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Autoregressive vibrational-dephasing analysis of the ν2 band of liquid methyl iodide in nanoporous glass
10 December 1999
Chemical Physics Letters 314 Ž1999. 459–464 www.elsevier.nlrlocatercplett
Autoregressive vibrational-dephasing analysis of the n 2 band of liquid methyl iodide in nanoporous glass Tracy R. Bryans a , Richard E. Wilde a , Mark W. Holtz b, Edward L. Quitevis a
a,b,)
Department of Chemistry and Biochemistry, Texas Tech UniÕersity, Lubbock, TX 79409, USA b Department of Physics, Texas Tech UniÕersity, Lubbock, TX 79409, USA Received 30 June 1999; in final form 1 October 1999
Abstract
˚ The spontaneous Raman spectrum of the n 2 methyl-deformation mode of liquid methyl iodide confined in 50-A-diameter pores of silica sol–gel glass was measured at room temperature. The n 2 band was broader for the microconfined liquid than for the bulk liquid. Vibrational correlation functions obtained by Fourier transforming the isotropic bands were modeled by using a memory-function procedure based on the Zwanzig–Mori formalism. The memory functions were analyzed by stochastic time series using an autoregressive model. Based on this analysis, the increase in the line width is largely attributed to inhomogeneous broadening due to site inhomogeneities at the pore surface. q 1999 Elsevier Science B.V. All rights reserved. 1. Introduction The dynamical structure of liquids confined in nanoporous materials has received a great deal of attention both experimentally and theoretically in recent years w1–3x. An understanding of microconfined liquids has broad technological implications in heterogeneous catalysis, membrane separations, lubrication, and oil recovery. Because of the large surface-to-volume ratio, high connectivity of the pores, and narrow pore size distribution, silica sol– gel glasses have been used extensively as host materials in studies of microconfined liquids. These materials have made it possible to obtain an understanding of microconfined liquids at the molecular level using NMR w4–6x, Raman scattering w7–12x, and ) Corresponding author. Tel.: q1-806-742-3066; fax: q1-806742-1289; e-mail: [email protected]
optical Kerr effect ŽOKE. spectroscopy w13–15x. In Raman studies of liquids in nanoporous silica glasses, the bandwidth of the isotropic band is used to monitor the effect of confinement on vibrational relaxation. The bandwidths increase with decreasing pore size, therefore implying that the rate of vibrational relaxation increases with decreasing pore size. This increase in the bandwidth has been previously attributed to collisional deactivation with the pore walls which causes faster vibrational relaxation w9,10x. This approach to the analysis of Raman data however assumes that the bands are mainly homogeneously broadened. In nanoporous glasses, one expects a large inhomogeneous contribution to broadening due to surface inhomogeneity. In this Letter, we describe a procedure that separates the contributions of inhomogeneous and homogeneous broadening in the Raman lineshapes of microconfined liquids. The analysis involves modeling
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 1 8 4 - 7
460
T.R. Bryans et al.r Chemical Physics Letters 314 (1999) 459–464
of the vibrational correlation function obtained from the isotropic band in terms of the memory function based on the Zwanzig–Mori formalism w16–20x. An autoregressive analysis of the memory function yields the inhomogeneous second moment, M2inh . The difference between the isotropic second moment, M2iso , and M2inh gives the homogeneous contribution to the spectral moment, M2homo . This analysis is applied for the first time to the Raman spectrum of the n 2 methyl-deformation mode of liquid methyl iodide confined in a nanoporous glass at room temperature. Methyl iodide was chosen because the Raman spectra of the bulk liquid w21x and microconfined liquid w9,10x have been previously studied. Furthermore, recent OKE experiments w15x, which probe reorientational dynamics, on CH 3 I in nanoporous glass indicate that CH 3 I is a weakly wetting liquid. Changes in the dephasing dynamics should therefore be due primarily to confinement effects and not surface interactions.
