Theoretical and experimental study of excess quantities in glass-forming systems

Journal of Non-Crystalline Solids 355 (2009) 2634–2639

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Theoretical and experimental study of excess quantities in glass-forming systems S. Magazù *, F. Migliardo Dipartimento di Fisica, Università di Messina, P.O. Box 55, I-98166 Messina, Italy

a r t i c l e

i n f o

Article history: Received 4 April 2009 Received in revised form 27 August 2009 Available online 21 October 2009 PACS: 61.43.Fs 61.05.fg 65.20.Jk 66.20.Ej Keywords: Neutron diffraction/scattering Transport properties – liquids Glass transition Glasses Thermal properties Thermodynamics Fragility Rheology Viscosity

a b s t r a c t The present work deals with the correlation among a macroscopic quantity, i.e. kinematic viscosity, and some microscopic quantities, namely the mean-square displacement and the free volume. In primis, some definitions of ‘fragility’, a quantity which allows to rank different liquids on the basis of the temperature dependence of kinematic and/or thermodynamic quantities, are recalled. Successively, the temperature dependence of viscosity, mean-square displacement and free volume is taken into account for three glass-forming systems and is analyzed in order to highlight the linear relationships between the logarithm of viscosity and the excess mean-square displacement and between the logarithm of viscosity and the excess free volume. Finally, on the basis of the observed correlations, two ‘fragility’ parameters, operatively evaluated by elastic neutron scattering and positron annihilation lifetime spectroscopy, are discussed and compared. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction It is well known that the different behavior of transport properties shown by different glass-forming liquids in approaching the glass transition is usually described in terms of fragility [1–5], which refers to the molecular structural sensitivity of a glass forming liquid to temperature changes in approaching the glass transition temperature. The fragility concept, introduced by Angell [5] to address the problem of viscous slowdown [6–8], describes, in its kinetic version, how fast the structural relaxation time (sa) increases with decreasing temperature on approaching Tg, defined as the temperature where sa becomes equal to 102 s. In this case the evaluated quantity refers to a limit value at a given temperature and not to a global behavior and, hence, has a local character. ‘Strong’ systems show a T dependence of sa(T), that can be described by an Arrhenius law sa ðTÞ ¼ s1 expðD=kB TÞ, whereas ‘fragile’ systems show close to Tg a much faster T dependence of the relaxation time, which is also markedly non-Arrhenius, and hence a T dependence of the activation energy D [5–8]. The ‘Angell’s kinetic fragility’, mA, is: * Corresponding author. Tel.: +39 0906765025; fax: +39 090395004. E-mail address: [email protected] (S. Magazù). 0022-3093/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2009.09.008

mA ¼

 d logðgðTÞ=g0 Þ ; dðT g =TÞ T¼T g

ð1Þ

where g0, the limiting high-temperature viscosity, and Tg are conventionally fixed to the value of g0 ffi 104 Poise and g(Tg) = 1013 Poise respectively [5–8]. Other indexes of fragility, instead of being associated to the limit behavior of a transport property in approaching Tg from higher temperature values, are based on the global temperature behavior of the viscosity. The whole pattern can be reproduced quite well by variation of the one parameter in a modified version of the famous Vogel–Fulcher or Vogel–Tamman–Fulcher (VTF) equation [9]. By writing the original equation in the form:

gðTÞ ¼ g0 exp



 DT 0 ; T  T0

ð2Þ

where g0 , D and T0 are system-dependent parameters. In this equation it is the parameter D which controls how closely the system obeys the Arrhenius law (D = 1) and T0, the VTF temperature, is the result of an extrapolation of experimental data. Malomuzh et al. [10] have shown that, assuming that the viscosity is satisfactory described by the formula gðTÞ ¼ g0 expðkEBaT Þ, Ea being the activation energy, indicating with Ts the Stickel

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temperature at which the asymptotes of log g as a function of 1/T in low and highly viscous regions intersect. The temperature dependence of Ea and b can be approximated by the formula: ðlÞ ðtÞ Ea ðWÞ ¼ EðhÞ a 1  W þ Ea W þ Ea Wð1  WÞ þ . . .

