Note: Optimization of the numerical data analysis for conductivity percolation studies of drying moist porous systems J. K. Moscicki, D. Sokolowska, L. Kwiatkowski, D. Dziob, and J. Nowak Citation: Review of Scientific Instruments 85, 026102 (2014); doi: 10.1063/1.4863321 View online: http://dx.doi.org/10.1063/1.4863321 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dielectric characteristics of polyvinylidene fluoride-polyaniline percolative composites up to microwave frequencies Appl. Phys. Lett. 103, 192902 (2013); 10.1063/1.4828795 Fundamentals of ionic conductivity relaxation gained from study of procaine hydrochloride and procainamide hydrochloride at ambient and elevated pressure J. Chem. Phys. 136, 164507 (2012); 10.1063/1.4705274 Conductivity Studies On PEO: AgCF 3 SO 3 Electrolyte System With Nanoporous Fillers AIP Conf. Proc. 1313, 177 (2010); 10.1063/1.3530482 Percolating ion transport in binary mixtures with high dielectric loss Appl. Phys. Lett. 88, 214103 (2006); 10.1063/1.2201556 Properties of the constant loss in ionically conducting glasses, melts, and crystals J. Chem. Phys. 110, 10576 (1999); 10.1063/1.478989
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 026102 (2014)
Note: Optimization of the numerical data analysis for conductivity percolation studies of drying moist porous systems J. K. Moscicki,1 D. Sokolowska,1 L. Kwiatkowski,2 D. Dziob,1 and J. Nowak1 1
Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland Department of Econometrics and Operations Research, Cracow University of Economics, Rakowicka 27, 31-510 Krakow, Poland 2
(Received 29 September 2013; accepted 13 January 2014; published online 3 February 2014) A simplified data analysis protocol, for dielectric spectroscopy use to study conductivity percolation in dehydrating granular media is discussed. To enhance visibility of the protonic conductivity contribution to the dielectric loss spectrum, detrimental effects of either low-frequency dielectric relaxation or electrode polarization are removed. Use of the directly measurable monofrequency dielectric loss factor rather than estimated DC conductivity to parameterize the percolation transition substantially reduces the analysis work and time. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863321] DC conductivity, σ DC , is a principal ion transport property of aqueous systems.1 It informs on (i) ions presence, number, and diversity in a system, and (ii) their mobility. Low frequency dielectric spectroscopy is routinely used to study different aspects of DC conductivity in the systems.2 Observation of changes in σ DC to study electrical continuity of water molecular network in the course of dehydration of porous media, is one of exotic but clever applications (Ref. 3 and references therein). In recent years, we performed DC conductivity percolation studies on drying yeast,4 algae,5 and different forms of thermally3, 6 or chemically7 conditioned granular silica. In the spectral low-frequency region (in our case, ∼f ≤ 100 kHz), the dielectric loss spectrum, ε (ω), is dominated by σ DC contribution, manifested in ε (ω) ∼ = σDC εo−1 ω−1 , or logε (ω) ∼ = −logω + log[σDC /εo ]
(1)
dependence,8 where εo and ω = 2π f are the permittivity of the vacuum and the electric field angular frequency. Such idealistic situation is, however, usually encountered only in a portion of the dielectric loss spectrum, since this “DC conductivity” frequency window is usually limited on the low frequency side by electrode polarization effects, and by relaxation processes at the higher frequencies.9 Thus, although the theoretical background of the data analysis is elementary, the processing of experimental data is usually cumbersome time and labor intensive. After visual inspection of ε (ω) and identification of the relevant frequency range, this section of the spectrum is then fit in the log-log representation to Eq. (1). If either hydration3–5 or time6 is used as the argument in the conductivity percolation scaling, procedure must be repeated all over again and again for all hydrations of the sample along the dehydration course. This amounts to hundreds of dielectric loss spectra, and the same amount of frequency range identification inspections with subsequent linear regression fits, before one ultimately obtains the desired variation of σ DC with hydration (or with time), a quite time consuming task. Once conductivity vs. hydration is obtained, the data are next visually inspected to find the hydration range where 0034-6748/2014/85(2)/026102/3/$30.00