2. Experimental section Tetramethoxysilane ŽTMOS. and methyl iodide from Aldrich and spectroscopic grade methanol from Mallincrodt were used as received. The procedure used in the preparation of nanoporous silica glass is an adaptation of the fast sol–gel method w22x. To a clean 3-dram vial with a magnetic stir bar were added 1.53 g of TMOS, 0.71 g of deionized water, 2.48 g of methanol, and 0.02 g of 1 N ammonium hydroxide. This corresponds to an initial methanol:water:TMOS:catalyst mole ratio of 8:4:1:0.002. The vial was held in a water bath by a 1.25-cm thick Styrofoam ring. The top of the liquid in the vial was made even with the top of the Styrofoam ring. The temperature of the water bath was maintained at f 708C by a stirrer-hot plate. The reaction mixture was stirred slowly to avoid splashing during the vigorous hydrolysis. After 10 min, f 0.7 mL of water was added to regain the initial volume. After 10 min more, f 1 mL methanol was added to regain the volume. The vial was lowered so that the top of the vial was slightly above the ring, and the temperature of the bath was increased to 808C. The reaction mixture was carefully watched for signs of thickening and increased viscosity. Bubbles sluggishly ris-
ing toward the liquid surface were used as a sign of thickening. When the stir bar movement slowed, the syrupy liquid was quickly removed from the bath and poured into a cylindrical mold and sealed with parafilm. The samples gelled within minutes of being poured into molds. A small hole was poked in the parafilm after the gels had shrunk from the sides of the mold, which occurred within 1–2 weeks. The gels were then placed in a warm environment Ž25– 308C. to age for 1–2 more weeks. The surface area, average pore diameter, and pore-size distributions were determined by using the Brunauer–Emmett– Teller ŽBET. method on a NOVA 1000 instrument ŽQuantachrome.. Dried gels were outgassed for two days at 2308C before the N2 adsorption and desorption or Raman measurements were performed. The adsorption–desorption isotherms showed hysteresis characteristic of mesopores. The BET analysis yielded specific areas of 500–700 m2rg for sol–gel glasses. Pore distributions determined from the desorption isotherm were narrow and gave an average ˚ The sol–gel monoliths were pore size of 50 A. allowed to stand in CH 3 I vapor for 2 h and then fully immersed in liquid CH 3 I for 48 h. The liquiddoped sol–gel glasses were then sealed in a cuvet filled with the liquid. The Raman spectra were recorded at 2.0–2.2 cmy1 resolution using an apparatus and procedures described previously w17x. The 488-nm line of the argon laser was focused along the long axis of the sample. The sample was masked so that only scattered light from the liquid-doped sol–gel glass was collected and imaged onto the slit of the monochromator. The hot-band problem was addressed by using the high frequency side of the n 2 band w19x. Any error is restricted to the band center, which has little influence on the analysis.
3. Results and discussion Fig. 1 illustrates the polarized and depolarized components of the n 2 band, with the hot band removed, for both bulk and microconfined CH 3 I. Microconfinement of the liquid caused the peak to shift from 1241.2 to 1242.5 cmy1 , the band width of the polarized component to increase from 4.9 to 7.2 cmy1 , and the depolarization ratio, rdepol , to increase
T.R. Bryans et al.r Chemical Physics Letters 314 (1999) 459–464
461
influencing the polarizability change of the n 2 vibrational mode. Future work will focus on this interesting observation. The isotropic spectrum is obtained from the polarized intensity, Ivv Ž v ., and depolarized intensity, I vh Ž v .: Iiso Ž v . s I vv Ž v . y 43 Ivh Ž v .
H
r
I vv Ž v . y 43 Ivh Ž v . d v
Ž 1.
Fourier transformation of the isotropic spectrum gives the vibrational correlation function: C Ž t . s Iiso Ž v . exp Ž yi v t . d v
H
Ž 2.
In the Zwanzig–Mori formalism, C Ž t . is related to the memory function, K Ž t ., through a generalized Langevin equation: dC Ž t . rdt s y K Ž t . C Ž t y t . dt
H
Fig. 1. The polarized Žsolid line. and depolarized Ždashed line. components of the n 2 band of bulk and microconfined liquid CH 3 I at room temperature with the hot band removed.
from 0.09 to 0.22. The peak shift has been previously attributed to surface interactions and pure confinement effects w9,10x. Although CH 3 I is a weakly wetting liquid, the slight peak shift is not surprising, given that some degree of interaction should occur between the polar silanol groups at the pore surface and the polarizable CH 3 I molecules. Resonant intermolecular vibrational coupling studies as a function of pore diameter indicate that the shift to higher vibrational frequencies partly arises from increased orientational order in the liquid induced by confinement w10x. Although the peak shift and band broadening in CH 3 I have been previously reported w9,10x, the effect of confinement on the depolarization ratio has not. We speculate that the electric field of the highly polar groups on the pore surface may be
Ž 3.
The experimental vibrational correlation and memory functions were obtained using the method of Cohen and Wilde w23x. As shown in Fig. 2, C Ž t . exhibits a faster decay for the microconfined liquid than for the bulk liquid. The vibrational relaxation time, tvib , calculated from the time integral of C Ž t ., went from 2.9 ps in the bulk liquid to 1.8 ps in the microconfined liquid. The decrease in tvib is consistent with previous work w9,10x. The value of t vib calculated in this way however does not necessarily reflect dephasing from dynamical processes, because C Ž t ., in principle, is given by the product of inhomogeneous and homogeneous autocorrelation functions, CinhŽ t . C homo Ž t .. An alternative means of achieving this separation can be found by resolving K exp Ž t . into a sum of inhomogeneous and homogeneous parts w16x: K exp Ž t . s K inh Ž t . q K homo Ž t .
