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Inelastic neutron scattering experiments on the dynamics of a glass-forming material in mesoscopic confinement
Journal of Non-Crystalline Solids 307–310 (2002) 547–554 www.elsevier.com/locate/jnoncrysol
Inelastic neutron scattering experiments on the dynamics of a glass-forming material in mesoscopic confinement Reiner Zorn a
a,*
, Lutz Hartmann b, Bernhard Frick c, Dieter Richter a, Friedrich Kremer b
Institut fuer Festkoerperforschung, Forschungszentrum J€ulich GmbH, D-52425 J€ulich, Germany b Fakult€at f€ur Physik und Geowissenschaften, Universit€at Leipzig, D-04103 Leipzig, Germany c Institut Laue-Langevin, F-38042 Grenoble, France
Abstract In this article we present results on the microscopic dynamics of a glass-forming liquid (salol) confined in microporous silica glass (Gelsil) obtained by inelastic neutron scattering. By combining different methods we are able to cover a large dynamical range, from the low-frequency vibrations to the a-relaxation. The most prominent effect was observed on the ‘Boson peak’ in the vibrational spectrum. A strong reduction of the modes at lowest frequencies could be observed. This result can be explained qualitatively by a cut-off of long wavelength sound modes but yet defies a ) did not show the expected quantitative description. Especially, a series of samples with different pore sizes (25–200 A scaling of the cut-off frequency with the diameter. The main effect on the a-relaxation is a broadening which we can explain by a tentative model of inhomogeneous relaxation time in the pores. According to this model the relaxation time in the core would be smaller. But due to the distribution the relaxation appears broadened and there is even a fraction of molecules which is ‘frozen’ on the time scale of quasielastic neutron scattering. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.12.)q; 61.20.Lc; 63.50.þx; 64.70.Pf
1. Introduction The dynamics of liquids and polymers confined in mesoscopically small (1–10 nm) pores, tubes or lamellae have become a field of increasing interest in experimental physics. The main reason for these studies are recent theories which explain the rapid increase of the viscosity in liquids when ap-
*
Corresponding author. Tel.: +49-2461 61 4681/5275; fax: +49-2461 61 2610. E-mail address: [email protected] (R. Zorn).
proaching the glass transition by a diverging ‘cooperativity length’ [1]. It should be possible to scrutinise such theories by imposing an artificial limit to this length by geometrical confinement of the liquid. Recent results [2–4] obtained by dielectric spectroscopy on a confined low-molecular glass former (salol) indicate that the a-relaxation dynamics is faster in the confinement compared to the bulk material at temperatures close to the glass transition. This finding is in agreement with the expectation from cooperativity length arguments. In addition these experiments show a broadening
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of the spectra with respect to those of the bulk material. The same material was studied by inelastic neutron scattering for its microscopic dynamics [5]. In that study the neutron time-of-flight technique was used which only allows access to fast motions (t < 20 ps). Here we complement these measurements by neutron backscattering spectra which allow access up to 2 ns. In this range the arelaxation can be observed and thus a direct comparison with the dielectric experiments is possible.
2. Experimental As a glass-forming liquid salol (phenylsalicylate, C13 H10 O3 , Tg ¼ 219 K) was used. The confining material was a controlled porous glass – Gelsil – produced by Geltech Inc. with pore sizes of 25, 50, . Because dielectric experiments [6] 75, and 200 A showed that direct soaking of the glass with salol leads to a formation of a strongly bound surface layer the internal glass surface was silanized [2]. In order to minimize neutron scattering by the silane layer deuterated dimethyldisilazane was used. Except for this isotopic substitution the samples were the same as used in [2]. The neutron time-of-flight experiments were done on the spectrometer IN6 at Institut Laue– Langevin in Grenoble, France. The instrument 1 was used with an incident wavelength ki ¼ 5:1 A resulting in a resolution of 100 leV (full width at half maximum) and elastic scattering vectors up to 1 . Q ¼ 2:05 A The backscattering spectrometer IN16 was used in the standard set-up with unpolished Si(1 1 1) monochromator/analyzer. This corresponds to a and an energy resolution of wavelength of 6.271 A 1 leV (full width at half-maximum, FWHM). The energy range available by Doppler shift was 14 leV and the highest available scattering vector 1 . Q ¼ 1:93 A Because of the high incoherent neutron crosssection of protons most of the neutron scattering is incoherent from the hydrogen atoms of salol. In order to correct for the residual scattering of the confining matrix for each sample a reference con-
taining the silanized gelsil glass only was subtracted.
3. Results 3.1. Time-of-flight spectroscopy Fig. 1 shows representative IN6 spectra of salol confinement at low temperature. The effect in 25 A of the subtraction of the empty gelsil matrix scattering can be seen from the difference between the small and big symbols. It is smaller for the other samples having a more favourable surface to volume ratio. Different detectors have been interpo1 , a scattering vector lated to constant Q ¼ 1:64 A where a large number of detectors is available on IN6 enabling good statistics. This comparison already shows that there is a large difference of the low-energy excitations induced by the confinement. While the bulk sample only shows a vague Boson peak at about 1 meV, for the confined sample the inelastic scattering below about 1.3 meV drops sharply towards the elastic line. Nevertheless, from the comparison with the 2 K spectrum it becomes clear that a significant part of
confined () and bulk () Fig. 1. Scattering function of 25 A salol at T ¼ 100 K obtained from IN6 at constant Q ¼ 1:64 1 . The continuous line shows the spectrum at 2 K which is A nearly identical to the resolution function of the instrument on the (neutron) energy gain side. The fine dots represent the spectrum of the gelsil–salol composite before subtracting the empty matrix.
