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Glass transition in 2- and 3-dimensionally confined liquids
Journal of Non-Crystalline Solids 235±237 (1998) 444±449
Glass transition in 2- and 3-dimensionally con®ned liquids P. Pissis a
a,* ,
A. Kyritsis a, G. Barut b, R. Pelster b, G. Nimtz
b
Department of Physics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece b II Physikalisches Institut der Universit at zu K oln, Z ulpicher str. 77, 50937 K oln, Germany
Abstract Broadband dielectric relaxation spectroscopy and thermally stimulated depolarization currents (TSDC) measurements were employed to investigate eects of con®nement on the glass transition of the hydrogen bonded liquid propylene glycol (PG) and the non-associating liquid N-methyl-e-caprolactam (NMEC). The liquids were con®ned 2dimensionally in the pores of porous glasses with mean pore diameter 2:5 6 d 6 20:0 nm and 3-dimensionally con®ned in butyl rubber with mean droplet diameter 7 6 d 6 11 nm. The data provide evidence for both the cooperativity concept and the existence of two states (interfacial layer and liquid). With decreasing d the a relaxation associated with the glass transition of the liquid becomes faster and broader and the glass transition temperature decreases. These eects are larger for 3- than for 2-dimensional con®nement. The cooperativity length, n, at Tg is determined to n 6 6 nm for PG and n 6 12 nm for NMEC. Ó 1998 Elsevier Science B.V. All rights reserved. PACS: 64.70.P; 77.40
1. Introduction The investigation of eects on the glass transition of glass-forming liquids induced by con®nement in nanometer sized voids may provide a test of theories and models of the glass transition [1], for which there is no generally accepted theory at present [2±4]. Size eects on the glass transition, as a direct consequence of cooperativity eects [1,5], should appear when the con®ning length becomes smaller than the cooperativity length (characteristic length of the glass transition) n. Moreover, the experimental observation of size effects (e.g. in measurements with varying con®ning length) would yield n most directly [1].
* Corresponding author. Tel.: +30-1-7722986; fax: +30-17722932; e-mail: [email protected].
0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 5 1 2 - 2
Dierential scanning calorimetry (DSC) [6], dielectric relaxation spectroscopy (DRS) [5,7±9], thermally stimulated depolarization currents (TSDC) techniques [5] and solvation dynamic techniques [10] have been employed to study con®nement eects on the glass transition in glassforming liquids 2-dimensionally con®ned in the pores of porous glasses (CPG), the results being rather controversial. Here we extend these studies to include, for the ®rst time, con®nement of liquids in butyl rubber (BR) containing hydrophylic inclusions, where the liquid is 3-dimensionally con®ned in droplets [11]. It should be noted that in liquids con®ned in microemulsions and studied by DSC [12] con®nement was 3-dimensional. In addition, we vary systematically the con®ning length d (diameter of pores/droplets), to obtain estimates for the cooperativity length n. The liquids studied include propylene glycol (PG), as a representative of
P. Pissis et al. / Journal of Non-Crystalline Solids 235±237 (1998) 444±449
445
hydrogen bonded liquids, and N-methyl-e-caprolactam (NMEC), as a representative of non-associating liquids. 