- Email: [email protected]
Can one learn about glasses from advances in granular materials?
Journal of Non-Crystalline Solids 293±295 (2001) 279±282
www.elsevier.com/locate/jnoncrysol
Can one learn about glasses from advances in granular materials? S.F. Edwards Polymers and Colloids Group, Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 OHE, UK
Abstract It is argued that the crudest view of the cooling of a glass is found if one divides its degrees of freedom into slow, con®gurational degrees of freedom, and fast such as vibrations and electronic excitations if appropriate. The latter can be regarded as in thermal equilibrium de®ning the temperature, whereas the former have a distribution which is not in equilibrium and could relax, for example, to an earlier temperature in the cooling. The crudest view of the con®gurational modes is to view the atoms as if they are grains in a granular material and the glass as the mode mixture where the slow try to catch up with the fast. The argument leads to the de®nition of three temperatures: TG the glass transition temperature for in®nitely slow cooling, Tg for cooling rate s_ and T0 for the temperature at which cooling starts, and their relation is T0 TG ; Tg T0 _ 0 TG 1 ls T where l is the constant of the material. Ó 2001 Published by Elsevier Science B.V.
1. Introduction There have been advances in our knowledge of granular materials which seem to us to oer some new ideas on glasses. The simplest granular system is that of perfectly rough grains which are incompressible and form contacts with their nearest neighbours. In general the con®guration of such packing depends on how the material has been created i.e., how it was deposited, how it was stressed, whether there are any other phases like pore ¯uid and gas present, all questions which a civil engineer, for example, knows are critical in applications. However, recent experiments by Nagel and co-workers [1,2] show that granular E-mail address: [email protected] (S.F. Edwards).
systems can behave under the appropriate conditions in a way showing them amenable to the laws of statistical mechanics. The crucial experiments show that external vibrations lead to a slow approach of the packing density to a ®nal steadystate value. Depending on the initial conditions and the magnitude of the vibration acceleration, the system can either reversibly move between steady-state densities, or can become irreversibly trapped into metastable states; that is, the rate of compaction and the ®nal density depend sensitively on the history of vibration intensities that the system experiences (see Fig. 1). This suggests that one might imagine a crude model of a glass in which the heavy atoms are like a granular material and the shaking or tapping of the above-mentioned experiment is provided by the vibrations and any other fast degrees of free-
0022-3093/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 8 3 0 - 4
280
S.F. Edwards / Journal of Non-Crystalline Solids 293±295 (2001) 279±282
Fig. 3. Two-dimensional continuous random network: a sketch of a three-fold co-ordinated elemental glass, where the continuous random network is formed by lines joining the centres of atoms. Fig. 1. Dependence of the packing density of a tapped granular material on the history of vibration intensities [1].
dom that are present. Papers are now appearing in the literature to develop such concepts [3±9] and we give a brief outline of one of them here. Although it is not necessary a `visual aid', it comes from the analysis of solvability of Newton's equations for static packings of hard rough grain analysis [10±12]. One ®nds (within certain provisos) that each grain has d 1 contacts, where d is the dimension of the system (see Fig. 2). The 2-D
picture then looks like a three-fold co-ordinated elemental glass (see Fig. 3). We now need a formulation for the glass.
2. Compactivity We argue that subject to N grains of the Chicago experiment ®lling a volume V, and when at rest (i.e., after tapping has ®nished) have contacts which ®x their positions in space, all con®gurations are equally likely [6]. This is just as energy possessing ergodic systems have a probability distribution P e
S=k
d E
H ;
1
W fRgHfRg;
2
grains have P e
Fig. 2. Two-dimensional continuous random network: a sketch of a packing of three-fold co-ordinated grains of irregular shape. Black circles denote the contact points and dashed lines, joining the centroids of contacts, form the continuous random network.
S=k
d V
where W fRg is the function of contact points fRg which leads to the volume V and HfRg means all the grains are in contact. Expressions for W are given in [13]. Such distributions are in overall agreement with the Chicago experiment, and with simulations by Kurchan and co-workers [8,9] which contain more detailed analysis than experiment can at present. Just as in thermodynamics
S.F. Edwards / Journal of Non-Crystalline Solids 293±295 (2001) 279±282
T
oE ; oS
3
in the granular system the key quantity is the compactivity X
oV : oS
4
A formal connection with thermal physics has 1 oP 5 X oT V T !0; which we use in Section 3.