conductivity shows signs of the percolation transition (the sample hydration is traditionally parameterized by the mass ratio of the wet and dry samples, h). Successive linear regression analysis in the log-log scale, cf. Eq. (2), yields the best estimates of the threshold and of the conductivity scaling exponent in either hydration, μh , or time-to-failure, μt , or both:6 ∗ (σDC − σDC ) ∝ (h − h∗ )μh ∝ (t ∗ − t)μt , ∗
(2)
∗
for h ≥ h and t ≤ t , asterisks denoting values at the percolation threshold. During the experimental data processing, two steps are troublesome. The first is the reliable choice of the frequency range in which the DC conductivity contribution dominates in the dielectric spectrum, so this spectrum section can be fit to Eq. (1). The second is a cumulative time effort needed to estimate σ DC vs. h from the batch of collected spectra.4 In what follows, we describe two improvements in the data handling which efficiently improve and simplify conductivity percolation studies based on dielectric spectroscopy: a simple standardized procedure for choosing the right frequency range, and a simplified method for determining and parameterization of the conductivity percolation transition. The last decade witnessed increasing interest in observation and providing reasonable explanation of low-frequency relaxation processes in hydrogen bonding liquids (Ref. 9 and references therein). The major difficulty has been the removal of the conductivity contribution obscuring observations of relaxation peaks. Among other methods, Kramers-Kronig relation was used to filter out this contribution from the dielectric loss spectrum:9 , 10 ε (ω) −
∂ε (ω) σDC ≈ − π2 . εo ω ∂lnω
(3)
One should note that the relation in Eq. (3) can be also brought into play for exactly the opposite task, to isolate the bare DC conductivity contribution to the dielectric loss spectrum,11 i.e.,
85, 026102-1
(ω) = ε (ω) + εKK
π 2
σDC ∂ε (ω) ≈ . ∂lnω εo ω
(4)
© 2014 AIP Publishing LLC
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Moscicki et al.
026102-2
Rev. Sci. Instrum. 85, 026102 (2014)
5
9
a b c
3
ha=1.0123 hb=0.2395 hc=0.0956
a
7
ha=0.5325 hb=0.0612
b
5 3
log
1
1 -1
Algae
-1
A150
-3
-3 0
2
4
6
8
10
0
2
4
6
8
10
log
FIG. 1. The overlay of the genuine dielectric loss spectrum [gray, ε (ω)] and calculated DC contribution, [black, ε KK (ω)], for Aerosil A150 and Algae, at distinctively different hydration levels each. “c” in the A150 panel corresponds to the close proximity of the percolation threshold.
To evaluate the outcome of such the procedure, the bare DC contribution, ε KK (ω) was calculated according to Eq. (4), for few representative complex permittivity spectra collected during dehydration of Aerosil A1503 and algae.5 Figure 1 shows overlays of the genuine dielectric loss spectrum and the calculated one (2nd-order Lagrange interpolation through three adjacent [lnωi ,ε (ωi )] data points, followed by analytical differentiation) at several different hydration levels. Two features are worth a note. First, the most important for the present considerations is the persistent presence of a central frequency range in which both, ε KK (ω) and ε (ω), overlap to within experimental error. Overlaying ε KK (ω) and ε (ω) could be then exploited as a convenient frequency range selection rule to find out the frequency range where the dielectric loss spectrum bears the most precise information on σ DC . Second, ε KK (ω) follows Eq. (1) over a substantially wider range of ω than ε (ω) does. The latter would become of particular importance very close to the percolation threshold since it could help to estimate σ DC in the critical moments of water network percolation, when DC conductivity contribution becomes erratic, cf. A150 case “c” in Fig. 1. Second, but not least issue we would like to touch upon in this Note, is reduction of labor and time necessary for quantifying conductivity evolution with hydration. The amount of time spent on that task can be substantially reduced if one skips altogether calculation of σ DC from fitting the appropriate section of the dielectric loss spectrum to Eq. (1). Note then that the conductivity dominance in ε (ω) spans usually over several decades of frequency, so the dielectric loss factor measured at any fixed frequency f from that range, in the course of drying should bear quite precise information on σ DC variation with h, cf. Fig. 2. The percolation scaling equation can be thus recast as
(εf − εf∗ ) ∝ (h − h∗ )με ,
for h ≥ h∗ ,
FIG. 2. Typical changes in the dielectric loss spectrum of a moist granular sample (Aerosil A150) in the course of drying. To emphasize the DC conductivity contribution to the spectrum, f × ε (f) rather than ε (f) is used for the ordinate axis, cf. Eq. (1). Blackened part of the surface corresponds to the maximum spectral range of constant σ DC μ f × ε (f); (LSF: R2 > 0.9, slope < 0.001), at each given instance of time, t, in the course of drying.