2. Experimental PG and NMEC, both 99.9%, were obtained from Sigma, and used without further treatment. BR with hydrophilic components, similar to that used in a previous investigation on the properties of mesoscopic water droplets [11] was provided by Bayer. For ®lling BR samples with a liquid, dry samples (circular sheets of 15 mm diameter and 1 mm thickness) were placed in a liquid bath in an autoclave. A diusion process leads to the formation of liquid droplets, the size of which is limited by the counter pressure of the elastic matrix. With increasing ®lling factors f, de®ned as liquid volume in the sample divided by sample volume (h 6 f 6 30%), the droplet diameter increases from 7 to 11 nm as determined by means of small angle X-ray scattering (SAXS) [13]. The CPG used were Vycor glass (Corning, No 7930) with mean pore diameter d 4.0 nm (sheets of 15 mm diameter and about 1 mm thickness) and Gelsil samples also of cylindrical shape with 10 mm diameter and about 1 mm thickness with nominal pore diameters 2.5, 5.0, 7.5 and 20 nm. The CPG samples were ®lled to saturation by immersion. For comparison some of the Vycor glass samples used in TSDC measurements on NMEC were chemically treated using hexamethyldisilazane [6,8], this treatment making the glass more hydrophobic. For broadband DRS measurements, 5 Hz±2 GHz, two network analyzers (HP 3577B and HP 8510B) were used. TSDC measurements, which correspond to measurements of dielectric losses against temperature at ®xed frequencies of 10ÿ2 ± 10ÿ4 Hz [14], were carried out in the temperature range 77±300 K. Details of preparation of samples and of measurements are given elsewhere [13]. 3. Results and analysis Fig. 1 shows in a log±log plot the dielectric loss e00 against frequency m of PG con®ned in BR at
Fig. 1. Log±log frequency plots of dielectric loss, e00 m, for PG bulk and con®ned in BR (®lling factor f 16.4%, mean droplet diameter d 10.2 nm) at 253 K, details in text.
f 16.4% and d 10.2 nm at 253 K. For bulk PG we observe a loss peak at high frequencies due to the a relaxation associated with the glass transition (Tg 167 K) and dc conductivity at low frequencies. For con®ned PG we observe, in addition to the a process, two relaxations at lower frequencies, Rel I and Rel II. The plot is typical for 3-dimensionally con®ned glass-forming liquids, whereas 2-dimensionally con®ned liquids have an additional (dc) contribution (a conductivity wing) at lower frequencies [7±9]. Measurements on dry BR and un®lled CPG samples for comparison show negligible losses (despite an a process in BR with Tg around 200 K). The two-shape-parameter Havriliak±Negami (HN) expression [15] e m e1 h
De 1 im=m0 1ÿa
ic
1
gives satisfactory ®ts to the a process in both the bulk and the con®ned liquid. In Eq. (1) e e0 ÿ ie00 is the complex permittivity, De and m0 are the intensity and the position of the relaxation process on the frequency scale, respectively, e1 e0 m m0 , a and c are shape parameters. In general, the one-shape-parameter symmetric Cole±Cole expression (c 1 in Eq. (1)) describes
446
P. Pissis et al. / Journal of Non-Crystalline Solids 235±237 (1998) 444±449
satisfactorily Rel I and Rel II. Rel I and Rel II are assigned to a slow liquid surface layer and to interfacial Maxwell±Wagner±Sillars polarization, respectively [8], in agreement with the results of detailed interfacial polarization calculations [13]. In what follows we consider the a process. Fig. 2 shows a plot of log mmax , the frequency of maximum dielectric loss, as a function of 1/T for the a process of PG bulk and PG con®ned in Vycor glass (d 4 nm). The relaxation becomes faster in the con®ned liquid, in agreement also with Fig. 1, the dierence decreasing with increasing temperature. Measurements in Gelsil and BR show that the shifts, compared to the bulk response, increase with decreasing pore/droplet diameter, d. The Vogel±Fulcher±Tamman (VFT) expression [2±4] B 2 mmax A exp ÿ T ÿ T0 with temperature-independent empirical parameters A, B and T0 was ®tted to the data in Fig. 2. The glass transition temperature, Tg , of bulk and con®ned liquids was determined from plots similar to those in Fig. 2 by the condition s(Tg ) 100 s
Fig. 3. TSDC thermograms normalized to the unit height for NMEC bulk (±±±±±±) and con®ned in BR, f 30% and d 7.6 nm (á á á). The insert shows the corresponding scaling plots.