1 X
oP oT
281
;
6
V
but in general there is a non-vanishing value 1 oP ZX : 7 oT V We can expect Z to look like (see Fig. 5) and a simple derivation leads to Tg T0
T0 TG ; _ 0 TG 1 ls T
8
where l is the characteristic of the material. In Fig. 6 it is the rate at which the cooling curve deviates from the vertical near T0 .
3. Cooling of a glass We aim to develop a theory shown in Fig. 4 where we plot the inverse viscosity as a function of _ the temperature in terms of the rate of cooling s, temperature T0 at which cooling starts and which is equal to the glass transition temperature if s_ ! 1, _ Tg , the glass temperature for a ®nite cooling rate s, and the glass temperature for s_ 0 TG . To derive such a diagram, we make what is probably far too crude an approximation but one that leads to quite reasonably looking curves. We argue that when the slow con®gurational and fast modes are in equilibrium
Fig. 5. The expected time dependence of the Z X _ given for decreasing values of the cooling rate s.
Fig. 4. The inverse viscosity g 1 as a function of temperature T given for dierent values of the cooling rate asymptotically approaching in®nity.
Fig. 6. Dependence of the function Z t on the ratio of temperature dierence and the cooling rate.
oP 1 oT V
282
S.F. Edwards / Journal of Non-Crystalline Solids 293±295 (2001) 279±282
4. Conclusion Crude as this formula (Eq. (8)) is, it does show some of the features of real glasses and suggests that it is worthwhile to produce a better developed theory based on the relaxation of con®gurational entropy as compared to a thermal system. Acknowledgements I wish to acknowledge the ®nancial support of Leverhulme Foundation and thank Dr D.V. Grinev for helpful discussions. References [1] E.R. Nowak, J.B. Knight, M.L. Povinelli, H.M. Jaeger, S.R. Nagel, Powder Technol. 94 (1997) 79.
[2] E.R. Nowak, J.B. Knight, E. Ben-Naim, H.M. Jaeger, S.R. Nagel, Phys. Rev. E 57 (1998) 1971. [3] S.F. Edwards, Ann. NY Acad. Sci. 371 (1981) 210. [4] S.F. Edwards, J. Phys.: Condens. Matter 2 (1990) SA63. [5] S.F. Edwards, in: Disorder in Condensed Matter Physics, Oxford, 1991. [6] S.F. Edwards, R.B.S. Oakeshott, Physica A 157 (1989) 1080. [7] A. Coniglio, M. Nicodemi, Physica A 296 (2001) 451. [8] A. Barrat, J. Kurchan, V. Loreto, M. Sellitto, Phys. Rev. Lett. 85 (2000) 5034. [9] A. Barrat, J. Kurchan, V. Loreto, M. Sellitto, Phys. Rev. E 63 (2001) 051301. [10] S.F. Edwards, D.V. Grinev, Phys. Rev. Lett. 82 (1999) 5397. [11] S.F. Edwards, D.V. Grinev, Physica A 294 (2001) 451. [12] R.C. Ball, R. Blumenfeld, ArXiv cond-matt/0008127. [13] C.C. Moun®eld, S.F. Edwards, Physica A 210 (1994) 279.
www.elsevier.com/locate/jnoncrysol
Can one learn about glasses from advances in granular materials? S.F. Edwards Polymers and Colloids Group, Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 OHE, UK
Abstract It is argued that the crudest view of the cooling of a glass is found if one divides its degrees of freedom into slow, con®gurational degrees of freedom, and fast such as vibrations and electronic excitations if appropriate. The latter can be regarded as in thermal equilibrium de®ning the temperature, whereas the former have a distribution which is not in equilibrium and could relax, for example, to an earlier temperature in the cooling. The crudest view of the con®gurational modes is to view the atoms as if they are grains in a granular material and the glass as the mode mixture where the slow try to catch up with the fast. The argument leads to the de®nition of three temperatures: TG the glass transition temperature for in®nitely slow cooling, Tg for cooling rate s_ and T0 for the temperature at which cooling starts, and their relation is T0 TG ; Tg T0 _ 0 TG 1 ls T where l is the constant of the material. Ó 2001 Published by Elsevier Science B.V.
