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Estimation of Neck Size Distribution in Ordered Cagelike Materials Using Nitrogen and Water Desorption Isotherms Kunimitsu Morishige J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b08091 • Publication Date (Web): 14 Nov 2017 Downloaded from http://pubs.acs.org on November 21, 2017

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The Journal of Physical Chemistry

Estimation of Neck Size Distribution in Ordered Cagelike Materials Using Nitrogen and Water Desorption Isotherms

Kunimitsu Morishige* Department of Chemistry, Okayama University of Science, 1-1 Ridai-cho, Kita-ku, Okayama 700-0005, Japan

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ABSTRACT: To estimate the neck size distributions in ordered cagelike materials KIT-5 with prolonged hydrothermal treatments, we measured the desorption isotherms of water at 293 K on the cagelike materials, and then calculated the desorption isotherms of nitrogen at 77 K and water at 293 K based on the model pore networks with normal distributions of neck sizes that mimic the pore structure of KIT-5. First it was found that the position of the desorption isotherms does not at all reflect the mean size of necks unless the width of the neck size distribution is fixed and thus the simple analysis of the desorption isotherms in the absence of networking effects leads to erroneous distribution of neck sizes. The gradual desorption isotherms of nitrogen and water observed for KIT-5 samples were not well reproduced based on the model pore networks with no correlation in the spatial distribution of neck sizes. On the other hand, inclusion of correlation in the spatial distribution of neck sizes led to exceedingly better fits between the experimental and calculated desorption isotherms of nitrogen and water on KIT-5. From the point of view of estimation of neck size distributions in the ordered cagelike materials, the water adsorption method at 293 K is superior to the nitrogen adsorption method at 77 K.


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I.

Introduction A nitrogen adsorption method at 77 K is most frequently used in characterization of

mesoporous materials that have been utilized as adsorbents, catalysts, etc.1 Capillary condensation and evaporation of nitrogen in the mesopores take place on adsorption and desorption, respectively, at different pressures in most cases. When the materials possess constrictions in the pores, it would be better to estimate pore size distributions using the adsorption isotherms2 because the presence of a small number of constrictions may alter significantly the shape of desorption isotherms. On the other hand, in principle it is possible to obtain information about the constrictions from the desorption isotherms. For some of conventional mesoporous materials,3-5 the desorption isotherms are steep, although the adsorption isotherms are gradual and thus the solids possess wide distributions of the mesopore sizes; that is, the isotherms exhibit hysteresis loops of type H2(a) in the IUPAC classification.6 Wall and Brown,3 and other workers4,7-9 have applied a concept of percolation to explain the steep desorption isotherms, where the pore space is treated as a lattice of cavities interconnected by necks in a three-dimensional (3D) network. In the application, the cavities and the necks in the porous medium are regarded as the sites and the bonds, respectively, of percolation theory. Bond percolation in the 3D lattice may lead to the steep desorption isotherm, even though the neck sizes are widely distributed. In recent years, a wide variety of ordered mesoporous materials have been synthesized by using ionic and nonionic surfactants as structure-directing agents.10-12 Among them, the materials with cagelike pores such as SBA-1611 and KIT-513 consist of almost spherical cavities (cages) arranged in 3D lattices such as body-centered cubic (bcc) and face-centered cubic (fcc) ones and connected through narrow necks. The pore structures are essentially identical to the 3 ACS Paragon Plus Environment


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model pore systems that were assumed in the application of percolation theory to the desorption process of nitrogen from the conventional mesoporous materials. However, the ordered cagelike materials that were synthesized by prolonged hydrothermal treatments at relatively high temperatures have almost always shown very gradual desorption isotherms,13-17 in contrast with the steep desorption isotherms observed for the conventional mesoporous materials. The hysteresis loops are now classified into type H2(b).6 The cagelike structures of the ordered solids seem to be still maintained after such hydrothermal treatments.16,17 Previously, the gradual desorption isotherms were explained based on the very broad distributions of neck sizes in the absence of percolation effects2,14,15 but not substantiated. The problem still remains to be solved.17-19 In the application of percolation theory to desorption, the size distribution of the necks interconnecting the large cavities can be related to the desorption isotherm through the percolation probability that depends on the coordination number z of the porous medium and the bond occupation probability q.3,4,7-9 Therefore, the connection between the size distribution of necks and the desorption isotherm is not straightforward. A conventional analysis of the desorption isotherm due to an independent-pore model gives an incorrect distribution of neck sizes in the interconnected pore system. In the past, many workers have tried to determine the coordination number and the size distribution of necks in the pore networks from the desorption isotherms for the mesoporous materials.3,4,7-9,20 As only part of the size distribution of necks may contribute to the desorption isotherm, however, the whole part of the distribution can be never determined unless the shape of the distribution is presumed. In addition, all these studies are based on the assumption that the necks of different sizes are randomly distributed throughout the sample particle. When this assumption is invalid, one cannot apply the percolation theory 4 ACS Paragon Plus Environment

