Viscous liquids and glasses under high pressure

84

Journal of Non-Crystalline Solids 131-133 (1991) 84-87 North-Holland

Viscous liquids and glasses under high pressure W . F . Oliver, C.A. H e r b s t a n d G . H . W o l f Department of Chemistry, Arizona State Unioersity, Tempe, A Z 85287, USA

The isomeric pentane system, as well as other systems, was studied as a function of pressure via Brillouin scattering. R o o m temperature experiments were carried out at different scattering angles in a Merrill-Bassett diamond anvil cell. The scattered light was analyzed using a nine-pass tandem F a b r y - P e r o t interferometer. Isopentane and mixtures such as 1 : 1 p e n t a n e / isopentane vitrify at 7.4 G P a as determined by a change in the slope of the sound velocity vs. pressure curve, and in some cases by the onset of shear modes. From the sound velocity data and thermodynamic considerations, one can extend the equations of state for these systems to high pressure. When the acoustic shear mode is measureable, it is possible to calculate the high frequency shear modulus from the data. These calculations combined with the measured viscosity allow the determination of the shear relaxation time over a broad range of pressures.

1. Introduction

measured from the Brillouin spectra are then converted to acoustic sound velocities as follows:

Brillouin scattering has been performed in a diamond anvil cell to obtain information on the hypersonic acoustic properties of various liquid systems through their pressure induced glass transitions [1]. From the longitudinal acoustic (LA) and transverse acoustic (TA) frequency shifts and widths, we obtain the corresponding acoustic sound velocities and attenuations as a function of pressure. All data reported here were recorded at room temperature. These data are then used to determine the equation of state and to calculate the moduli and shear relaxation time of the system under study.

v = ?,p/2 s i n ( 0 / 2 ) ,

(1)

2000 P=6.06 GPa

1600 ~ 1200

I

Z

1 ~ ~TA

_=0oo.o 400.0 TA

2. Experimental techniques

0.000 -20

I

I

I

I

I

I

I

-15

-10

-5

0

5

10

15

FREQUENCY

Brillouin spectra were recorded using a spectrometer consisting of five-pass and four-pass F a b r y - P e r o t interferometers in a tandem configuration [2]. By using an equal-angle forward scattering geometry [3], we can write n sin(~/2) = sin(8/2), where n is the index of refraction, and where ~ and 8 are the internal and external scattering angles, respectively. The frequency shifts

SHIFT

20

(GHz)

Fig. 1. Brillouin spectra of a 1:1 mixture of pentane and isopentane at 6.06 G P a showing both the longitudinal acoustic (LA) and transverse acoustic (TA) modes. A scattering angle of 32 ° and an equal-angle forward scattering geometry was used in our experiments. Also shown in this figure are the diamond T A modes (DTA). The strong features between the LA and D T A modes are artifacts of the tandem instrument called ghosts. The region of the T A modes has been scaled up by a factor of 10.

0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

W.F. Oliver et al. / Viscous liquids and glasses under high pressure

where h is the wavelength of the laser in vacuum and I, is the acoustic mode frequency. In this case the pressure dependence of the index of refraction is not needed in order to calculate the sound velocities. The very high contrast (10 ]7 ) of the nine-pass tandem system also allows us to observed the extremely weak TA modes in some systems. Figure 1 shows a typical spectrum taken at a pressure of 6.06 GPa (just below the glass transition) in 1 : 1 pentane/isopentane. Merrill-Bassett diamond anvil cells (DACs) were used to achieve pressures of up to 12 GPa [4]. Backing plates with slits were used to allow for a greater range of scattering angles and to avoid vignetting of the collection aperture. The pressure in the diamond cell was determined by ruby fluorescence techniques [5].

85

0.40

10.00 x x HWHM 0 0 O<>

x

E

6.00

0.35

o

0o

8.00

0.30

@o

VLA

>. xx

0 0 4.00 .J u.I > 2.00

0

× x

VTA

O x

0

x x

;x

o

~. "I-

0.20

lg

0.15

o

0.10

x

x

×

xxx

0.00

o

o 0 x

o

0.25

I

I

|

× Ix

2

4

6

8

PRESSURE

0.05 ×1

10

0.00 12

(GPa)

Fig. 2. Sound velocity as a function of pressure for both the LA (O) and T A ( o ) modes in a 1 : 1 mixture of pentane and isopentane. Also shown is the LA mode H W H M ( x ) as a function of pressure (referred to the fight-hand ordinate).

