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Density profile of nitrogen in cylindrical pores of MCM-41
Chemical Physics Letters xxx (2017) xxx–xxx
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Research paper
Density profile of nitrogen in cylindrical pores of MCM-41 Alan K. Soper ⇑, Daniel T. Bowron ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxon OX11 0QX, UK
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 21 January 2017 In final form 21 March 2017 Available online xxxx
A straightforward approach using radiation scattering (X-ray or neutron) combined with atomistic modelling is used to accurately assess the pore dimensions in the porous silica, MCM-41. The method is used to calculate the density profile of nitrogen absorbed in this material at a variety of fractional pressures, p/ p0, where p0 is the saturated vapour pressure, up to p/p0 = 0.36 at T = 87 K in the present instance. At this pressure two distinct layers of liquid nitrogen occur on the silica surface, with a relatively sharp gasliquid interface. It is suggested surface tension effects at this interface strongly influence the growth of further layers. Ó 2017 Elsevier B.V. All rights reserved.
Keywords: MCM-41 Porous silica Neutron scattering Pore dimensions Absorbed nitrogen layers Density profile EPSR modelling
1. Introduction The question of how to characterise the surface and pore dimensions of a porous material has been the subject of a debate which extends over many decades, and which even now is not fully resolved [1–7]. The so-called ‘BET’ method [1] involves measuring the absorbed volume of gas, v, as a function of applied relative pressure, p/p0, where p0 is the saturated vapour pressure at the p temperature of the experiment, then plotting the value of v ðpp 0Þ versus
p , p0
for moderate values of
p p0
6 0:3. Then, by making some
simplifying assumptions about how gas is adsorbed on the surface, the slope and intercept of this graph give the values of vm, the volume of a complete single molecule layer, and a parameter c which is related to the difference in heat energy released when laying down the first molecular layer, compared to that for subsequent layers. In principle, the value of vm, is a measure of the surface area available for adsorption if an assumption about the thickness of the first layer is available. Subsequently, Shull [2] discusses the limitations of this approach as the pressure approaches the saturation vapour pressure in narrow capillaries, and discusses a modification to the Kelvin equation for capillary condensation originally proposed by Cohan [8]. The latter author also gives a cogent explanation of the hysteresis that occurs when absorbing and desorbing fluids from cylindrical pores. At this stage it becomes clear that BET the-
⇑ Corresponding author. E-mail address: [email protected] (A.K. Soper).
ory will underestimate the pressure, and therefore overestimate the layer thickness, at which capillary condensation would occur. Shull further goes on to show how experimental gas adsorption data on ‘large’ crystals can be used to make up for the deficiencies in the BET theory. Later, Barrett, Joyner and Halenda (BJH) [3] showed how, for cylindrical pores, the adsorption/desorption isotherms might be used to estimate the radius of the pores. Implicit in all of these approaches is the idea that gas adsorbs on the surface of the pore in a two-stage process, namely ‘physical adsorption’ in which one or more molecular layers of adsorbate are built up, presumably at liquid-like density, on the substrate surface, followed by capillary condensation at high enough pressure, in which the pore becomes completely filled with liquid. The volume of gas emitted when this capillary condensate desorbs, combined with the volume of liquid remaining as physically absorbed, is then used to estimate the volume of the pore, and hence its dimensions. Both Shull and BJH use this information further to estimate the pore size distribution approximately. Given that the theories being used here take no account of the likely fluid-fluid and fluid-wall interactions (other than via the surface tension of the liquid-gas interface), and that real pore walls are unlikely to be smooth, it is surprising that these methods persist to the present day as standard procedures for characterising porous materials. Much more recently, approaches to characterising the properties of a fluid on contact with a wall which take some account of the forces of interaction in a mean field sense have been developed using density functional theory [4–7,9]. These methods appear to give a much more realistic and quantitative account of
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experimental absorption data, and point to the limitations of the Kelvin equation at small pore radii. In particular, Ravikovitch [5] observes that the BJH method can underestimate pore diameters by as much as 10 Å or more for smaller pore sizes (see Fig. 9 of that paper), a conclusion which is born out by the present work. Unfortunately these more sophisticated methods do not lend themselves readily to non-expert execution and interpretation and so it is likely the simpler BET/BJH methods will remain in vogue for the purposes of elementary laboratory analysis. As a result confusion has continued right up to the present time on the true nature of the pores in MCM-41. For example a series of experiments that were aimed at exploring the structural and dynamical properties of water confined in MCM-41 repeatedly states the pore diameter to be of order 15 Å [10–13]. Yet the evidence obtained by analysing neutron diffraction data from the Bragg peaks that arise from the hexagonal arrangement of pores in these materials [14,15], and see below, suggests that the pores must be larger than the stated BJH values in reality. Equally uncertain is the degree of micro-porosity in these materials. This relates to the extent to which fluid molecules can move into the underlying silica matrix, perhaps even exchange with molecules in neighbouring pores. One study [16] used n-nonane to block the possible micropores in MCM-41 and found essentially no change to the adsorption/desorption isotherms as a result. The conclusion was that MCM-41 is likely to have negligible microporosity. Yet in attempts to build atomistic models of the amorphous silica in MCM-41, it proves almost impossible to exclude some degree of fluid penetration into the substrate [17] if the latter is assumed amorphous. In this context, the n-nonane test might be relevant for demonstrating that larger molecules do not penetrate the amorphous silica substrate, but might not be relevant for smaller molecules like H2O and N2. Another factor in these materials is the likely roughness of the pore surface. The pore template in MCM-41 is laid down by a cylindrical aqueous micelle, and atomistic models of spherical micelles highlight the relatively rough nature of the surfaces that are created by this technique [18]. Atomistic computer simulation of the pores in MCM-41 also confirms they are likely to be relatively rough, and hence BET analysis might give incorrect results [19]. In fact the conclusions of this latter paper closely matches the conclusions of the present work to be presented below. What therefore emerges in this analysis is that despite the extensive studies that have been done over many years, there remains a great deal of uncertainty about the nature of these porous materials at the molecular level. Most likely each instance of MCM-41 has a slightly different morphology to the next, and this will make obtaining consistent results from gas absorption and similar experiments hard to obtain. Most important of all however is the fact that the pore diameter and surface structure needs to be carefully ascertained before any experiments are performed with these materials. To this end we propose here that a (relatively) simple procedure using X-ray or neutron diffraction to measure the low-wave-vector (Q) and high wave-vector diffraction pattern, followed by atomistic modelling of the measured data, can go a long way towards determining the pore volume, likely structure of the substrate, and degree of pore-size distribution. Typically in a good sample of MCM-41 one observes about 4 main Bragg diffraction peaks [5,20] in the low Q region, while the high Q region is sensitive to the local arrangement of silica. Atomistic fitting to these data already gives a lot of direct information on the structure of these materials and eliminates much of the uncertainty and speculation that comes from more indirect methods. At the very least it should corroborate the indirect evidence from gas adsorption/desorption isotherms. In the sections that follow, we describe the experimental and atomistic modelling techniques that were employed here, then
perform a study of dry MCM-41 to obtain the pore diameter. This is followed by a study of nitrogen absorbed on MCM-41, where a clear layering of the nitrogen is observed. The paper finishes with some concluding remarks. 2. Experimental Samples of MCM-41 were obtained from Sigma-Aldrich and used without further purification. We understood these had been synthesized with C16TAB surfactant. Nitrogen adsorption and desorption isotherms were interpreted using the BJH method, from which it was concluded the average nominal pore diameter by this method was 2.24 nm, which was within the specification supplied by the manufacturer. A mass of 0.611 g of MCM-41 was placed in a flat plate container made from an alloy of Ti and Zr in the mole ratio 0.676:0.324. This alloy contributes almost zero coherent scattering to the diffraction pattern because of the near cancellation of the Ti and Zr neutron scattering lengths at this composition. The wall thickness was 1 mm, and the space between the walls occupied by the sample was 2 mm. The lateral dimensions of the container, which was placed at right angles to the neutron beam, was 40 mm high by 35 mm wide. A gas handling rig was connected to the cell to control the amount of nitrogen absorbed, and the whole assembly was mounted on a closed cycle helium refrigerator to control the temperature within ±1 K. Total neutron scattering measurements were performed on the NIMROD diffractometer [21] at the ISIS Spallation Neutron Source. The experimental protocol was as follows. With the sample initially under vacuum, its temperature was lowered to 123 K and held there for around 5½ h while the neutron diffraction pattern of the dry sample was measured. Then nitrogen was introduced at 1 bar, corresponding to a relative pressure p/p0 = 0.035 and the diffraction pattern monitored repeatedly for short periods of around 6 min each. Subsequently, holding the nitrogen pressure stable in the range 0.73–1.0 bar, the temperature was ramped down to 87 K over a period of approximately 4 h, corresponding to a final relative pressure of p/p0 = 0.368. These relative pressures are calculated on the basis of the stated saturated vapour pressure of nitrogen at these temperatures [22]. Finally the sample was held under these conditions for a further 5 h, during which time there was only very slight further absorption of nitrogen. Fig. 1 (top) shows the temperature and the relative nitrogen pressure as a function of elapsed time. Unfortunately no data are available on the actual amount of nitrogen absorbed by the sample in this case. The neutron scattering data were subject to the standard corrections for background scattering, multiple scattering, container scattering, and self attenuation, and put on an absolute scale of differential scattering cross section (DCS) per atom of material, using the scattering from a standard 3 mm vanadium slab [23]. In the present instance, because the material is a powder, while the pore diameter and the atomic number density are only known approximately, it is necessary to supply a correction factor to the data which is derived from the expected value of the DCS at high Q, that is the single atom scattering. This value is given by the expression for the total DCS of a material:-
X 2 X dr ðQ Þ ¼ c a ba þ ð2 dX a a;bPa Z dab Þca cb ba bb 4pq
0
1
r 2 ðg ab ðrÞ 1Þ
sinQr dr Qr
ð1Þ
where ca and ba are the atomic fraction and neutron scattering length of atom a, q is the total atomic number density of the material respectively, and gab(r) the site-site radial distribution function between sites a and b, with Q the wave vector change in the scattering experiment. The first term on the LHS of (1) is the single atom
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A.K. Soper, D.T. Bowron / Chemical Physics Letters xxx (2017) xxx–xxx
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Fig. 2. Interference differential cross sections for dry MCM-41 (a), plus three nitrogen absorptions listed in legend (b)–(d). The numbers give the ratios of nitrogen atoms per mole of substrate, Si0.28O0.61H0.11 that were used in the EPSR simulations. These values are different from those obtained from Fig. 1, but within the likely uncertainties, in order to get the best possible fits to the scattering data.
