Langmuir 2000, 16, 10193-10197
10193
Spectroscopic Mesopore Size Characterization and Diffusion Measurement in Closed Porosity by Xenon NMR F. Cros, J.-P. Korb,* and L. Malier Laboratoire de Physique de la Matie` re Condense´ e, CNRS UMR 7643, Ecole Polytechnique, 91128 Palaiseau, France Received March 3, 2000. In Final Form: September 28, 2000 Nuclear magnetic resonance measurements of 129Xe chemical shifts in a large variety of nonionic calibrated microporous and mesoporous silica glasses with open porosity provide a calibration of spectroscopic mesopore size measurements. The pore size dependence of the xenon chemical shift is quantitatively interpreted by considering both fast exchange and van der Waals interactions at the surface, which cause chemical shifts. The temperature dependence of the chemical shift supports the proposed model and characterizes the enthalpy of adsorption at the pore surface. The xenon NMR measurement is also useful for very low or even quasi-closed porous systems and may provide assessment of the macroscopic xenon diffusion coefficient in xerogels.
Introduction NMR of 129Xe gas has been widely used to characterize the pore size of microporous materials.1-3 129Xe is chemically inert and has a relatively high gyromagnetic ratio and 26.2% natural abundance, which provide good sensitivity even for gas-phase samples. The large chemical shift range results in an effective separation of the NMR signals: xenon trapped in the pores and free xenon outside the pores of microporous materials. Other recent NMR experiments use hyperpolarized xenon.4 However, these experiments become inefficient in most porous media because of rapid spin-lattice relaxation T1 and poor xenon penetration within the relaxation time of the nonBoltzmann polarization. This penetration is almost equal to x(DT1). Most 129Xe studies have been reported on zeolites for which the interpretation of chemical shift is difficult because of electric fields induced by the numerous ionic species present in the zeolite cavities.1-3 Here, we extend the usual domain of application of this method to the simpler situation of nonionic mesoporous materials, using amorphous but pore-size-calibrated materials in the 20250 Å range. The sol-gel and colloid precipitation techniques provide a collection of glass samples, whose individual pore size distribution is narrow. We study experimentally the variation of the xenon chemical shift with the pore size and temperature. On the basis of these experiments, we propose a theoretical model that accounts for the experimental data without adjustable parameters. We believe that this method could be useful to give a reliable spectroscopic characterization of the pore size in a large variety of nonionic microporous and mesoporous materials and provides information about the adsorption coefficient. For a quasi-closed porous system, interpreta* Author for correspondence. E-mail:
[email protected]. Telephone: 33 1 69 33 47 39. Fax: 33 1 69 33 30 04. (1) Ito, T.; Fraissard, J. J. Chem Phys. 1982, 76, 5225. (2) Demarquay, J.; Fraissard, J. Chem. Phys. Lett. 1987, 136, 314. (3) Fraissard, J. Encyclopedia of Nuclear Magnetic Resonance; Grant, M., Harris, R., Eds.; John Wiley & Sons: Chichester, 1996; Vol. 5, p 3058. (4) Brunner, E.; Haakel, M.; Pines, A.; Reimer, J. A.; Seydoux, R. Magnetic Resonance and related phenomena, I, Proceedings of the Joint 29th Ampere and 13th ISMAR international conference, Berlin; Ziessow, D., Lubitz, W., Lendzian, F., Eds.; Technische Univ. Berlin: 1998; p 208.
tion of the 129Xe chemical shift is problematic, but it is still possible to measure the macroscopic xenon diffusion coefficient in xerogels with a very low or even quasi-closed porosity. Experimental Section Samples and NMR Measurements. We used a large set of calibrated porous silica glasses prepared by the sol-gel or silica colloids precipitation method. The pore size distributions were characterized by the nitrogen adsorption-desorption isotherm at 77 K (Figure 1). The typical shape of the isotherm found (BET isotherm of class IV)6 is consistent with a cylindrical pore geometry. Introducing the volume-to-surface ratio, one has the relation Vp/Sp ) R/2 defining the average pore radius R. The average pore radii of sol-gel samples extend from 14 to 81 Å, and those of silica colloids extend from 34 to 131 Å. The samples were heated at 150 °C during 4 h at a pressure of 1 mbar to desorb water and adsorbed gases, and then they were filled with xenon in a 10 mm NMR tube to a pressure of 5 bar at room temperature. Contrary to the nuclear relaxation rate, the 129Xe chemical shift is not very sensitive to the change of pressure. NMR measurements were made on a Bruker MSL 360, at a frequency of 99.3 MHz. The duration of a π/2 pulse was 17.5 µs. For a sample of 1 cm3 at 5 bar we have a signal-to-noise ratio for the trapped xenon of at least 3 after 100 accumulations without any apodization. The spin-lattice relaxation times T1 have been measured with the saturation recovery method. Following a temperature change, samples were allowed to equilibrate for 45 min. We show in Figure 2 a typical spectrum of xenon confined in a calibrated microporous silica glass. We note a chemical shift of 100 ppm between the NMR signals of trapped and free xenon, the latter being considered as the reference of the chemical shift.
