Water Filling and Emptying Kinetics in Two-Dimensional Hexagonal

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Water Filling and Emptying Kinetics in Two-Dimensional Hexagonal Mesoporous Silica of Same Pore Diameter but Different Pore Lengths Junho Hwang, Kosuke Yanagita, Kazuki Sakamoto, WeiLun Hsu, Sho Kataoka, Akira Endo, and Hirofumi Daiguji Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b01261 • Publication Date (Web): 25 Jul 2019 Downloaded from pubs.acs.org on July 28, 2019

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Water Filling and Emptying Kinetics in TwoDimensional Hexagonal Mesoporous Silica of Same Pore Diameter but Different Pore Lengths Junho Hwang1, Kosuke Yanagita1, Kazuki Sakamoto2, Wei-Lun Hsu1, Sho Kataoka3, Akira Endo3 and Hirofumi Daiguji1,2* 1Department

of Mechanical Engineering, Graduate School of Engineering, The University of

Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2Division

of Environmental Studies, Graduate School of Frontier Sciences, The University of

Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8563, Japan 3National

Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba

Central 5-2, 1-1-1 Higashi, Tsukuba, Ibaraki 305-8565, Japan

*Corresponding author Email: [email protected]

1 ACS Paragon Plus Environment

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ABSTRACT The effect of pore length on the water filling and emptying rates was studied using mesoporous silica (MPS) of same pore diameter but different pore lengths. The pore diameter of the synthesized MPS was ~8 nm, while the average pore lengths were 460, 1,770, and 4,000 nm. The gravimetric method was employed to record the time course of the adsorbed mass of water in the MPS at 298 K and 1 atm. In both the filling and emptying processes, the relaxation curves (time course of adsorbed mass of water per unit mass of sample) were not significantly related to the pore length. This independence of the initial adsorption and desorption rates on the pore length suggests that the surface of the MPS aggregates is the bottleneck in the overall adsorption and desorption processes and that the initial mass flux in each nanopore is inversely proportional to the pore length. Furthermore, because the relaxation times to reach the equilibrium states were independent of the pore length, the mass flux of water uptake, release, and transport probably increase with an increase in the pore length during the entire adsorption and desorption processes. A transport model to describe these phenomena was proposed.

KEYWORDS: capillary condensation, hydrophilic nanopores, vapor diffusion, water columns, particle aggregation


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INTRODUCTION A detailed understanding of the water filling and emptying processes in nanoporous media is of great importance in various fields of science and technology including dehumidification, evaporative cooling, food/textile preparation, energy conversion, and storage applications.1‒3 Many studies on water filling in nanoporous media have mainly focused on liquid water flow based on the competing action of the capillary and viscous drag forces (Lucas‒Washburn Law) when either end of the nanopores is in contact with a liquid water bath.4,5 On the other hand, when the nanoporous media are exposed to water vapor, the water filling process is more complex.6,7 Thus, both the vapor and liquid flows and phase-change phenomena must be considered inside nanoporous media to totally understand the water filling process. However, the detailed kinetics and major contributions to the entire process remain unclear, especially when the nanoporous media have complicated structures that include interconnected nanopores and wide pore size distributions. Recently, our gravimetric measurement of water vapor adsorption into mesoporous silica (MPS) with well-ordered two-dimensional hexagonal pore arrays, via a stepwise change in the relative humidity of a supplied gas from the onset points of capillary condensation, revealed that the adsorption rate gradually decreased with time even if the initial adsorption rate was much lower than that predicted by the Lucas‒Washburn law.8,9 This suggested that as time progressed, the water transport inside the nanopores became the rate-controlling step in the overall adsorption process, even if the initial adsorption rate was much lower than the intrinsic transport rate of the nanopores. The exact mechanism to limit the transport rate inside the nanopores has yet to be clarified.


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Although it is well known that the onset points of capillary condensation and evaporation depend on the pore size (Kelvin equation) and surface properties,10,11 the kinetic studies of capillary condensation and evaporation in nanopores are limited. Our recent molecular simulations on the capillary condensation of water in hydrophilic nanopores revealed that water vapor condenses near the entrance of the nanopores and the condensed water moves into the inner part of the nanopores.12,13 Furthermore, as previously mentioned, our recent gravimetric measurement studies also indicated that when the magnitude of a stepwise change in the relative humidity from the onset point of the capillary condensation or evaporation is sufficiently large, the time course of the adsorbed amount can be well fitted by the square root function [ f  t   t , where t is time].8,9 This suggests that both the filling and emptying processes should be related to the “classical Stefan problem,” which describes the moving interface between the vapor and liquid phases.14 According to this theory, the filling rate should be controlled by the transport rate of the vapor-liquid interface and the rate should be determined by the difference in the chemical potential between the initial and final relative humidities.15,16 In a continuum fluid, the flow rate (volume per unit time) is proportional to the pressure gradient and thus, it is inversely proportional to the pore length at a fixed pressure drop between the two ends of the pore. The Hagen‒Poiseuille and Darcy flows obey this rule. However, in capillary condensation and evaporation in nanopores, the flow rate will not always be inversely proportional to the pore length because the entire pore is not always filled with liquid and the local and instantaneous force balance is not as simple as that of a continuum fluid.17‒20 The effect of the pore length on the flow rate has not yet been clarified. The rapid development of nanofabrication techniques has recently led to the successful synthesis of MPS materials with well-ordered twodimensional hexagonal pore arrays and their filling dynamics have been quantitatively