3.3. 1 : 1 V / V pentane / isopentane 3. Systems studied We have studied the pentane isomeric system [6] and more recently we have begun studies of glycerol and propanol. In this paper, we will concentrate on the p e n t a n e / i s o p e n t a n e system. 3.1. Pentane Pentane has an equilibrium melting point of PM -- 1.8 GPa. It is, however, superpressed easily to approximately 3.8 GPa, above which a polycrystalline sample is formed. On decompression, it was observed to melt at PM- A jump in the sound velocity of about 14% is observed when crystallization occurs.

Static ruby fluorescence measurements reveal a glass transition in a 1 : 1 mixture of p e n t a n e / isopentane at P g - 7.4 GPa, in good agreeement with earlier work [5]. Since this system vitrifies at a relatively high pressure, it has often been used to maintain hydrostatic conditions in DAC studies of solids. We studied this mixture up to 11.72 G P a using Brillouin scattering at angles of 32 ° and 60 °. Slope changes in p 2 o r 02 were observed at Pg, particularly in the 32 ° data. Extremely weak shear acoustic modes were also observed above about 4.5 GPa. Figure 2 shows the pressure dependences of the sound velocities for the LA and TA modes as well as the full width at half maximum for the LA mode.

3.2. Isopentane 4. Equation of state Crystallization is difficult to achieve in isopentane. LA velocities were measured up to 7.12 GPa. No indications of a glass transition were observed in the ruby fluorescence spectra at this point. In a recent Raman experiment [7] a glassy state is indicated above about 7 GPa. Isopentane was superpressed in this experiment to - 1 0 GPa, after which a rapid decompression to a pressure greater than the thermodynamic melting point of PM -- 2.4 GPa produced a polycrystalline sample.

From the LA sound velocities, the equation of state (EOS) of these systems can be calculated by integrating any of the following equivalent thermodynamic relations: Cp/Cv -Y~ I T

v2

,

(Op 1 +Oa(3T ~ - f f ) r = v-~ )--P)s'

(2) (3)

W.F. Oliver et al. / Viscous liquids and glasses under high pressure

86 1,30

/

c{ constant

1,20 /

E

~

shows a much smaller pressure dependence. Thus, the assumption that c a is constant gives a least an upper bound on the actual p(P), which is substantiated by comparing the calculations with the data of Houck [9].

/

/

1.10

/-

/

/

/Cp/C

v constant / / constant

/

/

1.00

/

///~J~-+

+ ~ +

~- 0 . 9 0

5. Relaxation times

Z ~ 0.80

5.1. Shear relaxation time

0.70

0.60

~, , . ,

I , , , , I , | , ,

0.5

1

I,,

1.5

,,

I , , , ,

2

I , , , ,

2.5

3

3.5

PRESSURE (GPa) Fig. 3. Equations of state for the 1:1 p e n t a n e / i s o p e n t a n e mixture obtained by integrating eqs. (2)-(4). Also shown ( + ) are experimental measurements of the EOS at lower pressures from ref. [91.

(ap)

1 pgcp[ar 2

T=-~ +--T--~ aP is.

Power law or linear fits were made to the o(P) data. We can then calculate the longitudinal modulus M = P/)LA, 2 the shear modulus G s = p0EA, and the volume or bulk modulus K s = M _ 4 Gs" Combining the relaxed or infinite frequency shear modulus Goo, i.e., that above P g - 7.4 GPa, with the published viscosity data [10] *I(P), we can calculate the shear relaxation time ~-s(P) from the Maxwell relation: or t)2(P)

I , , , .