[5,8,9], where s is the dielectric relaxation time, s 1= 2pm0 . Fig. 3 shows TSDC measurements in the temperature region of the glass transition for NMEC bulk and con®ned in BR. The peak temperature Tm is a good measure of Tg [5]. Tg determined by TSDC and by DRS may depend on the method of measurement and the de®nition, the dependence being less pronounced for the shift of Tg , DTg Tg (bulk) ) Tg (con®ned). Figs. 4 and 5 show DTg against con®ning length d for PG and for NMEC, respectively. Both the DRS (Figs. 1 and 2) and the TSDC data (Fig. 3) show that the a process becomes faster in the con®ned liquid. For the same system the eects were found to become larger with decreasing con®ning length d and, for the same d, with decreasing temperature. They are larger in BR than in CPG (3- vs. 2-dimensional con®nement). 4. Discussion
Fig. 2. Semilogarithmic plot of frequency of maximum dielectric loss, mmax , against reciprocal temperature, 1/T, (Arrhenius plot) for the a process in PG bulk (h) and PG con®ned in Vycor glass (s, mean pore diameter d 4.0 nm). The lines are VFT (Eq. (2)) ®ts to the data.
The interpretation of our data is based on a two-states model [9], as a direct consequence of the results shown in Fig. 1: a relatively immobile surface part with dynamics mainly determined by
P. Pissis et al. / Journal of Non-Crystalline Solids 235±237 (1998) 444±449
447
Fig. 6. Normalized amplitude In of the TSDC peak of the a process of PG in BR vs. ®lling factor f. The lines are to guide the eye. Fig. 4. Glass transition temperature depression, DTg , vs. nominal pore/droplet diameter, d, for PG in BR (uncorrected, h; corrected, n; details in text) and in Gelsil (s). n and , denote TSDC data for PG in BR and in Vycor glass, respectively. Vertical bars indicate experimental errors. The lines are to guide the eye.
Fig. 5. DTg vs. d for NMEC con®ned in dierent geometries: Gelsil, d; Vioran [8], s; Vycor glass (TSDC), n; Vycor glass silanized (TSDC), ,; BR, m; BR (TSDC), ..
liquid-wall interactions and a volume part which experiences pure con®nement eects [8]. Thus, in a sense, chemical eects are separated from physi-
cal ones. A direct consequence of that is that the a process studied in detail in our work refers to the inner mobile layer only, i.e. to dimensions smaller than the nominal pore/droplet diameter d. Support for the two-states model comes from the dependence of the normalized magnitude In of the TSDC a peak of PG in BR on ®lling factor f in Fig. 6. In is a measure of the relaxation strength of the TSDC peak [5,14]. In in Fig. 6 increases overlinearly with increasing f. Bearing in mind that the mean droplet diameter d increases with increasing f, this dependence can be understood in terms of the two-states model: with increasing f and d the relative amount of liquid in the immobile interfacial layer, which does not contribute to the a process, decreases, with the result that De and In increase at a rate greater than linearly. Considering now the a process, the relaxation rates are faster for the con®ned liquid (Figs. 1±3), decreasing with increasing d towards those measured on the bulk liquid. The rates for bulk and con®ned liquids approach each other with increasing temperature, indicating weakening and disappearance of con®nement eects at temperatures Tg . However, the relaxation rates measured refer to the composite material (eective values [16,17]), not to the con®ned liquid. Eective medium theories can be used to regain the
448
P. Pissis et al. / Journal of Non-Crystalline Solids 235±237 (1998) 444±449