Ž 4.
Because vibrational dephasing is a stochastic process, the theory of stochastic time series w24x can be applied to the memory function associated with the
T.R. Bryans et al.r Chemical Physics Letters 314 (1999) 459–464
462
The function f ŽB. can be treated as a polynomial in Ž i s 1, 2, . . . , p .. Hence f ŽB. can B with roots Gy1 i be factored as p
f Ž B . s Ł Ž 1 y Gi B . .
Ž 9.
is1
If the stochastic process F Ž t . is stationary, the elements K tyi in Eq. Ž6. form a positive definite matrix and the roots Gy1 of f ŽB. must lie outside the unit i < circle, i.e.
Kks
Ý A i Gik .
Ž 10 .
is1 y1 If a pair of roots Gy1 is complex, the sum i , Gj k k A i Gi q A j G j behaves as a damped cosine. The solution to Eq. Ž6. is known as a pth-order autoregressive analysis. Eq. Ž6. can be rearranged to the form w16,24x:
pr2
K calc Ž t . s
Ý bi exp Ž yhi t . cos Ž V i t q u i .
Ž 11 .
is1
Fig. 2. The experimental vibrational correlation and memory functions of the n 2 band of bulk and microconfined liquid CH 3 I at room temperature.
dephasing process. Consider a random process F Ž t . that is linearly dependent on p previous values of this process whose input is white noise a t :
where bi , hi , V i , and u i are parameters determined from the experimental memory function, K exp Ž t . using the pth-order autoregressive analysis. This is not a fitting procedure, but is a means of obtaining the frequency dependence of K exp Ž t . w19x. If a trial K Ž t . were calculated using a given set of values of bi , hi , V i and u i , only the V i would be reproduced by AR analysis. Regretfully, AR analysis is unable to resolve the memory function into homogeneous and inhomogeneous parts. Nonetheless, the
p
Ft s a t q
Ý Ftyi fi Ž t s 1, 2, . . . . .
Ž 5.
is1
If the correlation function ² Ft F1 : is formed p
² Ft F1 : ' K t s
Ý K tyi fi Ž t s 1, 2, . . . . .
Ž 6.
is1
Let B be a backward-shift operator such that B K t s K ty1 .
Ž 7.
Using this operator, Eq. Ž6. can be written as
Ž 1 y f 1 B y f 2 B 2 y . . . yfp B p . K t ' f Ž B . K t s 0.
Ž 8.
Fig. 3. A plot showing the relationship between V 1 , V 2 , V 3 and M inh 2 .
T.R. Bryans et al.r Chemical Physics Letters 314 (1999) 459–464
463
Table 1 Autoregressive analysis of the n 2 band of bulk liquid CH 3 I Term
1
b hrpsy1 Vrrad psy1 urrad
0.12 0.42 0.34 0.04
2
3
4
5
6
0.11 0.44 0.94 y0.13
0.10 0.39 1.58 y0.17
0.08 0.37 2.26 y0.15
0.09 0.29 2.98 y0.05
0.07 0.32 3.70 y0.10
degree to which inhomogeneous dephasing or homogeneous dephasing contribute to the line broadening can be estimated through the use of second moments in the following way. In the limit of inhomogeneous broadening corresponding to a Gaussian distribution of static environments, the vibrational correlation function is represented by a Gaussian function: Cinh Ž t . s exp Ž yM2inh t 2r2 .
Ž 12 .