Fig. 1. (a) Evolution of temperature (blue, left-hand axis) and nitrogen relative pressure (orange, right-hand axis) in the present experiment. (b) Estimated absorbed number of nitrogen atoms per mole of substrate, assumed to be Si0.28O0.61H0.11, as explained in the text. Nitrogen molecule fractions will be half of these values. Each dot corresponds to the end of a neutron scan (except for the leftmost dot which corresponds to the start of the first scan), but note that these times do not necessarily reflect directly the amount of neutron beam actually received for each scan. The letters and vertical dashed lines indicate the regions used for subsequent structural analysis. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
scattering which is a known quantity if the atomic fractions are known and independent of Q. Hence it can be used to put the scattering data onto an absolute scale. The second term on the LHS of (1), involving a sum over pairs of atom types, will be referred to here as the ‘interference’ scattering and denoted by F(Q). Although the precise composition of these samples is unknown, it is widely believed they consist primarily of amorphous SiO2 with a number of silanol groups included. Hence for the present analysis the same composition as used in a previous analysis is assumed here as well, namely Si0.28O0.61H0.11 [24]. Assigning the composition means that the expected high Q level of the dry MCM-41 data can be calculated and this was used to assign the final scaling factor on the data: in the present case this factor was 3.98 with an assumed atomic number density of 0.0296 atoms/Å3. Then, for the samples where nitrogen absorption occurred, the same scaling factor and number density were assumed, so that the change in high Q scattering level would correspond approximately to the number of atoms of nitrogen absorbed per Si0.28O0.61H0.11 unit. These values are shown in Fig. 1 (bottom). It can be seen that, as expected, the amount of nitrogen absorbed depends directly on the relative pressure of the nitrogen, but that for the lowest relative pressures the scattering level of the data is indistinguishable from the dry sample due to the uncertainties in this method of absorption determination, believed to be on the order of 0.1 in the atomic fraction of absorbed nitrogen. However changes to the interference scattering as soon as nitrogen is added suggest that some absorption has in fact taken place even at the lowest relative pressure. The interference scattering, F(Q), from 4 representative scans of the sample during nitrogen absorption are shown in Fig. 2. These scans correspond to the four groups of scans highlighted by the let-
ters (a) – (d) in Fig 1 (bottom). Note that the scans at relative times near 08:00 could not be used because this was a period where the nitrogen content changed significantly, but there were no distinct scans in this period. Clearly there is pronounced Porod (low Q) scattering in these materials, arising from the graininess of these materials, but this will not be discussed or analysed further here since its origins are likely to be complex and related to the morphology of the samples at the microscopic level. Also clearly obvious are the Bragg peaks arising from the hexagonal lattice of cylindrical pores, the first four of which are indexed. It is notable that the higher order peaks steadily decrease in amplitude as the nitrogen absorbs, but the (1 0 0) peak remains pronounced. This rules out any idea that capillary condensation has occurred in these materials, even though the nitrogen relative pressure is in the region expected for such condensation [5,20]. We can say this with some confidence since if the pores had filled with liquid, the contrast between the substrate and the pore region would have changed substantially, and so significantly reduced the amplitude of the Bragg peaks. For the substrate the scattering length density is 0.283 fm/Å3 while for liquid nitrogen it is 0.325 fm/Å3. Since to a first approximation the height of the (1 0 0) Bragg peak depends on the square of the difference between these two values then that peak should decrease in intensity by a factor of (0.283– 0.324)2/0.2832 = 0.021 if the pore fills with liquid. In practice there is no sign of such a pronounced decrease at any composition, so the nitrogen must have either gone into the substrate itself, or been absorbed only on the wall of the pore. (In practice of course the intensity of the Bragg peaks will also depend sensitively on exactly how the nitrogen is arranged in the pore, but this factor is traditionally ignored in the literature.) It is likely of course that because we are well above (87 K = 0.7 Tc) the liquid nitrogen boiling point (77.4 K) this has prevented capillary condensation. The lowest sample temperature was deliberately chosen to be above the liquid nitrogen boiling point to ensure that no liquid nitrogen condensed on the outer surfaces of the MCM-41.
3. Empirical potential structure refinement (EPSR) The method used here follows closely that described already in a previous publication [17]. Initially a single hexagonal unit cell is constructed of dimension (a b c) of (46.068 Å 46.068 Å 148.000 Å), with the angle between a and b set to 120°. The a and b dimensions are set by the position of
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the (1 0 0) peak Fig. 2. A row of non-scattering ‘Q’ atoms is placed at the centre of the unit cell parallel to the c axis. These atoms are used only to define the pore and do not contribute to the calculated scattering. Si4+, O2 and OH units are now added at the required level, assuming an atomic number density in the substrate region of 0.066 atoms/Å3, that is the same as in bulk amorphous silica. The stoichiometry is kept the same as defined in Section 2, with small adjustments to ensure electrical neutrality of the substrate. A minimum distance constraint is applied between all substrate atoms and the Q atoms to ensure the pore dimensions are maintained as the simulation proceeds. 3.1. Determining the pore diameter
Fig. 4. Quality of fit in real space.