R. Zorn et al. / Journal of Non-Crystalline Solids 307 (2002) 547–554
549
the observed Sobs ðQ; EÞ below 2 meV is due to the resolution broadening of the elastic line. In order to separate the parts from the resolution and from low-energy vibrations we use the standard expression for the one-phonon scattering function [7]: Q2 gðxÞ h 2m x 1 ! hx 1 exp ; kB T
SðQ; xÞ ¼ e2W ðQÞ dðxÞ þ
ð1Þ
where expð2W ðQÞÞ is the Debye–Waller factor and m the average mass of an atom. The observed scattering law is the convolution of Eq. (1) with the instrumental resolution function RðQ; xÞ. Because the Boson peak is a broad feature compared to the resolution we can neglect the convolution effect on the inelastic term and get Sobs ðQ; xÞ ¼ RðQ; xÞ SðQ; xÞ 2
Q gðxÞ h 2m x 1 ! hx 1 exp : kB T
e2W ðQÞ RðQ; xÞ þ
ð2Þ
Used with spectra at two different temperatures, Eq. (2) gives a system of two linear equations from which the vibrational density of states (VDOS) gðxÞ and RðQ; xÞ can be calculated. 1 The Debye– Waller factors are mainly important to determine the amount of the resolution correction near the elastic line and were chosen to minimise the fluctuations of the calculated gðxÞ there. In all cases the higher temperature was 100 K and the lower ‘reference’ temperature 2 K. This calculation has been done only for the high angle detectors (h > 82°) because at smaller angles
1 Usually, SðQ; xÞ at the lowest temperature T 0 is directly taken as the resolution function RðQ; xÞ. The procedure here has two advantages: (1) SðQ; xÞ contains inelastic scattering due to zero point motion even at T ¼ 0 on the neutron energy loss side making it different from the true resolution RðQ; xÞ. (2) The procedure can be applied also if the lower temperature is significantly higher than zero because it calculates gðxÞ from the change of the spectra with temperature.
Fig. 2. Vibrational density of states (VDOS) at T ¼ 100 K for ( ), 50 A (.), 75 A (.), and 200 A (j) the samples with 25 A pore size and the bulk salol sample (d). The arrows indicate the frequencies where the cut-off is expected from simple sound wave considerations.
the elastic line contained small-angle scattering from the pore structure leading to a wrong normalisation. The gðxÞ values calculated from different detectors conincided within error bounds, Fig. 2 shows their averages as gðxÞ=x2 for the different pore sizes. 3.2. Backscattering Fig. 3 shows the backscattering spectra of salol confinement at different temperatures toin 25 A gether with the time-of-flight spectra from the same material. It can be seen that due to the resolution differing by two orders of magnitude the spectra are difficult to be compared directly. Also due to the smaller detected neutron intensities on a backscattering instrument the statistical errors are much larger in the IN16 data. Therefore, a Fourier transform technique has been used in order to combine the spectra in the time domain [8]. After an inverse Fourier transform the convolution of Eq. (2) is converted into a product and the intermediate scattering function can be obtained by division: SðQ; tÞ ¼ Sobs ðQ; tÞ=RðQ; tÞ:
ð3Þ
Here RðQ; tÞ is the resolution function. Neglecting quantum mechanical effects it can be obtained by
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Fig. 3. (a) Neutron backscattering spectra from IN16 of 25 A confined salol (normalised to the elastic peak value, after sub1 . Temperatures: 2 traction of matrix scattering) at Q ¼ 1:81 A K (resolution), 245, 260, 280, 300, 320, and 339 K (bottom to top curve). Detectors in an angular range of 20° are averaged. Outside the peak region a running average of 11 energy channels is displayed. (b) Time-of-flight spectra from IN6 for comparison (normalised to the elastic peak value, after subtraction of matrix scattering). Temperatures: 2 K (resolution), 100, 150, 200, 225, 245, 270, 300, and 339 K (bottom to top curve). No averaging has been done here.
inverse Fourier transform of spectra at lowest temperatures (here: 2 K). Fig. 4 shows the results of this procedure for confinement. The SðQ; tÞ bulk salol and that in 25 A values for t 6 20 ps originate from the time-of-flight experiment those for t P 150 ps from backscattering. In some cases the prefactor of SðQ; tÞ had to be adjusted in order to ensure limt!0 SðQ; tÞ ¼ 1 and the continuity between time-of-flight and backscattering data. The reasons for this were (1) an
Fig. 4. (a) Intermediate scattering function SðQ; tÞ of bulk salol determined by Fourier transform. The filled symbols (150–2000 ps) are derived from backscattering data, the empty symbols from time-of-flight data (0.1–20 ps). Temperatures: 245, 260, 280, 302, 319, and 339 K (top to bottom). The curves indicate the fits with a Kohlrausch function. (b) The same representation pores. Temperatures: 245, of SðQ; tÞ for salol confined in 25 A 260, 270, 280, 300, 320, and 339 K (top to bottom). The fits here include an elastic fraction in addition to the Kohlrausch function.
intensity reduction in the bulk spectra on IN6 due to a sample loss ( 30%), (2) geometry changes between the IN16 spectra (5%).
