   dS 2ðF/ - Fsat S0 Þ η/ boundary condition 2 : ¼ dη η ¼ η Fsat Equation 3 with constant S* and η* in the first boundary condition is a transient 1D diffusion equation, and its solution is found by assuming P = dS/dη subjected to P = 0 as η f ¥, which results in P = dS/dη = C1 exp(-η2), which in turn results in S = C2 erf(η) þ C3 subjected to S = S0 as η f ¥ and S = S* as η = η*. From here, the final solution was S - S/ erfcðηÞ ¼ 1erfcðη/ Þ S0 - S/

ð4Þ

the value of x at equilibrium. A procedure starting from S* =0 and continuing with time is applied to this solution to model film thickness dynamics that ultimately end at the equilibrium value of x from eq 1. II.2. Dynamics of Capillary Condensation. For the capillary condensation depicted in Figure 2d, initially vapor condenses at the corner and a liquid-vapor interface forms and growth occurs according to the Kelvin equation, where the curvature κ is related to vapor pressure near the interface   - σκðtÞ cos Θ ð6Þ P/ ðtÞ ¼ Psat exp ðF/ - FÞRT where R is the gas constant, T is the absolute temperature, F* is the liquid density, σ being the liquid surface tension, Θ is the contact angle between liquid and solid surfaces (Θ = 0 for calculations in this study), and F is the vapor density. Considering the geometry in Figure 2d, changes in vapor saturation in the gas phase are explained by the diffusion equation subjected to initial and boundary conditions as follows:

Substituting S from eq 4 into the second boundary condition of eq 3 results in 1 pffiffiffi Fsat ðS0 - S/ Þ H π ¼ pffiffiffi ð5Þ η/ expðη2/ Þ erfcðη/ Þ ¼ F/ - Fsat S0 π

1 ∂S ¼ r2 S in AðtÞ D ∂t initial condition : S ¼ S0 at t ¼ 0 boundary condition 1 : S ¼ S/ ðtÞ on BðtÞ boundary condition 2 : zero flux normal to EðtÞ

where H=[Fsat(S0-S*)/F*]-FsatS0 is a constant that is solely dependent on ambient conditions. Equation 5 is solved for η* to obtain

The moving boundary B(t) of Figure 2d renders the solution transient, for which Philip5 developed a solution by replacing the

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ð7Þ

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Figure 3. Kinetics of capillary condensation in terms of (a) x*(t) and (b) r*(t) for water at 20 C and 1 atm in an initially dry, wedge-shaped pore with R = π/12 for different indicated values of ambient saturation level S0.

moving meniscus with an equivalent circular arc, centered on the apex with radius r* inscribing the same liquid area as inscribed by B(t). The 3D transient problem of eq 7 will be replaced by a 1D transient problem similar to eq 2 with slightly different boundary conditions on the interface that has been described in detail in the original work of Philip.5 The stationary distribution of vapor saturation as a function of nondimensionalized radial coordinate η is S - S/ Eið - η2 Þ ¼ 1Eið - η/ 2 Þ S0 - S/

ð8Þ

where Ei is the exponential integral function. The transcendental equation for η* similar to eq 5 is - η/ 2 expðη/ 2 Þ Eið - η/ 2 Þ ¼

Fsat ðS0 - S/ Þ ¼ H F/ - Fsat S0

ð9Þ

Equation 9 is solved for η*2 to determine the interface position. Having the solution of eq 9 for constant S*, one can initiate S* = 0 for a dry pore introduced to ambient air with vapor saturation S0, calculate H and η* to calculate the new value for S*, and proceed with time until S* and S0 satisfy the Kelvin equation indicating that the interface has attained equilibrium with the ambient air. Converting capillary condensation and film adsorption dynamics with the SYL model (Figure 2b) to changes in mass liquid water in a sample requires the conversion of the 2D representation to a 3D representation and application to a population of rough grains. II.3. Dynamics of Adsorptive and Capillary Condensation: Characteristic Equilibration Times. The condensation process has been decomposed into capillary condensation and film adsorption. In Figure 3, we compare the dynamics of each Langmuir 2010, 26(17), 13924–13936