f ðrÞ ¼
1 2p q 2
Z Q 2 FðQ Þ
sinQr dQ : Qr
ð2Þ
This leads to a different quality of fit Fig. 4. Now the fits with a diameters of 27 Å and 41 Å are notably worse than at intermediate diameters, but there is a broad minimum in the region 37–40 Å. Combining these two results it seems the likely pore diameter is around 40 Å but is not larger than this value. Based on the same graphs, we estimate the uncertainty in these estimates to be of order 1 Å. To confirm this, we show in Fig. 5 the fits to the dry MCM-41 data for pore diameters of 40 Å and 27 Å. It now becomes clear that
F(Q) [barns/sr/atom]
1.2 100
(a) d = 40Å 1
0.8
10
1
10
1 0.1
1
10
Q [1/Å] 1.2
F(Q) [barns/sr/atom]
The first step is to determine the optimum pore diameter from the dry MCM-41 data. To achieve this a series of EPSR simulations were set up using the above prescription for a series of nominal pore diameters covering the range 27–41 Å, and the structure refined to give the best fit to the data in each case. The quality of fit is assessed by means of a statistic (v2) which measures how close the simulation gets to the data. This is plotted as a function of the assumed pore diameter in Fig. 3. It is seen that there is a clear minimum in this function at a nominal pore diameter of 40 Å. A pore diameter of that size would correspond to a wall thickness between pores of 6.1 Å, which is not unreasonable compared to other estimates [5,20]. However the quality factor is also diminishing as the pore diameter gets smaller, below 36 Å. It might be thought therefore that at even lower pore diameters, below 27 Å, the fit would improve even further. When assessing the quality of fit, one has to bear in mind the enormous difference in intensity of the (1 0 0) Bragg peak, compared to the other peaks Fig. 2. Hence the fit to this peak completely dominates the quality of fit value. Another way to assess the quality of fit is to compare the simulation with the data in real space (r-space), where there is much less sensitivity to one particular Bragg peak. To do this both data and simulation have to be Fourier transformed to r-space according to
100
(b) d = 27Å 1
0.8
10
1
10
1 0.1
1
10
Q [1/Å] Fig. 5. EPSR fits (lines) to the dry MCM-41 data (dots) for the case (a) of an assumed pore diameter of d = 40 Å and (b) d = 27 Å. The insets show the wider Q region 1 – 20 Å1.
whilst both simulations capture the (1 0 0) peak quite accurately, the higher order Bragg peaks are much better described with the larger diameter pore, particularly as regards the diffuse scattering between them. In particular the (1 1 0) Bragg peak is almost completely absent when d = 27 Å, and the fit in the wider Q region is generally better for the larger diameter pore. Both fits tend to over-estimate the heights of the higher order Bragg peaks, which may be a result of the lack of a pore size distribution in this model. 3.2. Simulation of nitrogen absorption
Fig. 3. Quality of fit in Q-space to dry MCM-41 data (curve (a) in Fig. 2) for EPSR simulations using a variety of assumed pore diameters.
Having decided on the pore diameter (40 Å) the unit cell of dry MCM-41 was used to create three new 2 2 1 supercells, giving an array of 4 pores on which to absorb nitrogen in each case. Nitrogen molecules (intramolecular spacing 1.11 Å) were introduced to give the three atom mole ratios shown in Fig 2, curves (b), (c) and (d). The Supplementary Information gives the full details of the
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A.K. Soper, D.T. Bowron / Chemical Physics Letters xxx (2017) xxx–xxx
EPSR simulations. Nitrogen molecules were inserted at random and with random orientations into the substrate and pores at any place where the interaction energy with the substrate was zero or less. As a result some penetration of the N2 into the substrate did occur, although it was small compared to nitrogen in the pores. Once these new simulations were at equilibrium (energy stable) the empirical potential was switched on and allowed to grow in amplitude to give the best possible fit to the data. Note that the positions of the substrate atoms continued to be refined against the data after nitrogen was added, as well as those of the nitrogen molecules. The results are shown in Fig. 6 for all three cases. Note that when the nitrogen was present it was necessary, to get the best fits, to keep the nitrogen away from the centre of the pore. This was achieved by imposing Q-N minimum distances of 16 Å for case (b), 15 Å for case (c) and 12.4 Å for case (d). These values were determined by trial and error as the simulation proceeded. If the nitrogen was allowed to drift too much to the centre of the pore, then the intensity of the simulated (1 0 0) Bragg peak would drop, causing the quality of fit to get noticeably worse. This reinforces the idea that the nitrogen in these systems stays close to the silica surface.
F(Q) [barns/sr/atom]
(a) 0.173 N 100
10
1.6 1.4 1.2 1 0.8 0.6 1
10
1 0.1
1
10
Q [1/Å]
F(Q) [barns/sr/atom]
(b) 0.254 N 100
10
1.6 1.4 1.2 1 0.8 0.6 1
10
1
10
Q [1/Å]
F(Q) [barns/sr/atom]
(c) 0.487 N 100
10
1.6 1.4 1.2 1 0.8 0.6 1
10
1 0.1
1
It can be seen that, generally, good fits to the data over 2 orders of magnitude in intensity, and over a Q range that spans more than 2 orders of magnitude of distance scale, are obtained. The higher order Bragg peaks for case (c) are the least satisfactory in this regard, suggesting again that this model has too much long range order in general. On the other hand, the fits to the local structure, as seen from the insets to Fig. 6, are good for all three compositions, suggesting that the assumed compositions in these simulations are close to those that occurred in the experiment. Further interrogation of these simulations can be performed by measuring the density of the silicon and nitrogen as a function of distance across the pore, measured from the central pore axis, for all four cases Fig. 7. These graphs show clearly that as nitrogen is absorbed by the pore, it develops first one, then two distinct layers near the silica surface. What happens when capillary condensation occurs will require further experiments at greater nitrogen absorption, and probably temperatures closer to the liquid nitrogen boiling point. It is also clear that the silica is quite structured near the surface. Finally we can get an idea of the possible orientations of the nitrogen molecules in these layers by measuring the distribution of angles the nitrogen molecule axis makes with the Q – N axis, that is the axis from the centre of the pore to one of the atoms on any nitrogen (see Fig. 8). It is observed that the layer of molecules closest to the substrate, (a) in Fig. 7, is rather loosely orientated, though with a slight preference for the nitrogen molecule to lie at right angles to the QN axis, and therefore flat on the substrate surface. This distribution is also slightly asymmetric with respect to the 90° position due to the way the angle is defined: the angle is with respect to one or other ends of the nitrogen molecule, not its centre, so that angles where the free end points towards the substrate will not necessarily be symmetric with those where it points away from the substrate. In this case it seems the nitrogen molecule has a slight tendency to point away from the surface. On the other hand for the second layer, further away from the substrate, there is a clear tendency for the nitrogen molecules to point along the Q-N axis, that is lie radially with respect to the pore axis.
4. Discussion
1 0.1
5
10
Q [1/Å] Fig. 6. EPSR fits (lines) to the MCM-41 plus nitrogen neutron scattering data (dots) at nitrogen atom ratios (per mole of substrate) of (a) 0.173, (b) 0.254, and (c) 0.487 as indicated in Fig 1 (bottom). The insets show the fits at wider Q in more detail.
Based on the preceding analysis, it becomes clear that the combination of total radiation scattering and empirical potential structure refinement gives important information about the nature of the substrate as well as the way any adsorbate is configured on the pore surface in MCM-41 cylindrical pores. Studying the dry MCM-41 Figs. 3–5, it seems the combined approach can give quite accurate accounts of the likely pore diameter to an accuracy of around 1 Å. These measures are completely independent of any gas adsorption isotherm data that may be available, but would have to be consistent with it. Hence in the present case the estimated pore diameter (40 Å) is clearly much larger than that derived from the gas adsorption data (22.7 Å), and this can only mean the interpretation of the latter does not take correct account of the nature of the surface. The same conclusion was found when comparing BJH with a density functional interpretation of the gas adsorption data [5]. The present conclusions are in line with those observed in a previous computer simulation of MCM-41 [19], and in another (smaller pore diameter) sample of MCM-41 [14]. They are also in line with a previous X-ray diffraction study of SBA porous silicas [25], and with independent studies of similar materials [26,27], which also concluded the pore sizes from the BJH method were too small. Indeed the conclusions from Ref. [19] and in
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A.K. Soper, D.T. Bowron / Chemical Physics Letters xxx (2017) xxx–xxx
0.06
0.0066
Si
(a) 0.0 N
0.0062 0.006
0.04
0.0058 0.02 0
5
10
15
20
25
30
35
40
0.0052 0.005 0.0048
3 ρ(r) [atoms/Å ]
0.08
0.0046
Si N
(b) 0.173 N
0.06
0
5
10
15
20
25
30
3
ρ(r) [atoms/Å ]
35
60
80
100
Si N
(c) 0.254 N
0.06
120
140
160
180
160
180
Q−N−N
0.008
40
p(θ)
0.08
0.007 0.006 0.005 0.004
0.04
0.003 0.02
0
20
40
60
80
100
120
140
θ [deg.] 0
5
10
15
20
25
30
35
40
r [Å] 0.08 3
40
(b) 12.6−15.8 Å
0.009
r [Å]
0.04 0.02 0
5
10
Fig. 8. Distribution of nitrogen molecule orientations with respect to the Q-N axis for (a) the first and (b) the second layer of nitrogen molecules shown in Fig. 7 case (d). The labels indicate the distances of each layer from the central row of Q atoms. In this case the first layer is furthest from the centre of the pore.