As shown previously w18x, K inhŽ t . corresponding to the Gaussian in Eq. Ž12. can be resolved by AR analysis into 3 damped cosines, the frequencies of which are functions of M2inh . Fig. 3 shows the dependence of V 1 , V 2 , V 3 on M2inh up to 50 cmy2 . Because broadening contributions are additive for K Ž t ., the three lowest frequencies, V 1 , V 2 , V 3 , in the AR analysis of K exp Ž t . must give the inhomogeneous component w18x. AR analysis of the experimental memory functions in Fig. 2 gives the frequency dependence shown in Tables 1 and 2. The AR analysis of bulk CH 3 I is a 24th-order process employing a lag of 0.38 ps. This analysis is in agreement with previous work w17x. In contrast, AR analysis of microconfined CH 3 I is a 16th-order process employing a lag of 0.35 ps. It is seen that microconfinement of the liquid leads to an increase in the values of the three lowest frequencies. Use of Fig. 3 gives M2inh s 9–11 cmy2 for the bulk liquid and M2inh s 25–28 cmy2 for the microconfined liquid. From the experimental data, we
7
8
0.05 0.38 4.42 0.06
0.06 0.34 5.08 0.01
9 0.06 0.33 5.75 y0.01
10 0.08 0.26 6.42 y0.07
11
12
0.08 0.21 7.13 y0.03
0.09 0.13 7.80 y0.14
calculate the value of M2iso to be 33 cmy2 for the bulk liquid and 59 cmy2 for the microconfined liquid. This leads to M2homo being equal to 22–24 cmy2 for the bulk liquid and 31–34 cmy2 for the microconfined liquid. The ratio M2inh Žconfined.r M2inh Žbulk. is f 2.6, whereas the ratio M2homo Žconfined.rM2homo Žbulk. is f 1.4. Based on this analysis, microconfinement of the liquid has a considerably larger effect on the inhomogeneous second moment than on the homogeneous second moment by a factor of f 1.9. The implication of this is that the observed increase in the Raman line width in going from bulk liquid to microconfined liquid is largely attributed to inhomogeneous broadening. This is not unexpected given that the molecules see a broad distribution of environments due to the nonuniform nature of the silica pore surface. The key premise that underlies this analysis is the clear separation of time scales between homogeneous and inhomogeneous processes. This work has shown that V 1 , V 2 , V 3 for the microconfined liquid are consistent with a single value of M2inh and, consequently, with a separation of homogenous and inhomogeneous dephasing time scales. This result also argues for a common environment for all the methyl iodide molecules. If, in the sol–gel glass, there were two or more environments having different intermolecular force fields, there would be two or more sets of inhomogeneous frequencies with a resultant mixing of these frequencies. No unique inhomogeneous second moment could then be deter-
Table 2 Autoregressive analysis of the n 2 band of CH 3 I in nanoporous glass Term
1
b hrpsy1 Vrrad psy1 urrad
0.20 0.51 0.55 0
2 0.14 0.58 1.56 y0.22
3 0.10 0.64 2.69 0
4 0.12 0.55 3.75 y0.03
5 0.12 0.55 4.88 0.13
6
7
8
0.12 0.55 5.82 y0.07
0.12 0.38 6.71 y0.29
0.10 0.12 7.61 y0.31
464
T.R. Bryans et al.r Chemical Physics Letters 314 (1999) 459–464
mined w20x. It can be concluded that, as far as the n 2 mode is concerned, all the molecules, whether in the interior of the pores or at the pore surface, must experience the same dephasing processes and hence must experience the same intermolecular force fields. Given that CH 3 I is a weakly wetting liquid, these results can be largely attributed to a pure confinement effect. Clearly, if Raman scattering is used to probe the dynamical structure of microconfined liquids, inhomogeneous broadening must be taken into account before the effect of confinement on vibrational relaxation can be quantified. Future work in our laboratories will focus on obtaining Raman data for liquid CH 3 I within sol–gel glasses with different pore sizes to determine the dependence of the homogeneous and inhomogeneous second moments and the depolarization ratio on pore diameter. Acknowledgements This work was supported by grants from the Texas Higher Education Coordinating Board through the Advanced Research Program Ž003644-057. and the Robert A. Welch Foundation ŽD-1019.. We thank Prof. D. Casadonte for the use of the NOVA 1000 in carrying out the BET measurements. References w1x J.M. Drake, J. Klafter, R. Kopelman, D.D. Awschalom ŽEds.., Dynamics in Small Confining Systems, Materials Research Society, Pittsburgh, PA, 1993.