contribution and its evolution toward equilibrium with ambient vapor-saturation values. The results describe condensation dynamics for a wide range of S0 values from 0.1 to vapor saturated air with S0 = 1. For capillary condensation in a wedge-shaped pore, Philip5 introduced a timelike dimensional group termed the characteristic equilibration time defined as  3 - σβ 50 50r0 3 ðF/ - FÞRT ln S0 ¼ teq, cap ¼ 50θeq, cap ¼ DFsat σβ DL ðF/ - Fsat ÞðF/ - FÞRT  2 σβ F/ - Fsat ð10Þ ¼ 50 ðF/ - FÞRT DFsat jln S0 j3 which characterizes the process dynamics. We derive a similar characteristic equilibrium time for the adsorption process. The explicit relationship between t and x* is obtained from the asymptotic expansion of the complementary error function as ¥ X pffiffiffi x2 ð2nÞ! ð - 1Þn ð11Þ πxe erfcðxÞ  1 þ 2n η!ð2xÞ n¼1 From eq 5 one obtains " # ¥ X 1 ð2nÞ! n η/ expðη/ Þ erfcðη/ Þ  pffiffiffi 1 þ ð - 1Þ π η!ð2η/ Þ2n n¼1 2

H ¼ pffiffiffi π

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Figure 5. Grain size distributions of the samples used in the experiments. These curves were obtained using a laser-diffraction particle size analyzer (LS 13 320, Beckman Coulter, CA).

and L ¼

Figure 4. Experimental setup consisting of a (1) dew point generator, (2) climate-controlled chamber, (3) accurate balance, (4) high-precision temperature and relative humidity sensor, (5) temperature-controlled water bath, (6) measurement and control data logger, and (7) computer for data acquisition and control.

An expression for η*2 (obtained from the truncation of the series) as an explicit function of H = [Fsat(S0 - S*)]/[F* - FsatS0] and valid for sufficiently small η* is 1 ð13Þ η/2 ¼ 2ð1 - HÞ We thus obtain from eqs 1 and 5 F ðS0 - S/ Þ H ¼ sat F -F ( /  sat  Fsat Asvl H ¼ exp F/ - Fsat 6πðF/ - FÞRTx0 3 " #) Asvl - exp 6πðF/ - FÞRTx/3   Asvl considering S ¼ exp 6πðF/ - FÞRTx3

ð14Þ

Using a Taylor expansion of the exponential function reduces eq 14 to   1 1 ð15Þ H ¼ L x/ 3 x0 3 where x0 3 ¼

Asvl 6πðF/ - FÞRT ln S0

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ð16Þ

- Fsat Asvl 6πðF/ - Fsat ÞðF/ - FÞRT

ð17Þ

Inserting eqs 13 and 15 into the integral relation among t, x*, and η = x/(4Dt)1/2 yielded an explicit relation between t and x*: ! Z Z 1 x/ x dx 1 x/ L L ¼ t ¼ x 1 - 3 þ 3 dx 2D 0 η2 ðxÞ D 0 x x0 " #   1 L 1 2 L ¼ ð18Þ þ x/ 1 þ 3 D x/ 2 x0 A timelike dimensional group is dominant at large t as θeq,adp = x05/LD, and leads to a characteristic equilibrium time for adsorption according to 2:5  108 x50 teq, adp ¼ 2:5  108 θeq, adp ¼ LD  2=3 A F/ - Fsat svl 8 ¼ 2:5  10 6πðF/ - FÞRT DFsat jln S0 j5=3

ð19Þ

with 2.5  108 being a proportionality constant to establish the equality. Different dynamics of capillary condensation and film adsorption are reflected in the differences between eqs 10 and 19. For example, whereas teq,cap increases with increasing S0, teq,adp decreases with increasing ambient vapor saturation S0. Different powers of ln S0 in the denominators of eqs 10 and 19 induce a change in the slope of the S0 = 1 curve for adsorption whereas it is constant for capillary condensation. These characteristic times also demonstrate that capillary condensation is a faster process than film adsorption and menisci evolution is completed sooner than film evolution for similar vapor saturation in a wedge. In Part IV, we derive characteristic timescales for capillary and film adsorption for an assembly of wedge-shaped pores and grain packs (whose geometric properties and other characteristics will be explained in the next section). II.4. Modeling an Assemblage of Partially Saturated Rough Grains. II.4.1. Roughness Modeling. A rough surface is described by the so-called Wenzel factor14 (roughness factor (14) Wenzel, R. N. Resistance of solid surfaces to wetting by water. Ind. Eng. Chem. 1936, 28, 988-994.

