DOI 10.1140/epje/i2003-10055-1
T HE E UROPEAN
P HYSICAL J OURNAL E
Confinement of molecular liquids:Consequences on
thermodynamic, static and dynamical properties of benzene and toluene
C. Alba-Simionesco1,a, G. Dosseh1, E. Dumont1, B. Frick2, B. Geil3, D. Morineau1,b, V. Teboul1,c, and Y. Xia1,d
1 Laboratoire de Chimie Physique, CNRS-UMR 8000, Bˆatiment 349, Universit´e de Paris-Sud, F-91405 Orsay, France
2 ILL 38042, Grenoble cedex, France
3 University of Darmstadt, Germany Received 1 January 2003 /
Published online: 30 October 2003 – cEDP Sciences / Societ`aItalianadi Fisica/ Springer-Verlag 2003 Abstract. We relate the dynamical behavior of molecular liquids confined in mesoscopic cylindrical pores to the thermodynamic properties, heat capacity and density and to the static structure by combining different experimental methods (H-NMR, calorimetry, elastic and inelastic neutron scattering, numerical simulations). The crystallization process is greatly reduced or avoided by confinement under standard cooling conditions, instead a glass transition temperatureTg at the 1000 s time scale can be observed. The pore averaged local structure of the confined liquid is not noticeably affected when “excluded-volume”
corrections are carefully applied, but follows the density changes reflected by the Bragg peak intensities of the porous matrices. The pore size dependence of Tg is dominated by two factors, surface interaction and finite-size effect. For the smallest pores (d ≤10σ,σbeing the van der Waals radius of a molecule), one observes an increase ofTg and a broadening of the transition region, related to the interaction with the surface that induces a slowing-down of the molecules close to the wall. This is confirmed by neutron scattering experiments and molecular-dynamics simulations at shorter time scales and higher temperatures, which indicate a remaining fraction of frozen molecules. For larger pore sizes, taking the decrease of density under confinement conditions into account, a decrease ofTgis observed. This could be related to finite-size effects onto the putative cooperativity length that is often invoked to explain glass formation. However, no quantitative determination of this length (not to mention its T-dependence) can be extracted, since the interaction with the wall itself introduces an additional length that adds to the complexity of the problem.
PACS. 64.70.Pf Glass transitions – 65.20.+w Thermal properties of liquids: heat capacity, thermal ex- pansion, etc. – 61.12.-q Neutron diffraction and scattering
1 Introduction
Materials confined in nanoscale dimensions possess prop- erties that differ from the corresponding bulk phases due to their reduced dimensionality and large interface effects.
For example, phase transition pressures and temperatures are often shifted from the bulk values and new phases can appear due to surface forces [1]. Restricted geometries
a e-mail:[email protected]
b Present address: Groupe Mati`ere Condens´ee et Mat´eriaux, CNRS-UMR 6626, Bˆatiment 11A, Universit´e de Rennes 1, F-35042 Rennes, France.
c Present address: Laboratoire des Propri´et´es Optiques des Mat´eriaux et Applications, CNRS-UMR 6136, Universit´e d’Angers, F-49045 Angers, France.
d Present address: School of Chemistry, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom.
have then significant consequences on first-order phase transitions (such as melting/freezing or solid-solid), but also in the glass formation. There is a need to develop a fundamental understanding of these confinement effects, both because of basic scientific interest, and because of potential applications.
The properties of molecular systems confined in meso- porous media is a topic of current interest with regard to the anomalous dynamics of deeply supercooled flu- ids [2]. The issue is to arrive at quantitative statements for the concept of cooperativity, an ingredient of most glass transition theories and outcome of some recent experi- ments [3]. Restricted geometries imply “cut-off” effects on the growth of any correlation or cooperativity length which is bounded by the pore size. Hence, the way the structural relaxation of a confined liquid slows down with decreasing temperature should shed light on the nature of the dynamical processes. Unfortunately, this simple view
gets complicated by additional mechanisms, such as sur- face effects, low-dimensionality and disorder, mechanisms that are matter of current debate [4, 5]. Moreover, when confined within a pore with a typical size of a few molec- ular diameters, both static and dynamical properties can be strongly altered. Despite the large interest in this field and numerous studies, no complete understanding of the thermal, structural and dynamical properties of nanoscop- ically confined phases has been achieved so far. The static and dynamical changes of confined, compared to bulk, su- percooled liquids should therefore be addressed simulta- neously. Consequently, our approach is to use several com- plementary techniques and various systems to disentangle the effects and to overcome the particularities of each ex- perimental probe or porous geometry. Significant advances have also been made by recent computer simulations [6], which should lead to a better microscopic understanding of fluids in confinement.
The remarkable progress in the synthesis of mesostruc- tured porous silicates offers the opportunity to tackle these questions on samples with well-defined, ordered porous ge- ometries and tunable surfaces. Here we have used MCM- 41 and SBA-15 mesoporous matrices as a model porous material for the study of confinement effects on the phase properties of benzene and toluene, some of the simplest glass-forming molecular systems [7–9].