Si N
(d) 0.487 N
0.06
0
20
0.01
0.02
0
0
θ [deg.]
0.04
0
0.0056 0.0054
0
r [Å]
ρ(r) [atoms/Å ]
Q−N−N
(a) 15.8−19.2 Å
0.0064
p(θ)
3 ρ(r) [atoms/Å ]
0.08
15
20
25
30
35
40
r [Å] Fig. 7. Density profile of silicon (red line) and nitrogen (blue line) as a function of distance from the centre of the pore for dry MCM-41 Fig. 5(a), plus the three cases of increasing nitrogen adsorption shown in Fig. 6. The dashed lines give the atomic number density of bulk liquid nitrogen. The secondary peak (s) for r > 25 Å for nitrogen in these figures arises from nitrogen absorbed in neighbouring pores. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
another study of water in MCM-41 [28] closely parallel the present observations for nitrogen in MCM-41. This is important because it has been stated several times that water ice cannot form in these pores because the pores are too small [29]. Once it is realised that the actual pore dimensions are most likely much larger than the nominal values obtained from gas adsorption data, and that the surfaces themselves might have a complex topography, these statements can no longer be justified and might be too simplistic for the real material. When nitrogen is introduced at 1 bar pressure, it is clear Fig 7, that nitrogen does indeed initially absorb in distinct layers, as was often supposed in the BET and BJH theories [1,3], but that capillary condensation does not occur in these experiments, in spite of conditions apparently being correct for such condensation [5]. This, however, may be a consequence of the sample temperature being too high [30], or the nature of the pore surface in this case. Yet the gas-liquid surface inside the pore is apparently quite sharp Fig. 7, spanning a region of 2 Å, which poses an interesting conundrum.
The radius of this surface is between 12.5 and 17 Å, measured from the centre of the pore, depending on whether two layers or one layer is adsorbed respectively. Being a curved surface it will presumably manifest surface tension effects. Because the liquid surface is concave with respect to the gas phase it encloses, there must be a pressure drop across the surface in the liquid phase, implying the adsorbed liquid is at a negative pressure compared to the gas phase (0.1 MPa). Assuming the surface tension of nitrogen is c = 6.8 mN/m at 87 K [31], this translates to a pressure change across the surface of c/r = 3.9 MPa across the first surface after it has formed. In other words the liquid on the surface of the pore has to be in a state of tension. After the second surface forms, the pressure drops further, due to the smaller radius, to 5.4 MPa. Hence the adsorbed liquid has to be in a state of significant tension, presumably sustained by the attractive force for the gas molecules onto the substrate. It could be speculated that this negative pressure might inhibit further layers of nitrogen to form on to the surface, because to do so would cause the pressure of the adsorbed liquid to become even more negative which might induce cavitation in the liquid and so prevent further adsorption. A similar concept has been used to measure the strain that the MCM lattice undergoes when fluid is absorbed in the capillary [32]. This idea could now be easily tested, simply by lowering the temperature of the sample towards 77 K, while monitoring the diffraction pattern: lowering the temperature causes the surface tension to rise quite markedly [27], which, on the above scenario, would allow greater tension in the liquid and hence more adsorption. Based on what has been shown here, the diffraction data are highly sensitive to how much gas is adsorbed and how it is arranged on the surface. This method is much more reliable than simply measuring the volume of gas adsorbed, since the latter data give no indication where the adsorbed gas has actually gone on the surface, whereas the diffraction data in this case are definitive.
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It is interesting to note that the physics of this surface tension effect appears to be built naturally into the density functional theories already discussed [4,5], in the form of the grand potential functional, but it does not appear in either the BET or BJH theories. This might explain why the two approaches give such markedly different results [5]. 5. Conclusion The combined use of radiation total scattering (neutron radiation in the present instance) and realistic computer simulation data modelling (EPSR in the present instance) on the porous material MCM-41 shows that a reliable and consistent picture can be obtained both of the pore dimensions, and the way adsorbates grow on the surface of the pore. The relative pressure used in the present measurements was apparently not large enough at the final temperature of the measurements (87 K) to instigate capillary condensation, but two distinct layers of nitrogen appear on the surface with little if any gas appearing at the centre of the pore. At this level the results are consistent with early theories about the way gases adsorb onto surfaces. The lack of capillary condensation in our experiment, even though the conditions were apparently sufficient to allow this to happen, leads us to speculate this might be a consequence of the surface tension at the gas-liquid interface which would cause the adsorbed nitrogen to be in a state of significant tension, and so inhibit further adsorption, until the temperature is lowered to raise the surface tension and so allow greater adsorption. It appears there is much work to do to better characterise this important system for the study of materials in confinement. Acknowledgements The authors would like to thank the Ionic Liquids Group at Queen’s University Belfast for the BJH estimates of the pore diameters of our samples. Neutron beam time at ISIS was provided by the Science and Technology Facilities Council Direct Access route. We thank a referee for useful additional references. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.cplett.2017.03. 060. References [1] S. Brunauer, P.H. Emmett, E. Teller, Adsorption of gases in multimolecular layers, J. Am. Chem. Soc. 60 (1938) 309–319, http://dx.doi.org/ 10.1021/ja01269a023. [2] C.G. Shull, The determination of pore size distribution from gas adsorption data, J. Am. Chem. Soc. 70 (1948) 1405–1410, http://dx.doi.org/ 10.1021/ja01184a034. [3] E.P. Barrett, L.G. Joyner, P.P. Halenda, The determination of pore volume and area distributions in porous substances. I. Computations from nitrogen isotherms, J. Am. Chem. Soc. 73 (1951) 373–380, http://dx.doi.org/ 10.1021/ja01145a126. [4] R. Evans, U.M.B. Marconi, P. Tarazona, Capillary condensation and adsorption in cylindrical and slit-like pores, J. Chem. Soc. Faraday Trans. 2 (82) (1986) 1763, http://dx.doi.org/10.1039/f29868201763. [5] P.I. Ravikovitch, S.C.O. Domhnaill, A.V. Neimark, F. Schueth, K.K. Unger, Capillary hysteresis in nanopores: theoretical and experimental studies of nitrogen adsorption on MCM-41, Langmuir 11 (1995) 4765–4772, http://dx. doi.org/10.1021/la00012a030. [6] J. Landers, G.Y. Gor, A.V. Neimark, Density functional theory methods for characterization of porous materials, Colloids Surf. A Physicochem. Eng. Asp. 437 (2013) 3–32, http://dx.doi.org/10.1016/j.colsurfa.2013.01.007. [7] I. Hitchcock, S. Malik, E.M. Holt, R.S. Fletcher, S.P. Rigby, Impact of chemical heterogeneity on the accuracy of pore size distributions in disordered solids, J. Phys. Chem. C 118 (2014) 20627–20638, http://dx.doi.org/10.1021/jp505482p.
