Dielectric responses in disordered systems: From molecules to materials

Journal of Non-Crystalline Solids 351 (2005) 2716–2722 www.elsevier.com/locate/jnoncrysol

Dielectric responses in disordered systems: From molecules to materials Ranko Richert

*

Department of Chemistry and Biochemistry, Arizona State University, Tempe, AZ 85287-1604, USA Available online 27 July 2005

Abstract With many current technologies, non-crystalline materials are required to perform on increasingly limited spatial scales approaching several 10 nm. The dynamics of molecules are an important aspect of disordered materials as they define the glass transition. Dielectric relaxation techniques are capable of observing the dynamics in glass-forming systems across 18 orders of magnitude, but the typical experimental approaches address macroscopic effects in bulk samples. Here, we explore the various possibilities of applying dielectric relaxation and related techniques in order to gain insight into the dynamics on a nanoscopic or even molecular level. Ó 2005 Elsevier B.V. All rights reserved. PACS: 64.70.Pf; 77.22.Gm

1. Introduction The time scales involved in the dynamics of molecules in condensed matter cover particularly wide ranges in the case of glass-forming materials, where the transition into the solid state occurs by a gradual slowing down of molecular motion instead of crystallization terminating the liquid phase at much higher temperatures [1]. In the supercooled regime, i.e. below the melting temperature Tm two characteristic features of viscous materials become most obvious: non-Arrhenius temperature dependence of some characteristic relaxation time, and non-exponential relaxation behaviour. Temperature dependences of viscosities or average relaxation times are successfully approximated by the empirical law of Vogel–Fulcher–Tammann (VFT) [2]: log10 ðs=sÞ ¼ A þ B=ðT
T 0 Þ;

*

ð1Þ

Tel.: +1 480 727 7052; fax: +1 480 965 2747. E-mail address: [email protected]

0022-3093/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2005.03.065

which implies a kinetic arrest if extrapolated to a finite temperature T = T0 (instead of diverging at zero temperature, as in the Arrhenius case). Because of the finite cooling rate or time window of a given experiment, the solidification sets in well above T0. In practice, the liquid-to-glass transition temperature at Tg is determined by the viscosity reaching gg = 1013 Poise or by a relaxation time as long as sg = 100 s [3]. Within the equilibrium liquid state, isothermal measurements of relaxation phenomena related to the structure of the system are non-exponential, i.e. a single time constant is not sufficient for the characterization of the correlation function under study [4]. Again an empirical law, the Kohlrausch–Williams–Watts (KWW) [5] or stretched exponential has been very successful in the description of time dependent correlation functions: "
bKWW # t UðtÞ ¼ U0 exp
; ð2Þ sKWW with 0 < bKWW 6 1 being a measure for the departure from exponential behaviour. It turns out that the degree of non-exponential relaxation and non-Arrhenius

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temperature dependence are correlated quantities [6]. Regarding the origin of such dispersive relaxations, our understanding has improved significantly in terms of heterogeneous dynamics within otherwise homogeneous materials [7–9]. As a result of a variety of experiments on this matter, fast and slow contributions to the dynamics are seen to be independent modes (giving rise to spectral selectivity) with different time scales being spatially clustered. The sizes of these dynamically distinct domains are assumed to be of the order of several nm [10,11], and therefore similar to the length scales of typically 3 nm near Tg assigned to the cooperatively rearranging regions (CRR) of the Adam–Gibbs approach [12]. There are a number of motivations for approaching the nanoscopic and molecular dimensions with techniques which provide information on the dynamics in disordered systems. A more quantitative assessment of the length and time scales involved in heterogeneous dynamics in one of the topics that has driven the development of novel techniques [8,9]. Another field of interest in the context of nanoscale dynamics is determined by the effects of confinement and interfaces on the motional degrees of freedom in viscous liquids [13–15]. Given the length scales of several nanometres inherent in the dynamics of glass-forming materials, it is not immediately clear how the properties of viscous liquids will change upon reducing the spatial dimensions of a sample [16]. Additionally, it may be expected that the effects of geometrical confinement will depend on the chemical and physical interactions at the liquid/solid interface. Therefore, many dielectric studies are devoted to improving our understanding of the diverse phenomenology of liquids subject to geometrical restrictions and/or interfacial effects. The aim of this paper is to provide an overview over the possibilities of approaching nanoscale dynamics with dielectric and related experimental methods.