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Article Table 1. Sample and Chamber Conditions of Different Experiments

A B C D

average temperature (C)

average relative humidity (%)

grain size (μm)

dew point generator temperature set point (C)

chamber

sample

chamber

sample

sample weight (g)

100-300 100-300 600-700 600-700

30 32 32.5 35

28.7 30.0 30.2 33.4

28.5 29.6 29.7 32.9

84.0 97.5 88.8 98.6

85.25 99.6 91.0 99.7

100.53 99.49 198.76 201.23

Figure 6. Water mass deposition due to condensation under different ambient conditions and sample grain sizes obtained by direct measurement (markers) and results of numerical simulations (lines): (a) 100 g of fine sand under low relative humidity, (b) 100 g of fine sand under high relative humidity, (c) 200 g of coarse sand under low relative humidity, and (d) 200 g of coarse sand under high relative humidity. (Chamber and sample conditions are summarized in Table 1.)

or rugosity), which is defined as the ratio of rough surface to smooth surface. Grain surface roughness is represented by cone-shaped pits protruding into a sphere. For a sphere with radius R and n cones with height hi and apex angle 2Ri, the Wenzel factor is n area of rough surface 1 X hi 2 tan Ri WF ¼ ¼ 1þ 2 area of smooth surface 4R i ¼ 1 cos Ri "  # n 1X - 1 hi tan Ri ð20Þ 1 - cos sin 2 i¼1 R In Wenzel’s original work,14 a value between 1.00 and 2.24 has been reported for WF. Robertson15 reported values of silica sand rugosity between 1.05 and 1.4. Borkovec and Wu16 found a fractal description of the specific surface area for a wide range of soil particles to be a ¼ Cλ2 - Ds rs Ds - 3

ð21Þ 2 - Ds

with fractal dimension Ds=2.4, C = 0.54 cm /g, and λ 3

= 0.111

(15) Robertson, R. H. S.; Em€odi, B. S. Rugosity of granular solids. Nature 1943, 152, 539-540.

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for the best fit values to the experimental data. Equation 21 correlates the specific surface area a (m2/g) to the particle radius rs (m). With a known particle density of the grains, WF is related to the spherical particle radius according to WF ¼ 9:8rs 0:4

ð22Þ

where rs is in centimeters. Thus for particle size varying between 100 μm and 1 mm, the Wenzel factor changes from 1.55 to 3.90. In summary, a normal distribution of the Wenzel factor between 1.05 and 4.00 is realistic for soil particles of interest in this work. II.4.2. Contact Points, Coordination Number, and Condensation Sites. The contact points between sand particles form condensation sites, the total number of which in a sample must be estimated. The average number of contacts per particle, known as the coordination number z, is strongly correlated with porosity.17 The minimal average coordination number for the static packing (16) Borkovec, M.; Wu, Q.; Degovics, G.; Laggner, P.; Sticher, H. Surface area and size distributions of soil particles. Colloids Surf., A 1993, 73, 65-76. (17) Donev, A.; Cisse, I.; Sachs, D.; Variano, E. A.; Stillinger, F. H.; Connelly, R.; Torquato, S.; Chaikin, P. M. Improving the density of jammed disordered packings using ellipsoids. Science 2004, 303, 990-993.

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Shahraeeni and Or Table 2. Numerical Values Used in the Simulations quantity

symbol

unit

value

vapor diffusivity vapor density liquid density density jump on the interface surface tension contact angle Hamaker constant specific gas constant of water vapor

D Fsat F* F* - F σ Θ Asvl R

cm2/s g/cm3 g/cm3 g/cm3 mN/m  (degree) J J/g K

0.257 1.73  10-5 1 ∼1 72.75 0 1.9  10-19 0.4615

Figure 7. Comparison of simulation (lines) and experimental results (symbols) for evaporation from pendular water meniscus held between two silica spheres describing the evolution of the airwater interfacial radius of curvature with time (experimental data of Cutts and Burns23). The dashed line describes the nonequilibrium condition by relaxing the end value of the meniscus radius to the experimentally measured curvature. (Note that the predicted characteristic equilibrium time is about 2.5 h instead of the reported value of 30 min in the experiment.)