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A. 104 (2007) 9570–9574, http://dx.doi.org/10.1073/pnas.0701352104. [13] Z. Wang, K. Ito, J.B. Leão, L. Harriger, Y. Liu, S.-H. Chen, Liquid-liquid phase transition and its phase diagram in deeply-cooled heavy water confined in a nanoporous silica matrix, J. Phys. Chem. Lett. 6 (2015) 2009–2014, http://dx. doi.org/10.1021/acs.jpclett.5b00827. [14] A.K. Soper, Density profile of water confined in cylindrical pores in MCM-41 silica, J. Phys. Condens. Matter Phys. Condens. 24 (2012) 64107, http://dx.doi. org/10.1088/0953-8984/24/6/064107. [15] A.K. Soper, J. Teixeira, T. Head-Gordon, Is ambient water inhomogeneous on the nanometer-length scale?, Proc Natl. Acad. Sci. U. S. A. 107 (2010) E44, http://dx.doi.org/10.1073/pnas.0912158107, author reply E45. [16] A. Silvestre-Albero, E.O. Jardim, E. Bruijn, V. Meynen, P. Cool, A. SepúlvedaEscribano, J. Silvestre-Albero, F. Rodríguez-Reinoso, Is there any microporosity in ordered mesoporous silicas?, Langmuir 25 (2009) 939–943, http://dxdoi. org/10.1021/la802692z. [17] A.K. Soper, Radical re-appraisal of water structure in hydrophilic confinement, Chem. Phys. Lett. 590 (2013) 1–15, http://dx.doi.org/10.1016/j. cplett.2013.10.075. [18] R. Hargreaves, D.T. Bowron, K. Edler, Atomistic structure of a micelle in solution determined by wide Q-range neutron diffraction, J. Am. Chem. Soc. 133 (2011) 16524–16536, http://dx.doi.org/10.1021/ja205804k. [19] C.G. Sonwane, C.W. Jones, P.J. Ludovice, A model for the structure of MCM-41 incorporating surface roughness, J. Phys. Chem. B 109 (2005) 23395–23404, http://dx.doi.org/10.1021/jp051713c. [20] G. Lelong, S. Bhattacharyya, S. Kline, T. Cacciaguerra, M.A. Gonzalez, M.L. Saboungi, Effect of surfactant concentration on the morphology and texture of MCM-41 materials, J. Phys. Chem. C 112 (2008) 10674–10680, http://dx.doi. org/10.1021/jp800898n. [21] D.T. Bowron, A.K. 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Research paper
Density profile of nitrogen in cylindrical pores of MCM-41 Alan K. Soper ⇑, Daniel T. Bowron ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxon OX11 0QX, UK
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 21 January 2017 In final form 21 March 2017 Available online xxxx
A straightforward approach using radiation scattering (X-ray or neutron) combined with atomistic modelling is used to accurately assess the pore dimensions in the porous silica, MCM-41. The method is used to calculate the density profile of nitrogen absorbed in this material at a variety of fractional pressures, p/ p0, where p0 is the saturated vapour pressure, up to p/p0 = 0.36 at T = 87 K in the present instance. At this pressure two distinct layers of liquid nitrogen occur on the silica surface, with a relatively sharp gasliquid interface. It is suggested surface tension effects at this interface strongly influence the growth of further layers. Ó 2017 Elsevier B.V. All rights reserved.
Keywords: MCM-41 Porous silica Neutron scattering Pore dimensions Absorbed nitrogen layers Density profile EPSR modelling
1. Introduction The question of how to characterise the surface and pore dimensions of a porous material has been the subject of a debate which extends over many decades, and which even now is not fully resolved [1–7]. The so-called ‘BET’ method [1] involves measuring the absorbed volume of gas, v, as a function of applied relative pressure, p/p0, where p0 is the saturated vapour pressure at the p temperature of the experiment, then plotting the value of v ðpp 0Þ versus
p , p0
for moderate values of
p p0
6 0:3. Then, by making some
simplifying assumptions about how gas is adsorbed on the surface, the slope and intercept of this graph give the values of vm, the volume of a complete single molecule layer, and a parameter c which is related to the difference in heat energy released when laying down the first molecular layer, compared to that for subsequent layers. In principle, the value of vm, is a measure of the surface area available for adsorption if an assumption about the thickness of the first layer is available. Subsequently, Shull [2] discusses the limitations of this approach as the pressure approaches the saturation vapour pressure in narrow capillaries, and discusses a modification to the Kelvin equation for capillary condensation originally proposed by Cohan [8]. The latter author also gives a cogent explanation of the hysteresis that occurs when absorbing and desorbing fluids from cylindrical pores. At this stage it becomes clear that BET the-
⇑ Corresponding author. E-mail address: [email protected] (A.K. Soper).
ory will underestimate the pressure, and therefore overestimate the layer thickness, at which capillary condensation would occur. Shull further goes on to show how experimental gas adsorption data on ‘large’ crystals can be used to make up for the deficiencies in the BET theory. Later, Barrett, Joyner and Halenda (BJH) [3] showed how, for cylindrical pores, the adsorption/desorption isotherms might be used to estimate the radius of the pores. Implicit in all of these approaches is the idea that gas adsorbs on the surface of the pore in a two-stage process, namely ‘physical adsorption’ in which one or more molecular layers of adsorbate are built up, presumably at liquid-like density, on the substrate surface, followed by capillary condensation at high enough pressure, in which the pore becomes completely filled with liquid. The volume of gas emitted when this capillary condensate desorbs, combined with the volume of liquid remaining as physically absorbed, is then used to estimate the volume of the pore, and hence its dimensions. Both Shull and BJH use this information further to estimate the pore size distribution approximately. Given that the theories being used here take no account of the likely fluid-fluid and fluid-wall interactions (other than via the surface tension of the liquid-gas interface), and that real pore walls are unlikely to be smooth, it is surprising that these methods persist to the present day as standard procedures for characterising porous materials. Much more recently, approaches to characterising the properties of a fluid on contact with a wall which take some account of the forces of interaction in a mean field sense have been developed using density functional theory [4–7,9]. These methods appear to give a much more realistic and quantitative account of
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experimental absorption data, and point to the limitations of the Kelvin equation at small pore radii. In particular, Ravikovitch [5] observes that the BJH method can underestimate pore diameters by as much as 10 Å or more for smaller pore sizes (see Fig. 9 of that paper), a conclusion which is born out by the present work. Unfortunately these more sophisticated methods do not lend themselves readily to non-expert execution and interpretation and so it is likely the simpler BET/BJH methods will remain in vogue for the purposes of elementary laboratory analysis. As a result confusion has continued right up to the present time on the true nature of the pores in MCM-41. For example a series of experiments that were aimed at exploring the structural and dynamical properties of water confined in MCM-41 repeatedly states the pore diameter to be of order 15 Å [10–13]. Yet the evidence obtained by analysing neutron diffraction data from the Bragg peaks that arise from the hexagonal arrangement of pores in these materials [14,15], and see below, suggests that the pores must be larger than the stated BJH values in reality. Equally uncertain is the degree of micro-porosity in these materials. This relates to the extent to which fluid molecules can move into the underlying silica matrix, perhaps even exchange with molecules in neighbouring pores. One study [16] used n-nonane to block the possible micropores in MCM-41 and found essentially no change to the adsorption/desorption isotherms as a result. The conclusion was that MCM-41 is likely to have negligible microporosity. Yet in attempts to build atomistic models of the amorphous silica in MCM-41, it proves almost impossible to exclude some degree of fluid penetration into the substrate [17] if the latter is assumed amorphous. In this context, the n-nonane test might be relevant for demonstrating that larger molecules do not penetrate the amorphous silica substrate, but might not be relevant for smaller molecules like H2O and N2. Another factor in these materials is the likely roughness of the pore surface. The pore template in MCM-41 is laid down by a cylindrical aqueous micelle, and atomistic models of spherical micelles highlight the relatively rough nature of the surfaces that are created by this technique [18]. Atomistic computer simulation of the pores in MCM-41 also confirms they are likely to be relatively rough, and hence BET analysis might give incorrect results [19]. In fact the conclusions of this latter paper closely matches the conclusions of the present work to be presented below. What therefore emerges in this analysis is that despite the extensive studies that have been done over many years, there remains a great deal of uncertainty about the nature of these porous materials at the molecular level. Most likely each instance of MCM-41 has a slightly different morphology to the next, and this will make obtaining consistent results from gas absorption and similar experiments hard to obtain. Most important of all however is the fact that the pore diameter and surface structure needs to be carefully ascertained before any experiments are performed with these materials. To this end we propose here that a (relatively) simple procedure using X-ray or neutron diffraction to measure the low-wave-vector (Q) and high wave-vector diffraction pattern, followed by atomistic modelling of the measured data, can go a long way towards determining the pore volume, likely structure of the substrate, and degree of pore-size distribution. Typically in a good sample of MCM-41 one observes about 4 main Bragg diffraction peaks [5,20] in the low Q region, while the high Q region is sensitive to the local arrangement of silica. Atomistic fitting to these data already gives a lot of direct information on the structure of these materials and eliminates much of the uncertainty and speculation that comes from more indirect methods. At the very least it should corroborate the indirect evidence from gas adsorption/desorption isotherms. In the sections that follow, we describe the experimental and atomistic modelling techniques that were employed here, then