critical parameter values and uncertainties should not be that severe. To explore consequences of such data analysis simplification for the percolation threshold and exponent values, we revisited and reanalyzed some of the dielectric data collected in the previous studies. For example, for Aerosil samples,
(a)
(b)
(5)
where ε f is the monochromatic dielectric loss factor value measured at the frequency f. Unavoidable penalty for using Eq. (5) instead of Eq. (2) rests in fact that relative uncertainty of a measured ε f value is larger than that of least square fit (LSF) averaged σ DC . However, since ε f is measured in the proximity of the percolation threshold over multiple discrete hydration levels, the penalty in the estimated percolation
FIG. 3. Comparison of the percolation scaling of ε f with h, in drying moist Aerosil A90, (a) measured at four distinctively different frequencies, and (b) for three different Aerosil samples, with ε f measured at the same frequency 1 kHz; for comparison the scaling of σ DC vs. h is also shown. Asterisks denote values at the percolation threshold.
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026102-3
Moscicki et al.
where dielectric loss spectra due to DC contribution continues over 2–3 decades of the frequency,3 it is possible to follow ε f scaling with h at frequencies separated from each other even by a factor of 10, cf. Fig. 3(a). At three lower frequencies shown, the scaling is pretty close to that observed with the classical σ DC vs. h approach. The fourth, highest frequency (1 MHz) is just outside the DC dominance range, and the percolation exponent is visibly lower, cf. the table (inset) in Fig. 3(a). Figure 3(b) shows analogous comparison but this time for percolation scaling followed at the same frequency (1 kHz) in samples of three different Aerosils: A90, A150, and A380. Again, agreement between scaling parameters obtained from σ DC vs. h, and from ε f vs. h is excellent; cf. the table (inset) in Fig. 3(b). In summary, we propose here a sequence of two data processing steps, the identification of the frequency range where DC conductivity contribution unequivocally dominates the dielectric loss spectrum, and the use of the monofrequency dielectric loss factor to follow conductivity percolation in the course of dehydration of porous materials, to significantly simplify and shorten data analysis. This, in turn, opens possibility for electrogravimetric technique3–5 for becoming an alternative and convenient source of information about porosity and water accessible
Rev. Sci. Instrum. 85, 026102 (2014)
surface of—most importantly—biomaterials in vivo, and solids. We appreciate the comments of the reviewer. Support for this work from NSC (Poland) under Grant No. NN202 105836 is gratefully acknowledged. 1 G.