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analyzed.21,22 Such MPS materials can also be employed as bulk porous materials for many fundamental studies of large-scale mass transport and phase transition phenomena in nanopores, including surface-charge-governed ion/proton transport23,24 and nanofluidic energy conversion25 devices. In this study, to clarify the effect of pore length on the water filling and emptying rates, we synthesized MPS particles of same pore diameter (~8 nm) but three different pore lengths (460, 1,770, and 4,000 nm) and employed the gravimetric method to construct their water adsorption and desorption relaxation curves at 298 K and 1 atm. In both the filling and emptying processes, the relaxation curves (time course of adsorbed mass of water per unit mass of the sample) were not significantly related to the pore length. This suggests that an increase in the pore length results in a larger amount of water uptake or release in each pore and that the water is transported faster inside the pore. Finally, a model to describe the transport of water in the MPS particles was proposed.

EXPERIMENTAL METHODS Synthesis of MPS Particles MPS particles having the same pore diameter but different lengths (rod-like mesoporous SBA15 silica) were synthesized by two methods proposed by Ding et al.26 and Wang et al.:27 Ding’s method26 uses the same procedure reported by Stucky et al.28 but with no addition of inorganic salts (KCl). Thus, SBA-15 rods of various pore lengths were produced by simply tuning the HCl concentration without any additives. Specifically, 1.2 g triblock copolymer Pluronic P123 (EO20PO70EO20) was added to 60 mL HCl aqueous solution of different HCl concentrations: 0.5, 1.5, 2.5 M. The mixture was stirred at 40 °C until the polymer dissolved completely. Next, 2.5 mL


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(0.0113 mol) tetraethyl orthosilicate (TEOS) was added to this solution under vigorous stirring for 5 min, after which the mixture was maintained under static conditions at 40 °C for 20 h. After the reaction, the mixture was transferred to an autoclave for hydrothermal treatment at 100 °C for 24 h. The resultant white precipitates were filtered and dried in air. Surfactants were removed by calcination in air at 550 °C for 6 h. Wang’s method27 followed the same procedure as Ding’s method but in the presence of glycerol. Specifically, equal amounts (1.8 g) of P123 and glycerol were dissolved in 69 g HCl aqueous solutions (HCl concentrations: 0.5, 1.5, 2.5 M) and then stirred at 35 °C to attain a transparent solution. Next, 3.87 g (0.0186 mol) TEOS was added to the solution under vigorous stirring for 5 min. Subsequently, the mixture was maintained under static conditions at 35 °C for 24 h, followed by aging at 100 °C for 24 h. The solid products were collected by filtration, washed with water, and dried at 80 °C overnight in air. The resulting powders were calcined at 550 °C for 5 h to remove the surfactant.

Characterization of MPS Samples The well-ordered MPS structure with two-dimensional hexagonal pore arrays was confirmed by scanning electron microscopy (SEM; S4800, Hitachi High-Technologies, Japan), field emission scanning electron microscopy (FE-SEM; SU9000, Hitachi High-Technologies, Japan), and X-ray diffractometer (XRD; D8 Advance, Bruker AXS, Germany) operated at 40 kV and 40 mA and CuKα radiation (λ = 0.154 nm). The MPS pore structure was characterized by measuring the adsorption and desorption isotherms of nitrogen at 77 K using an automatic adsorption measurement apparatus (BELSORP-max, MicrotracBEL Corp., Japan). The sample was degassed at 573 K for 8 h below 8×10−3 Pa prior to the adsorption measurements.29