ðhÞ

ðlÞ

ð3Þ

ðtÞ

bðWÞ ¼ b ð1  WÞ þ b W þ b Wð1  WÞ . . .

ð4Þ

The activation energy E and b depend on the variable W, which has the meaning of the effective number of the degrees of freedom which turn out to be responsible for the formation of highly viscous states [10]. The parameter b(i), which is proportional to the distance of the Stickel point from the curve log g(T) corresponding to the experimental data, can be treated as a measure of the fragility. In Fig. 1 the temperature dependence of viscosity together with the Stickel temperature for glycerol is shown. The dashed lines are the two asymptotes of log g as a function of 1/T in low and highly viscous regions whose intersection identifies the Stickel temperature. The fragility parameter m connected with the activation energy and with the thermal expansion b(h) is defined by the relation [10]:

m¼

1 ðhÞ ðhÞ ðE þ b T g Þ: T g lnð10Þ 0

ð5Þ

Finally, it is well known that the behavioral properties of a glass-forming system can be described in terms of the (3N + 1)dimensional potential energy landscape (PEL) in the configurational space [5,6]. It is important to stress that while the PEL does not depend on T the exploration of the PEL (i.e. which parts of the surface are explored) is strongly T dependent. The system explores an energy hyper-surface, whose complexity can be correlated with the density of the minima of the hyper-surface, characterized by a degeneracy DCp(Tg) and a distribution of the barrier heights between minima Dl. In other words the fragility of a system DC ðT Þ ð/ Dp l g Þ, i.e. its structural sensitivity to temperature changes in approaching the glass transition, is determined by these topological features of energy hyper-surface [5,6]. An Arrhenius behavior of viscosity in the Tg-scaled plot and a small heat capacity variation DCp(Tg) characterize the strongest systems, whereas a large departure from Arrhenius law and a large heat capacity variation DCp(Tg) characterize the most fragile ones; intermediate behaviors can be interpreted in terms of different kinetic (g) and thermodynamic contributions (DCp(Tg)). Typical examples of intermediate systems

are represented by hydrogen-bonded systems, such as glycerol (m = 50 [11]) and trehalose/H2O mixture (weight fraction = 0.5, m = 68 [12]). The leading idea of this work is to establish a bridge between the mean-square displacement and the free-volume microstructure of a system and the kinematic viscosity and to compare the ‘fragility’ degree of some glass-forming systems of different ‘strength’, such as glycerol, PB and OTP, through the analysis of elastic neutron scattering (ENS) and positron annihilation lifetime spectroscopy (PALS) findings. 2. Experimental techniques and materials The backscattering spectrometer IN13 at the Institute Laue Langevin (Grenoble, France) is characterized by a relatively high energy of the incident neutrons (16 meV) which makes it possible to span a wide range of momentum transfer Q (65.5 Å1) with a very good energy resolution (8 leV). Neutron scattering is particularly suited to the study of thermal molecular motions which have been shown to be correlated with the ability of a protein to undergo functional conformational changes [13], because neutrons of 1 Å wavelength have an energy close to 1 kcal/mol. Therefore neutron scattering experiments on IN13 [13–16] provide information on the motions of the sample hydrogens in a space–time window of 1 Å and 0.1 ns given by its scattering vector modulus, Q, range and energy resolution and allow to characterize both flexibility (obtained from the fluctuation amplitudes) and rigidity (obtained from how fluctuations vary with temperature and expressed as a mean environmental force constant) [13–16]. In the present paper a comparison among ENS data on meansquare displacements of glycerol [11], polybuthadiene (PB) [17], and ortoterphenyl (OTP) [18] and PALS data on the free volume of glycerol [19], PB [20] and OTP [21], obtained in the same instrumental configuration, is presented. ENS measurements were carried out across the glass transition temperature in a temperature range of 20–300 K. The instrumental configuration of IN13 used for all the presented data was the same: incident wavelength 2.23 Å; Q range 0.3–4.3 Å1; elastic energy resolution (FWHM) 8 leV. Raw data were corrected for cell scattering and detector response and normalized to unity at Q = 0 Å1. In the ENS spectra the statistical error affecting the raw data channels has been estimated lower than 2%, while the mean-square displacement has been derived in the Gaussian approximation according to:

hu2 i ¼ 3

 dfln½Selincoh ðQ Þg  2  dQ

Q ¼0

log η (in Pas)

9

6

3

0

3

4

TS

5 -1

1000/T(K ) Fig. 1. Temperature dependence of viscosity for glycerol together with the Stickel temperature. The dashed lines are the two asymptotes of log g as a function of 1/T in low and highly viscous regions whose intersection identifies the Stickel temperature.