perform a study of dry MCM-41 to obtain the pore diameter. This is followed by a study of nitrogen absorbed on MCM-41, where a clear layering of the nitrogen is observed. The paper finishes with some concluding remarks. 2. Experimental Samples of MCM-41 were obtained from Sigma-Aldrich and used without further purification. We understood these had been synthesized with C16TAB surfactant. Nitrogen adsorption and desorption isotherms were interpreted using the BJH method, from which it was concluded the average nominal pore diameter by this method was 2.24 nm, which was within the specification supplied by the manufacturer. A mass of 0.611 g of MCM-41 was placed in a flat plate container made from an alloy of Ti and Zr in the mole ratio 0.676:0.324. This alloy contributes almost zero coherent scattering to the diffraction pattern because of the near cancellation of the Ti and Zr neutron scattering lengths at this composition. The wall thickness was 1 mm, and the space between the walls occupied by the sample was 2 mm. The lateral dimensions of the container, which was placed at right angles to the neutron beam, was 40 mm high by 35 mm wide. A gas handling rig was connected to the cell to control the amount of nitrogen absorbed, and the whole assembly was mounted on a closed cycle helium refrigerator to control the temperature within ±1 K. Total neutron scattering measurements were performed on the NIMROD diffractometer [21] at the ISIS Spallation Neutron Source. The experimental protocol was as follows. With the sample initially under vacuum, its temperature was lowered to 123 K and held there for around 5½ h while the neutron diffraction pattern of the dry sample was measured. Then nitrogen was introduced at 1 bar, corresponding to a relative pressure p/p0 = 0.035 and the diffraction pattern monitored repeatedly for short periods of around 6 min each. Subsequently, holding the nitrogen pressure stable in the range 0.73–1.0 bar, the temperature was ramped down to 87 K over a period of approximately 4 h, corresponding to a final relative pressure of p/p0 = 0.368. These relative pressures are calculated on the basis of the stated saturated vapour pressure of nitrogen at these temperatures [22]. Finally the sample was held under these conditions for a further 5 h, during which time there was only very slight further absorption of nitrogen. Fig. 1 (top) shows the temperature and the relative nitrogen pressure as a function of elapsed time. Unfortunately no data are available on the actual amount of nitrogen absorbed by the sample in this case. The neutron scattering data were subject to the standard corrections for background scattering, multiple scattering, container scattering, and self attenuation, and put on an absolute scale of differential scattering cross section (DCS) per atom of material, using the scattering from a standard 3 mm vanadium slab [23]. In the present instance, because the material is a powder, while the pore diameter and the atomic number density are only known approximately, it is necessary to supply a correction factor to the data which is derived from the expected value of the DCS at high Q, that is the single atom scattering. This value is given by the expression for the total DCS of a material:-
X 2 X dr ðQ Þ ¼ c a ba þ ð2 dX a a;bPa Z dab Þca cb ba bb 4pq
0
1
r 2 ðg ab ðrÞ 1Þ
sinQr dr Qr
ð1Þ
where ca and ba are the atomic fraction and neutron scattering length of atom a, q is the total atomic number density of the material respectively, and gab(r) the site-site radial distribution function between sites a and b, with Q the wave vector change in the scattering experiment. The first term on the LHS of (1) is the single atom
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Fig. 2. Interference differential cross sections for dry MCM-41 (a), plus three nitrogen absorptions listed in legend (b)–(d). The numbers give the ratios of nitrogen atoms per mole of substrate, Si0.28O0.61H0.11 that were used in the EPSR simulations. These values are different from those obtained from Fig. 1, but within the likely uncertainties, in order to get the best possible fits to the scattering data.
Fig. 1. (a) Evolution of temperature (blue, left-hand axis) and nitrogen relative pressure (orange, right-hand axis) in the present experiment. (b) Estimated absorbed number of nitrogen atoms per mole of substrate, assumed to be Si0.28O0.61H0.11, as explained in the text. Nitrogen molecule fractions will be half of these values. Each dot corresponds to the end of a neutron scan (except for the leftmost dot which corresponds to the start of the first scan), but note that these times do not necessarily reflect directly the amount of neutron beam actually received for each scan. The letters and vertical dashed lines indicate the regions used for subsequent structural analysis. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
scattering which is a known quantity if the atomic fractions are known and independent of Q. Hence it can be used to put the scattering data onto an absolute scale. The second term on the LHS of (1), involving a sum over pairs of atom types, will be referred to here as the ‘interference’ scattering and denoted by F(Q). Although the precise composition of these samples is unknown, it is widely believed they consist primarily of amorphous SiO2 with a number of silanol groups included. Hence for the present analysis the same composition as used in a previous analysis is assumed here as well, namely Si0.28O0.61H0.11 [24]. Assigning the composition means that the expected high Q level of the dry MCM-41 data can be calculated and this was used to assign the final scaling factor on the data: in the present case this factor was 3.98 with an assumed atomic number density of 0.0296 atoms/Å3. Then, for the samples where nitrogen absorption occurred, the same scaling factor and number density were assumed, so that the change in high Q scattering level would correspond approximately to the number of atoms of nitrogen absorbed per Si0.28O0.61H0.11 unit. These values are shown in Fig. 1 (bottom). It can be seen that, as expected, the amount of nitrogen absorbed depends directly on the relative pressure of the nitrogen, but that for the lowest relative pressures the scattering level of the data is indistinguishable from the dry sample due to the uncertainties in this method of absorption determination, believed to be on the order of 0.1 in the atomic fraction of absorbed nitrogen. However changes to the interference scattering as soon as nitrogen is added suggest that some absorption has in fact taken place even at the lowest relative pressure. The interference scattering, F(Q), from 4 representative scans of the sample during nitrogen absorption are shown in Fig. 2. These scans correspond to the four groups of scans highlighted by the let-
ters (a) – (d) in Fig 1 (bottom). Note that the scans at relative times near 08:00 could not be used because this was a period where the nitrogen content changed significantly, but there were no distinct scans in this period. Clearly there is pronounced Porod (low Q) scattering in these materials, arising from the graininess of these materials, but this will not be discussed or analysed further here since its origins are likely to be complex and related to the morphology of the samples at the microscopic level. Also clearly obvious are the Bragg peaks arising from the hexagonal lattice of cylindrical pores, the first four of which are indexed. It is notable that the higher order peaks steadily decrease in amplitude as the nitrogen absorbs, but the (1 0 0) peak remains pronounced. This rules out any idea that capillary condensation has occurred in these materials, even though the nitrogen relative pressure is in the region expected for such condensation [5,20]. We can say this with some confidence since if the pores had filled with liquid, the contrast between the substrate and the pore region would have changed substantially, and so significantly reduced the amplitude of the Bragg peaks. For the substrate the scattering length density is 0.283 fm/Å3 while for liquid nitrogen it is 0.325 fm/Å3. Since to a first approximation the height of the (1 0 0) Bragg peak depends on the square of the difference between these two values then that peak should decrease in intensity by a factor of (0.283– 0.324)2/0.2832 = 0.021 if the pore fills with liquid. In practice there is no sign of such a pronounced decrease at any composition, so the nitrogen must have either gone into the substrate itself, or been absorbed only on the wall of the pore. (In practice of course the intensity of the Bragg peaks will also depend sensitively on exactly how the nitrogen is arranged in the pore, but this factor is traditionally ignored in the literature.) It is likely of course that because we are well above (87 K = 0.7 Tc) the liquid nitrogen boiling point (77.4 K) this has prevented capillary condensation. The lowest sample temperature was deliberately chosen to be above the liquid nitrogen boiling point to ensure that no liquid nitrogen condensed on the outer surfaces of the MCM-41.