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relaxation times [18]. Whenever spectral selectivity is observed, the faster and slower degrees of freedom must be independent and the dynamics are assumed heterogeneous. More recently, the spatial aspect of heterogeneous dynamics has been emphasized by novel multidimensional 13C solid-state exchange NMR experiments [10,11]. Typical dielectric relaxation data does not go beyond the information of an ensemble averaged two-time correlation function [19], from which no information on heterogeneous versus homogeneous dynamics can be derived. However, this limitation holds only for the regime of linear responses, and the non-linear technique of dielectric hole-burning (DHB) has been devised to further our insight into the field of heterogeneous dynamics [20,21]. DHB exploits spectral selectivity in a dynamically heterogeneous system, where large amplitude electric fields modify the subsequent small signal response of the sample. The original explanation of the effect rests on a frequency dependent loss e00 (x) > 0 and the resulting absorption of energy if a time dependent field is applied [22]. At sufficiently high sinusoidal ÔburnÕ fields, Eb sin(xbt), the absorbed energy Q / e00 ðxb ÞE2b leads to an increase in the fictive temperature, Tf > T. The question is whether the effect is equivalent to heating the entire sample or energy is absorbed selectively by those relaxing units whose time constants coincide with the inverse burn frequency xb. In the former case, the relaxation curve will shift uniformly along the log(t) scale, whereas selective burning modifies the response in a certain time range while the remaining curve is not affected. Based on this DHB approach, the signatures of heterogeneous dynamics have been observed for the a-process of supercooled liquids [21], for the slow b-relaxation in the glassy state [23], and for the diffusivity of charges in an ionic conductor [24]. Additionally, recent DHB results on viscous glycerol across 4.5 decades in frequency could be reproduced quantitatively if it is assumed that thermal and dielectric relaxation times are locally correlated [25,26].

2. Discussion 2.2. Solvation dynamics 2.1. Dielectric hole-burning The macroscopic responses observed for structural relaxations in liquids, secondary processes in glasses, and conductivity relaxations are typically non-exponential. These dispersive dynamics can be modelled in two different ways: homogeneous dynamics with elementary relaxing units contributing identically to the ensemble average, or heterogeneous dynamics with practically exponential local responses combined with a spatial distribution of characteristic time scales [17]. One way of discriminating these two pictures rests on spectral selectivity, where the experiment attempts to measure or modify a fraction of the distribution or spectrum of

In a solvation experiment, the liquid under study is doped at low concentrations with chromophores, i.e. molecules which can be excited selectively by a laser pulse [27]. Generally, the excited solutes will exhibit a permanent dipole moment (lE) which is different from that in the ground state (lG). This situation is outlined schematically in Fig. 1. Such an optical transition changes the electric field surrounding the chromophore and the dipoles within a polar liquid are no longer in equilibrium with the field of the probe molecule. Analogous to the result of applying a field in a dielectric relaxation experiment, the solvent molecules in the immediate vicinity of the solute will tend to align in order to

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t<0: µ = µG

t>0: µ =µE Fig. 1. Schematic representation of the change in the dipole moment of a chromophore (center circle) from the ground state (lG) to the excited state (lE) value upon laser excitation. Interactions with the dipoles of the liquid constituents (smaller circles) are spatially limited to the shaded area, within which dipoles tend to realign to the electric dipole field of the probe molecule. This is the basis for solvation dynamics experiments, employed as a molecular probing of dielectric polarization.

polarization P(t) / hpi(t)i, whereas the solvation technique yields the average in terms of the emission peak wavenumber hmi(t) as well as the variance hp2i ðtÞi
2 hpi ðtÞi in terms of the gaussian emission line widths r(t). The basic result of such an experiment is portrayed in Fig. 2, indicative of a peak within the r(t) curve, which can be rationalized quantitatively if heterogeneous dynamics is assumed [34,35]. Within the picture of homogeneous dynamics, a time invariant width r1 is predicted instead. This signature of heterogeneity has been observed in the viscous regime of a glass-forming liquids with relaxation times in the range 54 s to 1.5 ms, but also in the fluid state with few ns relaxation times [36], which is at temperatures exceeding the critical value Tc of the mode-coupling theory [37]. Obviously, heterogeneity is not just an alternative approach to the nature of the dynamics in liquids, because the assumption of dynamic homogeneity fails to comply with an increasing number of experimental and simulation results [8,9,38]. 2.3. Nanoconfined systems

As an alternative to non-linear dielectric approaches to the molecular details of the dynamics, triplet state solvation experiments have been used in order to provide additional information based upon the molecular dielectric technique outlined in the previous section [32]. Although optical techniques are employed in order to monitor the solvent responses, the information is equivalent to the access to higher moments of the spatial distribution of local polarization effects pi(t) [33]. A dielectric experiment is limited to the volume averaged

3 -1 〈ν (t) [ 10 cm ]

a

〈ν + σ

21.3

〈ν − σ 〈

21.1 20.9 190

σ (t) [ cm -1]

ð3Þ

21.5 〈

hxðtÞi
hxð1Þi and hxð0Þi
hxð1Þi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rðtÞ ¼ hx2 ðtÞi
hxðtÞi .