A. Voth, Acc. Chem. Res. 39, 143 (2006).
2 F. Bordi, C. Cametti, and R. H. Colby, J. Phys.: Condens. Matter 16, R1423
(2004). Sokolowska, D. Dziob, U. Gorska, B. Kieltyka, and J. K. Moscicki, Phys. Rev. E 87, 062404 (2013). 4 D. Sokolowska, A Krol-Otwinowska, and J. K. Moscicki, Phys. Rev. E 70, 052901 (2004). 5 D. Sokolowska, A. Krol-Otwinowska, M. Bialecka, L. Fiedor, M. Szczygiel, and J. K. Moscicki, J. Non-Cryst. Solids 353, 4541 (2007). 6 J. K. Moscicki and D. Sokolowska, Appl. Phys. Lett. 103, 263701 (2013). 7 J. Nowak, D. Dziob, M. Rutkowska, L. Chmielarz, D. Sokolowska, and J. K. Moscicki, “Conductivity percolation in the course of drying a water-wet mesoporous silica SBA-15,” J. Non-Cryst. Solids (submitted). 8 I. I. Popov, R. R. Nigmatullin, A. A. Khamzin, and I. V. Lounev, J. Appl. Phys. 112, 094107 (2012). 9 W. H. Woodward, A. J. Pasztor, T. Chatterjee, and A. I. Nakatani, Rev. Sci. Instrum. 84, 085109 (2013). 10 M. Wübbenhorst and J. van Turnhout, J. Non-Cryst. Solids 305, 40 (2002). 11 E. Axelrod, A. Givant, J. Shappir, Y. Feldman, and A. Sa’ar, Phys. Rev. B 65, 165429 (2002). 3 D.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Sun, 21 Dec 2014 21:44:14
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Sun, 21 Dec 2014 21:44:14
REVIEW OF SCIENTIFIC INSTRUMENTS 85, 026102 (2014)
Note: Optimization of the numerical data analysis for conductivity percolation studies of drying moist porous systems J. K. Moscicki,1 D. Sokolowska,1 L. Kwiatkowski,2 D. Dziob,1 and J. Nowak1 1
Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland Department of Econometrics and Operations Research, Cracow University of Economics, Rakowicka 27, 31-510 Krakow, Poland 2
(Received 29 September 2013; accepted 13 January 2014; published online 3 February 2014) A simplified data analysis protocol, for dielectric spectroscopy use to study conductivity percolation in dehydrating granular media is discussed. To enhance visibility of the protonic conductivity contribution to the dielectric loss spectrum, detrimental effects of either low-frequency dielectric relaxation or electrode polarization are removed. Use of the directly measurable monofrequency dielectric loss factor rather than estimated DC conductivity to parameterize the percolation transition substantially reduces the analysis work and time. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863321] DC conductivity, σ DC , is a principal ion transport property of aqueous systems.1 It informs on (i) ions presence, number, and diversity in a system, and (ii) their mobility. Low frequency dielectric spectroscopy is routinely used to study different aspects of DC conductivity in the systems.2 Observation of changes in σ DC to study electrical continuity of water molecular network in the course of dehydration of porous media, is one of exotic but clever applications (Ref. 3 and references therein). In recent years, we performed DC conductivity percolation studies on drying yeast,4 algae,5 and different forms of thermally3, 6 or chemically7 conditioned granular silica. In the spectral low-frequency region (in our case, ∼f ≤ 100 kHz), the dielectric loss spectrum, ε (ω), is dominated by σ DC contribution, manifested in ε (ω) ∼ = σDC εo−1 ω−1 , or logε (ω) ∼ = −logω + log[σDC /εo ]
(1)
dependence,8 where εo and ω = 2π f are the permittivity of the vacuum and the electric field angular frequency. Such idealistic situation is, however, usually encountered only in a portion of the dielectric loss spectrum, since this “DC conductivity” frequency window is usually limited on the low frequency side by electrode polarization effects, and by relaxation processes at the higher frequencies.9 Thus, although the theoretical background of the data analysis is elementary, the processing of experimental data is usually cumbersome time and labor intensive. After visual inspection of ε (ω) and identification of the relevant frequency range, this section of the spectrum is then fit in the log-log representation to Eq. (1). If either hydration3–5 or time6 is used as the argument in the conductivity percolation scaling, procedure must be repeated all over again and again for all hydrations of the sample along the dehydration course. This amounts to hundreds of dielectric loss spectra, and the same amount of frequency range identification inspections with subsequent linear regression fits, before one ultimately obtains the desired variation of σ DC with hydration (or with time), a quite time consuming task. Once conductivity vs. hydration is obtained, the data are next visually inspected to find the hydration range where 0034-6748/2014/85(2)/026102/3/$30.00