3. Empirical potential structure refinement (EPSR) The method used here follows closely that described already in a previous publication [17]. Initially a single hexagonal unit cell is constructed of dimension (a b c) of (46.068 Å 46.068 Å 148.000 Å), with the angle between a and b set to 120°. The a and b dimensions are set by the position of
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A.K. Soper, D.T. Bowron / Chemical Physics Letters xxx (2017) xxx–xxx
the (1 0 0) peak Fig. 2. A row of non-scattering ‘Q’ atoms is placed at the centre of the unit cell parallel to the c axis. These atoms are used only to define the pore and do not contribute to the calculated scattering. Si4+, O2 and OH units are now added at the required level, assuming an atomic number density in the substrate region of 0.066 atoms/Å3, that is the same as in bulk amorphous silica. The stoichiometry is kept the same as defined in Section 2, with small adjustments to ensure electrical neutrality of the substrate. A minimum distance constraint is applied between all substrate atoms and the Q atoms to ensure the pore dimensions are maintained as the simulation proceeds. 3.1. Determining the pore diameter
Fig. 4. Quality of fit in real space.
f ðrÞ ¼
1 2p q 2
Z Q 2 FðQ Þ
sinQr dQ : Qr
ð2Þ
This leads to a different quality of fit Fig. 4. Now the fits with a diameters of 27 Å and 41 Å are notably worse than at intermediate diameters, but there is a broad minimum in the region 37–40 Å. Combining these two results it seems the likely pore diameter is around 40 Å but is not larger than this value. Based on the same graphs, we estimate the uncertainty in these estimates to be of order 1 Å. To confirm this, we show in Fig. 5 the fits to the dry MCM-41 data for pore diameters of 40 Å and 27 Å. It now becomes clear that
F(Q) [barns/sr/atom]
1.2 100
(a) d = 40Å 1
0.8
10
1
10
1 0.1
1
10
Q [1/Å] 1.2
F(Q) [barns/sr/atom]
The first step is to determine the optimum pore diameter from the dry MCM-41 data. To achieve this a series of EPSR simulations were set up using the above prescription for a series of nominal pore diameters covering the range 27–41 Å, and the structure refined to give the best fit to the data in each case. The quality of fit is assessed by means of a statistic (v2) which measures how close the simulation gets to the data. This is plotted as a function of the assumed pore diameter in Fig. 3. It is seen that there is a clear minimum in this function at a nominal pore diameter of 40 Å. A pore diameter of that size would correspond to a wall thickness between pores of 6.1 Å, which is not unreasonable compared to other estimates [5,20]. However the quality factor is also diminishing as the pore diameter gets smaller, below 36 Å. It might be thought therefore that at even lower pore diameters, below 27 Å, the fit would improve even further. When assessing the quality of fit, one has to bear in mind the enormous difference in intensity of the (1 0 0) Bragg peak, compared to the other peaks Fig. 2. Hence the fit to this peak completely dominates the quality of fit value. Another way to assess the quality of fit is to compare the simulation with the data in real space (r-space), where there is much less sensitivity to one particular Bragg peak. To do this both data and simulation have to be Fourier transformed to r-space according to
100
(b) d = 27Å 1
0.8
10
1
10
1 0.1
1
10
Q [1/Å] Fig. 5. EPSR fits (lines) to the dry MCM-41 data (dots) for the case (a) of an assumed pore diameter of d = 40 Å and (b) d = 27 Å. The insets show the wider Q region 1 – 20 Å1.
whilst both simulations capture the (1 0 0) peak quite accurately, the higher order Bragg peaks are much better described with the larger diameter pore, particularly as regards the diffuse scattering between them. In particular the (1 1 0) Bragg peak is almost completely absent when d = 27 Å, and the fit in the wider Q region is generally better for the larger diameter pore. Both fits tend to over-estimate the heights of the higher order Bragg peaks, which may be a result of the lack of a pore size distribution in this model. 3.2. Simulation of nitrogen absorption
Fig. 3. Quality of fit in Q-space to dry MCM-41 data (curve (a) in Fig. 2) for EPSR simulations using a variety of assumed pore diameters.
Having decided on the pore diameter (40 Å) the unit cell of dry MCM-41 was used to create three new 2 2 1 supercells, giving an array of 4 pores on which to absorb nitrogen in each case. Nitrogen molecules (intramolecular spacing 1.11 Å) were introduced to give the three atom mole ratios shown in Fig 2, curves (b), (c) and (d). The Supplementary Information gives the full details of the
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A.K. Soper, D.T. Bowron / Chemical Physics Letters xxx (2017) xxx–xxx
EPSR simulations. Nitrogen molecules were inserted at random and with random orientations into the substrate and pores at any place where the interaction energy with the substrate was zero or less. As a result some penetration of the N2 into the substrate did occur, although it was small compared to nitrogen in the pores. Once these new simulations were at equilibrium (energy stable) the empirical potential was switched on and allowed to grow in amplitude to give the best possible fit to the data. Note that the positions of the substrate atoms continued to be refined against the data after nitrogen was added, as well as those of the nitrogen molecules. The results are shown in Fig. 6 for all three cases. Note that when the nitrogen was present it was necessary, to get the best fits, to keep the nitrogen away from the centre of the pore. This was achieved by imposing Q-N minimum distances of 16 Å for case (b), 15 Å for case (c) and 12.4 Å for case (d). These values were determined by trial and error as the simulation proceeded. If the nitrogen was allowed to drift too much to the centre of the pore, then the intensity of the simulated (1 0 0) Bragg peak would drop, causing the quality of fit to get noticeably worse. This reinforces the idea that the nitrogen in these systems stays close to the silica surface.
F(Q) [barns/sr/atom]
(a) 0.173 N 100
10
1.6 1.4 1.2 1 0.8 0.6 1
10
1 0.1
1
10
Q [1/Å]
F(Q) [barns/sr/atom]
(b) 0.254 N 100
10
1.6 1.4 1.2 1 0.8 0.6 1
10
1
10
Q [1/Å]
F(Q) [barns/sr/atom]
(c) 0.487 N 100
10
1.6 1.4 1.2 1 0.8 0.6 1
10
1 0.1
1
It can be seen that, generally, good fits to the data over 2 orders of magnitude in intensity, and over a Q range that spans more than 2 orders of magnitude of distance scale, are obtained. The higher order Bragg peaks for case (c) are the least satisfactory in this regard, suggesting again that this model has too much long range order in general. On the other hand, the fits to the local structure, as seen from the insets to Fig. 6, are good for all three compositions, suggesting that the assumed compositions in these simulations are close to those that occurred in the experiment. Further interrogation of these simulations can be performed by measuring the density of the silicon and nitrogen as a function of distance across the pore, measured from the central pore axis, for all four cases Fig. 7. These graphs show clearly that as nitrogen is absorbed by the pore, it develops first one, then two distinct layers near the silica surface. What happens when capillary condensation occurs will require further experiments at greater nitrogen absorption, and probably temperatures closer to the liquid nitrogen boiling point. It is also clear that the silica is quite structured near the surface. Finally we can get an idea of the possible orientations of the nitrogen molecules in these layers by measuring the distribution of angles the nitrogen molecule axis makes with the Q – N axis, that is the axis from the centre of the pore to one of the atoms on any nitrogen (see Fig. 8). It is observed that the layer of molecules closest to the substrate, (a) in Fig. 7, is rather loosely orientated, though with a slight preference for the nitrogen molecule to lie at right angles to the QN axis, and therefore flat on the substrate surface. This distribution is also slightly asymmetric with respect to the 90° position due to the way the angle is defined: the angle is with respect to one or other ends of the nitrogen molecule, not its centre, so that angles where the free end points towards the substrate will not necessarily be symmetric with those where it points away from the substrate. In this case it seems the nitrogen molecule has a slight tendency to point away from the surface. On the other hand for the second layer, further away from the substrate, there is a clear tendency for the nitrogen molecules to point along the Q-N axis, that is lie radially with respect to the pore axis.
4. Discussion
1 0.1
5
10
Q [1/Å] Fig. 6. EPSR fits (lines) to the MCM-41 plus nitrogen neutron scattering data (dots) at nitrogen atom ratios (per mole of substrate) of (a) 0.173, (b) 0.254, and (c) 0.487 as indicated in Fig 1 (bottom). The insets show the fits at wider Q in more detail.