(4)

where Cp and c v are the constant pressure and constant volume specific heats, respectively, p is the density, a is the coefficient of thermal expansion, T is temperature, and S is entropy. In these relations the adiabatic sound velocity 02 = V2A 4 2 --3OTA as measured by Brillouin scattering is used. Equation (3) is obtained by making adiabatic to isothermal corrections to eq. (2). Values for ( A T / A P ) s are available in the literature [8] and were extrapolated to higher pressures (at high pressure ( A T / A P ) s has a very small pressure dependence for these liquids). Ideal mixing was assumed in the case of the mixture to determine values of P0 (density at 1 bar), Cp and (OT/OP)s. Equations (2)-(4) were then numerically integrated, assuming constant cp/c v, cp, and a, respectively, to obtain p(P). Figure 3 shows the calculations obtained from integrating each of the above three thermodynamic relations, as well as measurements of the EOS at lower pressures [9]. The discrepancies in the different EOS calculations lie in the validity of the different assumptions made. In general, Ce/Cv, co, and a all decrease with pressure. Both a and Cp/Cv typically show a rapid exponential decrease at low P before levelling off, whereas Cp usually

,s(p )

n(P)

G.

(5)

Figure 4 shows the results of these calculations between atmospheric pressure and Pg.

5.2. Longitudinal relaxation time We are also able to fit the LA mode lineshapes with Lorentzians, but only after convolving the 2 ×

0 1:1

PENTANE/ISOPENTANE

-2

x

-4 ×

o~ - 6

x

O ,--I

x

x

X

x

xx

xxX I x Pg

x

x xxx ×

-8 ×x

x x

××

-10

×× ××

×x

× xX

-12 -14

0.0

I

I

1.0

2.0

I

I

I

I

I

3.0 4,0 5.0 6.0 7.0 8.0 P R E S S U R E (GPa) Fig. 4. Shear relaxation time as a function of pressure calculated from the Maxwell relation and our experimentally measured high frequency shear modulus.

W.F. Oliver et aL / Viscous liquids and glasses under high pressure

aperture b r o a d e n e d Lorentzian with the instrumental function. The fits are excellent and yield b o t h the acoustic m o d e frequencies and widths. In b o t h of the pentane isomers and in the mixture, a peak in the F W H M is observed at 2.25 G P a (see fig. 2) indicating a longitudinal relaxation time of - 19.9 ps. A similar peak is observed in glycerol, but at slightly negative pressures. The glycerol glass transition is at about 3.0 GPa.

6. Conclusion We are developing in our laboratory several techniques for probing l i q u i d / g l a s s behavior at high pressures. Brillouin scattering in a D A C enables us to measure the EOS for these systems as well as the moduli (or liquid incompressibilities) and the shear and longitudinal relaxation times as a function of pressure. W e have recently begun to apply Brillouin and R a m a n scattering to other systems such as glycerol, propanol, etc. We h o p e also to extend our high pressure techniques to include d y n a m i c light scattering, direct optical measurements of the equation of state, and N M R .

87

This work was supported in part by N S F contract n u m b e r E A R 8657437 and b y O N R contract n u m b e r N00014-87-K0471. I n addition, the authors would like to thank Professors Austin Angell and Stuart Lindsay for helpful discussions regarding this work.

References [1] C.A. Herbst, W.F. Oliver, S.M. Lindsay and G.H. Wolf, Bull. Am. Phys. Soc. 35 (1990) 402. [2] S.M. Lindsay, M.W. Anderson and J.R. Sandercock, Rev. Sci. Instrum. 52 (1981) 1478. [3] S.A. Lee, D.A. Pinnick, S.M. Lindsay and R.C. Hanson, Phys. Rev. B34 (1987) 2799. [4] A. Jayaraman, Rev. Mod. Phys. 55 (1983) 65. [5] G.J. Piermarini, S. Block and J.D. Barnett, J. Appl. Phys. 44 (1973) 5377. [6] W.F. Oliver, C.A. Herbst, S.M. Lindsay and G.H. Wolf, in progress. [7] C.A. Herbst, W.F. Oliver and G.H. Wolf, in progress. [8] R. Boehler and G.C. Kennedy, J. Appl. Phys. 48 (1977) 4183. [9] J.C. Houck, J. Res. Nat. Bur. Stand. Phys. and Chem. 78A (1974) 617. [10] J.D. Barnett and C.D. Bosco, J. Appl. Phys. 40 (1969) 3144.