4. Discussion 4.1. Boson peak The most prominent effect of the confinement on the fast dynamics of salol is the reduction of low frequency modes in the vibrational spectrum. Although the deviation from the Debye law
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gðxÞ / x2 in bulk salol being a characteristic of the Boson peak indicates that sound is not the sole origin of the low frequency VDOS a qualitative understanding of the reduction can be given in terms of a cut-off of sound waves in a resonator: If v is the sound velocity and l a characteristic dimension of the resonator (also depending on its shape) standing waves below a frequency mmax ¼ v=2l cannot exist. If we simply assume l to be the diameter of the pores and extrapolate the transverse sound velocity to T ¼ 100 K from light scattering data [9] yielding v ¼ 1200 m/s we obtain xmax ¼ 1:5, 0.75, 0.5, and 0:2 ps1 for the 25, 50, confinements respectively. 75, and 200 A The arrows in Fig. 2 show qualitative agreement but nevertheless the shift of the cut-off is much weaker than the 1 : 1=2 : 1=3 : 1=8 ratio predicted by the above considerations. In addition a quantitative calculation with spherical shapes does not give the correct values of the VDOS reduction with respect to the bulk [10]. This result is qualitatively similar to that of a computer simulation of amorphous silica [11]. Nevertheless, the size scaling property was fulfilled for that simulation data. The main difference between simulation and experimental data is that the former just change the borders of the periodic boundary conditions while in the experiment one needs a ‘real wall’ made from a different material. This makes a surface model plausible where the VDOS is reduced in a layer of thickness d close to the surface of the pores. In this case one would expect that the VDOS gR ðxÞ observed for pores of a radius R is an average of the bulk VDOS gbulk ðxÞ and that at the surface gsurf ðxÞ Vsurf Vsurf gR ðxÞ ¼ gsurf ðxÞ þ 1 gbulk ðxÞ; ð4Þ Vpore Vpore weighted by the respective volume fractions. In this case one would expect the ratio gbulk ðxÞ gR ðxÞ ; gbulk ðxÞ g12:5A ðxÞ
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Fig. 5. Ratios of the VDOS difference according to Eq. (5). The main graph shows the averages in the range x ¼ 0:2–1:8 ps1 and the inset the underlying x dependent values. The continuous curve is a fit assuming cylindrical pores with a surface , the dashed curve stands for spherical pores layer of d ¼ 13 A . The empty circle denotes the value from pores with d ¼ 9 A diameter which neither model can describe. with 200 A
The inset in Fig. 5 shows that the constancy of the ratio (5) is fairly well fulfilled. The average ratios can be fitted by the expected surface fraction for cylinders with radius R and surface layer or spheres with d ¼ 9 A for all thickness d ¼ 13 A pore diameters except 200 A (Fig. 5). The remaining suppression of VDOS in that case is double as much as one would expect from the surface ratio. 4.2. a-Relaxation As can be seen from Fig. 4(a) the intermediate scattering function of bulk salol can be well described by the Kohlrausch function 2 SðQ; tÞ ¼ f ðT Þ expððt=sK ðT ÞÞb Þ;
ð6Þ
with b ¼ 0:589 0:006. Here and in the following we assume that b does not depend on temperature.
ð5Þ
to be independent of x and proportional to the surface fraction Vsurf =Vpore . For a given d this ratio can be calculated for different geometries, e.g. spheres and cylinders.
2 When judging the quality of fits of SðQ; tÞ calculated by Fourier transform one has to take into account that correlated errors may be introduced if data with varying errors are transformed. Therefore, a seemingly ‘systematic’ deviation does not rule out the fit. All fits lie within the error bounds which are 0.01 for IN16 (approximately symbol size).
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This assumption (similar to the time–temperature superposition principle) allows a reliable determination of b by simultaneous fitting to all available temperatures. Surprisingly, the same fit is not possible for confined salol. Here one needs an additional constant part e to describe the data: SðQ; tÞ ¼ f ðT Þðexpððt=sK ðT ÞÞb Þ þ eÞ:
ð7Þ
Also with this modification the stretching parameter still assumes a very low value, b ¼ 0:253 0:002. The constant e ¼ 0:074 0:002 is equivalent to an elastic incoherent structure factor in the SðQ; xÞ spectra. This means (roughly speaking) that a fraction 7.4% of atoms is immobile on the time- and length-scale of the neutron experiments. Fig. 6 shows the temperature dependence of the resulting time constants. In order to make the values at such different stretching parameters comparable the average logarithmic relaxation time is shown. It can be seen that with regard to this quantity there is no big difference between confined and bulk system. If at all, one could state a smaller ‘fragility’ (i.e. deviation from Arrhenius
behaviour) for the confined material. A fit of the Vogel–Fulcher temperature dependence sðT Þ ¼ s0 expðB=ðT T0 ÞÞ
yields a fragility parameter B=T0 ¼ 3:2 (with T0 ¼ 207 K) for the bulk material and B=T0 ¼ 8:8 (T0 ¼ 171 K) for the confined. We note that in the temperature range investigated here the dielectric spectroscopy experiments [2] let not expect significant differences in the arelaxation time-scale. There the temperature dependencies of confined and bulk salol begin to differ significantly only below 240 K. Unfortunately, at those temperatures the relaxation is too slow to be observed reliably by neutron spectroscopy. One can only estimate from the extrapolation by the Vogel–Fulcher law shown in Fig. 6 that there is the same tendency as in the dielectric experiments, namely towards faster relaxation in the confinement. The necessity of a strictly elastic component in (7) is unsatisfactory because dielectric spectroscopy indicates that the immobilised surface layer present in native Gelsil pores is completely removed by the silanisation [2]. Also the extreme stretching parameter together with the more symmetric broadening of the dielectric relaxation time distribution seem to indicate that a simple Kohlrausch function is not a good model for the dynamical behaviour of salol in the pores. Therefore, we believe that in the explanation of the experiments an inhomogeneous model taking into account the distance of a liquid layer from the pore wall is necessary. This question has been discussed on computer simulation data [12] with the result that there is a strong (/ expðC=ðR rÞÞ, with R being the pore radius) dependence of the relaxation time on that distance R r. In order to fit our data simultaneously at all available temperatures we assume that the origin of this relaxation time increase is a shift of the Vogel–Fulcher temperature according to T0 ðrÞ ¼ T00 þ
Fig. 6. Average logarithmic relaxation times from the Kohlrausch fits of the SðQ; tÞ data in Fig. 4. The filled symbols rep confined salol, the empty symbols the resent the values for 25 A bulk reference. The curves are fits with the Vogel–Fulcher temperature dependence: (––) confined, (– – –) bulk.