2 Sample preparation and experimental details
Samples and matrices
MCM-41 and SBA-15 are synthesized porous silicates, with parallel cylindrical pores arranged on a honeycomb- type lattice [10]. They can be obtained with five differ- ent diameters, 2.4 and 3.5 nm for the MCM-41 and 4.7, 6.5, 8.7 nm for the SBA-15. We have determined the char- acteristics of the porous matrices by neutron diffraction and nitrogen adsorption experiments [11, 12]. The first ex- periments on toluene were performed with as-synthesized porous materials. In all other experiments a significant reduction of the incoherent scattering contribution to the neutron spectra, arising from hydrogen atoms, was achieved by a systematic deuteration of the silanol groups, with no further chemical treatment of the surface. Com- plete filling of the out-gased matrices was assured by an appropriate mass of liquid that corresponds the porous volume measured by volumetric adsorption.
Melting temperature determination
We combined DSC and NMR experiments to study phase changes in benzene confined in MCM-41 and SBA-15.
H1-NMR measurements provide a suitable method for the determination of phase-change and phase-mixture tem- peratures where DSC scans present very broad transition ranges like in confined systems. Moreover, depending on
the experimental conditions, NMR spectra may be quan- titative, allowing a precise determination of the amount of molecules in each phase.
Adiabatic calorimetry
The heat capacity of confined toluene has been measured with a laboratory-made adiabatic calorimeter and a pre- cision better than 0.1% on the total heat. We obtain the absolute heat capacity of confined liquids with an accu- racy of about 3%, after subtraction of the empty cell and matrix heat capacities and for a typical sample mass of only 300 to 500 milligrams [13]. Heat capacity measure- ments were performed following an intermittent heating procedure after a quench to 80 K at −10 Kmin−1. The glass temperature Tg is estimated from the maximum of the first derivative of the heat capacity (dCp/dT)Tgwith respect to temperature.
Neutron scattering for structure and density of confined liquids
Elastic neutron scattering experiments were carried out at the reactor Orph´ee of the L.L.B.1, on three different spectrometers (PAXE, G6-1, 7C2) so that an extended Q-range from 10−2˚A−1 to 101˚A−1 was covered. A full standard correction procedure was applied to the experi- mental diffraction intensities, as described in details in [13, 14] and the contribution from the matrix-matrix correla- tions was removed by subtraction of a properly weighted spectrum of the empty silicate. At smaller Q-values, the structure factor of the empty samples exhibits a Bragg peak arising from the honeycomb-like arrangement of the cylindrical pores. These Bragg peaks provide a unique way to measure the density of the confined liquids, because its intensity I is related to the square of the contrast, de- fined as the difference of the scattering length densityρ¯b between the silica matrix and the pore content:
I=A
ρSiO2¯bSiO2−ρfill¯bfill2, (1) whereAis a constant term, ¯bis the average coherent scat- tering length and ρis the number density of SiO2 or the filler (fill),e.g., C7D8. By comparing the intensities of the (100)-reflections of the filled and empty matrices, one can extract the density of the confined phase.
Molecular simulation
Monte Carlo molecular simulation of benzene and molecular-dynamic simulations for toluene have been per- formed in the canonical ensemble using the OPLS in- teraction potential with reaction field long-range electro- static interaction corrections [15]. Such simulations give very good agreement with structural and diffusion con- stant measurements as was shown in previous studies on
1 Laboratoire L´eon Brillouin CEA-CNRS, Saclay, France.
aromatic molecules [16, 17]. For the structural analysis with Monte Carlo methods, we have first simulated 800 molecules of bulk liquid at a temperature of 293 K in a cubic box with 49 ˚A side width and periodic boundary conditions. Then, two distinct porous geometries were de- fined: the first one with a smooth hard-wall repulsive cylin- der of 33 ˚A diameter and containing 285 molecules at the bulk density, corresponding to the ultimate case of con- finement by pure volume restriction, the second one is based on the requirement of identical molecule-molecule and molecule-surface interaction. The latter has been ob- tained by freezing the motion of the bulk molecules, which are located outside a cylindrical volume of diameter 33 ˚A, and only performing the simulation for the remaining molecules inside this cylinder. Thus an exact energetic balance between fluid-fluid and fluid-wall interaction is fulfilled with no further distinction between in-pore and out-pore molecules. Under this latter condition of confine- ment, the MD simulations were performed using a Predic- tor Corrector Gear Algorithm with toluene modeled as a rigid molecule. Equations of motions for the 507 molecules are solved within the quaternion formalism using classical mechanics. To take advantage of the cage effect in the dy- namics, a multiple time step approach was used, choosing 1 fs for the inner shell with a cut-off radius of 2.5σ and 10 fs for the outer shell with a cut-off radius of 3.8σ.
Neutron scattering for dynamical properties
Inelastic incoherent neutron scattering experiments were carried out on the time-of-flight (IN6) and backscatter- ing (IN16) spectrometers at the Institut Laue-Langevin at Grenoble, with a wavelength of 5.1 ˚A and 6.271 ˚A, re- spectively, covering a scattering vector range 0.2 < Q <
1.9 ˚A−1; however, due to the mentioned matrix Bragg peaks, only data at Q > 1 ˚A−1 were taken for the anal- ysis. The same hydrogenated toluene samples were used in both experiments with more than 80% of the incoher- ent scattering arising from H. The investigation of the methyl group dynamics in confinement is considered else- where [18]. Flat circular samples with SiO2 discs of 2 mm thickness were oriented with an angle of 135◦. The resid- ual contribution of the empty matrix was measured and subtracted.