Results and Discussion Biphasic Fast Exchange Model and Influence of Local Pore Shape and Curvature. Figure 3 shows the chemical shifts in both series of samples. The large domain of pore sizes explored permits comparison of our observations to the existing models which relate the chemical shift, δ, to the average pore radius R. Basically, there are two different models to interpret these data. The first model considers a fast chemical exchange between adsorbed and bulk phases of xenon: (5) Lide, D. R. Handbook of Chemistry and Physics; CRC Press: Boston, 1991; p D178. (6) Liu, G.; Li, Y.; Jonas, J. J. Chem. Phys. 1991, 95, 6892.
10.1021/la000322g CCC: $19.00 © 2000 American Chemical Society Published on Web 11/21/2000
10194
Langmuir, Vol. 16, No. 26, 2000
Cros et al.
Figure 1. Pore size distribution of a porous silica glass of average radius 51 Å, measured by the nitrogen adsorptiondesorption BET method.
Figure 2. Typical example of a colloidal silica of 118 Å at 240 K.
129Xe
NMR spectrum of a
Figure 4. Theoretical surface fraction of xenon according to Henry (eq 7) or Langmuir (eq 9) isotherms.
Equation 4 follows closely the experimental values of Figure 3 for the large pores but fails for the small pores. The best fit of our data gives kads ) 1.44 ( 0.16 × 1017 atoms/(bar m2), which is close to the results found in the literature for similar materials. The same fit gives a chemical shift value for the surface as δa ) 85 ( 3 ppm. We do not observe any improvement of the fit by considering in eq 4 the modifications induced by the size of the xenon itself.7 The second model proposed by Derouane8,9 takes into account the van der Waals interaction of a xenon close to a curved pore wall. Once integrated over the whole pore surface, this mode becomes
δ)
Figure 3. Experimental chemical shifts observed on sol-gel samples (b) and colloidal silica (Ο) at 295 K. The continuous line represents the best fit obtained according to eq 10 with only one free parameter: kads.
δ)
Naδa + Nvδv Na + N v
(1)
where Na, δa and Nv, δv are the xenon populations and chemical shifts of the surface and volume, respectively. Because the chemical shift for the volume corresponds to the gas at 5 bar outside the pores, one has δv ) 0 for the reference. For a pressure P < 100 bar the ideal gas equation is sufficient.5 Introducing the xenon density, F, at pressure P and temperature T, the xenon population Nv for a single pore of volume Vp and surface Sp is given by
Nv ) FVp )
P V kT p
(2)
The xenon population at the surface, Na, is given by the Henry’s law:
Na ) kads(T)PSp
(3)
where kads(T) is the adsorption coefficient at temperature T. The substitution of eqs 2 and 3 into eq 1 and the introduction of the radius R gives
δa
δ) 1+
R 2kads(T)kT
(4)
BU/R3 R- 1R
[ (
(5)
2 3
)]
Here B represents the screening hyperpolarizability factor essentially due to the less bonded electrons and U is the ionization potential. For xenon, one has BU ) 6492 ppm/ Å3 10 and ∼ 4.4 Å for the distance between the xenon and the pore surface (van der Waals radius of the physisorbed state). These factors have been previously measured for the pressure and temperature of our experiments; thus, this model does not require any adjustable parameters. However, this model overestimates the chemical shift and fits the data poorly. In fact, this second model assumes that all the confined xenon is adsorbed on the pore surface, that is, that all of the pore xenon is presumed to be at the surface. Thus, the chemical shift δa of an adsorbed xenon is predicted to be independent of the pore size. To account for nonsurface xenon in a pore, we include a biphasic exchange as an addition to the Derouane equation. We consider that δa in eq 4 varies as proposed by Derouane.10 This leads to the relation
δ)
BU/R3 R R- 1+ 1R 2kads(T)kT 1
[ (
2 3
)]
(6)
The best fit obtained with eq 6 for δ(R), with a single parameter kads, gives kads ) (1.73 ( 0.06) × 1017 atoms/ (bar m2) for the colloidal and sol-gel samples. Similar measurements give 2.28 × 1017 atoms/(bar m2) for silica gels. From these values, one obtains the fraction of xenon (∼80%) at the pore surface (Figure 4): (7) Ripmeester, J. A.; Ratcliffe, C. I. J. Phys. Chem. 1990, 94, 7652. (8) Derouane, E. G.; Andre´, J. M.; Lucas, A. A. Chem. Phys. Lett. 1987, 137, 336. (9) Derouane, E. G.; Nagy, J. B. Chem. Phys. Lett. 1987, 137, 341. (10) Jameson, C. J.; de Dios, A. C. J. Chem. Phys. 1992, 97, 417.