w2x J.M. Drake, J. Klafter, R. Kopelman, S.M. Troian ŽEds.., Dynamics in Small Confining Systems II, Materials Research Society, Pittsburgh, PA, 1995. w3x J.M. Drake, J. Klafter, R. Kopelman ŽEds.., Dynamics in Small Confining Systems III, Materials Research Society, Pittsburgh, PA, 1997. w4x G. Liu, Y. Li, J. Jonas, J. Chem. Phys. 95 Ž1991. 6892. w5x J. Zhang, J. Jonas, J. Phys. Chem. 97 Ž1993. 8812. w6x J.-P. Korb, A. Delville, S. Xu, G. Demeulenaere, P. Costas, J. Jonas, J. Chem. Phys. 101 Ž1994. 7074. w7x L. Nikiel, B. Hopkins, T.W. Zerda, J. Phys. Chem. 94 Ž1990. 7458. w8x L. Nikiel, T.W. Zerda, J. Phys. Chem. 95 Ž1991. 4063. w9x T.W. Zerda, in: L.L. Hench, J.K. West ŽEds.., Chemical Processing of Advanced Materials, John Wiley and Sons, New York, 1992, p. 103. w10x Y.T. Lee, S.L. Wallen, J. Jonas, J. Phys. Chem. 96 Ž1992. 7161. w11x S.L. Wallen, L. Nikiel, J. Yi, J. Jonas, J. Phys. Chem. 99 Ž1995. 15421. w12x J. Yi, J. Jonas, J. Phys. Chem. 100 Ž1996. 16789. w13x J. Warnock, D.D. Awschalom, M.W. Schafer, Phys. Rev. B 34 Ž1986. 475. w14x F.W. Deeg, G. Schwalb, in: J.M. Drake, J. Klafter, R. Kopelman, S.M. Troian ŽEds.., Dynamics in Small Confining Systems II, Materials Research Society, Pittsburgh, PA, 1994, p. 265. w15x B.J. Loughnane, J.T. Fourkas, J. Phys. Chem. B 102 Ž1998. 10288. w16x R.E. Wilde, Chem. Phys. Lett. 106 Ž1984. 166. w17x R.E. Wilde, T. Zyung, Mol. Phys. 55 Ž1985. 809. w18x R.E. Wilde, Mol. Phys. 57 Ž1986. 675. w19x R.E. Wilde, J. Yarwood, J. Raman Spectrosc. 19 Ž1988. 289. w20x Q. Feng, R.E. Wilde, J. Raman Spectrosc. 21 Ž1990. 277. w21x W.G. Rothschild, Dynamics of Liquids, John Wiley and Sons, New York, 1984. w22x Y. Haruvy, S.E. Webber, Chem. Mater. 3 Ž1991. 501. w23x S.S. Cohen, R.E. Wilde, J. Chem. Phys. 68 Ž1978. 1138. w24x G.E.P. Box, G.M. Jenkins, Time Series Analysis, HoldenDay, San Francisco, 1970.
Chemical Physics Letters 314 Ž1999. 459–464 www.elsevier.nlrlocatercplett
Autoregressive vibrational-dephasing analysis of the n 2 band of liquid methyl iodide in nanoporous glass Tracy R. Bryans a , Richard E. Wilde a , Mark W. Holtz b, Edward L. Quitevis a
a,b,)
Department of Chemistry and Biochemistry, Texas Tech UniÕersity, Lubbock, TX 79409, USA b Department of Physics, Texas Tech UniÕersity, Lubbock, TX 79409, USA Received 30 June 1999; in final form 1 October 1999
Abstract
˚ The spontaneous Raman spectrum of the n 2 methyl-deformation mode of liquid methyl iodide confined in 50-A-diameter pores of silica sol–gel glass was measured at room temperature. The n 2 band was broader for the microconfined liquid than for the bulk liquid. Vibrational correlation functions obtained by Fourier transforming the isotropic bands were modeled by using a memory-function procedure based on the Zwanzig–Mori formalism. The memory functions were analyzed by stochastic time series using an autoregressive model. Based on this analysis, the increase in the line width is largely attributed to inhomogeneous broadening due to site inhomogeneities at the pore surface. q 1999 Elsevier Science B.V. All rights reserved. 1. Introduction The dynamical structure of liquids confined in nanoporous materials has received a great deal of attention both experimentally and theoretically in recent years w1–3x. An understanding of microconfined liquids has broad technological implications in heterogeneous catalysis, membrane separations, lubrication, and oil recovery. Because of the large surface-to-volume ratio, high connectivity of the pores, and narrow pore size distribution, silica sol– gel glasses have been used extensively as host materials in studies of microconfined liquids. These materials have made it possible to obtain an understanding of microconfined liquids at the molecular level using NMR w4–6x, Raman scattering w7–12x, and ) Corresponding author. Tel.: q1-806-742-3066; fax: q1-806742-1289; e-mail: [email protected]
optical Kerr effect ŽOKE. spectroscopy w13–15x. In Raman studies of liquids in nanoporous silica glasses, the bandwidth of the isotropic band is used to monitor the effect of confinement on vibrational relaxation. The bandwidths increase with decreasing pore size, therefore implying that the rate of vibrational relaxation increases with decreasing pore size. This increase in the bandwidth has been previously attributed to collisional deactivation with the pore walls which causes faster vibrational relaxation w9,10x. This approach to the analysis of Raman data however assumes that the bands are mainly homogeneously broadened. In nanoporous glasses, one expects a large inhomogeneous contribution to broadening due to surface inhomogeneity. In this Letter, we describe a procedure that separates the contributions of inhomogeneous and homogeneous broadening in the Raman lineshapes of microconfined liquids. The analysis involves modeling
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 1 8 4 - 7
460
T.R. Bryans et al.r Chemical Physics Letters 314 (1999) 459–464
of the vibrational correlation function obtained from the isotropic band in terms of the memory function based on the Zwanzig–Mori formalism w16–20x. An autoregressive analysis of the memory function yields the inhomogeneous second moment, M2inh . The difference between the isotropic second moment, M2iso , and M2inh gives the homogeneous contribution to the spectral moment, M2homo . This analysis is applied for the first time to the Raman spectrum of the n 2 methyl-deformation mode of liquid methyl iodide confined in a nanoporous glass at room temperature. Methyl iodide was chosen because the Raman spectra of the bulk liquid w21x and microconfined liquid w9,10x have been previously studied. Furthermore, recent OKE experiments w15x, which probe reorientational dynamics, on CH 3 I in nanoporous glass indicate that CH 3 I is a weakly wetting liquid. Changes in the dephasing dynamics should therefore be due primarily to confinement effects and not surface interactions.