dielectric property of the con®ned liquid from data on the composite material. These theories are straightforward in BR where the liquid is con®ned in spherical droplets (SAXS measurements). Using Maxwell±Garnet theory [16,17] we calculated the corrected relaxation rates of the liquids con®ned in BR and, by means of plots, the corrected DTg values. The corrected DTg values are smaller than the uncorrected ones by about 3 K (Fig. 4). The corrections are expected to be less signi®cant in the case of CPG, because of both the shape of inclusions (pores) and the larger ®lling factors. Thus, the acceleration of the a process and the decrease of Tg as measured by DRS are partly due to the condition of measurements (eective medium) and partly due to con®nement eects. In the following we consider the con®nement eects. We can understand these eects on the basis of the concept of cooperativity and the Adam± Gibbs model [18]. In this model the cooperativity length n increases with decreasing temperature and diverges for T ® T0 . In a con®ned system n becomes comparable to the system size at a higher temperature (Tc in Fig. 7) than in an uncon®ned system. Then all the molecules in the
con®ned sample take part in the cooperative dynamics and will relax faster compared to bulk. For quantitative estimations it should be taken into account that the real sample size L, i.e. the size of the mobile inner layer, increases with increasing temperature towards the nominal con®ning length d (Fig. 7). In contrast to our results, Monte Carlo simulations show a slowing down of the relaxation of the orientation autocorrelation function of con®ned non-spherical particles [19]. DTg in Figs. 4 and 5 decrease with increasing d. For similar d, DTg values are signi®cantly larger for BR than for CPG in the case of PG (Fig. 4). We think that the larger DTg in BR is due to effects of the dimensionality of con®nement (3-dimensional in BR vs. 2-dimensional in CPG). The data on native and on silanized Vycor glass in Fig. 5 clearly show that DTg is larger for the silanized sample. From the results in Figs. 4 and 5 we determine the cooperativity length n to n 6 6 nm for PG and to n 6 12 nm for NMEC. n at Tg reported in the literature are a few nanometers [4,20]. Both the DRS (Fig. 1) and the TSDC response (Fig. 3) are broader for the con®ned liquids compared to the bulk, in agreement with the results of measurements on con®ned liquids [5,7±9]. This increase in width may be linked to increased local heterogeneity. The broadening of the response was found to be larger for 3- than 2-dimensionally con®ned liquids. 5. Conclusions
Fig. 7. Schematic diagram to illustrate the origin of con®nement eects on the dynamic of the glass transition in glassforming liquids. Con®nement eects appear for L=2 < n. For a given d and L/2 this occurs for T < Tc .
We conclude that our results are consistent with those expected for two liquids in the con®ned geometry: a relatively immobile layer close to the wall and the volume liquid in the inner layers. The a process of the volume liquid becomes faster and Tg shifts to lower temperatures compared to the bulk liquid. The eects are stronger for 3- than for 2-dimensional con®nement and disappear with increasing temperature or size. These results can be understood on the basis of the cooperativity concept and the Adam±Gibbs model and allow us to determine the cooperativity length.
P. Pissis et al. / Journal of Non-Crystalline Solids 235±237 (1998) 444±449
References [1] D. Sappelt, J. Jackle, J. Phys. A 26 (1993) 7325. [2] J. Jackle, Rep. Prog. Phys. 49 (1986) 171. [3] W. G otze, in: J.P. Hansen, D. Levesque, J. Zinn-Justin (Eds.), Liquids, Freezing and the Glass Transition, NorthHolland, Amsterdam, 1991, p. 291. [4] E. Donth, Relaxation and Thermodynamics in Polymers, Glass Transition, Akademie Verlag, Berlin, 1992. [5] P. Pissis, D. Daoukaki-Diamanti, L. Apekis, C. Christodoulides, J. Phys. 6 (1994) L325. [6] C.L. Jackson, G.B. McKenna, J. Non-Cryst. Solids 131± 133 (1991) 221. [7] J. Sch uller, Yu.B. Mel'nichenko, R. Richert, E.W. Fischer, Phys. Rev. Lett. 73 (1994) 2224. [8] J. Sch uller, R. Richert, E.W. Fischer, Phys. Rev. B 52 (1995) 15232. [9] M. Arndt, R. Stannarius, W. Gorbatschow, F. Kremer, Phys. Rev. E 54 (1996) 5377. [10] C. Streck, Yu.B. Mel'nichencko, R. Richert, Phys. Rev. B 53 (1996) 5341.