of d-dimensional frictionless spheres is zn = 2d,18 whereas for spheres with friction, it is zf = d þ 1.19 Hence, in three dimensions zn = 6 and zf = 4. Fournier et al.20 showed by simulations and experiment that the average number of capillary bridges in a pack of glass beads is six for a wide range of water content. We note that the number of liquid bridges or condensation sites per particle is, in general, higher than the coordination number because for a critical distance DC, condensation occurs even in the absence of contact between two grains21 DC ¼

2σ cos Θ Psat ηl kB T ln Pv

ð23Þ

where σ is the surface tension of a fluid interface (N/m), Θ is the contact angle of the liquid on the solid surface, nl is the number of molecules per unit volume of the liquid (5.5  104 mol/m3 at room temperature), kB is the Boltzmann constant (1.381  10-23J/K), T is temperature (K), Psat is saturation, and Pv is the vapor pressure. For subsequent applications, we consider z = 6 when estimating the number of condensation sites in a sample. (18) Alexander, S. Amorphous solids: their structure, lattice dynamics and elasticity. Phys. Rep. 1998, 296, 65-236. (19) Edwards, S. F. The equations of stress in a granular material. Physica A 1998, 249, 226-231. (20) Fournier, Z.; Geromichalos, D.; Herminghaus, S.; Kohonen, M. M.; Mugele, F.; Scheel, M.; Schulz, M.; Schulz, B.; Schier, C.; Seemann, R.; Skudelny, A. Mechanical properties of wet granular materials. J. Phys.: Condens. Matter 2005, 17, S477-S502. (21) Charlaix, E.; Ciccotti, M. Capillary Condensation in Confined Media. In Handbook of Nanophysics; Sattler, K., Ed.; CRC Press: Boca Raton, FL, in press, 2010; Vol. 1.

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Figure 8. Simulation results for dynamics of water mass changes due to capillary condensation (-) and film adsorption (---) for a 500-μm-diameter grain with WF = 2.5 for different ambient vapor-saturation levels. (a) S0 = 1, (b) S0 = 0.9, and (c) S0 = 0.8. Results illustrate how the dynamics of these two condensation mechanisms change with ambient vapor-saturation levels and the rate by which the capillary condensation contribution overtakes film adsorption with increasing vapor-saturation level.

II.4.3. Water Condensation in a Pack of Rough Spherical Grains. The volume of water deposited on a rough grain (known as WF), of radius R with n wedges and apex semiangles Ri and height hi that have been filled to the radius 1ri*(t) and covered with Langmuir 2010, 26(17), 13924–13936

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Figure 9. Simulation results for equilibrium water mass deposition per unit of apex angle and total amount of deposited water mass as a function of apex semiangle R under different ambient vapor saturation. (a) S0 = 0.99 and (b) S0 = 1.00.

an adsorbed film of thickness x*(t), is obtained from eqs 5 and 9 as V1 ðtÞ ¼ WF  4πR2 x/ ðtÞ þ

n X

4 Ri ri 3 ðtÞ 3 1 i¼1

ð24Þ

In each time step for each cone, we check for 1r*i < hi as a physical constraint (Figure 1e). Water deposited in pendular domains between grain contacts is approximated as the volume of revolution resulting from rotating the inscribed area between wedge sides and arc 2r* along the centerline of the grains (Figure 1f). This is designated 2r* to distinguish it from 1r* used for the interface radius of a substituted arc in wedge-shaped pits representing the roughness of the grains. Apex angles assumed to be equal to 22.5 as the first approximation of the circular grains with two hexadecagons are in contact along a common side and separated at the first corner by 45 as the apex angle. The volume of the body of revolution is pffiffiffi 4πð 2 - 1Þ 3 ð25Þ V2 ðtÞ ¼ 2 r ðtÞ 3 < R, where R is the minimum in the grain radii in contact, should be checked at each time step if the evolution of the interface does not exceed the borders of the confinement. II.4.4. Generation of a Pack of Sand Grains. For the numerical generation of a synthetic porous medium, we use a random packing algorithm described by Lehmann et al.22 The particle size distribution in terms of particle radius r is given by a log-normal distribution m(r).We estimated m(r) by fitting to a measured cumulative distribution function of experimental sand samples determined by a particle size analyzer. The fitted lognormal distribution m(r) was scaled by particle mass to estimate the number of particles per particle size (assuming constant particle density) and then integrated over all particle sizes and normalized with P(r) that denotes the probability of finding a particle of size equal to or smaller than r. To determine the size of modeled particles, a random number n between 0 and 1 is chosen and is interpreted as P(r), and a corresponding spherical particle with radius r is generated. The procedure was repeated until the volume 2r*