ð8Þ
k : Rr
ð9Þ
Using (8) one can calculate a dependence sK ðrÞ of the relaxation time on the position of a relaxing volume in the pore. Integration over the pore
R. Zorn et al. / Journal of Non-Crystalline Solids 307 (2002) 547–554
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Table 1 confined salol (Kohlrausch fit) and of the inhomogeneous model fit for 25 A confined salol Fit parameters of bulk and 25 A b T0ð0Þ (K) B (K) log10 ðs0 =psÞ K) k (A
Bulk
Confined (simple Kohlrausch)
Confined (inhomogeneous)
0.589 0.002 207 6 660 130 1.8 0.4 –
0.253 0.005 171 15 1500 400 3.7 0.9 –
0.322 0.003 179 1 831 5 1.35 0.01 50.3 1.1
shows this fit is nearly as good as the fit by Eq. (7) with a free temperature dependent sK ðT Þ (cf. Fig. 4(b)). It is also noteworthy that the parameter s0 now has a more reasonable value than in the former description.
5. Conclusions
Fig. 7. Fit of the intermediate scattering function SðQ; tÞ of pores with the relaxation time distribusalol confined in 25 A tion distribution described in the text, Eq. (10). Temperatures: 245, 260, 280, 300, 320, and 339 K (top to bottom).
volume yields the averaged relaxation function expected to be observed: 3 Z 2f ðT Þ R b SðQ; tÞ ¼ expððt=sK ðT ; rÞÞ Þr dr: R2 0 ð10Þ It has to be noted that this equation does not contain a strictly elastic part anymore. In this picture the apparent elastic part is due a fraction of the sample which is ‘effectively’ immobile on the time scale of the experiment. The description is based on five parameters (Table 1) determining SðQ; tÞ for all temperatures together with the temperature dependent prefactor f ðT Þ. As Fig. 7
3 The integral is shown here for the case of cylindrical pores which in our opinion is more probable than spherical. Nevertheless, the result is not much different for integration over a spherical volume.
A clear reduction of low-frequency vibrations constituting the Boson peak below x ¼ 1–2 ps1 (150–300 GHz) can be observed in confinements of . This can be qualitatively exall sizes 25–200 A pected from a simple sound wave picture for these modes. But because the expected scaling xcut-off / R1 cannot be observed it is clear that this explanation is oversimplified. A surface model assuming a reduced vibrational density of states close to the pore wall is more successful but is also not able to explain that the effect is still strong for pores with nominal diameter. 200 A This indicates that a deeper understanding of the vibrational modes of the Boson peak is necessary to achieve a quantitative understanding of its change due to confinement. Nevertheless, is cannot be ruled out that the irregularity of the confinements in case of the matrix made by a sol– gel process is a source of the unexpected behaviour. Therefore, experiments with more regular shaped pores (e.g. zeolites, MCM-41) should be performed. The broadening of the spectra observed by dielectric spectroscopy could be confirmed by the experiments here on a microscopic level. The intermediate scattering function can be described by a model with an explicit heterogeneity. The Vogel– Fulcher temperature is assumed to increase close to the pore wall. This leads to a strong increase of relaxation times. The slowest fraction of molecules
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appears frozen on the time scale of the experiments here. A weak point of the model is that it still needs a smaller value of the stretching parameter b in the confinement in addition to the broadening effect of the distribution. If any change at all one to higher b, i.e. more ‘fundamental’ behaviour, could be expected [13]. It may be that this problem can be resolved by assuming other T0 ðrÞ relations than (9). In the framework of this model the ‘core’ Vogel–Fulcher temperature T0 ðr ¼ 0Þ ¼ T00 þ k= R T00 is smaller than the bulk value by nearly 30°. Also the individual relaxation times in the centre of the pores are smaller at all temperatures. This would support the currently dominating opinion that the a-relaxation is faster in the confinement. Nevertheless, it can be seen from Fig. 6 that due to the averaging this is by no means clear from the original data. This can be taken as a cautious hint with respect to quick comparisons between characteristic times. It may be that part of the confusion whether confinement accelerates or decelerates the a-relaxation is due to different averaging schemes applied.