CðtÞ ¼

An entirely different approach to nanoscale dynamics of glass-forming systems is by geometrical confinement [39]. Very small sample geometries are achieved in various ways, e.g. free [40] or substrate supported [41,42] thin polymer films, polymer solutions [43], liquids imbibed in clays [44], liquids [45–48] and polymers [49]

〈

reestablish equilibrium with the excited state dipole of the probe. For the present purpose, the important aspect of the solvation technique is the spatially limitation of the solvent response (indicated by the shaded area in Fig. 1). The solvent reorientation is being monitored in terms of the time dependent emission spectra following electronic excitation of the chromophore, provided that the solvent reorganizes within the excited state lifetime of the probe [28,29]. Accordingly, long lived S0 T1 (0–0) phosphorescence is required in order to study the slow dynamics of supercooled liquids near Tg [30,31]. Because the individual vibronic transitions of inhomogeneously broadened emission spectra are well approximated by Gaussian lines, it is usually sufficient to focus on the first two moments of the transition frequencies x, which are related to the Stokes shift correlation function C(t) and linewidth r(t) via [30]

b

180 170

σ∞

160 10

-2

10

-1

0

10 t / τ KWW

10

1

10

2

Fig. 2. Master plot of the time-dependent Stokes shift hmi(t) (symbols, (a)) and of the inhomogeneous linewidth r(t) (symbols, (b)) versus t/sKWW for the solute QX in the solvent MTHF, measured at temperatures 91 K 6 T 6 97 K in steps of 1 K [34]. The solid line in (a) is a stretched exponential fit with bKWW = 0.5, the dashed lines indicate the ±r margins around the average hmi. The solid line in (b) is based upon a static distribution of exponential responses, whereas homogeneity would lead to r(t)  r1.

R. Richert / Journal of Non-Crystalline Solids 351 (2005) 2716–2722

confined to porous glasses, microemulsions [50], or highly regular geometries like zeolites [51] or templated materials such as SBA-15 or MCM-41 [52]. All these cases of spatially restricting the sample dimensions have been investigated by dielectric relaxation spectroscopy. Although length scales as small as a few nanometers are achieved, the dynamics are also highly sensitive to the particular chemical and physical interactions present at the interface between the sample material under study and the solid which determines the geometry [53]. As an example, solvation experiments on 3-methylpentane in porous glasses of 7.5 nm pore diameter have been performed with the probe molecules dissolved in the liquid and with the chromophores attached to the surface [54]. The results have revealed an increase of the pore volume averaged relaxation time by a factor of 40, while the interfacial layer is altered by a factor of 2000. In a recent combined dielectric and solvation study of propylene glycol (PG), it has been found that the glass transition is shifted to higher values by DTg = +4.5 K (increase of relaxation time relative to the bulk) when subject to the 10 nm diameter confinement of controlled porous glass [46]. In contrast, Tg is depressed by DTg =
7 K if PG is confined to 4.6 nm diameter droplets of a glass-forming PG/AOT/decalin microemulsion, where decalin is still in its fluid state near the Tg of PG [50]. This result suggests that the effect of finite size is less relevant than ÔhardÕ versus ÔsoftÕ confinement for the change in the relaxation behaviour relative to the bulk situation. As pointed out schematically in Fig. 3, the direction of the Tg shift is affected more by the relative viscosities of the two materials involved. Accordingly, it is suggested that focussing on the pure finite size effect is best accomplished by the situation of an ÔisoviscousÕ confinement, where the TgÕs of the confined and confining moieties are the same (center panel of Fig. 3). It should be emphasized that the influence of the extra-micellar viscosity (oil-phase) will depend on the strength of the interfacial interactions.