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Experimental Apparatus and Water Adsorption and Desorption Measurements In this study, a low temperature vapor adsorption measurement system (MSB-flow-SPT, MicrotracBEL Corp., Japan) integrated with a magnetic suspension balance (MSB; IsoSORP E10, Rubotherm, Germany) was employed to measure the adsorbed mass of water vapor.8,9, The detection limit was 10 μg. A schematic diagram of the adsorption measurement system is presented in Figure 1. This apparatus can measure the time course of the adsorbed mass of water by a gravimetric method in a flow system under atmospheric pressure and 298 K. The flow rate and absolute humidity of the carrier gas (He) were controlled by two mass flow controllers of branching tubes, one for the dried He gas and the other for water-saturated He gas. These gases were mixed at a constant flow rate of 200 standard cubic centimeters per minute (sccm; standard conditions: 273.15 K and 1 atm). Moist He gas at a fixed absolute humidity was supplied into the test section where an electrolytically polished stainless-steel basket containing the samples was suspended by the MSB. The sample weight in the basket was ~0.5 mg so that the transport of water vapor outside the sample was not the rate-controlling step (see Figure S1 in Supporting Information S1). The temperature of the test section was maintained at 298 K by a temperaturecontrolled oil bath. Before entering the test section, the supplied gas passed through a channel at the same temperature as the test section. The relative humidity of He gas was changed in a stepwise manner by altering the ratio of the dry-to-water-saturated He gas. Prior to the adsorption measurements, all the samples were evacuated at 573 K for 8 h below 8 × 10-3 Pa.29


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Figure 1. Schematic diagram of the low temperature vapor adsorption and desorption measurement system.

RESULTS AND DISCUSSION Synthesized MPS Samples Figure 2a–f displays the SEM images for the MPS samples synthesized with no additive and with glycerol as additive, respectively. For each set of synthetic conditions, the concentrations of the HCl aqueous solutions were 2.5 M, 1.5 M, and 0.5 M. We confirmed that all the synthesized MPS samples displayed two-dimensional hexagonal pore arrays and the pore length was equal to the grain size (see Figure S2 in Supporting Information S2 and Figure 8). The SEM images clearly reveal that the pore length decreases with an increase in the concentration of the HCl aqueous solutions; the pore length also decreases with the addition of glycerol.


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Figure 2. Scanning electron microscopy (SEM) images of the mesoporous silica (MPS) samples synthesized with (a, b, and c) no additive and (d, e, and f) glycerol as the additive at three different HCl aqueous solution concentrations: 2.5 M, 1.5 M, and 0.5 M, respectively. Magnification is identical for all the images (bar = 2 μm).

Figure 3a,b illustrates the nitrogen adsorption and desorption isotherms of the MPS samples synthesized with no additive and with glycerol as additive, respectively, at the three different HCl concentrations. Although the same template molecule was employed for all the MPS samples, the nitrogen adsorption and desorption isotherms of the different samples did not completely match. Thus, three out of the six samples: no additive at 2.5 M (Figure 2a), no additive at 1.5 M (Figure 2b), and with glycerol as additive at 0.5 M (Figure 2f) were selected and their nitrogen adsorption and desorption isotherms replotted as presented in Figure 3c. These three isotherms, labeled MPS 1 (short pore; pore length lp = 460 nm), MPS 2 (intermediate pore; lp = 1,770 nm), and MPS 3 (long pore; lp = 4,000 nm), respectively, well-matched each other. The average pore length was


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calculated from the FE-TEM images as shown in Figure S2. The structural properties of these three MPSs are summarized in Table 1.

Figure 3. Nitrogen adsorption and desorption isotherms of the mesoporous silica (MPS) samples synthesized with (a) no additive and (b) glycerol as additive at different HCl aqueous solution concentrations: 2.5 M, 1.5 M, and 0.5 M. (c) Replotted nitrogen adsorption and desorption isotherms of the MPS samples synthesized with: no addition at 2.5 M (MPS 1), no addition at 1.5 M (MPS 2), and with glycerol at 0.5 M (MPS 3).

Table 1 Pore structural properties of MPS samples synthesized with: no addition at 2.5 M (MPS 1), no addition at 1.5 M (MPS 2), and with glycerol at 0.5 M (MPS 3).

Sample

dpa

d100b

wc

ABETd

Vpe

lpf

(Synthesized condition)

(nm)

(nm)

(nm)

(m2 g-1)

(cm3 g-1)

(nm)

MPS 1 (2.5N)

7.6

8.7

2.5

869

1.10

460 ± 70

MPS 2 (1.5N)

7.9

9.6

3.2

870

1.09

1,770 ± 400

MPS 3 (G0.5N)

7.6

9.6

3.5

933

1.03

ca. 4,000

aPore

diameter calculated using the nonlocal density functional theory (NLDFT).


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b(100) cPore

interplanar spacing.

wall thickness obtained by subtracting the pore diameter, dp, from the unit cell dimension,

a, which represents the calculated d100 using the formula: a = d100 × (2/31/2). dBrunauer–Emmett–Teller ePore

(BET) specific surface area.

volume calculated from the αs-plot.

fAverage

pore length calculated from the FE-SEM images.