¼

N X

xa hr2a i½1  C a ðsÞ;

ð6Þ

a¼1

where C a ðtÞ can be assumed equal to 0 since it is a constant that rescales the observed mean-square displacement and xa can be assumed equal to 1 since the particles are assumed dynamically equivalent [13–16]. The mean-square displacement has been evaluated by using different Q ranges and finally choosing the Q range for whose the error for the mean-square displacement was lower than 5%. PALS measurements were performed in a temperature range of 100–300 K by the conventional fast–fast coincidence method as described in [19–21]. The time resolution of prompt spectra was 320 ps. A model-independent instrumental resolution function was obtained from the decay curve of 207Bi with a single lifetime of 186 ps. In the conventional three-component analysis the PATFIT-88 software package [23] was used. The application of the iterative Gold algorithm to experimental data on Doppler broadening of annihilation line eliminated a linear instability of a measuring equipment if one uses the 1274 keV [22] Na peak as the reference. This permitted the measurement of small changes of the annihilation peak (e.g. S-parameter) with high confidence [24].

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3. Theoretical background

14

hDu2 ðTÞi ¼



hhmi 2K force





 hhmi 1 : coth 2K B T

ð7Þ

On the other hand, it is important to notice that the experimentally determined mean-square displacement depends on the spectrometer resolution [25]. The energy resolution can be incorporated into the analysis as follows. At low Q the elastic scattering, by taking the slope of the natural logarithm of the elastic scattering as a function of Q2, it results [25]: 2

2

2

hDu iexp ¼ hDu iconv  hDu ires ;

ð8Þ

where hDu2 iconv is the long-time converged mean-square displacement (which is finite for a spatially confined system) while hDu2 ires arises from quasielastic scattering not resolved by the instrument and is therefore due to motions too slow to be detected [26]:

hDu2 ires ¼

X l>0

al

2

p

arctan

Dx : kl

ð9Þ

Here kl define characteristic timescales of relaxation processes (s1  1/kl), Dx is the width of the resolution function (half-width at full maximum), and al is the maximal contribution of relaxation process l to the mean-square displacement. Therefore the hu2 i values include all contributions to motions in the accessible space and time windows, from vibrational fluctuations (usually expressed as a Debye–Waller factor) as well as from diffusional motions [26]. The free-volume theory is based on the idea that molecules need ‘free’ volume in order to be able to rearrange, so, when a liquid contracts upon cooling, less free volume becomes available [27]. Cohen and collaborators [28] defined the free volume Vf as that part of the excess volume DV which can be redistributed without energy change. When applying the free-volume theory of Cohen and Turnbull to a structural relaxation process in a liquid, its frequency x is expressed by x ¼ C expðcV f =V f Þ, where Vf is the mean specific free volume of the liquid, cV f is the minimum specific free volume required for the occurrence of the process, and c = 0.5–1 [28]. It should be stressed that in order to have a structural relaxation (a relaxation) it is necessary not only to reach the condition of an appreciable local volume into the system but also to have a large enough mean-square displacement which allows a particle to escape from the shell of the first neighbouring particles. 4. Results In Fig. 2 the Angell plot of viscosity for glycerol, PB and OTP is shown. The different fragility degree m, as defined on the basis of viscosity measurements, has the value of 58 for glycerol, 64 for PB and 81 for OTP [6]. Fig. 3 shows the mean-square displacement as a function of temperature for glycerol, PB and OTP, as derived according to Eq. (6). For all the investigated systems it can be observed that a dynamical transition, marking a change from harmonic to anharmonic region in which new degrees of freedom are activated, oc-