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[11] R. Pelster, A. Kops, G. Nimtz, A. Enders, H. Kietzmann, P. Pissis, A. Kyritsis, D. Woermann, Ber. Bunsenges. Phys. Chem. 97 (1993) 666. [12] D.R. MacFarlane, C.A. Angell, J. Phys. Chem. 86 (1982) 1927. [13] G. Barut, PhD thesis, University of Cologne, Germany, 1996. [14] J. van Turnhout, in: G.M. Sessler (Ed.), Electrets, Springer, Berlin, 1980, p. 81. [15] C.J.F. B ottcher, P. Bordewijk, Theory of Electric Polarization, 2nd ed., vol. II, Elsevier, Amsterdam, 1978. [16] L.K.H. van Beek, in: J.B. Birks (Ed.), Progress in Dielectrics vol. 7, Heywood, London, 1967, p. 69. [17] S.S. Dukhin, in: E. Matijevic (Ed.), Surface and Colloid Science, vol. 3, Interscience, New York, 1971, p. 83. [18] G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. [19] C. Donati, J. J ackle, J. Phys. 8 (1996) 2733. [20] E.W. Fischer, E. Donth, W. Steen, Phys. Rev. Lett. 68 (1992) 2344.
Glass transition in 2- and 3-dimensionally con®ned liquids P. Pissis a
a,* ,
A. Kyritsis a, G. Barut b, R. Pelster b, G. Nimtz
b
Department of Physics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece b II Physikalisches Institut der Universit at zu K oln, Z ulpicher str. 77, 50937 K oln, Germany
Abstract Broadband dielectric relaxation spectroscopy and thermally stimulated depolarization currents (TSDC) measurements were employed to investigate eects of con®nement on the glass transition of the hydrogen bonded liquid propylene glycol (PG) and the non-associating liquid N-methyl-e-caprolactam (NMEC). The liquids were con®ned 2dimensionally in the pores of porous glasses with mean pore diameter 2:5 6 d 6 20:0 nm and 3-dimensionally con®ned in butyl rubber with mean droplet diameter 7 6 d 6 11 nm. The data provide evidence for both the cooperativity concept and the existence of two states (interfacial layer and liquid). With decreasing d the a relaxation associated with the glass transition of the liquid becomes faster and broader and the glass transition temperature decreases. These eects are larger for 3- than for 2-dimensional con®nement. The cooperativity length, n, at Tg is determined to n 6 6 nm for PG and n 6 12 nm for NMEC. Ó 1998 Elsevier Science B.V. All rights reserved. PACS: 64.70.P; 77.40
1. Introduction The investigation of eects on the glass transition of glass-forming liquids induced by con®nement in nanometer sized voids may provide a test of theories and models of the glass transition [1], for which there is no generally accepted theory at present [2±4]. Size eects on the glass transition, as a direct consequence of cooperativity eects [1,5], should appear when the con®ning length becomes smaller than the cooperativity length (characteristic length of the glass transition) n. Moreover, the experimental observation of size effects (e.g. in measurements with varying con®ning length) would yield n most directly [1].
* Corresponding author. Tel.: +30-1-7722986; fax: +30-17722932; e-mail: [email protected].