Adsorption and Desorption Isotherms of Water Figure 4a,b presents the adsorption and desorption isotherms of water obtained by the gravimetric and volumetric methods, respectively. The detailed data of the gravimetric measurements are shown in Figure S3 (Supporting Information S3). The adsorption and desorption isotherms from the gravimetric method were near-identical to those measured by the volumetric method. All the adsorption and desorption isotherms were classified as type V according to the IUPAC classification. The hysteresis between the adsorption and desorption isotherms was pronounced and the difference in the relative humidity between the adsorption and desorption isotherms was ~0.10 at an identical adsorbed mass. The adsorbed mass per unit mass of MPS samples 1, 2, and 3 at p/p0 = 0.90 were ~1.0, 0.99, and 0.85 g g-1, respectively. The adsorbed mass of MPS 3 near the pore-filling state was slightly smaller than that of MPS 1 and MPS 2. Because the pore volumes for MPS 1, 2, and 3 were 1.10, 1.09, and 1.03 cm3 g-1, respectively (Table 1), the water densities at p/p0 = 0.90 confined in the samples were calculated to be ~0.909, 0.908, and 0.825 g cm-3, respectively. This suggests that the water density in the nanopores was lower than that in the bulk (0.997 g cm-3 at 298 K and 1 atm).30,31 The onset points of capillary condensation


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and capillary evaporation were p/p0 = 0.73 and 0.75, respectively, for all three samples, implying that all three samples have the same pore size and surface properties.

Figure 4. Water adsorption and desorption isotherms measured by (a) gravimetric and (b) volumetric methods on the samples with no addition at 2.5 M (MPS 1), no addition at 1.5 M (MPS 2), and with glycerol at 0.5 M (MPS 3), respectively.

Effects of the Magnitude of the Stepwise Change in Relative Humidity on the Relaxation Curves Figure 5 presents a schematic illustration of water adsorption and desorption in a twodimensional hexagonal MPS by the stepwise change in relative humidity from the onset points of capillary condensation and evaporation. At the onset point of capillary condensation, water layers are formed on the inner surface of the nanopores. When the relative humidity increases stepwise from this point, capillary condensation occurs and the nanopores are filled with water. On the other hand, when the relative humidity decreases stepwise from this point, capillary evaporation occurs and only thin water layers remain on the inner surface of nanopores. In this study, to clarify the water filling and emptying rates of these processes, relaxation curves (adsorbed mass vs. time


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curves) were plotted for three different stepwise changes in relative humidity from the onset points of capillary condensation and evaporation, Δ(p/p0)Ads and Δ(p/p0)Des, as illustrated in Figure 5.

Figure 5. Schematic illustration of water adsorption and desorption in two-dimensional hexagonal mesoporous silica (MPS) by the stepwise change in relative humidity from the onset points of capillary condensation and evaporation.

Figure 6a–f displays the relaxation curves of the adsorption and desorption of water obtained upon three different stepwise changes in the relative humidity from the onset point of capillary condensation (0.73 to 0.80, 0.85, and 0.90) and evaporation (0.75 to 0.68, 0.65, and 0.60) on MPS 1, MPS 2, and MPS 3. In both the adsorption and desorption processes, the relaxation rates (dm/dt) increased with an increase in the magnitude of the stepwise change in the relative humidity. If the condensed phase is formed in the nanopores, then according to the classical Stefan Problem, which describes the moving interface between the vapor and liquid phases,32 the relaxation curves can be fitted by the square root function. Thus, the fitting function is given by m  t   ma  sgn  m  k  t  ta  ,

(1) 13

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where k, ma, and ta are the fitting parameters; Δm is the total mass change (Δm > 0 in the adsorption process and Δm < 0 in the desorption process); and sgn(x) is the signum function of x defined as sgn(x) = {–1 if x < 0; 0 if x = 0; 1 if x > 0}. The fitted parameters are summarized in Table S1–S3 (Supporting Information S4).


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Figure 6. Relaxation curves of (a, b, and c) adsorption and (d, e, and f) desorption of water on the samples with no addition at 2.5 M (MPS 1), no addition at 1.5 M (MPS 2), and with glycerol at 0.5 M (MPS 3), respectively, obtained upon three different stepwise changes of the relative humidity. At t = 0 the relative humidity changed from 0.73 to 0.80, 0.85, and 0.90, respectively, in the adsorption process and from 0.75 to 0.68, 0.65, and 0.60, respectively, in the desorption process.