References [1] E. Donth, The Glass Transition: Relaxation Dynamics in Liquids and Disordered Materials, Springer Series in Material Science, vol. 48, Berlin, Germany, 2001. [2] M. Arndt, R. Stannarius, H. Groothues, E. Hempel, F. Kremer, Phys. Rev. Lett. 79 (1997) 2077. [3] A. Huwe, F. Kremer, P. Behrens, W. Schwieger, Phys. Rev. Lett. 82 (1999) 2338. [4] F. Kremer, A. Huwe, M. Arndt, P. Behrens, W. Schwieger, J. Phys.: Condens. Matter 11 (1999) A175. [5] R. Zorn, D. Richter, L. Hartmann, F. Kremer, B. Frick, J. Phys. IV France 10 (2000) Pr 7–83. [6] M. Arndt, R. Stannarius, W. Gorbatschow, F. Kremer, Phys. Rev. E 54 (1996) 5377. [7] S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter, vol. 1, Oxford University, New York, 1987, p. 121. [8] R. Zorn, B. Frick, L.J. Fetters, J. Phys. Chem. 116 (2002) 845. [9] G. Li, W.M. Du, A. Sakai, H.Z. Cummins, Phys. Rev. A 46 (1992) 3343. [10] R. Zorn, L. Hartmann, D. Richter, F. Kremer, B. Frick, IFF Yearbook 1998, Forschungszentrum, J€ ulich, 1999. [11] J. Horbach, W. Kob, K. Binder, Eur. Phys.J. B 19 (2001) 531. [12] P. Scheidler, W. Kob, K. Binder, J. Phys. IV France 10 (2000) Pr 7–33. [13] K.L. Ngai, J. Phys. IV France 10 (2000) Pr 7–21.
Inelastic neutron scattering experiments on the dynamics of a glass-forming material in mesoscopic confinement Reiner Zorn a
a,*
, Lutz Hartmann b, Bernhard Frick c, Dieter Richter a, Friedrich Kremer b
Institut fuer Festkoerperforschung, Forschungszentrum J€ulich GmbH, D-52425 J€ulich, Germany b Fakult€at f€ur Physik und Geowissenschaften, Universit€at Leipzig, D-04103 Leipzig, Germany c Institut Laue-Langevin, F-38042 Grenoble, France
Abstract In this article we present results on the microscopic dynamics of a glass-forming liquid (salol) confined in microporous silica glass (Gelsil) obtained by inelastic neutron scattering. By combining different methods we are able to cover a large dynamical range, from the low-frequency vibrations to the a-relaxation. The most prominent effect was observed on the ‘Boson peak’ in the vibrational spectrum. A strong reduction of the modes at lowest frequencies could be observed. This result can be explained qualitatively by a cut-off of long wavelength sound modes but yet defies a ) did not show the expected quantitative description. Especially, a series of samples with different pore sizes (25–200 A scaling of the cut-off frequency with the diameter. The main effect on the a-relaxation is a broadening which we can explain by a tentative model of inhomogeneous relaxation time in the pores. According to this model the relaxation time in the core would be smaller. But due to the distribution the relaxation appears broadened and there is even a fraction of molecules which is ‘frozen’ on the time scale of quasielastic neutron scattering. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.12.)q; 61.20.Lc; 63.50.þx; 64.70.Pf
1. Introduction The dynamics of liquids and polymers confined in mesoscopically small (1–10 nm) pores, tubes or lamellae have become a field of increasing interest in experimental physics. The main reason for these studies are recent theories which explain the rapid increase of the viscosity in liquids when ap-
*
Corresponding author. Tel.: +49-2461 61 4681/5275; fax: +49-2461 61 2610. E-mail address: [email protected] (R. Zorn).
proaching the glass transition by a diverging ‘cooperativity length’ [1]. It should be possible to scrutinise such theories by imposing an artificial limit to this length by geometrical confinement of the liquid. Recent results [2–4] obtained by dielectric spectroscopy on a confined low-molecular glass former (salol) indicate that the a-relaxation dynamics is faster in the confinement compared to the bulk material at temperatures close to the glass transition. This finding is in agreement with the expectation from cooperativity length arguments. In addition these experiments show a broadening
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 4 8 5 - 0
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R. Zorn et al. / Journal of Non-Crystalline Solids 307 (2002) 547–554
of the spectra with respect to those of the bulk material. The same material was studied by inelastic neutron scattering for its microscopic dynamics [5]. In that study the neutron time-of-flight technique was used which only allows access to fast motions (t < 20 ps). Here we complement these measurements by neutron backscattering spectra which allow access up to 2 ns. In this range the arelaxation can be observed and thus a direct comparison with the dielectric experiments is possible.
2. Experimental As a glass-forming liquid salol (phenylsalicylate, C13 H10 O3 , Tg ¼ 219 K) was used. The confining material was a controlled porous glass – Gelsil – produced by Geltech Inc. with pore sizes of 25, 50, . Because dielectric experiments [6] 75, and 200 A showed that direct soaking of the glass with salol leads to a formation of a strongly bound surface layer the internal glass surface was silanized [2]. In order to minimize neutron scattering by the silane layer deuterated dimethyldisilazane was used. Except for this isotopic substitution the samples were the same as used in [2]. The neutron time-of-flight experiments were done on the spectrometer IN6 at Institut Laue– Langevin in Grenoble, France. The instrument 1 was used with an incident wavelength ki ¼ 5:1 A resulting in a resolution of 100 leV (full width at half maximum) and elastic scattering vectors up to 1 . Q ¼ 2:05 A The backscattering spectrometer IN16 was used in the standard set-up with unpolished Si(1 1 1) monochromator/analyzer. This corresponds to a and an energy resolution of wavelength of 6.271 A 1 leV (full width at half-maximum, FWHM). The energy range available by Doppler shift was 14 leV and the highest available scattering vector 1 . Q ¼ 1:93 A Because of the high incoherent neutron crosssection of protons most of the neutron scattering is incoherent from the hydrogen atoms of salol. In order to correct for the residual scattering of the confining matrix for each sample a reference con-
taining the silanized gelsil glass only was subtracted.