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data to the desorption isotherm. In our preceding studies,17-19 we have suggested the presence of correlation in the spatial distribution of neck sizes in the materials synthesized by prolonged hydrothermal treatments as the origin of the gradual desorption isotherms. The ordered cagelike materials are usually obtained as finely divided substances and thus the system size of pore network is relatively small. In addition, the coordination number is known from the pore structure. Therefore, the desorption process of a fluid from the interconnected pore system can be directly simulated based on a model pore system that mimics the pore structure of the material without resorting to percolation theory. The sizes of cages and necks in the ordered cagelike materials are tunable.13-17 The large cages are able to accommodate large molecules like enzymes, although the narrow necks actually control the uptake of the large molecules.21 In some preparations, the necks can be reduced to a size small enough to result in a molecular sieving effect among small molecules such as CH4 and CO2.22 The large cages can act as reaction vessels for various catalytic reactions. The size distribution of the necks and the networking effect are very important in transfer and transportation of molecules in the interconnected pore system. Therefore, these ordered cagelike materials have attracted a great deal of interest because of their unique pore structures and potential applications in immobilization of biomolecules, drug delivery, separation, and catalysis. On adsorption, nitrogen behaves quite independently in different parts of the pore system.23 With increasing pressure, necks of larger size are progressively filled with liquid nitrogen. After all the necks are filled, capillary condensation of nitrogen takes place in the large cages with further increase in pressure. In principle, high-resolution pore size analysis of the adsorption isotherm can give pore size distribution curves of the necks interconnecting the large cages.2,24 However, the sizes of the necks are widely distributed and the total volume of 5 ACS Paragon Plus Environment


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the necks is very small compared to that of the cages. In addition, there always exist complementary pores located inside the framework walls of the ordered cagelike materials.25 The complementary pores may be formed as a result of penetration of poly(ethylene oxide) chains of the triblock copolymer template within the silica framework of as-synthesized materials. Some of them are related to connecting pores (necks) between large cavities but the rest is unrelated to the necks. These factors make the precise determination of the neck size distribution from the adsorption isotherms almost impossible. When the largest of necks escaping from the cavity are smaller than ∼5 nm in diameter, cavitation of nitrogen capillary-condensed in the cavity takes place at a nearly constant pressure at 77 K with decrease in pressure, irrespective of neck size distributions.14,15,17,19,26 In such cases, one cannot obtain any further information about neck size distributions from the desorption isotherms. When the relative pressures p/p0 of cavitation are plotted as a function of reduced temperature T/Tc, all the data for nitrogen, oxygen, argon, and carbon dioxide confined in the large cavities of SBA-16 were represented by a common curve.27 Here, p is the equilibrium pressure and p0 the saturation vapor pressure of a gas at a temperature T, and Tc is the bulk critical temperature of the gas. The relative pressure of cavitation decreased with decrease in T/Tc. This implies that the occurrence of cavitation can be prevented by measuring a desorption isotherm at as low T/Tc as possible, irrespective of the kind of adsorptive. The values of T/Tc for nitrogen at 77 K and water at 293 K are ∼0.61 and 0.45, respectively. Therefore, it is inferred that the critical size of necks below which cavitation of confined liquids takes place may be decreased by measuring a desorption isotherm of water at 293 K, instead of nitrogen at 77 K. In the present study, first we show that measurements of desorption isotherm of water at 293 K can be used to detect a size distribution of necks with sizes smaller than for nitrogen at 77 6 ACS Paragon Plus Environment

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K. Next, we calculate the desorption isotherms of nitrogen at 77 K and water at 293 K based on the model pore systems with normal distributions of neck sizes that mimic the pore structures of the ordered cagelike materials and then examine the connection between the neck size distribution and the desorption isotherm of nitrogen at 77 K and water at 293 K for the model cagelike structures. Under the assumption of the random distribution of neck sizes normally distributed, a fitting program to extract neck size distributions from the experimental desorption isotherms of nitrogen at 77 K and water at 293 K on the ordered cagelike materials is constructed. As the result, the assumption of random distribution of neck sizes throughout the sample particle turned out to be invalid for the ordered cagelike silicas synthesized by the prolonged hydrothermal treatments. Finally, we deal with the model pore systems with spatially correlated distributions in neck size. It gave a much better fit to the experimental desorption isotherms. EXPERIMENTAL SECTION Materials and Measurements. KIT-5 was synthesized by using Pluronic F127 triblock copolymer as a structure-directing agent and benzene as a solubilizing agent. The synthesis and characterization of KIT-5 samples have been described in detail elsewhere.17 The obtained KIT-5 silicas are denoted KIT-5-x, where x corresponds to the aging time at 393 K. ASMS-3A was synthesized by using Pluronic F127 triblock copolymer as a structure-directing agent and mesitylene as a solubilizing agent without aging. The synthesis and characterization of a ASMS-3A silica has been also described in detail elsewhere.22,28 The adsorption-desorption isotherms of water at 293 K were measured gravimetrically on a Rubotherm (BEL Japan).29 All the measurements were conducted on the hydroxylated surfaces of silicas. 7 ACS Paragon Plus Environment