The strict corrections are delicate, because of cross- correlations terms, possible adsorption of the empty ma- trices and rotational dynamics of the functional groups (even deuterated) attached at the surface. Different de- tectors have been grouped around Q = 1.3 ˚A−1, where the center-of-mass diffusion is dominant [19]. We choose to perform Fourier transformation (FT) of the dynami- cal structure factor S(Q, ω), and divide by the resolution function to get the intermediate self-scattering function FS(Q, t).
3 Phase diagram
Despite its importance, the freezing and melting of con- fined fluids is not completely understood. There is exper-
Tm / K
in MCM-41/SBA-15 in C.P.G.
in A.C.F. Benzene
d /nm
T /K
1/d /nm-1 surface effects
finite size effects 300
280
260
240
220
180
160
140
120
100
1 10 100
0.0 0.1 0.2 0.3 0.4 0.5
100 10 5 3 2 d /nm
a)
b)
Tm bulk
Fig. 1.a) Phase diagram of benzene for melting temperatures versus pore diameter in different host matrices: in activated carbon fibers (ACF, diamonds) [9], in controlled porous glasses (CPG, triangles) [8] grafted at the surface with trimethylsilyl groups, in MCM-41 and SBA-15 (squares) [5] without surface treatment. The bulkTmis 278.5 K. b) Glass transition temper- atureTg of tolueneversus the inverse of the pore diameterd; mean glass transition temperature (filled circles) defined at the maximum of the heat capacity derivative; the arrows represent the widths of the transition region; the dashed line refers to Tg(∞) of the bulk toluene and the dotted line to the latter at the same density than that in the pores. Lines are guide for eyes and error bars are of the size of the symbols.
imental evidence that for large pores the reduction of the melting temperature with respect to its bulk value is well described by the Gibbs-Thomson equation. A phase dia- gram for the melting temperatureversus the pore size for confined benzene is presented in Figure 1a. The existence range of the crystalline phase decreases with pore size and below the pore size of 4.7 nm, which corresponds roughly to 10 molecular diameters, no crystallization is observed for cooling under standard conditions. Instead, unlike for the bulk material [7], a glassy state is easily reached. De- termination of the glass transition line allows to draw a
phase diagram for the glass domain such as Figure 1b for toluene or Figure 9 (see below).
Below a critical size of roughly 10 molecular diameters σ, this value depending on the pore size and the confined fluid itself, freezing does not occur at all (σ = 5 ˚A for benzene). However, the role played by the pore geome- try (pore shape, pore size distribution and pores intercon- nectivity) of the confining material, which is not explicit in the Gibbs-Thomson equation, seems crucial. For ex- ample, simulations show that the departures of the freez- ing and melting temperatures from the bulk value are in cylindrical pores always lower than in slit pores with the same walls properties. Thus, freezing of benzene [5, 8, 9]
may be observed in two-dimensional slit pores down to pore widths of 0.7 nm compared to about 5nm in cylin- ders (Fig. 1a).
Moreover, the departure from the Gibbs-Thomson re- lation is more or less pronounced depending on the hy- drophobic nature of the surface. Our experiments were carried out on benzene confined in MCM-41 and SBA- 15. In other experiments, in order to increase their hy- drophobicity, the surface properties of the porous mate- rials were modified by grafting trimethylsilyl groups [8]
onto the surface of the CPG samples or by dehydroxylat- ing the porous silica surface under vacuum at 723 K. In both cases the reduced number of OH groups at the sur- face leads to an increased wetting of the surface by the organic liquids and the melting point depression dimin- ishes with the hydrophobicity of the surface. The relative strength of the solid-fluid and fluid-fluid interactions has been used in molecular simulations to determine whether the freezing temperature of a confined fluid increases or decreases with respect to the bulk freezing temperature and was found to be in agreement with another kind of diagram, based on adsorbed-fluids wall interaction as pro- posed by Radhakrishnan et al.[4]. Moreover there is ex- perimental evidence that for many confined solids a liquid layer [5] persists next to the pore walls, whose thickness depends on both the confined liquid and the confining ma- terial. Because of this liquid layer, the question of how to calculate the phase transition enthalpy,∆tH, in confined systems must be addressed.
When crystallization is suppressed, one obtains a glass, the signature of which is a jump of the heat capacity and of the thermal expansion atTg as measured by adiabatic calorimetry and by the density of the confined fluids. In Figure 1b,Tgof toluene is plotted against the inverse pore diameter. Using the above-described experimental tech- niques, we have been able to extend the study of the struc- tural relaxation up to macroscopically long times and to characterize the thermodynamic properties of the liquid and glass in confined geometry. Contrast-matching exper- iments are used initially to validate the method of density measurement. As illustrated in Figure 2a for toluene and benzene in 3.5 nm pores, the change of slope in ρ versus T in the region around the glass transition reflects the change of the thermal expansion. Similar measurements for toluene indicate that in the smallest pore of 2.4 nm di- ameter, the glass transition occurs in a temperature region
ρ
T /K
0 50 100 150 200 250 300
0.00 0.05 0.10 0.15 0.20 0.25
msd of confined toluene /Å2
Tg bulk (117K)
Tg1 (~120K) toluene in 3.5nm
Tg2 (~150K) toluene in 2.4nm benzene in d=3.5nm
toluene in d=3.5nm
a)
b)
Bulk toluene
Fig. 2. a) Temperature dependence of the number densityρ of confined toluene (filled circles) (with reference to the bulk value (empty circles) and confined benzene (filled squares) (with reference to the bulk (empty squares)) in MCM-41 with a diameter of 3.5 nm. The glass transition is marked by the change of slope inρ(T),i.e.of the thermal expansion (lines are guides for eyes). The density of the liquid was deduced from the change of the (100) Bragg peak intensity of the MCM-41 (see Eq. (1)). b) Estimated mean square displacement (msd) of confined toluene calculated from the backscattering elastic scans of IN16 in the high-Qregime: in 2.4 nm (squares) and in 3.5 nm (circles). The glass transition is associated to a change in theT-dependence.