3. Results 3.1. Time-of-flight spectroscopy Fig. 1 shows representative IN6 spectra of salol confinement at low temperature. The effect in 25 A of the subtraction of the empty gelsil matrix scattering can be seen from the difference between the small and big symbols. It is smaller for the other samples having a more favourable surface to volume ratio. Different detectors have been interpo1 , a scattering vector lated to constant Q ¼ 1:64 A where a large number of detectors is available on IN6 enabling good statistics. This comparison already shows that there is a large difference of the low-energy excitations induced by the confinement. While the bulk sample only shows a vague Boson peak at about 1 meV, for the confined sample the inelastic scattering below about 1.3 meV drops sharply towards the elastic line. Nevertheless, from the comparison with the 2 K spectrum it becomes clear that a significant part of
confined () and bulk () Fig. 1. Scattering function of 25 A salol at T ¼ 100 K obtained from IN6 at constant Q ¼ 1:64 1 . The continuous line shows the spectrum at 2 K which is A nearly identical to the resolution function of the instrument on the (neutron) energy gain side. The fine dots represent the spectrum of the gelsil–salol composite before subtracting the empty matrix.
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the observed Sobs ðQ; EÞ below 2 meV is due to the resolution broadening of the elastic line. In order to separate the parts from the resolution and from low-energy vibrations we use the standard expression for the one-phonon scattering function [7]: Q2 gðxÞ h 2m x 1 ! hx 1 exp ; kB T
SðQ; xÞ ¼ e2W ðQÞ dðxÞ þ
ð1Þ
where expð2W ðQÞÞ is the Debye–Waller factor and m the average mass of an atom. The observed scattering law is the convolution of Eq. (1) with the instrumental resolution function RðQ; xÞ. Because the Boson peak is a broad feature compared to the resolution we can neglect the convolution effect on the inelastic term and get Sobs ðQ; xÞ ¼ RðQ; xÞ SðQ; xÞ 2
Q gðxÞ h 2m x 1 ! hx 1 exp : kB T
e2W ðQÞ RðQ; xÞ þ
ð2Þ
Used with spectra at two different temperatures, Eq. (2) gives a system of two linear equations from which the vibrational density of states (VDOS) gðxÞ and RðQ; xÞ can be calculated. 1 The Debye– Waller factors are mainly important to determine the amount of the resolution correction near the elastic line and were chosen to minimise the fluctuations of the calculated gðxÞ there. In all cases the higher temperature was 100 K and the lower ‘reference’ temperature 2 K. This calculation has been done only for the high angle detectors (h > 82°) because at smaller angles
1 Usually, SðQ; xÞ at the lowest temperature T 0 is directly taken as the resolution function RðQ; xÞ. The procedure here has two advantages: (1) SðQ; xÞ contains inelastic scattering due to zero point motion even at T ¼ 0 on the neutron energy loss side making it different from the true resolution RðQ; xÞ. (2) The procedure can be applied also if the lower temperature is significantly higher than zero because it calculates gðxÞ from the change of the spectra with temperature.
Fig. 2. Vibrational density of states (VDOS) at T ¼ 100 K for ( ), 50 A (.), 75 A (.), and 200 A (j) the samples with 25 A pore size and the bulk salol sample (d). The arrows indicate the frequencies where the cut-off is expected from simple sound wave considerations.
the elastic line contained small-angle scattering from the pore structure leading to a wrong normalisation. The gðxÞ values calculated from different detectors conincided within error bounds, Fig. 2 shows their averages as gðxÞ=x2 for the different pore sizes. 3.2. Backscattering Fig. 3 shows the backscattering spectra of salol confinement at different temperatures toin 25 A gether with the time-of-flight spectra from the same material. It can be seen that due to the resolution differing by two orders of magnitude the spectra are difficult to be compared directly. Also due to the smaller detected neutron intensities on a backscattering instrument the statistical errors are much larger in the IN16 data. Therefore, a Fourier transform technique has been used in order to combine the spectra in the time domain [8]. After an inverse Fourier transform the convolution of Eq. (2) is converted into a product and the intermediate scattering function can be obtained by division: SðQ; tÞ ¼ Sobs ðQ; tÞ=RðQ; tÞ:
ð3Þ
Here RðQ; tÞ is the resolution function. Neglecting quantum mechanical effects it can be obtained by
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Fig. 3. (a) Neutron backscattering spectra from IN16 of 25 A confined salol (normalised to the elastic peak value, after sub1 . Temperatures: 2 traction of matrix scattering) at Q ¼ 1:81 A K (resolution), 245, 260, 280, 300, 320, and 339 K (bottom to top curve). Detectors in an angular range of 20° are averaged. Outside the peak region a running average of 11 energy channels is displayed. (b) Time-of-flight spectra from IN6 for comparison (normalised to the elastic peak value, after subtraction of matrix scattering). Temperatures: 2 K (resolution), 100, 150, 200, 225, 245, 270, 300, and 339 K (bottom to top curve). No averaging has been done here.
inverse Fourier transform of spectra at lowest temperatures (here: 2 K). Fig. 4 shows the results of this procedure for confinement. The SðQ; tÞ bulk salol and that in 25 A values for t 6 20 ps originate from the time-of-flight experiment those for t P 150 ps from backscattering. In some cases the prefactor of SðQ; tÞ had to be adjusted in order to ensure limt!0 SðQ; tÞ ¼ 1 and the continuity between time-of-flight and backscattering data. The reasons for this were (1) an
Fig. 4. (a) Intermediate scattering function SðQ; tÞ of bulk salol determined by Fourier transform. The filled symbols (150–2000 ps) are derived from backscattering data, the empty symbols from time-of-flight data (0.1–20 ps). Temperatures: 245, 260, 280, 302, 319, and 339 K (top to bottom). The curves indicate the fits with a Kohlrausch function. (b) The same representation pores. Temperatures: 245, of SðQ; tÞ for salol confined in 25 A 260, 270, 280, 300, 320, and 339 K (top to bottom). The fits here include an elastic fraction in addition to the Kohlrausch function.
intensity reduction in the bulk spectra on IN6 due to a sample loss ( 30%), (2) geometry changes between the IN16 spectra (5%).