Mesopore Size Characterization by Xenon NMR
{
Langmuir, Vol. 16, No. 26, 2000 10195
}
Na R ) 1+ Na + N v 2kads(T)kT
-1
(7)
The surface fraction is thus larger than the bulk fraction even for the largest pores. According to eq 3, this fraction has been obtained providing that the surface population depends linearly on the pressure, that is, Henry’s law. However, the validity of this hypothesis may be doubtful at high coverage. For instance, the substitution into eq 3 of the adsorption coefficient kads ) 1.73 × 1017 atoms/(bar m2) found at 5 bar yields a surface density σ ) 8.65 × 1017 xenon/m2. This density stands well below σm ) 5.6 × 1018 xenon/m2, calculated at full coverage with an average surface area of 18 Å2. By taking into account the possibility of a saturation by a single layer of xenon at the surface, Langmuir introduced another isotherm,11 with a surface density defined as
kadsP σ ) σm σm + kadsP
Figure 5. Temperature dependence of xenon chemical shift in silica colloids of 118 Å pore radius. The dotted points (b) correspond to the experiments. The biphasic fast exchange model (continuous line) has allowed us to estimate the adsorption enthalpy, ∆Hads ) 31 ( 1 kJ/mol, in the high-temperature range. The 2 points represent the deviation from this model at low temperature, and the corresponding continuous line is the best fit obtained with a van der Waals surface interaction model.
(8)
Although numerous isotherm equations exist, we introduce here the simplest one that will be largely sufficient to explain our data. A modification of the surface fraction thus results:
{
Na ) 1+ Na + Nv
}
R kadsσm kT 2 σm + kadsP
-1
(9)
Figure 6. 129Xe NMR spectra recorded in the same porous colloidal silica of 118 Å at 225 K (a) and 255 K (b).
which appears to be very small at room temperature (Figure 4). However, it will present a large influence at low temperature, as described below. Equation 6 thus becomes
{
δ) 1+
}[
R kadsσm kT 2 σm + kadsP
-1
BU/R3 R- 1R
(
2 3
)]
(10)
where kads is the only free parameter. Application of eq 10 to the data in Figure 3 yields kads ) 2.05 × 1017 ( 0.1 atoms/(bar m2), which is sufficiently close to the previous one to justify the absence of any observed modification in the chemical shift in the range of pressure studied. Temperature Dependence of the Chemical Shift in Confinement. We have displayed in Figure 5 the temperature dependence between 210 and 310 K of the xenon chemical shift for a silica colloid sample with a pore size of 118 Å radius filled with xenon at a pressure of 5 bar at 295 K. We note the presence of two different lines in the xenon spectrum below 260 K (Figure 6b). This spectral resolution may reveal the existence of two different sites of adsorption. Below 230 K, the 129Xe line shape broadens and becomes structured, which is an indication of many adsorption sites (Figure 6a). To simplify our investigation, we limit our study to the temperature range where we have no more than two lines. Moreover, we have measured the quantity of xenon embedded in the porous sample for each temperature (Figure 7) obtained by measuring the integrated area of the line shape after correction of the temperature effects of the Curie law. The change in slope between 235 and 240 K indicates that xenon condensation occurs. Thus, the first layer of xenon is almost saturated at 240 K. Although the appearance of such a step intensity (11) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361.
Figure 7. Temperature dependence of xenon quantity in colloidal silica of 118 Å pore radius. The continuous line is just a guide for the eye.
increase depends on the pore filling, the transition temperature is reproducible. At 230 K the pressure of a perfect gas decreases to 3.9 bar. At this pressure the temperature of liquefaction is 192 K. Thus, we observe a condensation of xenon at a temperature well above that for xenon by itself. The complete explanation of this phenomenon is not our first concern here, we suggest that additionnal surface interactions or a capillary condensation occurs. Last, we observe that the quantity of confined xenon is not constant between 240 and 300 K. In the case where the surface fraction stays negligible compared to the volume fraction, this variation could be attributed to the quantity of free gas.12 However, we have seen above that one has a high surface fraction of xenon (Figure 4), so the observed increase in the quantity of confined xenon at low temperature should only be due to an enhancement of the adsorbed xenon and thus of the adsorption coefficient kads. Modification of the Mixed Model at Low Temperature. We suppose that the model of fast exchange (12) Pasquier, V.; Levitz, P.; Delville, A. J. Phys. Chem. 1996, 100, 10249.