roughly around 150 K,i.e., more than 30 K above the bulk glass transition (117 K) [20]. This behavior is confirmed by an analysis of the elastic temperature scans obtained on IN16 (see Fig. 2b), from which the mean square displace- ment (msd) is calculated in theQrange corresponding to intermolecular distances [21].
The thermal expansion of the liquid in confined geom- etry is smaller than in the bulk (up to a factor of 1.5) in the temperature range measured, leading to a smaller expansion jump at the glass transition. This implies that the density of the confined glass is smaller than that of the bulk glass even for pores as large as 4.7 nm,i.e., of about 10 molecular diameters; in this latter case, the effect is however very small with a relative change of less than 2%
0.1
0.0
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10 0.3
0.2
0.1
0.0
Q (Å-1)
Q (Å-1) S(Q) (molec.-1 sr-1 )S(Q) (molec.-1 sr-1 )
Liquid benzene at T=290 K
Bulk
D=65Å, W=36Å D=35Å, W=10Å D=24Å, W=9Å
Bulk
D=65Å, W=36Å D=35Å, W=10Å D=24Å, W=9Å Crystal and vitreous benzene at T=70 K
a)
b)
Fig. 3. a) Structure factor of bulk benzene at 290 K (dashed line) and experimental composite structure factors S(Q) of benzene confined in various matrices of diameter D (solid lines). Inset: S(Q) computed from an excluded-volume anal- ysis of the bulk spectrum at 290 K. b) Same as a) at 70 K, showing, on decreasing the pore size, an evolution from a bulk crystalline phase to a defective crystal and a disordered vitre- ous phase.
but reflecting a systematic trend. This has recently been confirmed by wide-angle neutron scattering experiments (see below).
The most important result of our study is the non- monotonic variation of the glass transition temperature with pore size and the extreme spreading (Fig. 1b) out of the dynamics of the confined liquid. This favors the in- terpretation in terms of a surface boundary-induced het- erogeneity of the confined fluid in the smallest pores. This interpretation is also retained in the most recent studies of the fast dynamics (see below and [22]). However, the de- crease of the glass transition temperature observed for the largest pore sizes suggests that finite-size effects dominate the dynamics of the supercooled liquids and its coopera- tive nature in this range, the most pronounced effect being observed for a pore size of the order of 10 molecular diame- ters for toluene. This decrease is however modest since the comparison with the bulk Tg must take the reduction of the density into account, and also requires the knowledge of the ρ-dependence ofTg. Additionally, the reduction of
0.1
0.0
0 1 2 3 4 5
0.1
0.0
0 1 2 3 4 5 Q (Å-1)
S(Q) (molec.-1 sr-1 )
Q (Å-1) S(Q) (molec.-1 sr-1 )
Vitreous toluene at T=70 K
Computation at T=70 K
Bulk
D=35Å, W=10Å D=24Å, W=9Å a)
b)
Bulk
D=35Å, W=10Å D=24Å, W=9Å
Fig. 4. a) Structure factor of bulk toluene at 70 K (dashed thin line). Experimental composite structure factors S(Q) of liquid toluene confined in various matrices of diameterD(solid lines), see equation (2). b) Same as a) butS(Q) computed from an excluded-volume analysis of the bulk spectrum at 70 K.
Tg(−6 K in the most favorable case) is always smaller than the reduction ofTm,i.e.a shorter supercooled range, but systematic and in the same direction. This point should be also considered when the configurational entropy is cal- culated and the Kauzmann paradox is emphasized.