conductivity shows signs of the percolation transition (the sample hydration is traditionally parameterized by the mass ratio of the wet and dry samples, h). Successive linear regression analysis in the log-log scale, cf. Eq. (2), yields the best estimates of the threshold and of the conductivity scaling exponent in either hydration, μh , or time-to-failure, μt , or both:6 ∗ (σDC − σDC ) ∝ (h − h∗ )μh ∝ (t ∗ − t)μt , ∗
(2)
∗
for h ≥ h and t ≤ t , asterisks denoting values at the percolation threshold. During the experimental data processing, two steps are troublesome. The first is the reliable choice of the frequency range in which the DC conductivity contribution dominates in the dielectric spectrum, so this spectrum section can be fit to Eq. (1). The second is a cumulative time effort needed to estimate σ DC vs. h from the batch of collected spectra.4 In what follows, we describe two improvements in the data handling which efficiently improve and simplify conductivity percolation studies based on dielectric spectroscopy: a simple standardized procedure for choosing the right frequency range, and a simplified method for determining and parameterization of the conductivity percolation transition. The last decade witnessed increasing interest in observation and providing reasonable explanation of low-frequency relaxation processes in hydrogen bonding liquids (Ref. 9 and references therein). The major difficulty has been the removal of the conductivity contribution obscuring observations of relaxation peaks. Among other methods, Kramers-Kronig relation was used to filter out this contribution from the dielectric loss spectrum:9 , 10 ε (ω) −
∂ε (ω) σDC ≈ − π2 . εo ω ∂lnω
(3)
One should note that the relation in Eq. (3) can be also brought into play for exactly the opposite task, to isolate the bare DC conductivity contribution to the dielectric loss spectrum,11 i.e.,
85, 026102-1
(ω) = ε (ω) + εKK
π 2
σDC ∂ε (ω) ≈ . ∂lnω εo ω
(4)
© 2014 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Sun, 21 Dec 2014 21:44:14
Moscicki et al.
026102-2
Rev. Sci. Instrum. 85, 026102 (2014)
5
9
a b c
3
ha=1.0123 hb=0.2395 hc=0.0956
a
7
ha=0.5325 hb=0.0612
b
5 3
log
1
1 -1
Algae
-1
A150
-3
-3 0
2
4
6
8
10
0
2
4
6
8
10
log
FIG. 1. The overlay of the genuine dielectric loss spectrum [gray, ε (ω)] and calculated DC contribution, [black, ε KK (ω)], for Aerosil A150 and Algae, at distinctively different hydration levels each. “c” in the A150 panel corresponds to the close proximity of the percolation threshold.
To evaluate the outcome of such the procedure, the bare DC contribution, ε KK (ω) was calculated according to Eq. (4), for few representative complex permittivity spectra collected during dehydration of Aerosil A1503 and algae.5 Figure 1 shows overlays of the genuine dielectric loss spectrum and the calculated one (2nd-order Lagrange interpolation through three adjacent [lnωi ,ε (ωi )] data points, followed by analytical differentiation) at several different hydration levels. Two features are worth a note. First, the most important for the present considerations is the persistent presence of a central frequency range in which both, ε KK (ω) and ε (ω), overlap to within experimental error. Overlaying ε KK (ω) and ε (ω) could be then exploited as a convenient frequency range selection rule to find out the frequency range where the dielectric loss spectrum bears the most precise information on σ DC . Second, ε KK (ω) follows Eq. (1) over a substantially wider range of ω than ε (ω) does. The latter would become of particular importance very close to the percolation threshold since it could help to estimate σ DC in the critical moments of water network percolation, when DC conductivity contribution becomes erratic, cf. A150 case “c” in Fig. 1. Second, but not least issue we would like to touch upon in this Note, is reduction of labor and time necessary for quantifying conductivity evolution with hydration. The amount of time spent on that task can be substantially reduced if one skips altogether calculation of σ DC from fitting the appropriate section of the dielectric loss spectrum to Eq. (1). Note then that the conductivity dominance in ε (ω) spans usually over several decades of frequency, so the dielectric loss factor measured at any fixed frequency f from that range, in the course of drying should bear quite precise information on σ DC variation with h, cf. Fig. 2. The percolation scaling equation can be thus recast as
(εf − εf∗ ) ∝ (h − h∗ )με ,
for h ≥ h∗ ,
FIG. 2. Typical changes in the dielectric loss spectrum of a moist granular sample (Aerosil A150) in the course of drying. To emphasize the DC conductivity contribution to the spectrum, f × ε (f) rather than ε (f) is used for the ordinate axis, cf. Eq. (1). Blackened part of the surface corresponds to the maximum spectral range of constant σ DC μ f × ε (f); (LSF: R2 > 0.9, slope < 0.001), at each given instance of time, t, in the course of drying.