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Pore Network Model. The pores of KIT-5 are modeled as a pore network that consists of a fcc array of large spherical cavities interconnected through narrow necks of cylindrical shape (see Figure 1). The network has an underlying coordination number of Z=12. In practice, some of the necks might not be present and some of those that are present might be too small to admit an

h (2,0,0)

(2,0,2) (2,1,1) (1,0,1) (2,2,2)

(2,2,0) (1,1,0) (0,0,0)

(1,1,2) (0,0,2)

(1,2,1)

ℓ

(0,1,1) (0,2,0)

(0,2,2)

k

Figure 1. Schematic illustration of spherical cavities arranged in a fcc lattice and interconnected through narrow necks.

adsorbate molecule. A cavity has 6 kinds of necks with independent coordinates and thus individual necks are assigned to the cavity in calculations of a desorption process in the pore network. Integer coordinates are assigned to the cavities in the pore network. We assume that the diameters of all cavities are appreciably larger than those of all necks and the probability for an arbitrary neck to have a given value of the diameter does not depend on the sizes of adjacent cavities and necks. In addition, the volume of the necks will be negligibly small compared to that of the cavities. The size distribution of the cavities can be determined from the adsorption isotherm of nitrogen at 77 K.24 Throughout the present study, the diameters of individual 8 ACS Paragon Plus Environment

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cavities were supposed to follow a normal distribution with mean value of 20 nm and standard deviation of 2 nm. When the pressure is decreased from the state of completely filled pores, all the cavities first become metastable. With further decrease in pressure, particular necks then become metastable if the pressure reaches their desorption pressures. However, desorption of liquid condensed in a cavity is blocked by a liquid that obstructs its neighboring necks even if both the liquid in the cavity and the liquid in the neighboring necks are metastable. For a particular cavity to empty, a meniscus must pass at least one of the neighboring necks. Only the cavities that are connected to the bulk vapor by a series of open necks empty. First percolation of vapor phase into the pore network was sequentially examined cavity-by-cavity within the surface layers of the particle. Then, percolation process was extended so as to include the whole network system. We calculated the fraction of unfilled cavities which will be accessible from the surface through open necks as a function of the relative pressure (p/p0). We considered a 3D fcc lattice of L×L×L unit cells with the outer necks connected to the bulk vapor that mimic the external surface of the porous material. Unless otherwise indicated, most of the calculations were performed for the lattice size of L=50 because the unit-cell lengths of KIT-5 are around 28 nm17 and the particle sizes are around 1 µm.13,22 The diameters of individual necks are supposed to follow a normal distribution with mean values Dneck and standard deviation σ. The total number of cavities is 515151. The calculations were performed using programs written in Fortran under the environment of free software Cygwin. The source code of main programs can be found in Supporting Information. For determination of metastability of the necks, we employed the relationship between the pore diameter and desorption pressure (p/p0) of nitrogen at 77 K in the cylindrical pores of silicas reported by Broekhoff and De Boer.30 We also 9 ACS Paragon Plus Environment


extracted the relationship between the pore diameter and desorption pressure of water at 293 K for the cylindrical pores of silicas from the isotherm data of Kittka et al.31 and Jarnert et al.32 The numerical relation is given by p/p0(H2O) = 0.2756 ln(D) + 0.118

(1)

where D is the pore diameter (in nm).

EXPERIMENTAL RESULTS N2 adsorption on KIT-5. The adsorption-desorption isotherms of nitrogen at 77 K on KIT-5 samples changed drastically in shape as the hydrothermal treatment at 393 K was prolonged.17 The amount of adsorption in experiments is composed of a surface film on the pore walls of unfilled cavities, a liquid in filled cavities, and a surface film on the external surfaces. In order to compare with the calculated results mentioned below, the effect of the surface film was eliminated from the experimental isotherms according to the procedure of Cimino et al.20 Figure2 shows the fractions of unfilled cavities along the adsorption-desorption isotherms for

1.4

Fraction unfilled

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KIT-5-1 day

1.2

KIT-5-3 days

KIT-5-7 days

KIT-5-5 days

1 0.8 0.6 0.4 0.2 0 0

0.5

1

0.5

1

0.5

1

0.5

1

p/p0

Figure 2. The fraction of unfilled cavities along the adsorption-desorption isotherms for nitrogen at 77 K on four kinds of KIT-5 samples. Open and closed symbols denote adsorption and desorption points, respectively.