4 Structural properties
4.1 Neutron diffraction experiments
Wide-angles neutron scattering is considered as a major experimental tool to analyze the local structure of con- densed matter. In the case of a confined fluid, it provides information about the nature of the phase, so that phase transitions are directly established. However, a quantita- tive description of the variations of the local structure of the confined fluid requires a much more refined analysis taking excluded-volume effects and cross-correlation terms into account. The experimental differential cross-section of a confined fluid arises from the superposition of three
terms. Two terms are related to the interference of neu- trons scattered by the matrix and the fluid by themselves whereas the third one is related to the cross-correlation between the matrix and the fluid. The contribution from the matrix-matrix correlations cancels after a properly weighted subtraction of the spectrum of the empty ma- trix, so as to get the “composite structure factor” S(Q) displayed in Figures 3a, 4b in the case of benzene and toluene, respectively. This function contains both fluid- fluid and fluid-matrix correlations, as expressed by equa- tion (1),
S(Q) =SMFluid(Q) + 2
XSiO2 XBz
ˆbSiO2
ˆbBz SMSiO2-Fluid(Q). (2) It has also to be stressed that these two contributions are subject to “excluded-volume” effects, which may sig- nificantly distort their shapes and should be taken into account for a quantitative structural description [23]. In- deed,S(Q) from the confined benzene at 70 K allows dis- criminating between crystalline and amorphous confined phases (cf. Fig. 3b). In the largest pore (D= 6.5 nm), the stable bulk crystalline phase is formed. A broadening of the Bragg peaks indicates some finite-size effects and de- fects. On decreasing the size of the pore, a highly defective structure is obtained (D= 3.3 nm) which finally turns to an essentially amorphous benzene phase for the smallest pore size (D = 2.4 nm). These results are consistent with the NMR and calorimetric measurements and the phase diagram displayed in Figure 1a. They prove that for suf- ficiently large pores, confined benzene crystallizes into a phase very similar to the one observed in bulk conditions, in opposition to other systems like water [24, 25]. More- over, the amorphous structure of the phase confined in the smallest pore agrees with the observation of a glassy dynamics and offers an exceptional opportunity to study the properties of one of the simplest molecule.
At room temperature benzene shows a liquid struc- ture factor for all pore sizes. However, some systematic differences between the four structure factors are observed below 4 ˚A−1 in Figure 3a. Above this Q-value, the struc- ture factor is essentially dominated by the intramolecu- lar form factor, which is constant for benzene. The two main features are i) a negative intensity in the low-Qpart for confined benzene, which prevents any extrapolation to the macroscopic compressibility and ii) a systematic decrease in intensity of the maxima of the main diffrac- tion peak (between 1.3 and 2 ˚A−1). The main diffraction peak is mostly sensitive to the local structure of the fluid.
However, it has been shown in confined geometry that both liquid-matrix cross-correlation terms and “excluded- volume” effects may also affect this part. A careful analysis is necessary, since these two last effects are not related to actually different local order and might lead to misleading conclusions. We have recently generalized the “excluded- volume” effect formalism to the case of inhomogeneous matrices such as MCM-41 and performed a computation of the different terms that contribute to S(Q) [14]. One shows that the structure factor can be written as equa- tion (3), where f1(Q) is the intramolecular form factor,
the first integral corresponds to fluid-fluid correlations and the second one to cross-correlations. X, ρand b refer to molar fractions, density and neutron scattering lengths, respectively:
S(Q) =f1(Q) +XFluid4π Qρ
× ˜gFluid(r)−1
gu(pp)intra(r) +gu(pp)(r)−1
×rsin(Qr)dr+ 2ˆbSiO2
ˆbFluidXSiO24π Qρ
× ˜gFluid-SiO2(r)g(pw)u (r)−1
rsin(Qr)dr. (3)
In bulk, the pair correlation function reflects the spa- tial correlations between the molecules associated with the average local order in the liquid. In a confined ge- ometry, this function is sensitive to both the local struc- ture of the liquid (intrinsic intermolecular correlations) and the requirement that a fraction of space is inacces- sible to the molecules. An appropriate description of the structure of the confined liquid can be obtained by a sep- aration of these intrinsic intermolecular correlations from those induced by the excluded-volume effect. In the case of a MCM-41, one introduces three uniform fluid pair cor- relations functions gupp(r),g(pp)introu (r), andgupw(r) which correspond to the pore-pore and pore-wall correlations within the matrix and depend only on the porous geome- try. The intrinsic fluid-fluid and fluid-wall correlations are contained in ˜gFluid(r) and ˜gFluid-SiO2(r) and can be di- rectly compared to the structure in bulk or in other con- fined geometries.
In the case of benzene, only very weak molecule-matrix interactions are expected and the porous geometry is pre- cisely defined so that the extent of the different contri- butions can be computed. In inset of Figure 3a the ex- pected different contributions to S(Q) are shown if one assumes that the intrinsic cross-correlations are negligible (˜gFluid-SiO2(r) = 1) and that the intrinsic local order in the confined liquid remains equal to the bulk one.
There results significant modifications of the as- computed composite structure factor S(Q), which essen- tially reflect the observed experimental features, namely the negative low-Q limit and the intensity of the max- ima of the main diffraction peak. Although it has not been possible to exclude, within experimental uncertain- ties, the possibility of an intrinsic difference of structure between bulk and confined liquid benzene, it is proved that some major experimental features can be rationalized by taking these additional correction terms into account. A similar analysis for vitreous toluene at T = 70 K is not sufficient to describe the evolution of the structure factor even qualitatively (cf. Fig. 4). The observed experimental features (Fig. 4a) for the two pore diameters are a shift to lower Q-values of the main diffraction peak (in the range from 1.3 ˚A−1 to 1.8 ˚A−1) and some modulations of the intensity at larger Q (from 2.2 ˚A−1 to 4 ˚A−1). The low experimental statistics prevent any further quantitative
analysis, although three essential sources can be retained for a possible interpretation. It may arise from a differ- ence of density of the confined glass. The shift to lower Q-values of the main diffraction peak is in fact consis- tent with an apparently lower density at low temperature measured by small-angle neutron scattering from contrast effects consideration in theQ-range of the Bragg peak of the mesoporous matrix (see Fig. 2a). Some specific cor- relations of the molecules with the interface, probably enhanced by the dipole-dipole interactions, would induce additional intensities in the cross-correlation term, which are not taken into account in this description. This would also affect S(Q) at relatively highQ-values, c.a. 2 ˚A−1 to 4 ˚A−1. Intrinsically different order within the fluid could also be invoked, like anisotropic structures or layering ef- fects. The conditions for the occurrence of such structures in confined phases may be examined by numerical simu- lations.