critical parameter values and uncertainties should not be that severe. To explore consequences of such data analysis simplification for the percolation threshold and exponent values, we revisited and reanalyzed some of the dielectric data collected in the previous studies. For example, for Aerosil samples,
(a)
(b)
(5)
where ε f is the monochromatic dielectric loss factor value measured at the frequency f. Unavoidable penalty for using Eq. (5) instead of Eq. (2) rests in fact that relative uncertainty of a measured ε f value is larger than that of least square fit (LSF) averaged σ DC . However, since ε f is measured in the proximity of the percolation threshold over multiple discrete hydration levels, the penalty in the estimated percolation
FIG. 3. Comparison of the percolation scaling of ε f with h, in drying moist Aerosil A90, (a) measured at four distinctively different frequencies, and (b) for three different Aerosil samples, with ε f measured at the same frequency 1 kHz; for comparison the scaling of σ DC vs. h is also shown. Asterisks denote values at the percolation threshold.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Sun, 21 Dec 2014 21:44:14
026102-3
Moscicki et al.
where dielectric loss spectra due to DC contribution continues over 2–3 decades of the frequency,3 it is possible to follow ε f scaling with h at frequencies separated from each other even by a factor of 10, cf. Fig. 3(a). At three lower frequencies shown, the scaling is pretty close to that observed with the classical σ DC vs. h approach. The fourth, highest frequency (1 MHz) is just outside the DC dominance range, and the percolation exponent is visibly lower, cf. the table (inset) in Fig. 3(a). Figure 3(b) shows analogous comparison but this time for percolation scaling followed at the same frequency (1 kHz) in samples of three different Aerosils: A90, A150, and A380. Again, agreement between scaling parameters obtained from σ DC vs. h, and from ε f vs. h is excellent; cf. the table (inset) in Fig. 3(b). In summary, we propose here a sequence of two data processing steps, the identification of the frequency range where DC conductivity contribution unequivocally dominates the dielectric loss spectrum, and the use of the monofrequency dielectric loss factor to follow conductivity percolation in the course of dehydration of porous materials, to significantly simplify and shorten data analysis. This, in turn, opens possibility for electrogravimetric technique3–5 for becoming an alternative and convenient source of information about porosity and water accessible
Rev. Sci. Instrum. 85, 026102 (2014)
surface of—most importantly—biomaterials in vivo, and solids. We appreciate the comments of the reviewer. Support for this work from NSC (Poland) under Grant No. NN202 105836 is gratefully acknowledged. 1 G.
A. Voth, Acc. Chem. Res. 39, 143 (2006).
2 F. Bordi, C. Cametti, and R. H. Colby, J. Phys.: Condens. Matter 16, R1423
(2004). Sokolowska, D. Dziob, U. Gorska, B. Kieltyka, and J. K. Moscicki, Phys. Rev. E 87, 062404 (2013). 4 D. Sokolowska, A Krol-Otwinowska, and J. K. Moscicki, Phys. Rev. E 70, 052901 (2004). 5 D. Sokolowska, A. Krol-Otwinowska, M. Bialecka, L. Fiedor, M. Szczygiel, and J. K. Moscicki, J. Non-Cryst. Solids 353, 4541 (2007). 6 J. K. Moscicki and D. Sokolowska, Appl. Phys. Lett. 103, 263701 (2013). 7 J. Nowak, D. Dziob, M. Rutkowska, L. Chmielarz, D. Sokolowska, and J. K. Moscicki, “Conductivity percolation in the course of drying a water-wet mesoporous silica SBA-15,” J. Non-Cryst. Solids (submitted). 8 I. I. Popov, R. R. Nigmatullin, A. A. Khamzin, and I. V. Lounev, J. Appl. Phys. 112, 094107 (2012). 9 W. H. Woodward, A. J. Pasztor, T. Chatterjee, and A. I. Nakatani, Rev. Sci. Instrum. 84, 085109 (2013). 10 M. Wübbenhorst and J. van Turnhout, J. Non-Cryst. Solids 305, 40 (2002). 11 E. Axelrod, A. Givant, J. Shappir, Y. Feldman, and A. Sa’ar, Phys. Rev. B 65, 165429 (2002). 3 D.
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