(22) Lehmann, P.; Wyss, P.; Flisch, A.; Lehmann, E.; Vontobel, P.; Krafczyk, M.; Kaestner, A.; Beckmann, F.; Gygi, A.; Fluhler, H. Tomographical imaging and mathematical description of porous media used for the prediction of fluid distribution. Vadose Zone J. 2006, 5, 80-97.

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of the generated particles equals the measured volume fraction of the solid phase (1 - j)V, with j being the measured porosity and V being the sample volume. For each grain, a random value of WF was assigned and grain roughness was formed by random wedgeshaped pits generated on the grain to satisfy the assumed WF. In a second step related to the spatial positioning of the spheres, particles are distributed randomly in space, starting with the largest particles. If a particle overlaps with another, then a new random position was chosen. To maximize particles in contact, particles with no contact with neighbors were rearranged by moving them in toward the center of the nearest neighbor. This procedure of redistribution was repeated until no contactless particle was found. After redistribution, distances from neighbors are checked for each particle and compared with DC from eq 23, and for those less than or equal to DC, they are considered to be condensation sites as well.

Part III: Model Validation, Experimental Setup, and Results To validate modeling results at the macroscale, we compared the dynamics of a simulated process of condensation in a virtual pack of rough grains exposed to known vapor-saturation levels with corresponding data from experiments using sand samples conducted under controlled ambient conditions. The experimental setup and procedure will be explained in detail in this section. At the microscale, experimental data for an evaporation test from recently published work by Cutts and Burns23 were compared with the simulation results of this work under similar conditions as another test case for the model. An experimental setup shown in Figure 4 was used to test key aspects of model predictions. Sand samples were placed on a temperature-controlled surface within a psychrometrically controlled chamber. The temperature and relative humidity were measured inside the chamber using an HMT337 Vaisala HUMICAP humidity and temperature transmitter (Vaisala, Finland). The measurement range of this sensor is 0-100% for relative humidity and -40-80 C for temperature. The accuracy of the relative humidity measurement including nonlinearity, hysteresis, and repeatability is (1% for RH between 0 and 90% and 1.7% for (23) Cutts, R. E.; Burns, S. E. Evolution of surface area-to-volume ratio for a water meniscus evaporating between contacting silica spheres. J. Colloid Interface Sci. 2010, 343, 298-300.

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RH between 90 and 100%. For a temperature of 20 C, the accuracy is (0.2 C. Two output channels of the sensor were connected to a CR1000 data logger (Campbell Scientific, Logan, UT, USA) that recorded chamber conditions and controlled the water bath temperature set point to keep the surface (and the sand sample) near the target temperature. The chamber was connected to the LI-610 dew point generator (LI-COR, Lincoln, NE, USA) that provided a vapor stream with precisely controlled temperature and 100% relative humidity. The relative humidity inside the chamber was calculated from the temperature set point of the dew point generator (DPG)24 RH ¼