Based on the preceding analysis, it becomes clear that the combination of total radiation scattering and empirical potential structure refinement gives important information about the nature of the substrate as well as the way any adsorbate is configured on the pore surface in MCM-41 cylindrical pores. Studying the dry MCM-41 Figs. 3–5, it seems the combined approach can give quite accurate accounts of the likely pore diameter to an accuracy of around 1 Å. These measures are completely independent of any gas adsorption isotherm data that may be available, but would have to be consistent with it. Hence in the present case the estimated pore diameter (40 Å) is clearly much larger than that derived from the gas adsorption data (22.7 Å), and this can only mean the interpretation of the latter does not take correct account of the nature of the surface. The same conclusion was found when comparing BJH with a density functional interpretation of the gas adsorption data [5]. The present conclusions are in line with those observed in a previous computer simulation of MCM-41 [19], and in another (smaller pore diameter) sample of MCM-41 [14]. They are also in line with a previous X-ray diffraction study of SBA porous silicas [25], and with independent studies of similar materials [26,27], which also concluded the pore sizes from the BJH method were too small. Indeed the conclusions from Ref. [19] and in
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A.K. Soper, D.T. Bowron / Chemical Physics Letters xxx (2017) xxx–xxx
0.06
0.0066
Si
(a) 0.0 N
0.0062 0.006
0.04
0.0058 0.02 0
5
10
15
20
25
30
35
40
0.0052 0.005 0.0048
3 ρ(r) [atoms/Å ]
0.08
0.0046
Si N
(b) 0.173 N
0.06
0
5
10
15
20
25
30
3
ρ(r) [atoms/Å ]
35
60
80
100
Si N
(c) 0.254 N
0.06
120
140
160
180
160
180
Q−N−N
0.008
40
p(θ)
0.08
0.007 0.006 0.005 0.004
0.04
0.003 0.02
0
20
40
60
80
100
120
140
θ [deg.] 0
5
10
15
20
25
30
35
40
r [Å] 0.08 3
40
(b) 12.6−15.8 Å
0.009
r [Å]
0.04 0.02 0
5
10
Fig. 8. Distribution of nitrogen molecule orientations with respect to the Q-N axis for (a) the first and (b) the second layer of nitrogen molecules shown in Fig. 7 case (d). The labels indicate the distances of each layer from the central row of Q atoms. In this case the first layer is furthest from the centre of the pore.
Si N
(d) 0.487 N
0.06
0
20
0.01
0.02
0
0
θ [deg.]
0.04
0
0.0056 0.0054
0
r [Å]
ρ(r) [atoms/Å ]
Q−N−N
(a) 15.8−19.2 Å
0.0064
p(θ)
3 ρ(r) [atoms/Å ]
0.08
15
20
25
30
35
40
r [Å] Fig. 7. Density profile of silicon (red line) and nitrogen (blue line) as a function of distance from the centre of the pore for dry MCM-41 Fig. 5(a), plus the three cases of increasing nitrogen adsorption shown in Fig. 6. The dashed lines give the atomic number density of bulk liquid nitrogen. The secondary peak (s) for r > 25 Å for nitrogen in these figures arises from nitrogen absorbed in neighbouring pores. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
another study of water in MCM-41 [28] closely parallel the present observations for nitrogen in MCM-41. This is important because it has been stated several times that water ice cannot form in these pores because the pores are too small [29]. Once it is realised that the actual pore dimensions are most likely much larger than the nominal values obtained from gas adsorption data, and that the surfaces themselves might have a complex topography, these statements can no longer be justified and might be too simplistic for the real material. When nitrogen is introduced at 1 bar pressure, it is clear Fig 7, that nitrogen does indeed initially absorb in distinct layers, as was often supposed in the BET and BJH theories [1,3], but that capillary condensation does not occur in these experiments, in spite of conditions apparently being correct for such condensation [5]. This, however, may be a consequence of the sample temperature being too high [30], or the nature of the pore surface in this case. Yet the gas-liquid surface inside the pore is apparently quite sharp Fig. 7, spanning a region of 2 Å, which poses an interesting conundrum.
The radius of this surface is between 12.5 and 17 Å, measured from the centre of the pore, depending on whether two layers or one layer is adsorbed respectively. Being a curved surface it will presumably manifest surface tension effects. Because the liquid surface is concave with respect to the gas phase it encloses, there must be a pressure drop across the surface in the liquid phase, implying the adsorbed liquid is at a negative pressure compared to the gas phase (0.1 MPa). Assuming the surface tension of nitrogen is c = 6.8 mN/m at 87 K [31], this translates to a pressure change across the surface of c/r = 3.9 MPa across the first surface after it has formed. In other words the liquid on the surface of the pore has to be in a state of tension. After the second surface forms, the pressure drops further, due to the smaller radius, to 5.4 MPa. Hence the adsorbed liquid has to be in a state of significant tension, presumably sustained by the attractive force for the gas molecules onto the substrate. It could be speculated that this negative pressure might inhibit further layers of nitrogen to form on to the surface, because to do so would cause the pressure of the adsorbed liquid to become even more negative which might induce cavitation in the liquid and so prevent further adsorption. A similar concept has been used to measure the strain that the MCM lattice undergoes when fluid is absorbed in the capillary [32]. This idea could now be easily tested, simply by lowering the temperature of the sample towards 77 K, while monitoring the diffraction pattern: lowering the temperature causes the surface tension to rise quite markedly [27], which, on the above scenario, would allow greater tension in the liquid and hence more adsorption. Based on what has been shown here, the diffraction data are highly sensitive to how much gas is adsorbed and how it is arranged on the surface. This method is much more reliable than simply measuring the volume of gas adsorbed, since the latter data give no indication where the adsorbed gas has actually gone on the surface, whereas the diffraction data in this case are definitive.
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A.K. Soper, D.T. Bowron / Chemical Physics Letters xxx (2017) xxx–xxx
It is interesting to note that the physics of this surface tension effect appears to be built naturally into the density functional theories already discussed [4,5], in the form of the grand potential functional, but it does not appear in either the BET or BJH theories. This might explain why the two approaches give such markedly different results [5]. 5. Conclusion The combined use of radiation total scattering (neutron radiation in the present instance) and realistic computer simulation data modelling (EPSR in the present instance) on the porous material MCM-41 shows that a reliable and consistent picture can be obtained both of the pore dimensions, and the way adsorbates grow on the surface of the pore. The relative pressure used in the present measurements was apparently not large enough at the final temperature of the measurements (87 K) to instigate capillary condensation, but two distinct layers of nitrogen appear on the surface with little if any gas appearing at the centre of the pore. At this level the results are consistent with early theories about the way gases adsorb onto surfaces. The lack of capillary condensation in our experiment, even though the conditions were apparently sufficient to allow this to happen, leads us to speculate this might be a consequence of the surface tension at the gas-liquid interface which would cause the adsorbed nitrogen to be in a state of significant tension, and so inhibit further adsorption, until the temperature is lowered to raise the surface tension and so allow greater adsorption. It appears there is much work to do to better characterise this important system for the study of materials in confinement. Acknowledgements The authors would like to thank the Ionic Liquids Group at Queen’s University Belfast for the BJH estimates of the pore diameters of our samples. Neutron beam time at ISIS was provided by the Science and Technology Facilities Council Direct Access route. We thank a referee for useful additional references. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.cplett.2017.03. 060. References [1] S. Brunauer, P.H. Emmett, E. Teller, Adsorption of gases in multimolecular layers, J. Am. Chem. Soc. 60 (1938) 309–319, http://dx.doi.org/ 10.1021/ja01269a023. [2] C.G. Shull, The determination of pore size distribution from gas adsorption data, J. Am. Chem. Soc. 70 (1948) 1405–1410, http://dx.doi.org/ 10.1021/ja01184a034. [3] E.P. Barrett, L.G. Joyner, P.P. Halenda, The determination of pore volume and area distributions in porous substances. I. Computations from nitrogen isotherms, J. Am. Chem. Soc. 73 (1951) 373–380, http://dx.doi.org/ 10.1021/ja01145a126. [4] R. Evans, U.M.B. Marconi, P. Tarazona, Capillary condensation and adsorption in cylindrical and slit-like pores, J. Chem. Soc. Faraday Trans. 2 (82) (1986) 1763, http://dx.doi.org/10.1039/f29868201763. [5] P.I. Ravikovitch, S.C.O. Domhnaill, A.V. Neimark, F. Schueth, K.K. Unger, Capillary hysteresis in nanopores: theoretical and experimental studies of nitrogen adsorption on MCM-41, Langmuir 11 (1995) 4765–4772, http://dx. doi.org/10.1021/la00012a030. [6] J. Landers, G.Y. Gor, A.V. Neimark, Density functional theory methods for characterization of porous materials, Colloids Surf. A Physicochem. Eng. Asp. 437 (2013) 3–32, http://dx.doi.org/10.1016/j.colsurfa.2013.01.007. [7] I. Hitchcock, S. Malik, E.M. Holt, R.S. Fletcher, S.P. Rigby, Impact of chemical heterogeneity on the accuracy of pore size distributions in disordered solids, J. Phys. Chem. C 118 (2014) 20627–20638, http://dx.doi.org/10.1021/jp505482p.
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