2. Experimental section Tetramethoxysilane ŽTMOS. and methyl iodide from Aldrich and spectroscopic grade methanol from Mallincrodt were used as received. The procedure used in the preparation of nanoporous silica glass is an adaptation of the fast sol–gel method w22x. To a clean 3-dram vial with a magnetic stir bar were added 1.53 g of TMOS, 0.71 g of deionized water, 2.48 g of methanol, and 0.02 g of 1 N ammonium hydroxide. This corresponds to an initial methanol:water:TMOS:catalyst mole ratio of 8:4:1:0.002. The vial was held in a water bath by a 1.25-cm thick Styrofoam ring. The top of the liquid in the vial was made even with the top of the Styrofoam ring. The temperature of the water bath was maintained at f 708C by a stirrer-hot plate. The reaction mixture was stirred slowly to avoid splashing during the vigorous hydrolysis. After 10 min, f 0.7 mL of water was added to regain the initial volume. After 10 min more, f 1 mL methanol was added to regain the volume. The vial was lowered so that the top of the vial was slightly above the ring, and the temperature of the bath was increased to 808C. The reaction mixture was carefully watched for signs of thickening and increased viscosity. Bubbles sluggishly ris-
ing toward the liquid surface were used as a sign of thickening. When the stir bar movement slowed, the syrupy liquid was quickly removed from the bath and poured into a cylindrical mold and sealed with parafilm. The samples gelled within minutes of being poured into molds. A small hole was poked in the parafilm after the gels had shrunk from the sides of the mold, which occurred within 1–2 weeks. The gels were then placed in a warm environment Ž25– 308C. to age for 1–2 more weeks. The surface area, average pore diameter, and pore-size distributions were determined by using the Brunauer–Emmett– Teller ŽBET. method on a NOVA 1000 instrument ŽQuantachrome.. Dried gels were outgassed for two days at 2308C before the N2 adsorption and desorption or Raman measurements were performed. The adsorption–desorption isotherms showed hysteresis characteristic of mesopores. The BET analysis yielded specific areas of 500–700 m2rg for sol–gel glasses. Pore distributions determined from the desorption isotherm were narrow and gave an average ˚ The sol–gel monoliths were pore size of 50 A. allowed to stand in CH 3 I vapor for 2 h and then fully immersed in liquid CH 3 I for 48 h. The liquiddoped sol–gel glasses were then sealed in a cuvet filled with the liquid. The Raman spectra were recorded at 2.0–2.2 cmy1 resolution using an apparatus and procedures described previously w17x. The 488-nm line of the argon laser was focused along the long axis of the sample. The sample was masked so that only scattered light from the liquid-doped sol–gel glass was collected and imaged onto the slit of the monochromator. The hot-band problem was addressed by using the high frequency side of the n 2 band w19x. Any error is restricted to the band center, which has little influence on the analysis.
3. Results and discussion Fig. 1 illustrates the polarized and depolarized components of the n 2 band, with the hot band removed, for both bulk and microconfined CH 3 I. Microconfinement of the liquid caused the peak to shift from 1241.2 to 1242.5 cmy1 , the band width of the polarized component to increase from 4.9 to 7.2 cmy1 , and the depolarization ratio, rdepol , to increase
T.R. Bryans et al.r Chemical Physics Letters 314 (1999) 459–464
461
influencing the polarizability change of the n 2 vibrational mode. Future work will focus on this interesting observation. The isotropic spectrum is obtained from the polarized intensity, Ivv Ž v ., and depolarized intensity, I vh Ž v .: Iiso Ž v . s I vv Ž v . y 43 Ivh Ž v .
H
r
I vv Ž v . y 43 Ivh Ž v . d v
Ž 1.
Fourier transformation of the isotropic spectrum gives the vibrational correlation function: C Ž t . s Iiso Ž v . exp Ž yi v t . d v
H
Ž 2.