10196
Langmuir, Vol. 16, No. 26, 2000
Cros et al.
Table 1. Enthalpy of Adsorption of Xenon on Different Surfaces material
∆Hads (kJ/mol)
tempa (K)
ref
Vycor Vycor silica gel Duran glass graphite sol-gel and silica colloids
-5.30 -3.16 from -8.60 to -20.90 -27.90 -15.80 -31.00
232-260 180-278 175-400 360-400 not communicated 275-300
12 15 16 17 18 this paper
a The enthalpy of adsorption being given by an Arrhenius plot for a given temperature range, this range is stated here.
(eq 4) describes the chemical shift variation at room temperature, where the temperature dependence of the adsorption coefficient follows a Langmuir model:11
b0
(
)
x
Figure 8. Temperature dependence of xenon chemical shifts in sol-gel samples of 41 and 80 Å pore radii. The continuous lines do not come from a fit but are the expected variations with the models and the parameters described in the text.
∆Hads 295 × ) kads(295 K) RbT T ∆Hads 1 1 exp (11) Rb 295 T
An estimation of r is thus needed. A semiempirical relation, based on an homogeneous xenon adsorption on the surface, might be
where b0 is a prefactor that can be calculated and ∆Hads is the adsorption enthalpy. We have just determined kads (2.05 ( 0.10 × 1017 atoms/(bar m2) at room temperature); thus, we deduce ∆Hads ) 31 ( 1 kJ/mol from the chemical shift variation around 295 K (Figure 5). This value is compatible with the results of the literature for different surfaces (Table 1). Introducing the static polarization, R(0), the predicted value obtained with this simple model is ∆Hads ) -R(0)U/82 ) -54 kJ/mol for a planar surface. This value is much too large and does not correspond to experiments, but this model does not consider the dielectric nature of the surface. We now compare the observed temperature dependence of the chemical shift (Figure 5) with the theoretical expression given in eqs 9 and 10. Here the chemical shift, δ(R), is given by the product of the fraction of xenon at the surface and the chemical shift at the surface:
Due to the bounded limit of the surface fraction, one predicts from eq 12 that δ(R) is limited to a maximal value δa(R), which is 78 ppm for a pore radius R ) 118 Å (from eq 5). We observe (Figure 5) that, below 260 K, the chemical shift is above such a predicted maximal value. Although the mixed model described the pore size dependence of the chemical shift nicely, it should be modified in the lowtemperature range. We assume that the chemical shift at low temperature comes from a process that does not occur at high temperature. To check a posteriori the validity of such a hypothesis, we have subtracted the values derived from the model above from the experimental values plotted in Figure 5. The difference obtained is shown as the triangles in Figure 5. We have seen above that, around 240 K, the surface density should reach its maximal coverage for an average distance of 4.4 Å between xenon atoms. Then one must take into account another source of chemical shift coming from the van der Waals interactions between two adsorbed xenon atoms separated by a mean distance r:10
where η ∼ 1 is a coefficient which takes into account the degree of compactness of coverage. In fact, such a parameter exhibits the property 〈1/r6〉 * 1/〈r〉6 when the average 〈 〉 is taken over the xenon pairwise correlation function. To take into account all the effective interactions with a given adsorbed xenon, we fix arbitrarily to five the mean number of neighbors. This number is just between the cases found for a square and hexagonal lattices. Of course η is strongly correlated with the average number of neighbors surrounding a given xenon at the surface. However, neither η nor the average number of neighbors is an important parameter. Only the power law (eqs 13 and 14) matters. The best fit of the triangular points of Figure 5 leads to the value η ) 0.81. To check a posteriori our previous statements, we have considered other samples of different average pore sizes. In Figure 8, we present the observed temperature dependence of the xenon chemical shifts in sol-gel samples of 41 and 80 Å pore radii. The continuous lines present in this figure are computed on the basis of our present model with the parameters described in the text and without any parameter adjustments. This result shows that our model may account for the data from room temperature to 220 K. We can therefore use the chemical shift dependence for one sample, in this case the 118 Å pore radius sample, to obtain the parameters η and ∆Hads, which allow us to predict the chemical shift behavior for other pore sizes. Application to the Measurement of the Diffusion Coefficient of Xenon in a Quasi-closed Porosity. Xenon NMR may be used to measure the diffusion coefficient in a disordered porous medium with a closed or quasi-closed porosity. The ratio of this diffusive coefficient to the one of the free gas gives information about the tortuosity factor and, in consequence, about the accessibility of the pore network. We used the usual synthesis routes of the sol-gel process to obtain a glass with a closed porosity.13 An organic-inorganic xerogel was prepared from the methyltriethoxysilane alkoxyde percursor which has a non-hydrolizable organic part.14 The main interest of this material is to possibly trap organic dye molecules in the closed porosity of the glass and finally use it as a filter in nonlinear optics. We checked
R(0) 3 δ(r) - δ(∞) ) BU 6 4 r
(13) Cros, F. Ph.D. Thesis, Ecole Polytechnique, 1998. (14) Canva, M.; Dubois, A.; Georges, P.; Brun, A.; Chaput, F.; Ranger, A.; Boilot, J.-P. SPIE (Sol-Gel Opt. 3) 1994, 2288, 298.