Figure 7 illustrates the initial relaxation rates of adsorption and desorption as a function of the difference in chemical potential, Δµ, between the pre- and post-states of the stepwise change in relative humidity. The initial relaxation rate, (dm/dt)0, was calculated from the gradient of m with respect to t at t = 0. Assuming the ideal gas law for water vapor:


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  p p0 post   RT ln    p p0  pre 

 ,  

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(2)

where R and T are the gas constant and temperature, respectively. As shown in Figure 7, the difference in (dm/dt)0 attributed to the pore length was much smaller than the difference in (dm/dt)0 attributed to Δµ. This implies that the initial relaxation rates (dm/dt)0, defined as the initial rate of adsorbed mass of water per unit mass of sample, are not significantly related to the pore length. This suggests that the initial mass flux of water in each nanopore should be closely related to the pore length because at an identical sample mass the number of nanopores is inversely proportional to the pore length.

Figure 7. Initial relaxation rates of water adsorption and desorption as a function of the difference in chemical potential, Δµ, between the pre- and post-states of the stepwise change in relative humidity.

Water Filling and Emptying Kinetics in the MPS Particle Aggregates


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A relaxation curve is useful to evaluate the kinetic properties of the adsorbents. However, it does not always express the kinetic properties of nanopores with different pore dimensions. This study proposes a simple model to predict the mass flux of water in each nanopore during the water adsorption and desorption processes. The MPS particles (Figure 8a), which include well-aligned nanopores, form an aggregate (Figure 8b).

Figure 8. Field emission scanning electron miscroscopy (FE-SEM) images of (a) two-dimensional hexagonal mesoporous silica (MPS) particles sinthsized with no addition at 2.5 M (MPS 1) and (b) the MSP 1 aggregates. (c) Model to predict the mass flux of water vapor in the MPS aggregate, Jg, and the mass flux of water in the MPS nanopore, Jp, during the water adsorption and desorption processes.


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The shape of the MPS particle is assumed to be a polygonal prism and cylindrical pores are vertically aligned to the two parallel planes (“Particle” in Figure 8c). The area of the entire pore entrances on the two parallel planes per unit mass of the sample, Ap (cm2 g-1), is thus given by

Ap  2Vp lp ,

(3)

where Vp (cm3 g-1) is the specific pore volume and lp (cm) is the pore length. The pore number density per unit mass of the sample, Np (g-1), is given by

N p  Ap 2ai  Vp

 r l  , 2 p p

(4)

where ai (cm2) is the area of a single pore entrance and rp (cm) is the pore radius. It is also assumed that the entire inner surface of the nanopores is covered with an immobile water film of thickness t owing to the strong intermolecular interaction between the water and pore walls (“Nanopore” in Figure 8c). The effective area of the entire pore entrances on the polygonal plane per unit mass of the sample, Ap,eff (cm2 g-1), is calculated from Ap and t (= 0.5 nm).8 The calculated parameters Np, Ap, and Ap,eff (Table 2) are much larger for MPS 1 than those of MPS 2 and MPS 3.

Table 2 Calculated parameters of Np, Ap, and Ap,eff in MPS samples with no addition at 2.5 M (MPS 1), no addition at 1.5 M (MPS 2), and with glycerol at 0.5 M (MPS 3). Sample

Np (g-1)

Ap (cm2 g-1)

Ap,eff (cm2 g-1)

MPS 1 (2.5N)

5.28 × 1016

4.78 × 104

3.60 × 104

MPS 2 (1.5N)

1.26 × 1016

1.23 × 104

9.36 × 103

MPS 3 (G0.5N)

4.42 × 1015

3.90 × 103

2.94 × 103

Furthermore, it is assumed that the MPS particles form the aggregate (“Aggregate” in Figure 8c). The entire surface area of the MPS aggregates per unit mass of the sample is given by 18 ACS Paragon Plus Environment

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Aa  N a aa ,

(5)

where Na (g-1) is the number density of the aggregates per unit mass of the sample and aa (cm2) is the surface area of the aggregate. If the radius of the spherical aggregate ra is 25 μm and its density ρa is 0.5 g cm-3, Aa = 4πra2 / [(4/3)πra3ρa] = 2.4×103 (cm2 g-1). The density of the aggregate ρa can be determined by the following equation:

a  f a  vp  vSiO  ,

(6)