Glycerol PB OTP

12 10

log η (in poise)

It is known that the mean-square displacement of a set of quanmi hhmi ( K B being the Debye tempertized harmonic oscillators for T < hh 2K B ature, KB being the Boltzmann constant, hmi being the average frequency of a set of oscillators considered as an Einstein solid) is almost a constant equal to the zero-point fluctuations 2Khhmi (where force Kforce is the average force field constant of a set of oscillators considered as an Einstein solid), whereas it linearly increases with mi . The mean-square displacements origthe temperature for T > hh 2K B inating from harmonic vibrational motions can be expressed by:

8 6 4 2 0 -2 -4

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Tg /T Fig. 2. Angell plot of viscosity for glycerol, PB and OTP.

curs. The procedure to fit the mean-square displacement behavior in the harmonic region has been performed in agreement with Eq. (7). The fitting procedure results are also shown in Fig. 3. By defining hu2iloc as the difference between the mean-square displacement of the ordered and disordered phase [12,29]:

hu2 iloc ¼ hu2 ianharm  hu2 iharm

ð10Þ

one obtains for the viscosity:

g ¼ g0 exp½u20 =hu2 iloc :

ð11Þ

In Fig. 4 the linear dependence of viscosity on the local meansquare displacement for glycerol, PB and OTP is shown together with the results of the fitting procedure performed according to Eq. (11).On the other hand, the free volume can be expressed by [28,30]:

V f ¼ V  V occ

ð12Þ

and denoted as the hole free volume. This volume appears due to dynamic disorder in the liquid in excess of an occupied volume Vocc. In the free-volume theory [28,30], viscosity is related to free volume through the relation:

g ¼ g0 expðcV 0 =V f Þ;

ð13Þ

where V0 is the critical value for free volume and c is a numerical factor introduced by taking into account the overlap of free volume and can be neglected, obtaining the expression:

g ¼ g0 expðV 0 =V f Þ:

ð14Þ

In this frame the local mean-square displacement and the hole free volume, in a first approximation, can be connected by the relation V f ¼ ð4p=3Þðhu2 iloc Þ3=2 . Analogously to the mean-square displacement, in Fig. 5 the free volume behavior as a function of temperature for glycerol, PB and OTP is reported, while Fig. 6 shows the linear dependence of logarithm of viscosity on the free volume for glycerol, PB and OTP, together with the results of the fitting procedure performed according to Eq. (14). It is evident that the linear dependence of log g versus hu2 i1 loc as evaluated from ENS measurements finds a correspondence with the linear dependence of log g versus the total free volume as evaluated from PALS measurements. 5. Discussion The interpretative model proposed to describe the observed behavior of the mean-square displacements moves by the follow-

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(a)

0.20

(a)

Glycerol

Glycerol

0.15

log η (in Poise)

5

2

(Å )

6

2

0.10

0.05

4

3

0.00 100

150

200

250

2

300

5

10

T (K)

(b)

(b)

0.8

20

4

PB

PB

3

2

log η (in Poise)

0.6

(Å )

15 2

1/loc (a. u .)

2

0.4

0.2

2

1

0 0.0 50

100

150

200

250

300

1

T (K)

(c)

2

0.20

(c)

4

10

OTP

OTP

0.15

log η (in Poise)

8

2

(Å )

3 2

1/loc (a. u .)

2

0.10

0.05

6

4

2 0.00 50

100

150

200

250

300

T (K)

0

5

6

7

8

9

10

11

12

2

Fig. 3. Mean square displacements for (a) glycerol, (b) PB and (c) OTP as a function of temperature. The dashed straight lines are the results of the fitting procedure performed following Eq. (7).

ing picture for the elementary flow process (the a-relaxation): an atom jumps in the fast processes (b-relaxation motions) with a Gaussian probability distribution characterized by a mean-square amplitude hu2iloc [29]. If the amplitude of the fast motion exceeds a critical displacement u0, a local structural reconfiguration (arelaxation) takes place. Assuming the time scale of the fast motion to be independent on temperature, the waiting time for the occurrence of an a-process at a given atom is proportional to the probability to find the atom outside of the sphere with radius u0. In a previous work [12], we introduced an operative definition for fragility, based on the evaluation by neutron scattering of the