In the Zwanzig–Mori formalism, C Ž t . is related to the memory function, K Ž t ., through a generalized Langevin equation: dC Ž t . rdt s y K Ž t . C Ž t y t . dt
H
Fig. 1. The polarized Žsolid line. and depolarized Ždashed line. components of the n 2 band of bulk and microconfined liquid CH 3 I at room temperature with the hot band removed.
from 0.09 to 0.22. The peak shift has been previously attributed to surface interactions and pure confinement effects w9,10x. Although CH 3 I is a weakly wetting liquid, the slight peak shift is not surprising, given that some degree of interaction should occur between the polar silanol groups at the pore surface and the polarizable CH 3 I molecules. Resonant intermolecular vibrational coupling studies as a function of pore diameter indicate that the shift to higher vibrational frequencies partly arises from increased orientational order in the liquid induced by confinement w10x. Although the peak shift and band broadening in CH 3 I have been previously reported w9,10x, the effect of confinement on the depolarization ratio has not. We speculate that the electric field of the highly polar groups on the pore surface may be
Ž 3.
The experimental vibrational correlation and memory functions were obtained using the method of Cohen and Wilde w23x. As shown in Fig. 2, C Ž t . exhibits a faster decay for the microconfined liquid than for the bulk liquid. The vibrational relaxation time, tvib , calculated from the time integral of C Ž t ., went from 2.9 ps in the bulk liquid to 1.8 ps in the microconfined liquid. The decrease in tvib is consistent with previous work w9,10x. The value of t vib calculated in this way however does not necessarily reflect dephasing from dynamical processes, because C Ž t ., in principle, is given by the product of inhomogeneous and homogeneous autocorrelation functions, CinhŽ t . C homo Ž t .. An alternative means of achieving this separation can be found by resolving K exp Ž t . into a sum of inhomogeneous and homogeneous parts w16x: K exp Ž t . s K inh Ž t . q K homo Ž t .
Ž 4.
Because vibrational dephasing is a stochastic process, the theory of stochastic time series w24x can be applied to the memory function associated with the
T.R. Bryans et al.r Chemical Physics Letters 314 (1999) 459–464
462
The function f ŽB. can be treated as a polynomial in Ž i s 1, 2, . . . , p .. Hence f ŽB. can B with roots Gy1 i be factored as p
f Ž B . s Ł Ž 1 y Gi B . .
Ž 9.
is1
If the stochastic process F Ž t . is stationary, the elements K tyi in Eq. Ž6. form a positive definite matrix and the roots Gy1 of f ŽB. must lie outside the unit i < circle, i.e.
Kks
Ý A i Gik .
Ž 10 .
is1 y1 If a pair of roots Gy1 is complex, the sum i , Gj k k A i Gi q A j G j behaves as a damped cosine. The solution to Eq. Ž6. is known as a pth-order autoregressive analysis. Eq. Ž6. can be rearranged to the form w16,24x:
pr2
K calc Ž t . s
Ý bi exp Ž yhi t . cos Ž V i t q u i .
Ž 11 .
is1
Fig. 2. The experimental vibrational correlation and memory functions of the n 2 band of bulk and microconfined liquid CH 3 I at room temperature.
dephasing process. Consider a random process F Ž t . that is linearly dependent on p previous values of this process whose input is white noise a t :
where bi , hi , V i , and u i are parameters determined from the experimental memory function, K exp Ž t . using the pth-order autoregressive analysis. This is not a fitting procedure, but is a means of obtaining the frequency dependence of K exp Ž t . w19x. If a trial K Ž t . were calculated using a given set of values of bi , hi , V i and u i , only the V i would be reproduced by AR analysis. Regretfully, AR analysis is unable to resolve the memory function into homogeneous and inhomogeneous parts. Nonetheless, the
p
Ft s a t q
Ý Ftyi fi Ž t s 1, 2, . . . . .
Ž 5.
is1
If the correlation function ² Ft F1 : is formed p
² Ft F1 : ' K t s
Ý K tyi fi Ž t s 1, 2, . . . . .
Ž 6.
is1
Let B be a backward-shift operator such that B K t s K ty1 .
Ž 7.
Using this operator, Eq. Ž6. can be written as
Ž 1 y f 1 B y f 2 B 2 y . . . yfp B p . K t ' f Ž B . K t s 0.
Ž 8.
Fig. 3. A plot showing the relationship between V 1 , V 2 , V 3 and M inh 2 .
T.R. Bryans et al.r Chemical Physics Letters 314 (1999) 459–464
463
Table 1 Autoregressive analysis of the n 2 band of bulk liquid CH 3 I Term
1
b hrpsy1 Vrrad psy1 urrad
0.12 0.42 0.34 0.04
2
3
4
5
6
0.11 0.44 0.94 y0.13
0.10 0.39 1.58 y0.17
0.08 0.37 2.26 y0.15
0.09 0.29 2.98 y0.05
0.07 0.32 3.70 y0.10
degree to which inhomogeneous dephasing or homogeneous dephasing contribute to the line broadening can be estimated through the use of second moments in the following way. In the limit of inhomogeneous broadening corresponding to a Gaussian distribution of static environments, the vibrational correlation function is represented by a Gaussian function: Cinh Ž t . s exp Ž yM2inh t 2r2 .