0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 5 1 2 - 2
Dierential scanning calorimetry (DSC) [6], dielectric relaxation spectroscopy (DRS) [5,7±9], thermally stimulated depolarization currents (TSDC) techniques [5] and solvation dynamic techniques [10] have been employed to study con®nement eects on the glass transition in glassforming liquids 2-dimensionally con®ned in the pores of porous glasses (CPG), the results being rather controversial. Here we extend these studies to include, for the ®rst time, con®nement of liquids in butyl rubber (BR) containing hydrophylic inclusions, where the liquid is 3-dimensionally con®ned in droplets [11]. It should be noted that in liquids con®ned in microemulsions and studied by DSC [12] con®nement was 3-dimensional. In addition, we vary systematically the con®ning length d (diameter of pores/droplets), to obtain estimates for the cooperativity length n. The liquids studied include propylene glycol (PG), as a representative of
P. Pissis et al. / Journal of Non-Crystalline Solids 235±237 (1998) 444±449
445
hydrogen bonded liquids, and N-methyl-e-caprolactam (NMEC), as a representative of non-associating liquids. 2. Experimental PG and NMEC, both 99.9%, were obtained from Sigma, and used without further treatment. BR with hydrophilic components, similar to that used in a previous investigation on the properties of mesoscopic water droplets [11] was provided by Bayer. For ®lling BR samples with a liquid, dry samples (circular sheets of 15 mm diameter and 1 mm thickness) were placed in a liquid bath in an autoclave. A diusion process leads to the formation of liquid droplets, the size of which is limited by the counter pressure of the elastic matrix. With increasing ®lling factors f, de®ned as liquid volume in the sample divided by sample volume (h 6 f 6 30%), the droplet diameter increases from 7 to 11 nm as determined by means of small angle X-ray scattering (SAXS) [13]. The CPG used were Vycor glass (Corning, No 7930) with mean pore diameter d 4.0 nm (sheets of 15 mm diameter and about 1 mm thickness) and Gelsil samples also of cylindrical shape with 10 mm diameter and about 1 mm thickness with nominal pore diameters 2.5, 5.0, 7.5 and 20 nm. The CPG samples were ®lled to saturation by immersion. For comparison some of the Vycor glass samples used in TSDC measurements on NMEC were chemically treated using hexamethyldisilazane [6,8], this treatment making the glass more hydrophobic. For broadband DRS measurements, 5 Hz±2 GHz, two network analyzers (HP 3577B and HP 8510B) were used. TSDC measurements, which correspond to measurements of dielectric losses against temperature at ®xed frequencies of 10ÿ2 ± 10ÿ4 Hz [14], were carried out in the temperature range 77±300 K. Details of preparation of samples and of measurements are given elsewhere [13]. 3. Results and analysis Fig. 1 shows in a log±log plot the dielectric loss e00 against frequency m of PG con®ned in BR at
Fig. 1. Log±log frequency plots of dielectric loss, e00 m, for PG bulk and con®ned in BR (®lling factor f 16.4%, mean droplet diameter d 10.2 nm) at 253 K, details in text.
f 16.4% and d 10.2 nm at 253 K. For bulk PG we observe a loss peak at high frequencies due to the a relaxation associated with the glass transition (Tg 167 K) and dc conductivity at low frequencies. For con®ned PG we observe, in addition to the a process, two relaxations at lower frequencies, Rel I and Rel II. The plot is typical for 3-dimensionally con®ned glass-forming liquids, whereas 2-dimensionally con®ned liquids have an additional (dc) contribution (a conductivity wing) at lower frequencies [7±9]. Measurements on dry BR and un®lled CPG samples for comparison show negligible losses (despite an a process in BR with Tg around 200 K). The two-shape-parameter Havriliak±Negami (HN) expression [15] e m e1 h
De 1 im=m0 1ÿa
ic
1
gives satisfactory ®ts to the a process in both the bulk and the con®ned liquid. In Eq. (1) e e0 ÿ ie00 is the complex permittivity, De and m0 are the intensity and the position of the relaxation process on the frequency scale, respectively, e1 e0 m m0 , a and c are shape parameters. In general, the one-shape-parameter symmetric Cole±Cole expression (c 1 in Eq. (1)) describes
446
P. Pissis et al. / Journal of Non-Crystalline Solids 235±237 (1998) 444±449
satisfactorily Rel I and Rel II. Rel I and Rel II are assigned to a slow liquid surface layer and to interfacial Maxwell±Wagner±Sillars polarization, respectively [8], in agreement with the results of detailed interfacial polarization calculations [13]. In what follows we consider the a process. Fig. 2 shows a plot of log mmax , the frequency of maximum dielectric loss, as a function of 1/T for the a process of PG bulk and PG con®ned in Vycor glass (d 4 nm). The relaxation becomes faster in the con®ned liquid, in agreement also with Fig. 1, the dierence decreasing with increasing temperature. Measurements in Gelsil and BR show that the shifts, compared to the bulk response, increase with decreasing pore/droplet diameter, d. The Vogel±Fulcher±Tamman (VFT) expression [2±4] B 2 mmax A exp ÿ T ÿ T0 with temperature-independent empirical parameters A, B and T0 was ®tted to the data in Fig. 2. The glass transition temperature, Tg , of bulk and con®ned liquids was determined from plots similar to those in Fig. 2 by the condition s(Tg ) 100 s
Fig. 3. TSDC thermograms normalized to the unit height for NMEC bulk (±±±±±±) and con®ned in BR, f 30% and d 7.6 nm (á á á). The insert shows the corresponding scaling plots.