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nitrogen at 77 K on four kinds of KIT-5 samples. The shape of the desorption isotherm changed drastically as the hydrothermal treatment was prolonged. On the other hand, the shape of the adsorption isotherm remained almost unchanged. The sharp adsorption isotherms observed for all the samples indicated the presence of uniform cavities. For a sample KIT-5-1 day, the hysteresis loop closed sharply at p/p0 of 0.47 corresponding to the lower limit of the adsorption-desorption hysteresis. This indicates that desorption takes place via cavitation and the diameter of the narrow necks is smaller than ∼5 nm.19,26 However, for other samples, the hysteresis loops gradually closed above the lower limit of hysteresis, which indicates the neck diameters are above ∼5 nm. H2O adsorption on ASMS-3A and KIT-5. All the cagelike silicas used in the present study were preserved in a sample bottle for more than 6 years and thus the surfaces are thoroughly hydroxylated by water in the air. We performed multiple cycles of water adsorption on KIT-5-1 day. First, the adsorption-desorption isotherm of water was measured on the sample evacuated at 293 K. After the measurement, the sample was evacuated for 5 h at 673 K and the second isotherm was measured on the dehydroxylated sample. Finally, the sample was again evacuated at 293 K and then the third isotherm was measured on the rehydroxylated sample. The first and third isotherms were almost superimposed. Only the adsorption-desorption isotherms for water that were measured on the hydroxylated samples are given in the present study. For ASMS-3A, the hysteresis loop of water at 293 K closed sharply at p/p0 of 0.33. For two kinds of KIT-5 samples, on the other hand, the hysteresis loop gradually closed at higher p/p0 (Figures 1S-3S of the Supporting Information). Two basic mechanisms of desorption in the pore network of cagelike materials are distinguished as pore-blocking percolation and cavitation. It is well-understood that the 11 ACS Paragon Plus Environment


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percolation mechanism is observed in the pore networks with sufficiently large necks. If the neck diameter is smaller than a certain critical size at a given temperature and adsorptive, the mechanism of desorption from the cavities involves cavitation. The cavitation-induced desorption takes place abruptly at a certain pressure because the cavitation is the properties of a bulk liquid. The cavitation pressure at a given temperature and adsorptive is almost constant for a wide range of cavity diameter, except for small-pore cagelike materials with small hysteresis loop.33 The critical diameter of neck depends on temperature and the adsorptive. For nitrogen and argon adsorption at 77 and 87 K, respectively, the critical diameter of neck lies at 5-6 nm.2 If the neck size distribution of ASMS-3A is very narrow and desorption of water takes place by a pore blocking mechanism, the neck diameter can be estimated to be ∼2.2 nm using the relationship (1) between the pore diameter and desorption pressure of water at 293 K for the cylindrical pores of silicas. This size is significantly larger than the neck diameter of ∼0.4 nm that was estimated from a molecular sieving effect between CO2 and CH4.22 ASMS-3A does not at all adsorb nitrogen at 77 K because of steric hinderance.28 Therefore, it is clearly indicated that water confined in the large cavities of ASMS-3A desorbs at p/p0 of 0.33 with decreasing pressure, while water stays in the narrow necks. This is just cavitation-induced desorption. The critical diameter of neck for water adsorption at 293 K turns out to be ∼2.2 nm. Compared to this, desorption of water from KIT-5-1 day and KIT-5-7 days occurs very gradually at higher pressures, indicating that desorption of water from these two materials takes place via percolation (pore-blocking-controlled evaporation). For KIT-5-1 day, the desorption of water at 293 K proceeds by a pore-blocking mechanism, although for the same sample the desorption of nitrogen at 77 K takes place via cavitation. For estimation of neck size distributions in the ordered cagelike materials, the water adsorption method at 293 K is superior to the nitrogen 12 ACS Paragon Plus Environment

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adsorption method at 77 K. Figure 3 shows the fractions of unfilled cavities along the adsorption and desorption isotherms for water at 293 K on ASMS-3A and two kinds of KIT-5 samples.

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Fraction unfilled

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KIT-5-7 days

KIT-5-1 day

ASMS-3A

1.2 1 0.8 0.6 0.4 0.2 0 0

0.5

1

1

0.5

0.5

1

p/p0

Figure 3. The fraction of unfilled cavities along the adsorption-desorption isotherms for water at 293 K on three kinds of ordered cagelike silicas. Open and closed symbols denote adsorption and desorption points, respectively.

THEORETICAL RESULTS N2 desorption. With no correlation in the spatial distribution of neck sizes, the desorption isotherm of a liquid condensed in the model pore network of KIT-5 was affected significantly by the mean diameter and standard deviation of the necks with normal distribution in size, in accord with a previous work.34 Figure 4 shows the effect of the standard deviation on the desorption isotherm of nitrogen at 77 K in the model pore network that consists of necks normally distributed with the mean diameter of 3 nm. The position and shape of the desorption isotherm changed significantly with a change of the standard deviation in the normal distribution of neck sizes. Figure 5 shows the corresponding distributions of the neck size. It is 13 ACS Paragon Plus Environment


apparent that the position of the desorption isotherm does not at all reflect the mean size of necks unless the width of the neck size distribution is fixed. It is also useful to mention that a very wide distribution of neck sizes does not result in the gradual desorption isotherm. The values of larger than 1 nm in σ means that an appreciable fraction of necks have negative values 1.2

Fraction unfilled

1 0.8 0.6 0.2

0.4

1.0

3.0 4.0 nm

2.0

0.2 0 0.2

0.4

0.6

p/p0

0.8

Figure 4. The fraction of unfilled cavities along the desorption isotherms for nitrogen at 77 K in the model pore networks that consist of necks normally distributed with Dneck of 3.0 nm and various standard deviations and of cavities normally distributed with Dcavity of 20 nm and σ of 2 nm.