2

where vp is the MPS pore volume, vSiO2 is the specific volume of SiO2, and fa is the packing fraction of the MPS particles in the aggregate. Here, the parameters: vp = 1.10 cm3 g-1, vSiO2 = 0.377 cm3 g-1, and fa = 0.74, which are equal to those of hexagonal close-packed crystal structure, are employed. The data in Table 2 reveal that the effective area of the entire pore entrances on the polygonal planes per unit mass of the sample, Ap,eff, could be larger than the entire surface area of the MPS aggregates per unit mass of the sample, Aa, (Ap,eff > Aa). This suggests that the surface of the MPS aggregates is the bottleneck in the overall adsorption process from the bulk gas phase to the nanopores. The adsorption rate of water, dmw/dt (g s-1), can be expressed by the mass flux of the water vapor on the surface of the MPS aggregate, Jg(t) (g m-2 s-1), or the mass flux of water inside the nanopore, Jp(t) (g m-2 s-1), as follows: dmw  J g  t  aa  N a mMPS   J g  t  Aa mMPS , dt

(7)

dmw  J p  t   2ai,eff   N p mMPS   J p  t  Ap,eff mMPS , dt

(8)

In the relaxation curve measurement, the sample mass, mMPS, was so small that the initial rate (dmw/dt)0 was proportional to mMPS (Supporting Information S1). Furthermore, (dm/dt)0 (=(d(mw/mMPS)/dt)0) did not depend on the pore length at identical Δµ (Figure 7). Eq. 7 indicates 19 ACS Paragon Plus Environment

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that Jg(0) should be independent of the type of samples because Aa is independent of the type of samples. Moreover, if the transport inside the pore is the rate-controlling step at t = 0, Jp(0) should be independent of the sample type and thus, (dmw/dt)0 should increase with a decrease in the pore length (eq. 8). However, because the surface of the MPS aggregate is the bottleneck in the overall adsorption and desorption processes (Ap,eff > Aa) and the transport outside the pore is the ratecontrolling step at t = 0, Jp(0) is inversely proportional to Ap,eff and thus, (dmw/dt)0 does not depend on the pore length (eq. 8).

Water Filling and Emptying Kinetics in Nanopores with Different Pore Lengths More interestingly, not only Jp(0) but also Jp(t) (t > 0) was also inversely proportional to Ap,eff. In Figure 6a–f, the slopes of the relaxation curves (gradient of m with respect to t) indicate the rate of water adsorption and desorption. From eq. 8, the rate of water adsorption and desorption can be expressed as:

dm d  mw mMPS    J p  t  Ap,eff , dt dt

(9)

Figure 6a–f reveals that the relaxation curve does not depend greatly on the type of samples. If the range of the stepwise change in relative humidity is the same, the relaxation curves are close to each other. This suggests that Jp(t) is inversely proportional to Ap,eff in the entire adsorption and desorption processes (eq. 9). Figure 9a,b illustrates the relationship between the relaxation curve and the pore length. For simplicity, two samples with different pore lengths were considered: the second sample (Figure 9b) was half as long and twofold more than the first one (Figure 9a) so that the total masses of the two samples were identical. In the first sample (Figure 9a), the initial adsorption rate (dm/dt)0 is determined by the transport outside the nanopore. If this transport were the rate-controlling step 20 ACS Paragon Plus Environment

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throughout the entire adsorption process, the relaxation curve would be a straight line (red dotted line). However, because the rate-controlling step changed with time from the transport outside the pore to that inside the pore, the relaxation curve became non-linear. In the second sample (Figure 9b), the initial adsorption rate is determined by the transport outside the nanopore. The initial adsorption rate for each particle should be half as large as that for the first sample. Therefore, it is expected that the less amount of water will fill the shorter nanopores more smoothly. Thus, the relaxation curve will approach a straight line (red dotted line). However, the experimental results reveal that the relaxation curve does not greatly depend on the type of sample. This suggests that the mass flux of water in the longer pore is larger than that in the shorter one during the entire adsorption process.

Figure 9. Illustration of the relationship between the adsorption relaxation curves and the pore length: (a) one long porous particle and (b) two short porous particles of identical mass. Illustration of water transport inside (c) one long pore and (d) two short pores.


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Transport Model of Water Filling and Emptying in Nanopores with Different Pore Lengths In order to explain this phenomenon, the transport model of the proceeding and receding water column was next considered. It is assumed that the nanopores are located in moist air at a fixed relative humidity and that the water columns (water bridges) are formed near the entrances of the pore (Figure 9c,d). Although the water bridge could be formed at any place inside the pore during the filling process, we focused on the water bridge formed near the entrances of the pore to reproduce the relaxation process represented by the square root function of time. Schneider et al.14 reported two different scenarios for filling of capped nanocapillaries from the vapor phase in their molecular simulation study: 1) filling process occurred by the detachment of the meniscus from the cap and 2) spontaneous condensation of the liquid close to the pore opening and its subsequent growth toward the closed pore end. They showed that the relaxation curve (the time course of the adsorbed amount) can be well fitted by the square root function of time only in the second scenario. This model assumes that the time course of the adsorbed mass can initially be expressed by the linear function of time, is subsequently expressed by the square root function of time, and is finally expressed by a constant value in the equilibrium state (Supporting Information S5). If the relaxation curve is normalized by half the pore length (lp/2) and the relaxation time to reach the equilibrium state (teq), the normalized relaxation curve can be specified at a fixed normalized transport parameter C*[= K ΔP teq/(ϕk η (lp/2)2)], where K is the hydraulic permeability of the porous material, ϕk is the porosity based on the Kelvin radius, ΔP is the pressure difference between the two ends of the permeated liquid, and η is the viscosity of the liquid. If it is proved that C* remains constant under the assumption that teq is independent of lp, it will follow that the relaxation curve is independent of lp. From the equations for the permeated liquid transport in