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Regarding the interpretation of dielectric relaxation results obtained on geometrically confined samples, two notes of caution should be considered. First, the dielectric sample will consist of the material to be studied (filler) and the matrix which determines the geometry. Therefore, the composite signal obtained actually refers to the mixture of filler and matrix, which can alter the results of dielectric relaxation peaks significantly, even if the matrix does not exhibit any dielectric relaxation processes in the frequency and temperature window of the experiment [55]. The challenge is to extract the dielectric function of the filler, ef ðxÞ, from the measured composite data, ec ðxÞ. In order to achieve this, knowledge of the volume fraction and geometrical factors of the filler and the dielectric function of the matrix, em ðxÞ, are required but not necessarily sufficient quantities. For simple geometries and not too high filler fractions, the approximations provided by the Maxwell–Wagner–Sillars theory may turn out sufficient to disentangle real confinement from trivial electrostatic mixing effects [56]. In realistic cases, the uncertainty regarding the position and amplitude of dielectric loss peaks can be quite severe [57]. More elaborate treatments can improve the reliability of the procedure [58,59], but ignoring the dielectric heterogeneity of such samples is justified only if index matching conditions, ef ðxÞ  em ðxÞ, are fulfilled. A second possible misinterpretation originates in part from lacking information regarding the absolute amplitudes of dielectric loss signals under confinement relative to the bulk counterpart [60]. As shown by the calculated time and frequency domain relaxation curves of Fig. 4, it is easy to conclude erroneously on faster dynamics in a situation where the slower components are frozen but no acceleration is occurring. Although this problem is not specifically related to dielectric spectroscopy, frequency domain representations of relaxation features appear to be particularly vulnerable to these effects. For instance, the dashed line in Fig. 4(a) results from making the slow

Fig. 3. Hard, isoviscous, and soft confinement conditions for the study of nanoscale dynamics, shown here for the filler propylene glycol (PG) with Tg = 170 K. The matrix Tg for the hard confinement case refers to silica as in the case of porous glasses. Soft confinement has been studied in terms of microemulsions with PG as intramicellar ÔwaterÕ phase, AOT as surfactant, and decalin (DHN) as extramicellar ÔoilÕ phase with Tg = 135 K. The isoviscous case has not been realized yet.

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(a)

(b)

1

Φ (t)

ε'' (ω)

0.5

0.0

0 -2

-1 0 1 log10(ω /s-1)

2

-2

-1

0 1 log 10 (t /s)

2

Fig. 4. (a) Calculated dielectric loss for a bulk (—) and confined (- - -) situation, where confinement is assumed to separate the response into a contribution near the bulk peak frequency and a much slower one. Especially after normalization (. . .), the relaxation under confinement appears faster than in the bulk. (b) The same situation of (a) after transformation to the time domain. Comparing the bulk and confined results on an absolute amplitude scale indicates that there is no faster component in the confined case.

components of the solid curve shift outside the frequency window, i.e. none of the contributions actually become faster. Still, the loss component that remains visible displays a peak at higher frequencies relative to the solid curve, which could be misinterpreted as an acceleration of the dynamics. 2.4. Dielectric probe rotation A further dielectric route towards molecular rather than macroscopic dynamics can be envisioned by the following thought experiment: consider the possibility of tagging certain molecules that contribute to the dielectric signal while others remain dielectrically inert. This would facilitate the access to the motion of particular probes within a viscous medium which remains inactive with respect to the dielectric polarization, a situation indicated in Fig. 5. With optical experiments one

be pro

Fig. 5. Schematic view of a larger probe molecule within an ensemble of smaller liquid constituents. According to similar experimental situations, this case would lead to exponential probe rotation even within dispersive relaxation behaviour of the host. This guest-to-host size ratio of around a factor of 2 has been observed to result in probe rotation that is a factor of up to 100 slower (and therefore exponential) than the host relaxation time.

can exploit selectivity regarding absorption or emission wavelengths in order to pick up the effects of specific solutes in a liquid or polymer. The obvious method of selectivity in dielectric spectroscopy is the dipole moment of the molecules. Accordingly, one would dissolve polar molecules in a non-polar liquid and the polarization will be dominated by an even small amount of such dielectric probes. Experiments of this kind have been performed by Johari and Smyth [61] and by Williams and Hains [62], using decalin and ortho-terphenyl as the respective non-polar matrices for several different more polar dielectric probes. In these cases of guest molecules that do not exceed the size of the respective hosts, the probes have been observed to reflect the nonexponential dynamics of the matrices in terms of time scale and dispersion. With significantly larger probes, more exponential rotational correlation functions for the probe molecules are obtained [63]. More systematic variations of the relative sizes of guest and host molecules have been investigated by optical depolarization studies, which observe the single-particle (rank ‘ = 1) rotational correlation functions of the probes for a comparison with the dynamics of the host [64–66]. A typical approach towards probe rotation is by the Stokes–Einstein–Debye equation (or variants thereof), although the problem is not of a hydrodynamic nature [67]. In view of the heterogeneous nature of structural relaxation in supercooled liquids, molecular probes are embedded within dynamically distinct domains and the rotational time constant of a probe should reflect the particular dynamics or effective viscosity of its immediate vicinity. Accordingly, different probes will sample spatially distributed dynamics and the ensemble averaged rotational behaviour should be as dispersive as the host relaxation. This conclusion is supported by our recent optical depolarization studies, from which we were able to conclude that fast (slow) probes regarding their rotation time constant are locally correlated with fast (slow) host environments, i.e. the relaxation time distributions of guest and host dynamics are spatially correlated [68]. A prerequisite for this situation to occur is that the guest molecule is not significantly larger than the host molecules. Otherwise, the probe could average spatially or temporally over the existing heterogeneities [65]. Spatial averaging requires the probe to interact with a representative set of domains in the course of its rotation, while temporal averaging is based on the finite lifetime a domain memorizes its current time constant. Temporal averaging is promoted by an increasing probe size because larger probes reorient slower and the exponential limit is reached when the environmental variable has fluctuated sufficiently during the time window of the probe rotation process [68]. Again based on combining optical depolarization with solvation experiments, a transition to exponential probe rotation behaviour correlates with the probe rotation