0.5

0.4 sigma(0.2 nm)

dN/dD /nm-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.3

× 1/5

sigma(1.0 nm) sigma(2.0 nm)

0.2 sigma(3.0 nm) sigma(4.0 nm)

0.1

0 -10

-5

0

5

10

15

Diameter of neck / nm

Figure 5. Normalized distributions of neck size normally distributed with Dneck of 3.0 nm and various standard deviations.


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Fraction unfilled

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0.8 L=20 L=50 L=100

0.6 0.4 0.2 0 0.2

0.4

p/p0

0.6

0.8

Figure 6. The fraction of unfilled cavities along the desorption isotherms for nitrogen at 77 K in three kinds of the model pore networks that consist of necks normally distributed with Dmean of 3.0 nm and σ of 2.0 nm. The three kinds of the pore networks consist of a fcc array of cavities with 20×20×20, 50×50×50, and 100×100×100 unit cells, respectively.

as their diameters. In the calculations of the desorption isotherms, these necks are treated as closed pores. Figure 6 shows the effect of the network size on the desorption isotherm of nitrogen at 77 K in the model pore networks that consist of necks normally distributed with Dmean of 3.0 nm and σ of 2.0 nm. The network size affects considerably the shape of the isotherm in the beginning of desorption at higher p/p0, although the shape in the tail region of the isotherm at lower p/p0 is not at all influenced by it. On the other hand, the width of the neck size distribution affects considerably the shape in the tail region of the desorption isotherm, while its effect on the shape of the isotherm in the beginning of desorption is not so large. Figure 7 compares the desorption isotherms of nitrogen at 77 K calculated for three kinds of the model pore networks with Dmean = 5.3 nm and σ = 1.0 nm, Dmean = 3.0 nm and σ= 3.0 nm, and Dmean = 0.7 nm and σ= 5.0 nm, respectively. The positions of the desorption isotherms are almost identical, although the shapes of the isotherms vary appreciably. The slope of the 15 ACS Paragon Plus Environment


1.2 1

Fraction unfilled

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.8 0.6 D=3.0 nm, sigma=3.0 nm 0.4 D=5.3 nm, sigma=1.0 nm 0.2 D=0.7 nm, sigma=5.0 nm 0 0.2

0.4

0.6

0.8

p/p0

Figure 7. The fraction of unfilled cavities along the desorption isotherms for nitrogen at 77 K in three kinds of the model pore networks that consist of necks normally distributed with different pairs of Dmean and σ.

desorption isotherm became gradual with large increase in σ, although the change of the slope is not so large as compared to that expected simply from the large increase in width of the neck size distribution. This implies that if a normal distribution of neck sizes can be assumed, the neck size distribution can be estimated through a proper analysis of the experimental desorption isotherm as a whole. We constructed a least-squares fitting program for estimating a neck size distribution in KIT-5 with a fcc array of cavities from the desorption isotherm of nitrogen at 77 K. Calculations of the desorption isotherms mentioned above were also performed to determine the fraction of unfilled cavities along desorption isotherms as a function of the fractions of metastable necks q, that is, an accessibility curve of the cavities from the surfaces for the model network consisting of cavities in a fcc lattice. Any combinations of Dmean and σ gave identical curves. Figure 8 shows the accessibility curve for the model pore network consisting of a fcc array of cavities 16 ACS Paragon Plus Environment

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1.2 1

Accessibility

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0.8 0.6 calculated

0.4

approximate equation 0.2 0 0

0.1

0.2

0.3

0.4

q Figure 8. Accessibility of the cavities in the model pore network that consists of a fcc array of cavities with 50×50×50 unit cells as a function of the fractions of metastable necks.

with 50×50×50 unit cells. Approximate equations to the calculated data were obtained by the least-squares fitting, in order to use in the analysis of neck size distribution using the experimental desorption isotherms of nitrogen and water on the ordered cagelike materials. When q is increased beyond the bond percolation threshold (qc = 0.119) of a fcc lattice, the accessibility is rapidly increased and then approaches unity in the neighborhood of q = 0.4. This implies that the necks in more than the lower half of the size distribution do not at all contribute to the desorption isotherm, that is, the desorption isotherm is never affected by more than the lower half of the neck size distribution curve. It is well-known that the maze-like properties of the pore network in cagelike materials make a large part of the neck size distribution inaccessible to measurements.3,4,7-9,34 On the other hand, it is reasonable to consider that the neck sizes in the cagelike materials distribute randomly. A normal distribution is one of the most important distribution functions of random number. If we assume that the neck sizes follow the normal distribution, the whole distribution curve of the neck size can be assessed from the 17 ACS Paragon Plus Environment