4.2 Monte Carlo simulation
Performing Monte Carlo simulation with a realistic inter- action potential it has been possible to consider two ulti- mate models of confinement, within a cylindrical pore of diameter D= 33 ˚A. The two porous models differ by the nature of the surface interaction. The first case presents a strict geometric exclusion, modeled by a smooth repulsive wall. The second case satisfies the balance between fluid- fluid and fluid-wall interactions, modeled by a corrugated wall of frozen benzene molecules.
The occurrence of specific order in terms of layering and anisotropy can be quantified by the radial density pro- file displayed in Figure 5. It corresponds to the probability of finding a benzene molecule with its center of mass at a given distance from the wall and its molecular plane form- ing a given angle with the closest wall. Smooth repulsive wall induce strong layering from the interface to the center of the pore, leading to 6–7 layers across one pore diameter.
Moreover, the orientation average of the molecules within the first layer is not satisfied, as demonstrated by the two sharp peaks around 0 and 180 degrees. They correspond to strong preferential orientation of the planar molecule par- allel to the pore surface. However, this orientational order decays much more rapidly than the translational one, since the molecules arranged in the subsequent layers present essentially random orientations. This is conceivable given the weak orientation correlations of neighboring benzene molecules in bulk liquid, which could have promoted this surface effect deeper inside the pore. This local surface- induced orientation is absent in the case of the corrugated wall since it is constructed from the isotropic bulk liquid phase. Conversely to the first model, no extra surface in- teractions are encountered and the structure of the wall matches the original configuration of the confined phase.
In this case, however a weaker layering is noticeable.
This later case is obviously more illustrative of the experimental systems, since strong layering and orienta- tional order require unrealistic smooth porous geometries.
a)
b)
Fig. 5. Radial density profile of liquid benzene at 293 K con- fined in a cylindrical pore of diameter D = 33 ˚A, obtained by Monte Carlo simulation with two distinct surface inter- actions: a) the wall is modeled by a smooth repulsive wall;
b) modeled by acorrugated wall of frozen benzene molecules.
Molecule/wall angle is defined as the angle between the vec- tor perpendicular to the molecular plane and the radial pore vector.
5 Dynamics
The dynamic properties were studied on the same sam- ples, i.e. matrices, preparation, filling by hydrogenated or deuterated toluene, at long time scale by adiabatic calorimetry and at short time scale by quasielastic neutron scattering (here only the incoherent part is presented).
Complementary to these approaches, numerical simula- tions help to understand local mechanisms.
5.1 Long-time relaxation —Calorimetric glass transition
An important feature observed by calorimetric measure- ments onconfined toluene is the quite large broadening of the region of the glass transition (Fig. 1b) with decreasing pore size from a few degrees to several ten degrees. This temperature broadening is consistent with the weaker in- crease of the mean-square displacement observed at least in the smallest pores (Fig. 2b); it can be reformulated in terms of a distribution of relaxation times. Adiabatic calorimetry corresponds to a given time scale (of the order of 103s under adiabatic conditions or 102s by DSC) for the temperature scanning of the enthalpic response of the system. In this picture an extended temperature range for the calorimetric glass transition means a highly stretched relaxation function and a broad distribution of relaxation times. The slowest molecules freeze at a higher tempera- ture than the mean glass transition temperature, whereas the fastest freeze at a lower temperature; the dynamics of a confined fluid is then highly heterogeneous and this character is induced by the wall boundary conditions, an amount of molecules remaining trapped at the surface.
The observed broadening of the glass transition region continuously increases with decreasing pore size, which agrees well with this picture. Even in the case of the largest decrease of the mean glass transition temperature (with a pore diameter of 4.7 nm), the transition region ends about 10 K above the bulk one. For shorter time scale, it is also expected that the stretching of structural relaxation func- tion increases and requires several decades in time to decay entirely, as recently stressed in quasielastic neutron scat- tering studies of salol confined in CPGs [22] and toluene in MCM-41 and SBA-15 under the same conditions [21].
5.2 Short-time relaxation:incoherent quasielastic neutron scattering and molecular-dynamics simulation Incoherent quasielastic neutron scattering experiments were performed on a time-of-flight spectrometer in the time range of few tenths of picoseconds. The structural relaxation process is illustrated in Figure 6 by the de- cay of the intermediate scattering function, fitted by a Kohlrausch function and an additional background. While a small increase of the stretching parameter is observed, we do not find significant changes of the relaxation time compared to the bulk within experimental error bars; but the accessible time scale is quite short for seeing a de- parture from a super-Arrhenius behavior as pointed out by dielectric experiments [2]. But what is more important here is the increased amount of frozen molecules in con- finement compared to bulk and their contribution to the spectra when temperature decreases in agreement with the picture described above from calorimetric data. The combination of the two factors,i.e., the increasing number of arrested particles on the observation time scale and the increase of the relaxation time, does not allow any mode coupling theory analysis as it is conventionally applied for such experiments [26]. The similarity between our results
0.1 1 10 100
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
C7H8 confined (d=3.5nm)
Fig. 6. Incoherent intermediate scattering function (IISF) FS(Q, t) of toluene confined in MCM-41 (d= 3.5 nm) deter- mined by Fourier transformation of S(Q, ω) obta ined on the time-of-flight spectrometer IN6 atQ= 1.3 ˚A−1; different tem- peratures are plotted from 300 K down to 100 K. The dotted line is the fit of the IISF at 200 K with a Kohlrausch function and an additional background due to frozen molecules at the surface shown by an arrow. The point indicated by a simple arrow is located at the equivalent level of the IISF as defined by MD simulations at 200 K extracted from Figure 7.