4. Discussion 4.1. Boson peak The most prominent effect of the confinement on the fast dynamics of salol is the reduction of low frequency modes in the vibrational spectrum. Although the deviation from the Debye law
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gðxÞ / x2 in bulk salol being a characteristic of the Boson peak indicates that sound is not the sole origin of the low frequency VDOS a qualitative understanding of the reduction can be given in terms of a cut-off of sound waves in a resonator: If v is the sound velocity and l a characteristic dimension of the resonator (also depending on its shape) standing waves below a frequency mmax ¼ v=2l cannot exist. If we simply assume l to be the diameter of the pores and extrapolate the transverse sound velocity to T ¼ 100 K from light scattering data [9] yielding v ¼ 1200 m/s we obtain xmax ¼ 1:5, 0.75, 0.5, and 0:2 ps1 for the 25, 50, confinements respectively. 75, and 200 A The arrows in Fig. 2 show qualitative agreement but nevertheless the shift of the cut-off is much weaker than the 1 : 1=2 : 1=3 : 1=8 ratio predicted by the above considerations. In addition a quantitative calculation with spherical shapes does not give the correct values of the VDOS reduction with respect to the bulk [10]. This result is qualitatively similar to that of a computer simulation of amorphous silica [11]. Nevertheless, the size scaling property was fulfilled for that simulation data. The main difference between simulation and experimental data is that the former just change the borders of the periodic boundary conditions while in the experiment one needs a ‘real wall’ made from a different material. This makes a surface model plausible where the VDOS is reduced in a layer of thickness d close to the surface of the pores. In this case one would expect that the VDOS gR ðxÞ observed for pores of a radius R is an average of the bulk VDOS gbulk ðxÞ and that at the surface gsurf ðxÞ Vsurf Vsurf gR ðxÞ ¼ gsurf ðxÞ þ 1 gbulk ðxÞ; ð4Þ Vpore Vpore weighted by the respective volume fractions. In this case one would expect the ratio gbulk ðxÞ gR ðxÞ ; gbulk ðxÞ g12:5A ðxÞ
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Fig. 5. Ratios of the VDOS difference according to Eq. (5). The main graph shows the averages in the range x ¼ 0:2–1:8 ps1 and the inset the underlying x dependent values. The continuous curve is a fit assuming cylindrical pores with a surface , the dashed curve stands for spherical pores layer of d ¼ 13 A . The empty circle denotes the value from pores with d ¼ 9 A diameter which neither model can describe. with 200 A
The inset in Fig. 5 shows that the constancy of the ratio (5) is fairly well fulfilled. The average ratios can be fitted by the expected surface fraction for cylinders with radius R and surface layer or spheres with d ¼ 9 A for all thickness d ¼ 13 A pore diameters except 200 A (Fig. 5). The remaining suppression of VDOS in that case is double as much as one would expect from the surface ratio. 4.2. a-Relaxation As can be seen from Fig. 4(a) the intermediate scattering function of bulk salol can be well described by the Kohlrausch function 2 SðQ; tÞ ¼ f ðT Þ expððt=sK ðT ÞÞb Þ;
ð6Þ
with b ¼ 0:589 0:006. Here and in the following we assume that b does not depend on temperature.
ð5Þ
to be independent of x and proportional to the surface fraction Vsurf =Vpore . For a given d this ratio can be calculated for different geometries, e.g. spheres and cylinders.
2 When judging the quality of fits of SðQ; tÞ calculated by Fourier transform one has to take into account that correlated errors may be introduced if data with varying errors are transformed. Therefore, a seemingly ‘systematic’ deviation does not rule out the fit. All fits lie within the error bounds which are 0.01 for IN16 (approximately symbol size).