desorption data. Otherwise, we are not able to know the whole part of the neck size distribution. This validates the use of the known shape of the distribution curve. As long as the normal distribution of neck sizes is assumed, q can be analytically calculated as a function of neck diameter using an approximate equation of the cumulative normal distribution.35 Neck diameter in the normal distribution is related to desorption pressure of nitrogen at 77 K from the cylindrical necks of silicas using the Broekhoff-de Boer relationship.30 Therefore, a choice of a pair of values for Dmean and σ produces a unique desorption isotherm of nitrogen at 77 K on the model pore network that consists of a fcc array of cavities interconnected through narrow necks. The least-squares fitting was done using the program developed by us. The fitting parameters were the mean value of neck diameter (Dmean) and the standard deviation (σ) in normal distribution of neck sizes. Figure 9 compares the fitted desorption curves with two kinds of theoretical desorption isotherms calculated with Dmean of 5.3 nm and σ of 1.0 nm, and Dmean of 0.7 nm and σ of 5.0 nm, respectively. We obtained Dmean of 5.3 nm and σ of 1.1 nm, and Dmean of 0.3 nm and σ of 5.4 nm, respectively, from the least-squares fittings, which were shown by the solid lines in the figure. These are in fairly good agreement with the model parameters used 1.2 1

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0.8 0.6 simulation(D=0.7 nm, sigma=5.0 nm)

0.4

fitting(D=0.3 nm, sigma=5.4 nm) simulation(D=5.3 nm, sigma=1.0 nm)

0.2

fitting(D=5.3 nm, sigma=1.1 nm)

0 0.3

0.4

0.5

0.6

0.7

0.8

p/p0 Figure 9. A comparison of the fitted desorption curves with two kinds of calculated desorption isotherms.


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in calculations of the theoretical desorption isotherms. A difference between the best-fitted parameters and the model parameters used in calculations of the desorption isotherms seems to stem from a systematic error in the approximate equations for the accessibility curve. The desorption of nitrogen at 77 K on KIT-5-1 day occurred via cavitation and thus fitting based on the model pore network was not tried. Figure 10 compares the experimental desorption isotherm of nitrogen at 77 K on a sample KIT-5-7 days with two kinds of theoretical desorption isotherms. The first one was obtained by the least-squares fitting of the observed isotherm to the isotherm calculated based on the model pore system with a noncorrelated distribution of neck sizes. The best-fitted parameters were Dmean of − 4.4 nm and σ of 10.8 nm. The neck size distribution is very wide. However, it is apparent that a fit between the experimental and

1.2 1

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0.8 0.6 experiment

0.4

fitting(no spatial correlation) 0.2

fitting( spatial correlation)

0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p/p0

Figure 10. A comparison of the experimental desorption isotherm of nitrogen at 77 K on KIT-5-7 days with two kinds of calculated desorption isotherms. The explanation is given in the text.

calculated desorption isotherms is very bad. The gradual desorption isotherms of nitrogen at 77 K, which have been observed for the ordered cagelike materials with prolonged hydrothermal treatments, cannot be well accounted for by the very wide distribution in the pore network with 19 ACS Paragon Plus Environment


the spatially noncorrelated distribution of neck sizes. As far as we consider the spatially noncorrelated distribution of neck sizes throughout the sample particle, we cannot reproduce the gradual desorption isotherms experimentally observed because percolation threshold always exists for the pore network with the noncorrelated distribution of neck sizes. The second isotherm was obtained by simulation of a desorption process in the pore network system with a spatially correlated distribution in neck size, where necks are more enlarged in the outer layer of the sample particle. The fitting procedure was done by trial and error. The model pore network with the correlated distribution of neck sizes gave an exceedingly better fit to the experimental desorption isotherm. In the model pore network with the spatially correlated distribution in neck size, the mean diameter was changed with layer number, while the standard deviation was maintained at a fixed value of 1.0 nm. For two other samples of KIT-5-3 days and KIT-5-5 days, the fittings were also considerably improved by including a spatially correlated distribution in neck size. Figure 11 shows variations of Dmean against the layer number (L) for three kinds of KIT-5 samples that gave good fits between the experimental and calculated desorption

12 KIT-5-3 days(N2)

10

KIT-5-5 days(N2) KIT-5-7 days(N2)

8

Dmean / nm

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KIT-5-1 day(H2O)

6

KIT-5-7 days(H2O)

4 2 0 -2 -10

0

10

20

30

40

50

Layer No.

Figure 11. Plots of Dmean against the layer number for four kinds of KIT-5 samples that gave good fits between the experimental and calculated desorption isotherms. The explanation is given in the text.


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isotherms, where the standard deviation was fixed at 1.0 nm. L= -1 corresponds to the outermost layer, while L=49 corresponds to the innermost layer of the cubic particle. This figure also includes the results of analysis of the water desorption isotherms on two samples, KIT-5-1 day and KIT-5-7 days, which will be described in a subsequent section. H2O desorption. As the desorption isotherm of water on ASMS-3A shows, cavitation of water

confined in the cavities of silicas occurred at p/p0=0.33 at 293 K. Desorption of water on two samples KIT-5-1 day and KIT-5-7 days at 293 K occurred at p/p0 higher than the cavitation pressure. This indicates that these two systems can be analyzed based on the model pore network. Figure 12 compares the experimental desorption isotherm of water at 293 K on KIT-5-1 day with two kinds of simulated isotherms. The first one was obtained by the least-squares fitting of the experimental isotherm to the isotherm calculated based on the model pore system with the noncorrelated distribution of neck sizes. The best-fitted parameters are Dmean of 0.0 nm and σ of 2.8 nm. However, the least-squares fitting due to the non-correlated distribution of neck sizes did not at all give a good fit, similar to the desorption isotherms of 1.2 1