kads(T) )
xT
exp -
[
(
Na δ(R) ) δa(R) Na + Nv
)]
(12)
(13)
r ) η/xσ
(14)
Mesopore Size Characterization by Xenon NMR
Langmuir, Vol. 16, No. 26, 2000 10197
line in Figure 9 shows the best fit obtained with eq 15 with a diffusion coefficient D ) 10-11 m2/s. This value gives only an order of magnitude of D because of the noise in Figure 9. This is a very small value in comparison with the one, D ) 10-5 m2/s, calculated for a xenon free gas at 5 bar and room temperature. The large difference between the gas and the microporous diffusion is caused by the effects of a highly tortuous field. So, we are definitely, in this sample, in the condition of quasi-closed porosity.
Figure 9. Time dependence of the quantity of confined xenon entering inside the quasi-closed porosity of a methyltriethoxysilane (MTEOS) organic-inorganic xerogel within a xenon pressure of 5 bar.
that the closed porosity rules out using nitrogen adsorption-desorption methods for characterization. X-ray scattering methods could not be used due to the absence of contrast. We have prepared samples of 3 mm size in a NMR tube under 5 bar of pressure and have recorded 129Xe spectra every 20 min using 10 µs excitation pulses corresponding to a 50° tip angle. We collected 20 transients separated by a 60 s time delay. The chemical shift of the line corresponding to the embedded xenon does not vary with time. We plot the spectral intensity as a function of time for 40 h in Figure 9. To extract the diffusion coefficient, we consider our samples as spheres of diameter 2a ) 3 mm. From a theoretical point of view, the time dependence of the total quantity of adsorbed xenon C(t) is obtained from the solution of a diffusion equation in the limiting conditions of zero initial concentration and a constant surface concentration due to the permanent action of the external pressure:19
[
C(t) ) C∞ 1 -
6
∞
∑ 2 n)1
π
1
( )]
Dt exp -n2 n2 a2
(15)
In eq 15, C∞ represents the asymptotic quantity of embedded xenon at long times (Figure 9). The continuous
Conclusion We have presented measurements of 129Xe chemical shifts in a large variety of calibrated microporous and mesoporous silica glasses. The main results of our study are as follows: (1) The pore size dependences of xenon chemical shifts have been quantitatively interpreted by a model which includes both fast exchange and van der Waals interaction. (2) The temperature dependence of the chemical shifts supports the proposed model and provides information on the enthalpy of adsorption and homogeneity of the pore surface. This approach provides a method for a spectroscopic mesopore size characterization. (3) Finally, we extracted a direct measurement of the very small xenon diffusion coefficient in organic-inorganic xerogels which allows us to characterize a quasi-closed porosity. Acknowledgment. The authors thank Vivian Slager, Lance Ballard, Hai-Cho Chang, and Jiri Jonas from the University of Illinois, Urbana, and F. Chaput from Ecole Polytechnique, Palaiseau, for the preparation of samples. Special thanks to R. G. Bryant (University of Virginia) for stimulating discussions. LA000322G (15) Burguess, C. G.; Everett, D. H.; Nuttall, S. Langmuir 1990, 6, 1734. (16) Terskikh, V. V.; Mudrakovskii, I. L.; Mastikhin, V. M. J. Chem. Soc., Faraday Trans. 1993, 89, 4239. (17) Butscher, R.; Wa¨ckerle, G.; Mehring, M. J. Chem. Phys. 1994, 100, 6923. (18) Birgeneau, R. J.; Horn, P. M. Science 1986, 232, 329. (19) Crank, J. The Mathematics of diffusion; Clarendon Press: Oxford, 1975.