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porous materials (eq. S3) and the time-dependent length of the permeated liquid, h(t) (eq. S5) (Supporting Information S5), Jp(t) is given by

Jp t   

K P , k h  t 

(10)

where ρ is the permeated liquid density. For two samples with different pore lengths shown in Figure 9a–d, the twice longer pore should have twice larger values of Jp(t) and h(t). This suggests that the twice longer pore should have twice larger values of the transport parameter K/(ϕk η) and ΔP to keep C* constant under the assumption that teq is independent of lp. Here, it is assumed that ΔP can be expressed by the difference in capillary pressure between the two ends of the permeated liquid, ΔPv, which is given by  1 1 Pv  2   r  c,in rc,out

  , 

(11)

where γ is the surface tension at the liquid‒vapor interface and rc,in and rc,out are the curvature radii of the menisci facing the interior and exterior of the pore, respectively (see Supporting Information S5). Furthermore, it is assumed that the transport of the water bridge includes the activated process, that is, the water bridge moves only when the local energy is larger than the height of the energy barrier for transport. When the initial adsorption rate is (dm/dt)0 = 3.7×10-3 (g g-1 s-1) at Δµ = 0.52 (kJ mol-1) for MPS 3 (see Figure 7), the initial mass flux of water in the pore is Jp(0) = (dm/dt)0/Ap,eff = 1.25×10-5 (kg m-2 s-1). The initial mass flow rate of water in each pore is (dmwp/dt)0 = Jp(0)×ai,eff = 4.26×10-22 (kg s-1) = 14,240 (water molecules per second), that is, one water molecule enters the pore every 70.2 μs. If a water column of which density is ρ = 8.25×102 (kg m-3) proceeds inside the MPS 3 nanopore at a constant speed, the average velocity is vwp = (dmwp/dt)0/(ρai,eff) = 15.1 (nm s-1) and the relaxation time to reach the pore filling state is teq = ρ ai,eff (lp/2)/(dmwp/dt)0 = 135


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(s). On the other hand, according to the Lucas‒Washburn Law, the relaxation time to reach the pore filling state can be predicted to be teq = 2η (lp/2)2/(γ (rp – t)) = 7.1 (μs) and the average velocity is vwp = (lp/2)/teq = 0.29 (m s-1). Here, η = 10-3 (Pa s), γ = 7.2×10-2 (N m-1), and t = 0.5 (nm). This simple prediction suggests that the water bridge should stay in most of the time and move intermittently. Valiullin et al.6 reported that the relaxation process slows down dramatically within the hysteresis region and is dominated by activated rearrangement of the adsorbate density within the host material in their experimental and simulation study of the molecular dynamics during transient sorption of fluids in mesoporous materials. They assumed that liquid droplets appear at various places in the pore structure, forming bridges between the pore walls as the average density increases. They showed that the total relaxation can be represented by a linear combination of two relaxation processes: a diffusive one and an activated one. It should be noted that the relaxation curve was not fitted by the square root function of time. In this study, we considered the activated process for the proceeding water column formed near the entrances of the pore because the objective is to reproduce the relaxation process represented by the square root function of time. By considering these two assumptions, the transport model of the proceeding water column was further developed. Figure 10a–c illustrates the time expansion of the proceeding water column. The water bridge of which length is h1 starts moving toward the inner part of the nanopore when the gradient of pressure, ΔPv/h1, is larger than a threshold value (relaxation toward global equilibrium) (Figure 10a). However, the space that was occupied by the water bridge cannot be filled with water instantly because the nanopore is in contact with water vapor not with liquid water. As a result, the water bridge deforms (rc,out decreases) and no longer moves forward when ΔPv/h1 is smaller than a threshold value (relaxation toward local minimum of the free energy) (Figure 10b). Subsequently, water vapor adsorbs onto the outer surface of the water bridge and


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rc,out increases (relaxation toward vapor-liquid equilibrium) (Figure 10c) and the water bridge of which length is h2 (h2 > h1) moves forward again when the gradient of pressure, ΔPv/h2, is larger than the threshold value.