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0.8

experiment fitting(no spatial correlation)

0.6

fitting(spatial correlation) 0.4 0.2 0 0.2

0.4

0.6

0.8

1

p/p0

Figure 12. A comparison of the experimental desorption isotherm of water at 293 K on KIT-5-1 day with two kinds of calculated desorption isotherms. The explanation is given in the text.



nitrogen at 77 K on KIT-5-3 days, KIT-5-5 days, and KIT-5-7 days. The second isotherm was obtained by simulation of a desorption process in the pore network with the spatially correlated distribution in nick size. Similarly, a good fit was obtained by including the spatially correlated distribution in neck size. In the model pore network with the spatially correlated distribution in neck size, the necks are more enlarged in the outer layer of the sample particle, while σ is maintained at a fixed value of 1.0 nm. Figure 13 shows a comparison between the experimental desorption isotherm and two kinds of theoretical isotherms for water at 293 K on KIT-5-7 days. The first theoretical isotherm was obtained for the model pore network with the spatially correlated distribution of neck sizes that was determined from a comparison with the desorption isotherm of nitrogen at 77 K on the same material. Despite the different adsorbates, a fairly good fit was obtained, except for the region at lower p/p0. A fit in the lower p/p0 region between the experimental and theoretical desorption isotherms was improved by including a further narrowing of neck sizes in the inner region of the sample particle. Figure 11 also shows the plots of Dmean against L obtained from the analysis of the desorption isotherms of water on KIT-5-1

1.2 1

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0.8

experiment fitting(no spatial correlation)

0.6

fitting(spatial correlation) 0.4 0.2 0 0.2

0.4

0.6

0.8

1

p/p0

Figure 13. A comparison of the experimental desorption isotherm of water at 293 K on KIT-5-7 days with two kinds of calculated desorption isotherms. The explanation is given in the text.


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day and KIT-5-7 days. For KIT-5-1 day that possesses the least degree of correlation in the spatial distribution of neck sizes, the neck size distribution estimated with no spatial correlation was compared with that with the spatial correlation. For the model pore networks consisting of a fcc lattice of cavities with normal distribution of neck sizes, necks with sizes in the vicinity of a sum of Dmean and σ mainly participate in the percolation of vapor phase. For the non-correlated distribution, the corresponding diameters of necks are ∼2.8 nm. On the other hand, the effective diameters of necks with the correlated distribution are estimated by adding the value of σ (1 nm) to the curves shown in Figure 11. Therefore, the two distributions of neck sizes are roughly consistent with each other. DISCUSSION

Simple interpretation of the gradual desorption isotherms, which have often been observed for the ordered cagelike materials with prolonged hydrothermal treatments, may lead us to a conclusion that the size distributions of the necks are quite wide and any percolation effects do not operate upon desorption of liquid confined in the cavities.2,14,15 Previously, the absence of percolation effects was related to formation of an open pore structure in the cagelike materials such as the coalescence of adjacent necks and the development of large holes in pore walls.14,15 Indeed, in the ordered cagelike silica with prolonged hydrothermal treatment, formation of channel-like pores from coalescence of some adjacent connecting cavities has been confirmed by means of electron tomography.36 As long as these defect structures are randomly distributed throughout the sample particle, however, percolation effects will still occur in the desorption process, because percolation of the vapor phase into the defective regions needs formation of ordinary percolation pathways from the bulk vapor phase. When these defect structures are abundant and correlated to each other throughout the particle, earlier onset of the 23 ACS Paragon Plus Environment


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capillary evaporation could be observed in the desorption isotherm. The formation of a large number of the correlated defect structures such as channel structures throughout the sample particles should affect appreciably the small-angle X-ray scattering (SAXS) patterns and transmission electron microscopy (TEM) images because all these techniques including the nitrogen adsorption method furnish averaged information on the bulk sample through the materials. On the other hand, the 3D array structures of the cavities and the cavity diameter uniformity were maintained even after the prolonged hydrothermal treatments of the cagelike materials.13,14,16,17,36 As far as the normal distribution of neck sizes randomly distributed is assumed, the desorption isotherm of a fluid from the model pore network consisting of the fcc array of cavities interconnected through narrow necks is determined only by two parameters of the neck size distribution, that is, the mean diameter (Dmean) and standard deviation (σ) of the necks. The ordered cagelike silica KIT-5 possesses essentially the same pore structure as the model pore network examined here. However, the gradual desorption isotherms of nitrogen and water observed for KIT-5 samples were not well reproduced based on the model pore network that seemed to mimic the pore structure of KIT-5, as long as the pore network system has no correlation in the spatial distribution of neck sizes. On the other hand, inclusion of correlation in the spatial distribution of neck sizes led to an exceedingly better fit between the experimental and calculated desorption isotherms of nitrogen and water on KIT-5. The spatially correlated distribution of neck sizes does not affect appreciably both the SAXS pattern and the TEM image because the fcc structure of cavity arrangement is maintained.13,16,36 Desorption of nitrogen from KIT-5-1 day at 77 K occurred at the lower limit, which indicates that cavitation controlled desorption took place and thus the neck diameters are below 24 ACS Paragon Plus Environment