[5,8,9], where s is the dielectric relaxation time, s 1= 2pm0 . Fig. 3 shows TSDC measurements in the temperature region of the glass transition for NMEC bulk and con®ned in BR. The peak temperature Tm is a good measure of Tg [5]. Tg determined by TSDC and by DRS may depend on the method of measurement and the de®nition, the dependence being less pronounced for the shift of Tg , DTg Tg (bulk) ) Tg (con®ned). Figs. 4 and 5 show DTg against con®ning length d for PG and for NMEC, respectively. Both the DRS (Figs. 1 and 2) and the TSDC data (Fig. 3) show that the a process becomes faster in the con®ned liquid. For the same system the eects were found to become larger with decreasing con®ning length d and, for the same d, with decreasing temperature. They are larger in BR than in CPG (3- vs. 2-dimensional con®nement). 4. Discussion
Fig. 2. Semilogarithmic plot of frequency of maximum dielectric loss, mmax , against reciprocal temperature, 1/T, (Arrhenius plot) for the a process in PG bulk (h) and PG con®ned in Vycor glass (s, mean pore diameter d 4.0 nm). The lines are VFT (Eq. (2)) ®ts to the data.
The interpretation of our data is based on a two-states model [9], as a direct consequence of the results shown in Fig. 1: a relatively immobile surface part with dynamics mainly determined by
P. Pissis et al. / Journal of Non-Crystalline Solids 235±237 (1998) 444±449
447
Fig. 6. Normalized amplitude In of the TSDC peak of the a process of PG in BR vs. ®lling factor f. The lines are to guide the eye. Fig. 4. Glass transition temperature depression, DTg , vs. nominal pore/droplet diameter, d, for PG in BR (uncorrected, h; corrected, n; details in text) and in Gelsil (s). n and , denote TSDC data for PG in BR and in Vycor glass, respectively. Vertical bars indicate experimental errors. The lines are to guide the eye.
Fig. 5. DTg vs. d for NMEC con®ned in dierent geometries: Gelsil, d; Vioran [8], s; Vycor glass (TSDC), n; Vycor glass silanized (TSDC), ,; BR, m; BR (TSDC), ..