Figure 10. Illustration of the time expansion of the proceeding water column: (a) relaxation toward global equilibrium, (b) relaxation toward local minimum of the free energy around the water bridge and (c) relaxation toward vapor-liquid equilibrium.

This transport model considers two relaxation processes toward local equilibrium, that is, (a) the relaxation process to achieve the local minimum of the free energy around the water bridge and (b) the relaxation process to achieve the vapor-liquid equilibrium. By considering the relaxation processes, C*[= K ΔP teq/(ϕk η (lp/2)2)] could be constant under the assumption that teq is independent of lp because the intermittent flow could be regarded as an equivalent continuous flow with the same Jp(t) by assuming the effective values of the transport parameter K/(ϕk η) and ΔP. Without considering these relaxation processes, the normalized transport parameter C* should


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not be constant under the assumption that teq is independent of lp because all the parameters of K, ΔP, ϕk and η should be constant. If the surface of the MPS particle aggregate is the bottleneck in the overall adsorption processes, the increase in lp leads to the increase in the cross-sectional area of a gas flow passage for each nanopore, agas (= Aa/2Np = Aa ai,eff lp/2Vp), resulting in the increase in the mass flux of water vapor into each nanopore. It is expected that as the mass flux of water vapor into each nanopore increases, the frequency of the water bridge movement increases, and thus the effective value of the transport parameter K/(ϕk η) increases. Furthermore, as the mass flux of water vapor into each nanopore increases, the value of rc,out when the water bridge starts moving increases, and thus the effective value of ΔP increases. It is still difficult to quantify K/(ϕk η) and ΔP inside nanopores. Furthermore, it is also difficult to clarify the effect of water behavior existing in the interior of the pore on the water bridge proceeding. However, it is expected that this transport model will be further improved by considering the relaxation processes toward local equilibrium in more detail. A molecular dynamics simulation of water capillary condensation reproduces the advective flow of a permeated liquid induced by the difference in curvature radii between the two ends of the liquid.13 Furthermore, another molecular dynamics simulation of water capillary condensation shows that the calculated relaxation curves are reasonably explained by assuming that the effective viscosity decreases with increasing the mass flux.12 If the transport of water inside the nanopore is perfectly independent of the transport of water vapor outside the nanopore, this transport phenomenon cannot be explained by any macroscopic transport theory. By assuming a bottleneck outside nanopores in the entire gas flow passage, the model shown in Figure 10a–c can reasonablly exlain the pore filling process and the pore length


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independence of the adsorption rates. A similar discussion is valid for the receding water column in the desorption process.

CONCLUSIONS In this study, two-dimensional hexagonal MPS particles of same pore size but different pore lengths were successfully synthesized and the effect of pore length on the water filling and emptying rates was studied. The following conclusions can be drawn from this study: The water adsorption and desorption isotherms in the synthesized MPS samples, plotted from the gravimetric data at 298 K and 1 atm, are not significantly related to the pore length. Furthermore, the water adsorption and desorption relaxation curves (time course of adsorbed mass of water per unit mass of the sample) are also not significantly related to the pore length both in the filling and emptying processes. Because the initial adsorption and desorption rates are independent of the pore length, the surface of the MPS aggregates should be a bottleneck in the overall adsorption and desorption processes from the bulk gas phase to the nanopores and the effective area of the entire pore entrances should be larger than the entire surface area of the MPS aggregates per unit mass of sample. Furthermore, because the number of nanopores is inversely proportional to the pore length at identical sample mass, the initial mass flux of water in each nanopore should also be inversely proportional to the pore length at an identical sample mass. Because the relaxation time necessary to reach the equilibrium state is independent of the pore length, the mass flux of water in the longer pore is larger than that in the shorter one in the all the adsorption and desorption processes. This transport phenomenon can qualitatively be explained by a transport model of a proceeding and receding water column.


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ASSOCIATED CONTENT Supporting Information. Effect of sample weight on the initial adsorption and desorption rates, FE-SEM images of synthesized MPSs, detailed data of gravimetric measurements, fitted parameters of equation 1 for relaxation curves of water adsorption and desorption on MPSs 1, 2 and 3, and models of capillary flow in nanopores. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *Email: [email protected], Phone/Fax: +81 3 5841 6971. Funding Sources JST CREST Grant Number JPMJCR11C3 and JPMJCR17I3, Japan. Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT This work was supported by JST-CREST Grant Number JPMJCR11C3 and JPMJCR17I3, Japan and financial aid from Mitsubishi Electric Corporation.

REFERENCES 28 ACS Paragon Plus Environment

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Water Filling and Emptying Kinetics in Two-Dimensional Hexagonal

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