1/loc (a. u .) Fig. 4. Linear dependence of the logarithm of viscosity on the local mean-square displacement for (a) glycerol, (b) PB and (c) OTP. The dotted lines are the result of the fitting procedure performed according to Eq. (11).

mean-square displacement. The definition, differently from many other phenomenological parameters as, e.g. resilience [13], lies on a physical model and allows to link a macroscopic transport quantity, i.e. viscosity, with an atomic quantity, the nanoscopic mean-square displacement. Taking into account equations (10), (11), we introduced a numerical parameter in order to evaluate the ‘fragility’ degree of the investigated systems in a different way in respect to conventional procedures as follows:

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(a)

(a)

10

80 Glycerol

Glycerol

log η (in poise)

8

3

Vf (Å )

60

40

6

4

20 100

150

200

250

0.01

T (K)

(b)

0.03

0.02

0.04

1/Vf (a. u.)

50

(b)

PB

10

PB

40

log η (in poise)

3

Vf (Å )

8 30

20

10

0

6 4 2

0

50

100

150

200

0

250

0.02

T (K)

(c)

OTP

0.05

0.06

0.07

10 OTP

8

3

log η (in poise)

80

Vf (Å )

0.04

1/Vf (a. u.)

(c) 90

0.03

70

60

6 4 2

50 230

240

250

260

270

0 0.005

280

T (K)

0.010

ð15Þ

0.020

1/Vf (a. u.)

Fig. 5. Free volume for (a) glycerol, (b) PB and (c) OTP as a function of temperature.

 dðu20 =hu2 iloc Þ Mu ¼ : dðT g =TÞ T¼T þ

0.015

Fig. 6. Linear dependence of the logarithm of viscosity on the free volume for (a) glycerol, (b) PB and (c) OTP. The dotted lines are the result of the fitting procedure performed according to Eq. (14).

g

From the evaluation of the hu2iloc values we can evaluate a local fragility parameter [12]. The obtained values, shown in Fig. 7, indicate that the present operative definition for fragility furnishes a direct proportionality between Mu and m. Let us analyze the free volume behavior as a function of temperature. Assuming a Lennard–Jones potential function for a molecule within its cage in the condensed phase, it can be shown that at small DV considerable energy is required to redistribute the excess volume; however, at DV greater than some value DVg, most of the volume added can be redistributed freely. As a result, the transition from glass to liquid may be associated with the introduction of

appreciable free volume into the system [28]. On the other hand, Doolittle defined the free volume by subtracting the molecular volume defined by extrapolating the liquid volume to zero temperature, implying that Vf ? 0 only when T ? 0 [30]. On the basis of the analogy found between the relationships linking by a side viscosity and mean-square displacement and by the other side viscosity and free volume, we propose the following definition of local fragility:

MV ¼

 dðV 0 =V f Þ : dðT g =TÞ T¼T þ g

ð16Þ

S. Magazù, F. Migliardo / Journal of Non-Crystalline Solids 355 (2009) 2634–2639

500

OTP

400

Mu

200

PB

MV

0

tionality law between the Mu and MV fragility parameters and the m fragility parameter, it is possible to get information on the fragility degree of glass-forming systems using spectroscopic techniques such as ENS, EXAFS and PALS. References

300

100

2639

Glycerol

60

80

m Fig. 7. Proportionality law between the Mu and MV fragility parameters and the m fragility parameter. The dashed line is the result of a linear fitting procedure.

This definition implies that the excess free volume evaluation allows to estimate directly the fragility degree. The values obtained by using the above described procedure are almost the same obtained by using the mean-square displacement dependence on temperature, as shown in Fig. 7. Such a measure procedure can be useful to get information on the system and its fragility when experimental data are available at temperature values far from the glass transition temperature.

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

6. Conclusions [27]

In this work the link between viscosity and mean-square displacement and between viscosity and free volume is discussed. The approach finds a good agreement with the experimental data collected on glycerol, PB and OTP. Through the established propor-

[28] [29] [30]

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