liquid-wall interactions and a volume part which experiences pure con®nement eects [8]. Thus, in a sense, chemical eects are separated from physi-
cal ones. A direct consequence of that is that the a process studied in detail in our work refers to the inner mobile layer only, i.e. to dimensions smaller than the nominal pore/droplet diameter d. Support for the two-states model comes from the dependence of the normalized magnitude In of the TSDC a peak of PG in BR on ®lling factor f in Fig. 6. In is a measure of the relaxation strength of the TSDC peak [5,14]. In in Fig. 6 increases overlinearly with increasing f. Bearing in mind that the mean droplet diameter d increases with increasing f, this dependence can be understood in terms of the two-states model: with increasing f and d the relative amount of liquid in the immobile interfacial layer, which does not contribute to the a process, decreases, with the result that De and In increase at a rate greater than linearly. Considering now the a process, the relaxation rates are faster for the con®ned liquid (Figs. 1±3), decreasing with increasing d towards those measured on the bulk liquid. The rates for bulk and con®ned liquids approach each other with increasing temperature, indicating weakening and disappearance of con®nement eects at temperatures Tg . However, the relaxation rates measured refer to the composite material (eective values [16,17]), not to the con®ned liquid. Eective medium theories can be used to regain the
448
P. Pissis et al. / Journal of Non-Crystalline Solids 235±237 (1998) 444±449
dielectric property of the con®ned liquid from data on the composite material. These theories are straightforward in BR where the liquid is con®ned in spherical droplets (SAXS measurements). Using Maxwell±Garnet theory [16,17] we calculated the corrected relaxation rates of the liquids con®ned in BR and, by means of plots, the corrected DTg values. The corrected DTg values are smaller than the uncorrected ones by about 3 K (Fig. 4). The corrections are expected to be less signi®cant in the case of CPG, because of both the shape of inclusions (pores) and the larger ®lling factors. Thus, the acceleration of the a process and the decrease of Tg as measured by DRS are partly due to the condition of measurements (eective medium) and partly due to con®nement eects. In the following we consider the con®nement eects. We can understand these eects on the basis of the concept of cooperativity and the Adam± Gibbs model [18]. In this model the cooperativity length n increases with decreasing temperature and diverges for T ® T0 . In a con®ned system n becomes comparable to the system size at a higher temperature (Tc in Fig. 7) than in an uncon®ned system. Then all the molecules in the
con®ned sample take part in the cooperative dynamics and will relax faster compared to bulk. For quantitative estimations it should be taken into account that the real sample size L, i.e. the size of the mobile inner layer, increases with increasing temperature towards the nominal con®ning length d (Fig. 7). In contrast to our results, Monte Carlo simulations show a slowing down of the relaxation of the orientation autocorrelation function of con®ned non-spherical particles [19]. DTg in Figs. 4 and 5 decrease with increasing d. For similar d, DTg values are signi®cantly larger for BR than for CPG in the case of PG (Fig. 4). We think that the larger DTg in BR is due to effects of the dimensionality of con®nement (3-dimensional in BR vs. 2-dimensional in CPG). The data on native and on silanized Vycor glass in Fig. 5 clearly show that DTg is larger for the silanized sample. From the results in Figs. 4 and 5 we determine the cooperativity length n to n 6 6 nm for PG and to n 6 12 nm for NMEC. n at Tg reported in the literature are a few nanometers [4,20]. Both the DRS (Fig. 1) and the TSDC response (Fig. 3) are broader for the con®ned liquids compared to the bulk, in agreement with the results of measurements on con®ned liquids [5,7±9]. This increase in width may be linked to increased local heterogeneity. The broadening of the response was found to be larger for 3- than 2-dimensionally con®ned liquids. 5. Conclusions
Fig. 7. Schematic diagram to illustrate the origin of con®nement eects on the dynamic of the glass transition in glassforming liquids. Con®nement eects appear for L=2 < n. For a given d and L/2 this occurs for T < Tc .
We conclude that our results are consistent with those expected for two liquids in the con®ned geometry: a relatively immobile layer close to the wall and the volume liquid in the inner layers. The a process of the volume liquid becomes faster and Tg shifts to lower temperatures compared to the bulk liquid. The eects are stronger for 3- than for 2-dimensional con®nement and disappear with increasing temperature or size. These results can be understood on the basis of the cooperativity concept and the Adam±Gibbs model and allow us to determine the cooperativity length.
P. Pissis et al. / Journal of Non-Crystalline Solids 235±237 (1998) 444±449
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