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This assumption (similar to the time–temperature superposition principle) allows a reliable determination of b by simultaneous fitting to all available temperatures. Surprisingly, the same fit is not possible for confined salol. Here one needs an additional constant part e to describe the data: SðQ; tÞ ¼ f ðT Þðexpððt=sK ðT ÞÞb Þ þ eÞ:
ð7Þ
Also with this modification the stretching parameter still assumes a very low value, b ¼ 0:253 0:002. The constant e ¼ 0:074 0:002 is equivalent to an elastic incoherent structure factor in the SðQ; xÞ spectra. This means (roughly speaking) that a fraction 7.4% of atoms is immobile on the time- and length-scale of the neutron experiments. Fig. 6 shows the temperature dependence of the resulting time constants. In order to make the values at such different stretching parameters comparable the average logarithmic relaxation time is shown. It can be seen that with regard to this quantity there is no big difference between confined and bulk system. If at all, one could state a smaller ‘fragility’ (i.e. deviation from Arrhenius
behaviour) for the confined material. A fit of the Vogel–Fulcher temperature dependence sðT Þ ¼ s0 expðB=ðT T0 ÞÞ
yields a fragility parameter B=T0 ¼ 3:2 (with T0 ¼ 207 K) for the bulk material and B=T0 ¼ 8:8 (T0 ¼ 171 K) for the confined. We note that in the temperature range investigated here the dielectric spectroscopy experiments [2] let not expect significant differences in the arelaxation time-scale. There the temperature dependencies of confined and bulk salol begin to differ significantly only below 240 K. Unfortunately, at those temperatures the relaxation is too slow to be observed reliably by neutron spectroscopy. One can only estimate from the extrapolation by the Vogel–Fulcher law shown in Fig. 6 that there is the same tendency as in the dielectric experiments, namely towards faster relaxation in the confinement. The necessity of a strictly elastic component in (7) is unsatisfactory because dielectric spectroscopy indicates that the immobilised surface layer present in native Gelsil pores is completely removed by the silanisation [2]. Also the extreme stretching parameter together with the more symmetric broadening of the dielectric relaxation time distribution seem to indicate that a simple Kohlrausch function is not a good model for the dynamical behaviour of salol in the pores. Therefore, we believe that in the explanation of the experiments an inhomogeneous model taking into account the distance of a liquid layer from the pore wall is necessary. This question has been discussed on computer simulation data [12] with the result that there is a strong (/ expðC=ðR rÞÞ, with R being the pore radius) dependence of the relaxation time on that distance R r. In order to fit our data simultaneously at all available temperatures we assume that the origin of this relaxation time increase is a shift of the Vogel–Fulcher temperature according to T0 ðrÞ ¼ T00 þ
Fig. 6. Average logarithmic relaxation times from the Kohlrausch fits of the SðQ; tÞ data in Fig. 4. The filled symbols rep confined salol, the empty symbols the resent the values for 25 A bulk reference. The curves are fits with the Vogel–Fulcher temperature dependence: (––) confined, (– – –) bulk.
ð8Þ
k : Rr
ð9Þ
Using (8) one can calculate a dependence sK ðrÞ of the relaxation time on the position of a relaxing volume in the pore. Integration over the pore
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Table 1 confined salol (Kohlrausch fit) and of the inhomogeneous model fit for 25 A confined salol Fit parameters of bulk and 25 A b T0ð0Þ (K) B (K) log10 ðs0 =psÞ K) k (A
Bulk
Confined (simple Kohlrausch)
Confined (inhomogeneous)
0.589 0.002 207 6 660 130 1.8 0.4 –
0.253 0.005 171 15 1500 400 3.7 0.9 –
0.322 0.003 179 1 831 5 1.35 0.01 50.3 1.1
shows this fit is nearly as good as the fit by Eq. (7) with a free temperature dependent sK ðT Þ (cf. Fig. 4(b)). It is also noteworthy that the parameter s0 now has a more reasonable value than in the former description.
5. Conclusions
Fig. 7. Fit of the intermediate scattering function SðQ; tÞ of pores with the relaxation time distribusalol confined in 25 A tion distribution described in the text, Eq. (10). Temperatures: 245, 260, 280, 300, 320, and 339 K (top to bottom).
volume yields the averaged relaxation function expected to be observed: 3 Z 2f ðT Þ R b SðQ; tÞ ¼ expððt=sK ðT ; rÞÞ Þr dr: R2 0 ð10Þ It has to be noted that this equation does not contain a strictly elastic part anymore. In this picture the apparent elastic part is due a fraction of the sample which is ‘effectively’ immobile on the time scale of the experiment. The description is based on five parameters (Table 1) determining SðQ; tÞ for all temperatures together with the temperature dependent prefactor f ðT Þ. As Fig. 7
3 The integral is shown here for the case of cylindrical pores which in our opinion is more probable than spherical. Nevertheless, the result is not much different for integration over a spherical volume.
A clear reduction of low-frequency vibrations constituting the Boson peak below x ¼ 1–2 ps1 (150–300 GHz) can be observed in confinements of . This can be qualitatively exall sizes 25–200 A pected from a simple sound wave picture for these modes. But because the expected scaling xcut-off / R1 cannot be observed it is clear that this explanation is oversimplified. A surface model assuming a reduced vibrational density of states close to the pore wall is more successful but is also not able to explain that the effect is still strong for pores with nominal diameter. 200 A This indicates that a deeper understanding of the vibrational modes of the Boson peak is necessary to achieve a quantitative understanding of its change due to confinement. Nevertheless, is cannot be ruled out that the irregularity of the confinements in case of the matrix made by a sol– gel process is a source of the unexpected behaviour. Therefore, experiments with more regular shaped pores (e.g. zeolites, MCM-41) should be performed. The broadening of the spectra observed by dielectric spectroscopy could be confirmed by the experiments here on a microscopic level. The intermediate scattering function can be described by a model with an explicit heterogeneity. The Vogel– Fulcher temperature is assumed to increase close to the pore wall. This leads to a strong increase of relaxation times. The slowest fraction of molecules
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appears frozen on the time scale of the experiments here. A weak point of the model is that it still needs a smaller value of the stretching parameter b in the confinement in addition to the broadening effect of the distribution. If any change at all one to higher b, i.e. more ‘fundamental’ behaviour, could be expected [13]. It may be that this problem can be resolved by assuming other T0 ðrÞ relations than (9). In the framework of this model the ‘core’ Vogel–Fulcher temperature T0 ðr ¼ 0Þ ¼ T00 þ k= R T00 is smaller than the bulk value by nearly 30°. Also the individual relaxation times in the centre of the pores are smaller at all temperatures. This would support the currently dominating opinion that the a-relaxation is faster in the confinement. Nevertheless, it can be seen from Fig. 6 that due to the averaging this is by no means clear from the original data. This can be taken as a cautious hint with respect to quick comparisons between characteristic times. It may be that part of the confusion whether confinement accelerates or decelerates the a-relaxation is due to different averaging schemes applied.
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