R. Richert / Journal of Non-Crystalline Solids 351 (2005) 2716–2722

time being a factor of 20 or more slower than the host dynamics. Because exponential probe dynamics are observed already starting at guest/host size ratios of 1.3 (in terms of molecular weights), these results suggest that time rather than spatial averaging is at the origin of exponential probe rotation [68]. In any case, observing the transition from dispersive to exponential rotation of probes as a function of their size yields valuable information on the spatial extent and/or persistence time of the dynamically distinct domains [9]. In the case of a substantial time scale separation between guest and host dynamics, it should also be possible to discriminate the two contributions in a dielectric relaxation experiment. We have recently performed high resolution dielectric measurements on various solutes at the 1 wt% level in the non-polar glass-forming liquid 3methylpentane (3MP) with a Tg of 78.5 K. The solutes studied were 2-ethyl-1-hexanol, 5-methyl-2-hexanol, 2methyl-1-butanol, 1-propanol, and 2-methyl-THF [69]. In the case of 2-ethyl-1-hexanol (2E1H), different concentrations of 0.5, 1, and 2 wt% have shown a linear relation between signal amplitude and 2E1H weight fraction. The results for the 2 wt% 2E1H in 3MP mixture are portrayed in the 3D graph of Fig. 6, showing the temperature invariant offset of the 2E1H loss peak of 3.2 decades relative to the a-process of 3MP. The study clearly shows that the ratio of dielectric relaxation times for the guest (sG) and host (sH) varies in the range 1.05 6 sG/sH 6 1470 from the small probe 2-methylTHF to the largest one, 2E1H [69]. In the case of the hydroxyl probes in 3MP, hydrogen bond assisted clustering is responsible for the large hydrodynamic volumes. However, analogous experiments with polar, soluble, and rigid probes of varying size will be able to identify the transition from dispersive to exponential rotation behaviour and thereby provide valuable infor2 wt% 2E1H

98 wt% 3MP

3 10 ε''

6

4 2 2 0

80

90

100 T/K

)

3 110

/ (f

4

Hz

g 10

lo

Fig. 6. Dielectric loss e00 (x,T) versus logarithmic frequency and temperature for 2 wt% 2-ethyl-1-hexanol (2E1H) in 3-methylpentane (3MP) [69]. The temperature range is 75 K 6 T 6 115 K in steps of 1 K, the frequency range is 50 Hz 6 f 6 20 kHz or 1.7 6 log10(f/ Hz) 6 4.3. Independent of temperature, 2E1H relaxes a factor of 1470 slower than 3MP in this system.

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mation on the persistence time and/or spatial dimensions of dynamically distinct domains. The interesting feature is that nanoscale aspects of the dynamics, like the transition from molecular probe rotation to the hydrodynamic limit, become accessible with straightforward impedance measurements on macroscopic samples.

3. Concluding remarks Dielectric relaxation experiments provide detailed results on the dynamics of many materials with an exceptionally wide range of accessible time scales [70]. However, polarization data obtained from standard impedance measurements refer to the macroscopically averaged behaviour, with very limited information regarding the microscopic or heterogeneous nature of the dynamics in disordered system. This work has outlined selected dielectric and related experimental techniques which are being employed in order to approach the dynamics of glass-forming materials on spatial scales of nanometres and even on the molecular level. The dynamic heterogeneity observed in these systems and relaxation time gradients of one decade per nm seen near interfaces underline the importance of nanoscale approaches.

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