Ž 12 .
As shown previously w18x, K inhŽ t . corresponding to the Gaussian in Eq. Ž12. can be resolved by AR analysis into 3 damped cosines, the frequencies of which are functions of M2inh . Fig. 3 shows the dependence of V 1 , V 2 , V 3 on M2inh up to 50 cmy2 . Because broadening contributions are additive for K Ž t ., the three lowest frequencies, V 1 , V 2 , V 3 , in the AR analysis of K exp Ž t . must give the inhomogeneous component w18x. AR analysis of the experimental memory functions in Fig. 2 gives the frequency dependence shown in Tables 1 and 2. The AR analysis of bulk CH 3 I is a 24th-order process employing a lag of 0.38 ps. This analysis is in agreement with previous work w17x. In contrast, AR analysis of microconfined CH 3 I is a 16th-order process employing a lag of 0.35 ps. It is seen that microconfinement of the liquid leads to an increase in the values of the three lowest frequencies. Use of Fig. 3 gives M2inh s 9–11 cmy2 for the bulk liquid and M2inh s 25–28 cmy2 for the microconfined liquid. From the experimental data, we
7
8
0.05 0.38 4.42 0.06
0.06 0.34 5.08 0.01
9 0.06 0.33 5.75 y0.01
10 0.08 0.26 6.42 y0.07
11
12
0.08 0.21 7.13 y0.03
0.09 0.13 7.80 y0.14
calculate the value of M2iso to be 33 cmy2 for the bulk liquid and 59 cmy2 for the microconfined liquid. This leads to M2homo being equal to 22–24 cmy2 for the bulk liquid and 31–34 cmy2 for the microconfined liquid. The ratio M2inh Žconfined.r M2inh Žbulk. is f 2.6, whereas the ratio M2homo Žconfined.rM2homo Žbulk. is f 1.4. Based on this analysis, microconfinement of the liquid has a considerably larger effect on the inhomogeneous second moment than on the homogeneous second moment by a factor of f 1.9. The implication of this is that the observed increase in the Raman line width in going from bulk liquid to microconfined liquid is largely attributed to inhomogeneous broadening. This is not unexpected given that the molecules see a broad distribution of environments due to the nonuniform nature of the silica pore surface. The key premise that underlies this analysis is the clear separation of time scales between homogeneous and inhomogeneous processes. This work has shown that V 1 , V 2 , V 3 for the microconfined liquid are consistent with a single value of M2inh and, consequently, with a separation of homogenous and inhomogeneous dephasing time scales. This result also argues for a common environment for all the methyl iodide molecules. If, in the sol–gel glass, there were two or more environments having different intermolecular force fields, there would be two or more sets of inhomogeneous frequencies with a resultant mixing of these frequencies. No unique inhomogeneous second moment could then be deter-
Table 2 Autoregressive analysis of the n 2 band of CH 3 I in nanoporous glass Term
1
b hrpsy1 Vrrad psy1 urrad
0.20 0.51 0.55 0
2 0.14 0.58 1.56 y0.22
3 0.10 0.64 2.69 0
4 0.12 0.55 3.75 y0.03
5 0.12 0.55 4.88 0.13
6
7
8
0.12 0.55 5.82 y0.07
0.12 0.38 6.71 y0.29
0.10 0.12 7.61 y0.31
464
T.R. Bryans et al.r Chemical Physics Letters 314 (1999) 459–464
mined w20x. It can be concluded that, as far as the n 2 mode is concerned, all the molecules, whether in the interior of the pores or at the pore surface, must experience the same dephasing processes and hence must experience the same intermolecular force fields. Given that CH 3 I is a weakly wetting liquid, these results can be largely attributed to a pure confinement effect. Clearly, if Raman scattering is used to probe the dynamical structure of microconfined liquids, inhomogeneous broadening must be taken into account before the effect of confinement on vibrational relaxation can be quantified. Future work in our laboratories will focus on obtaining Raman data for liquid CH 3 I within sol–gel glasses with different pore sizes to determine the dependence of the homogeneous and inhomogeneous second moments and the depolarization ratio on pore diameter. Acknowledgements This work was supported by grants from the Texas Higher Education Coordinating Board through the Advanced Research Program Ž003644-057. and the Robert A. Welch Foundation ŽD-1019.. We thank Prof. D. Casadonte for the use of the NOVA 1000 in carrying out the BET measurements. References w1x J.M. Drake, J. Klafter, R. Kopelman, D.D. Awschalom ŽEds.., Dynamics in Small Confining Systems, Materials Research Society, Pittsburgh, PA, 1993.
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