and the previous ones on salol in GPGs [22] suggests that the regular or irregular shape of the pores is not the pre- vailing input for the glass formation as it is for first-order phase transitions, but that the chemical nature and the shape of the interface play the dominant role on the dy- namical process.
The incoherent intermediate scattering function (IISF) is an interesting quantity to investigate by MD simula- tions, because it corresponds directly to the experimental data obtained by inelastic neutron scattering (a quanti- tative comparison is shown in Fig. 6). This function de- scribes the autocorrelation of the density fluctuations at the wave vectorQ:
S(Q, ω) = 1 N
dω 2πeiωt
N j,k=1
eiQRj(0)e−iQRk(t)
and by F.T. : F(Q, t) =
∞
−∞
S(Q, ω)·cosωt·dω,
where Ri(t) is the position of the atoms (or the center of mass) of moleculesjandk(herej=k) at timet,N being the number of molecules.
In Figure 7 we plotted the IISF for toluene at 200 K andQ= 1.3 ˚A−1 in thez(cylindrical pore axis) or radial x-direction, respectively, averaged over all the molecules whatever their distance to the wall at a given time.
The Gaussian approximation calculation of the IISF fails systematically, in particular when we approach the wall; it leads to much slower relaxation functions than expected from simple mean-square displacement calcu- lations, in agreement with the fact that the correlation between molecules increases consistently under confining
Fig. 7.Incoherent intermediate scattering function at a wave vector 1.3 ˚A−1and 200 K calculated from MD simulations with the bulk density: continuous lines in thez-direction of the pore FSZ(Q, t) and dashed lines in thex-radial directionFSX(Q, t). In addition, the IISF representative for the center-of-mass motion (CoM) is plotted, showing a slower decay in all cases.
conditions. The increase of stretching of the IISF is also a result related to the presence of a surface and a highly heterogeneous dynamics, a molecule at the surface staying inside the cage of its immobile neighbors. So, it becomes interesting to investigate the IISF as a function of the wall distance.
The wall effect is clearly seen in Figure 8 where the structural relaxation time (τα) is displayedversus the dis- tance from the center of the pore at 200 K. The relaxation time is here defined as the value at which the IISF for the z-direction decays in time to e−1 of its initial value. The relaxation time is seen to be roughly constant near the cen- ter of the pore then increasing exponentially but continu- ously when approaching the wall. Near the wall the molec- ular motions are at least by one-order of magnitude slower than in the center. To investigate their motions, we have also calculated the van Hove correlation functionG(r, t), which represents the probability for a particle to be at time tat a distancerfrom its earlier position at timet= 0 and r = 0; we found a behavior typical of hopping processes involving the rotational motion of the molecule ring.
Due to the continuous variation ofτα, no correlation is established with the possible layers observed in the den- sity profiles (as for benzene in Fig. 5 or in [17]). This, however, may not be the case for different type of walls.
The molecules at the center behave in a more isotropic way, comparable to the bulk, and finite-size effects could be expected. However, the relaxation time is larger than in the bulk, at all distances and all directions, even at the
Fig. 8. Structural relaxation time on thez-direction (axis of the cylindrical pore) versus the distance from the centre of the pore at 200 K. The relaxation time is defined as the time it takes for the IISF average over thez-direction to decay to e−1 from its initial value. Dashed line: fit by A+Bexp(Cr), with adjusting constantsA,B andC, serving only as a guide to the eye.
center, and the rigid-surface effect persists in the whole pore with presumably an additional cooperativity length larger than the pore radius of 3σbuilt here. The surface induces the slowing-down, and no finite-size effects use- ful for the glass transition study are highlighted. Only at higher temperatures, a quantitative bulk-like time scale is reached, while at lower temperature, the cooperativity length induced by the immobile molecules at the surface increases and an even slower behavior is observed.
6 Conclusion
The only way to rationalize the non-trivial pore size de- pendence of Tg in the case of toluene is to invoke size ef- fects that are not quantifiable. One can expect such trends for so-called fragile liquids (as toluene), where dynamical heterogeneities in the bulk state are suspected. However, for intermediate or even stronger liquids (as benzene and Lennard-Jones fluids), size effects would be smaller and no decrease ofTgis expected as proposed in Figure 9 with the small number of data available. The broadness of the tran- sition is the second relevant feature of the present study;
it could be explained by an inhomogeneous distribution
0.0 0.1 0.2 0.3 0.4 0.5 0.8
1.0 1.2 1.4 1.6
Tgconf/Tgbulk
Fig. 9. Phase diagram of the glass transitionversus pore di- ameter and fragility: i) the behavior of fragile liquids such as toluene (circles), oTP etc. is plotted showing finite-size effects in competition with the surface effects (full line with surface effects and dotted line without surface effects); and ii) a less fragile behavior of benzene (squares) and Lennard-Jones liq- uids is represented, where the surface controls theT changes.