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∼5 nm. One cannot obtain any further details of the neck size distribution. On the other hand, desorption of water from the same material at 293 K occurred at p/p0 higher than the lower limit of hysteresis loop for water. This clearly reveals that measurements of water desorption isotherm at 293 K can be employed for estimation of the size distribution of necks with small sizes that are intractable by the nitrogen adsorption method at 77 K; that is, from the point of view of estimation of neck size distributions in the ordered cagelike materials, the water adsorption method at 293 K is superior to the nitrogen adsorption method at 77 K. For a sample KIT-5-1 day, Dmean of the necks in the most outer layers is ∼5 nm. Initially it decreases rapidly with increase in depth inside the particle and then slowly with further increase in depth, approaching ∼1 nm. For a sample KIT-5-7 days, the spatial distribution in neck size estimated from the water desorption isotherm is more reliable than that from the nitrogen desorption isotherm. All the necks with diameters less than 5 nm result in the occurrence of cavitation upon desorption of nitrogen at 77 K and thus they show almost the same values in the plots of Dmean against the layer number. This can be seen in the plots for the inner region that were obtained from the analysis of the desorption isotherms of nitrogen at 77 K on three kinds of KIT-5 samples, KIT-5-3 days, KIT-5-5 days, and KIT-5-7 days. For these three kinds of KIT-5 samples, appreciable fractions of the necks have diameters less than 5 nm, although the spatial correlation in neck size is present. The presence of the spatially correlated distributions in neck size estimated from the detailed analysis of the desorption isotherms is consistent with the results of successive adsorption of water and nitrogen previously reported by us.18,19 The neck sizes of the ordered cagelike materials increased with the prolonged hydrothermal treatment, accompanied by the increase in the spatial correlation of neck sizes. With increase in the hydrothermal treatment time, the necks in the outer region further increase in size, whereas neck sizes in the 25 ACS Paragon Plus Environment


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inner region decrease. For KIT-5-7 days, the necks in the inner region of the particles shrank with the prolonged hydrothermal treatment. This resulted in the appearance of long tail in the calculated desorption isotherm of nitrogen at 77 K at p/p0 below the lower limit of hysteresis loop, although such a long tail in the experimental desorption isotherm was not observed at 77 K. When the temperature was decreased below 77 K, however, such a long tail in the experimental desorption isotherm of nitrogen has been observed.17 This is because the critical size of the necks below which cavitation-controlled desorption takes place with decrease in pressure decreases with decrease in temperature.17,26 The appearance of the long tail in the desorption isotherm indicates trapping of nitrogen in the cavities with very small necks under suppression of cavitation. Most of ordered cagelike materials such as SBA-16,27 KIT-5,13,17 FDU-1,14,26 FDU-12,16,21 and KLE silicas24 consist of almost spherical cavities arranged in bcc or fcc lattice and connected through narrow neck, although cagelike materials of bcc structure are few. We have examined the model pore network of bcc structure in the same way. As a previous study34 shows, lowering the coordination number resulted in the shift of the desorption branch into lower pressures. The present study can be directly applied to most of the cagelike materials and easily extended to other cagelike materials of different structures.

CONCLUSIONS

The KIT-5 samples with prolonged hydrothermal treatments have the spatially correlated distributions of neck sizes; that is, necks in the outer region of the particles are enlarged by the prolonged hydrothermal treatments, while the necks in the inner region are shrank. Therefore, the gradual desorption isotherms of nitrogen and water observed for these KIT-5 samples were 26 ACS Paragon Plus Environment

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not well reproduced based on the model pore networks with no correlation in the spatial distribution of neck sizes that seemed to mimic the pore structure of KIT-5. On the other hand, the model networks with the spatially correlated distributions of neck sizes gave exceedingly better fits between the experimental and calculated desorption isotherms. However, we cannot rule out possibilities other than correlation in the spatial distribution of neck sizes as the origin of the gradual desorption branch. The measurements of water desorption isotherm at 293 K can be employed for estimation of the size distribution of necks with small sizes that are intractable by the nitrogen adsorption method at 77 K.

ASSOCIATED CONTENT Supporting Information

Adsorption-desorption isotherms of water on ASMA-3A, KIT-5-1 day, and KIT-5-7 days at 293 K, source code of main programs used in calculations of desorption isotherm of nitrogen at 77 K. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author

*Phone: +81-86-256-9494. Fax: +81-86-256-9757. E-mail: [email protected]. ORCID

Kunimitsu Morishige: 0000-0003-3874-5115 Notes

The author declares no competing financial interest.

ACKNOWLEDGMENTS



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We thank Mr. M. Tateishi for his technical assistance in the preparation of KIT-5 samples. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors

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