eðTchamber Þ  100% eðTDPG, set Þ 

17:502T eðTÞ ¼ 0:61365 exp 240:97 þ T

ð26Þ  ð27Þ

where e is the saturated air vapor pressure (kPa) at a given temperature T (C). Accurate control over the sample vapor density was indirectly achieved through temperature control using two temperature set points of the dew point generator and water bath. Equations 26 and 27 are used twice, once between the dew point generator and chamber and then between the chamber and sample. The CR1000 data logger was programmed to control the temperature set point of the water bath based on the desired saturation level calculated from eqs 26 and 27 for the chamber temperature and relative humidity. Equation 28 provides the relationships between the water bath temperature set point (Tsample), the desired saturation level of the sample (RHsample), the chamber temperature (Tchamber), and the chamber relative humidity (RHchamber): ! 17:502Tsample RHchamber 17:502Tchamber þ ¼ ln ð28Þ 240:97 þ Tsample RHsample 240:97 þ Tchamber For the experiments, we first set the chamber relative humidity close to the target sample saturation level and then attained equilibrium in the chamber by the dew point generator air flow. By recording the chamber temperature and relative humidity and for the desired vapor-saturation level of the sample, the temperature set point of the water bath is calculated from eq 28. In this way, we were able to control the sample vapor saturation to within (0.05% error. The water bath (Thermo Scientific NESLAB, Karlsruhe, Germany) provided the liquid temperature in the range of -25 to þ150 C, allowing sample temperature control to within (0.1 C. The chamber was placed on an accurate balance (3200 ( 0.01 g) connected to a computer to record the mass change in the sample. Sand samples with two different grain size distributions (coarse, 600-700 μm; fine 100-300 μm) shown in Figure 5 were used in experiments under two different relative humidity values for each grain size for 48 h each. Obtaining similar amount of water mass deposition requires more time for coarse than for fine sand to maintain a similar accuracy. Table 1 summarizes the sample and chamber conditions for different experiments. Sample mass changes recorded by the balance are shown and compared to analytical simulations in Figure 6. Numerical values used for the simulations are listed in Table 2. The agreement between analytical results and experimental data inspires confidence in the (24) LI-COR, I., LI-610, portable dew point generator, instruction manual. LI-COR, I.: Lincoln, NE, 1991.

13932 DOI: 10.1021/la101596y

Figure 10. Simulation of the variation in deposited water mass with grain size (all grains with WF=2.5) under different saturation levels: (a) total water mass change, (b) variations in the capillary condensation contribution with grain size, and (c) variations in the film adsorption contribution with grain size.

numerical procedure, despite small discrepancies attributed to differences between physical and numerical samples. Nonetheless, a better fit might always result from using a more sophisticated model considering a nonzero contact angle and secondary effects such as sequential evaporation-condensation and hysteresis owing to different contact angles during condensation and evaporation. Langmuir 2010, 26(17), 13924–13936

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Figure 11. Simulation of the variations in accumulated water mass with changes in mean grain diameter under different ambient saturation levels for an initially dry sand sample of 100 g: (a) the total water mass accumulated varies over 3 orders of magnitude by varying the mean grain diameter between 100 and 1000 μm; (b) distribution of grain diameters for the simulated sand sample with a mean grain diameter of 100 μm (smallest mean grain size); and (c) distribution of grain diameters for the simulated sand sample with a mean grain diameter of 950 μm (largest mean grain size).

Another form of model testing on the microscale was based on results from a recent experimental study that quantified the rate of water evaporation from receding pendular menisci between two silica spheres under isothermal conditions.23 Cutts and Burns23 reported the dynamics of air-water interface radius changes that could be simulated by reversing the process of condensation to equilibrium with prescribed vapor saturation. Model calculations based on initial and end values of air-water interface radii are depicted in Figure 7 by a solid line. The model predicts faster initial changes followed by a period of small variation in interfacial curvature. To match experimental results better, we relaxed the constraint on the end value of meniscus curvature and set it equal to its measured value (dashed line in Figure 7); the theoretical equilibrium time needed to reach this target curvature was longer than reported, requiring 2.5 h instead of the 0.5 h reported during the experiment. The deviations between model results and experimental data could be attributed to an incomplete equilibrium conditions at the end of the experiment (as suggested by the nonequilibrium curve) that may also reflect vapor diffusion limitations from the interface to bulk air and perhaps to geometrical simplifications of the pendular domain representation by the wedge approximation. Nevertheless, the general agreement at this pore scale provides support for model applicability. Langmuir 2010, 26(17), 13924–13936

Part IV: Discussion IV.1. Comparison of Different Contributions to Condensation. The amount of deposited water due to the two mechanisms of film adsorption and capillary condensation is different as depicted in Figure 8. In general, the capillary component of the chemical potential is dominant close to vapor saturation, whereas adsorptive processes dominate under dry conditions. Figure 8 illustrates that although the film adsorption is higher even for S0 = 0.8, capillary condensation increases by several orders of magnitude and overtakes film adsorption for vaporsaturation levels higher than 0.9. r* is on the order of a micrometer whereas x* changes on the nanometer scale near vapor saturation. Nevertheless, even under high vapor saturation, x* covers all surfaces regardless of medium rugosity and adsorption may remain significant especially for low WF surfaces. IV.2. Key Parameters Controlling Capillary Condensation and Film Adsorption. Capillary condensation and film adsorption dynamics are determined by porous medium characteristics specified primarily by the grain size distribution and surface rugosity and by ambient conditions represented by relative humidity and temperature. The interplay of these factors will be illustrated in the following sections. DOI: 10.1021/la101596y