Such representation would be in favour of a cooperative length existing in glass-forming liquids.
of relaxation times in the pore. Furthermore, the surface layer induced by confinement plays a crucial role here as for the melting point. The modification of the strength of the hydrogen bonds at the surface and the increase of the surface hydrophobicity by grafting organic function may be helpful for the modification of the confined fluid
—wall surface tension at constant topology. At this stage, no quantitative estimation of a supramolecular length or spatial information on the dynamical heterogeneities in the bulk supercooled liquid at the glass transition can be deduced until we understand the mechanism involved at the interface.
The authors are very grateful to Drs. M. Koza, J. Teixiera and I. Mirebeau for their help in neutron scattering experiments and helpful discussions.
References
1. L.D. Gelb, K.E. Gubbins, R. Radhakrishnan, M.
Sliwinska-Bartkowiak, Rep. Progr. Phys.62, 1573 (1999);
J. Dubochet, M. Adrian, J. Teixeira, C.M. Alba, R.K.
Kadiyala, D.R. MacFarlane, C.A. Angell, J. Phys. Chem.
88, 6727 (1984).
2. B. Frick, R. Zorn, H. B¨uttner (Editors), International Workshop on Dynamics in Confinement; J. Phys. IV10 (2000) and references therein; A. Sch¨onhalset al., this is- sue p. 173; A. Sch¨onha ls, R. Sta uga , J. Non-Cryst. Solids 235-237, 450 (1998).
3. H. Sillescu, J. Non-Cryst. Solids243, 81 (1999); M.D. Edi- ger, Annu. Rev. Phys. Chem.51, 99 (2000); B. Schiener, R. B¨ohmer, A. Loidl, R.V. Chamberlin, Science274, 752 (1996) and references therein.
4. R. Radhakrishnan, K.E. Gubbins, M. Sliwinska- Bartkowiak, J. Chem. Phys. 112, 11048 (2000); H.K.
Christensen, J. Phys. Condens. Matter13, R95 (2001).
5. G. Dosseh, Y. Xia, C. Alba-Simionesco, J. Phys. Chem. B 107, 6445 (2003).
6. See, H. Meyer, J. Baschnagel, this issue, p. 147.
7. C. Alba-Simionesco, J. Teixeira, C.A. Angell, J. Chem.
Phys.91, 395 (1989).
8. C.L. Jackson; G.B. McKenna, J. Chem. Phys. 93, 9002 (1990); R. Mu, V.M. Malhotra, Phys. Rev. B 44, 4296 (1991).
9. A. Watanabe, K. Kaneko, Chem. Phys. Lett. 305, 71 (1999).
10. C.T. Kresge, M.E. Leonowitz, W.J. Roth, J.C. Vartuli, J.S.
Beck, Nature359, 710 (1992); D. Zhao, J. Feng, Q. Huo, N. Melosh, G.H. Fredrickson, B.F. Chmelka, G.D. Stucky, Science279, 548 (1998).
11. M. Kruk, M. Jaroniec, A. Sayari, Chem. Matter 11, 492 (1999).
12. D. Morineau, Y. Xia, C. Alba-Simionesco, J. Chem. Phys.
117, 8966 (2002).
13. D. Morineau, C. Alba-Simionesco, J. Chem. Phys. 109, 8494 (1998).
14. D. Morineau, C. Alba-Simionesco, J. Chem. Phys. 118, 9389 (2003).
15. W.L. Jorgensen, E.R. Laird, T.B. Nguyen, J. Tirado- Rivers, J. Comp. Chem.14, 206 (1993).
16. D. Morineau, G. Dosseh, R.J.-M. Pellenq, M.-C. Bellissent- Funel, C. Alba-Simionesco, Mol. Simul.20, 95 (1997).
17. V. Teboul, C. Alba-Simionesco, J. Phys. Condens. Matter 14, 5699 (2002).
18. P. Scheidler, W. Kob, K. Binder, this issue p. 5.
19. C. Alba-Simionesco, A. T¨olle, D. Morineau, B. Farago, G.
Coddens, cond-mat/0103599.
20. C. Alba-Simionesco, J. Fan, C.A. Angell, J. Chem. Phys.
110, 5262 (1999).
21. C. Alba-Simionesco, D. Morineau, G. Dosseh, B. Geil, B.
Frick, in preparation.
22. R. Zorn, L. Hartmann, B. Frick, D. Richter, F. Kremer, J.
Non-Cryst. Solids (2001).
23. F. Bruni, M.A. Ricci, A.K. Soper, J. Chem. Phys. 109, 1478 (1998); A.K. Soper, F. Bruni, M.A. Ricci, J. Chem.
Phys.109, 1486 (1998).
24. K. Morishige, K. Ka wa no, J. Phys. Chem. B 103, 7906 (1999).
25. J. Dore, Chem. Phys.258, 327 (2000).
26. A. T¨olle, C. Alba-Simionesco, to be published in J. Chem.
Phys.