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IV.2.1. Role of Porous Medium Structural Parameters. IV.2.1.a. Effect of Surface Roughness. The rugosity of a grain in our model is based on the Wenzel factor and is a function of the grain size and apex angle. For a fixed grain size, one can deduce from eq 20 that the grain roughness increases with decreasing apex angles. For a single wedge with apex semiangle R as depicted in Figure 9, the mass deposition behavior per unit angle decreases monotonically with R because widening the apex angle weakens the capillarity; however, because the mass gain due to capillary condensation reflects the interplay between the available space for deposition and the strength of capillarity, the total deposited mass is not monotonous and for a certain angle the amount of deposited mass is maximized. For a given grain size distribution, there is an optimum rugosity that maximizes the mass deposition capacity of the medium under a given vapor-saturated condition. IV.2.1.b. Effect of Grain Size Distribution. The grain size distribution is a significant factor affecting the mass adsorption capacity, as shown in Figure 10, for mass deposition as a function of grain diameter. Except for very fine grains where the storage in surface roughness is reduced, the mass accumulating as a result of capillary condensation on a single grain is almost independent of grain size in a fixed WF (Figure 10b) whereas the accumulated mass due to film adsorption increases with the grain radius (Figure 10c). The total amount of water adsorbed by a single grain increases slightly with radius (Figure 10a), However, for the upscaled sample, decreasing the specific surface area with radius decreases the total amount of deposited water mass with increasing grain size. Figure 11 compares the total deposited water mass as a function of grain size for different ambient vapor saturations for a fixed sample weight. We considered different packings with a constant weight of 100 g generated as explained previously in Part II, section II.4.4 such that their mean grain diameter varied between 110 and 950 μm; for each value of the mean grain diameter, there is a normal distribution around the mean value (the first and last grain size distributions m(r) are illustrated in Figure 11b,c). These packs of grains were exposed to different levels of ambient vapor saturation, and the resulting water mass adsorption process was calculated using the analytical model. The final amount of deposited water after establishing equilibrium versus the mean grain diameter is shown in Figure 11. The results show that the mass adsorption capacity of a medium may vary over 3 orders of magnitude with changes in grain size within a common range of sand size. IV.2.2. Effects of Ambient Conditions. IV.2.2.a. Role of Ambient Temperature. Ambient temperature affects eqs 2 and 7 in different ways. It appears explicitly in eqs 1 and 6 relating interface curvature and film thickness to chemical potential. Second, temperature affects the surface tension σ and hence the capillarity behavior. E€otv€os’ rule is often used to represent surface tension-temperature dependency as25 σV 2=3 ¼ kðTc - TÞ

ð29Þ

where V is the molar volume of the substance, Tc is the critical temperature, and k is a constant that is valid for most substances and is equal to 2.1  10-7 (J/K 3 mol2/3). For water, it yields σ ¼ 72:75½1 - 0:002ðT - 291Þ

ð30Þ

Figure 12 depicts the dependency of mass change on temperature for a single 500-μm-diameter grain and a Wenzel factor equal (25) Paszli, I.; Laszlo, K. Molar surface energy and E€otv€os’s law. Colloid Polym Sci. 2007, 285, 1505-1508.

13934 DOI: 10.1021/la101596y

Figure 12. Simulation of the variations in the deposited water mass in equilibrium with ambient temperature for a 500-μm-diameter grain with WF = 2.5 at different vapor-saturation levels: (a) total water mass deposited, (b) changes in the capillary condensation contribution with temperature, and (c) changes in the film adsorption contribution with temperature.

to 2.5 at different saturation levels. It reveals the stronger dependency of the capillary contribution to temperature than to film absorption. As one might expect, temperature reduction increases the capillarity strength and enhances the amount of condensed water. IV.2.2.b. Role of Relative Humidity. The ambient relative humidity (vapor saturation) is the key controlling variable affecting both the amount of mass deposition and the dynamics. Figure 13 illustrates the dependency of capillary condensation and film adsorption on the relative humidity, confirming results in Figures 9, 10, and 12. For a fixed radius, increasing WF results in Langmuir 2010, 26(17), 13924–13936

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Figure 13. Simulation of the variations in deposited water mass on a single grain with ambient saturation for different Wenzel factors, grain diameters, and ambient temperatures: (a) changes in deposited water mass due to capillary condensation versus the ambient vapor-saturation level for a grain with 1 mm diameter at 10 C for three different Wenzel factors of 1.5, 2.5 and 3.5, (b) water mass deposition due to film adsorption versus the ambient vapor-saturation level for the same conditions as in part a, (c) water mass deposition due to capillary condensation versus the ambient vapor-saturation level for a grain with a Wenzel factor of 2.5 and three different diameter values of 100 μm, 550 μm, and 1 mm at a temperature of 35 C, (d) water mass deposition due to film adsorption for the same conditions as in part c, (e) water mass deposition due to capillary condensation versus the ambient vapor-saturation level for a grain with a Wenzel factor of 2.5 and a diameter of 550 μm at three different temperature (10, 35, and 60 C), and (f) water mass deposition due to film adsorption for the same conditions as in part e.

more deposited water by capillary condensation and film adsorption as seen in Figure 13a,b. As shown in Figure 10, capillary condensation is independent of the individual grain radius, as also depicted in Figure 13c. Mass deposition increases with relative humidity nonmonotonically with respect to grain radius. In contrast, mass deposited due to film adsorption continuously increases with increasing grain radius as shown in Figure 13d. Among the four controlling parameters, the temperature effect is the weakest in terms of the deposited mass as confirmed by the narrow range of deposited Langmuir 2010, 26(17), 13924–13936

mass variations for temperature variations in the range of 10 to 60 C for both capillary condensation and film adsorption as shown in Figure 13e,f.

Part V. Conclusions The study quantifies the dynamics and amount of vapor deposition and evaporation assisted by capillarity and surface adsorption onto natural soil and rock surfaces. These processes play an important role in the onset and rates of mobilization of dust and other wind erosion processes, travel distances, and DOI: 10.1021/la101596y

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residence times of emitted particles by various combustion processes. The results of this study are particularly important for modeling pore-scale phase change processes in unsaturated porous media to resolve issues related to postulated vaportransport enhancement mechanisms.26 We investigated key elements affecting the dynamics of condensation (and evaporation) on the pore scale and scaled up the results to simulate mass adsorption/desorption processes in a granular bed. At the pore scale, changes in ambient temperature and vapor density conditions induce phase changes at interfaces and establish a transient vapor flow regime resulting in the evolution of the interfaces and a change in mass due to liquid phase deposition. For dry conditions, capillary condensation in a single pore is extremely rapid (microseconds for RH = 50%) and tends to slow down for high vapor saturation (e.g., hours for RH = 99.99%). In contrast, water film adsorption (by van del Waals forces) exhibits complex dynamics with a rapid film deposition within microseconds that decreases with increasing RH, exhibiting the considerably slow attainment of equilibrium at high vapor saturation (e.g., about a day for RH = 99%). Figure 3 depicts (26) Philip, J. R.; De Vries, D. A. Moisture movement in porous materials under temperature gradients. Trans. Am. Geophys. Union, 1957, 38, 222-231.

13936 DOI: 10.1021/la101596y

Shahraeeni and Or

differences between capillary condensation and film adsorption dynamics as specified by eqs 10 and 19, indicating that capillary condensation attains equilibrium faster than film adsorption. At low vapor saturation, mass deposition by film adsorption is high, and with increasing vapor density, mass deposition by capillary condensation becomes dominant. Results of vapor condensation and mass changes in a scaled up pack of rough sphere models were compared with experimental data to evaluate model performance. Naturally, the dominant factor was vapor saturation that interacts with media structural properties such as grain size distribution and surface roughness to affect the dynamics and rates of mass deposition (or loss). The structure of deposited mass and the rate by which mass is exchanged between various domains in partially saturated porous medium provides the basis for the quantification of vapor transport due to capillary and temperature gradients that would be pursued in subsequent work. Acknowledgment. We gratefully acknowledge funding by Swiss National Science Foundation project 2000021-113676/1 and the generous assistance of Daniel Breitenstein and Dr. Peter Lehmann in various aspects of the study.

Langmuir 2010, 26(17), 13924–13936