1. Introduction There have been advances in our knowledge of granular materials which seem to us to oer some new ideas on glasses. The simplest granular system is that of perfectly rough grains which are incompressible and form contacts with their nearest neighbours. In general the con®guration of such packing depends on how the material has been created i.e., how it was deposited, how it was stressed, whether there are any other phases like pore ¯uid and gas present, all questions which a civil engineer, for example, knows are critical in applications. However, recent experiments by Nagel and co-workers [1,2] show that granular E-mail address: [email protected] (S.F. Edwards).
systems can behave under the appropriate conditions in a way showing them amenable to the laws of statistical mechanics. The crucial experiments show that external vibrations lead to a slow approach of the packing density to a ®nal steadystate value. Depending on the initial conditions and the magnitude of the vibration acceleration, the system can either reversibly move between steady-state densities, or can become irreversibly trapped into metastable states; that is, the rate of compaction and the ®nal density depend sensitively on the history of vibration intensities that the system experiences (see Fig. 1). This suggests that one might imagine a crude model of a glass in which the heavy atoms are like a granular material and the shaking or tapping of the above-mentioned experiment is provided by the vibrations and any other fast degrees of free-
0022-3093/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 8 3 0 - 4
280
S.F. Edwards / Journal of Non-Crystalline Solids 293±295 (2001) 279±282
Fig. 3. Two-dimensional continuous random network: a sketch of a three-fold co-ordinated elemental glass, where the continuous random network is formed by lines joining the centres of atoms. Fig. 1. Dependence of the packing density of a tapped granular material on the history of vibration intensities [1].
dom that are present. Papers are now appearing in the literature to develop such concepts [3±9] and we give a brief outline of one of them here. Although it is not necessary a `visual aid', it comes from the analysis of solvability of Newton's equations for static packings of hard rough grain analysis [10±12]. One ®nds (within certain provisos) that each grain has d 1 contacts, where d is the dimension of the system (see Fig. 2). The 2-D
picture then looks like a three-fold co-ordinated elemental glass (see Fig. 3). We now need a formulation for the glass.
2. Compactivity We argue that subject to N grains of the Chicago experiment ®lling a volume V, and when at rest (i.e., after tapping has ®nished) have contacts which ®x their positions in space, all con®gurations are equally likely [6]. This is just as energy possessing ergodic systems have a probability distribution P e
S=k
d E
H ;
1
W fRgHfRg;
2
grains have P e
Fig. 2. Two-dimensional continuous random network: a sketch of a packing of three-fold co-ordinated grains of irregular shape. Black circles denote the contact points and dashed lines, joining the centroids of contacts, form the continuous random network.
S=k
d V
where W fRg is the function of contact points fRg which leads to the volume V and HfRg means all the grains are in contact. Expressions for W are given in [13]. Such distributions are in overall agreement with the Chicago experiment, and with simulations by Kurchan and co-workers [8,9] which contain more detailed analysis than experiment can at present. Just as in thermodynamics
S.F. Edwards / Journal of Non-Crystalline Solids 293±295 (2001) 279±282
T
oE ; oS
3
in the granular system the key quantity is the compactivity X
oV : oS
4
A formal connection with thermal physics has 1 oP 5 X oT V T !0; which we use in Section 3.
1 X
oP oT
281
;
6
V
but in general there is a non-vanishing value 1 oP ZX : 7 oT V We can expect Z to look like (see Fig. 5) and a simple derivation leads to Tg T0
T0 TG ; _ 0 TG 1 ls T
8
where l is the characteristic of the material. In Fig. 6 it is the rate at which the cooling curve deviates from the vertical near T0 .
3. Cooling of a glass We aim to develop a theory shown in Fig. 4 where we plot the inverse viscosity as a function of _ the temperature in terms of the rate of cooling s, temperature T0 at which cooling starts and which is equal to the glass transition temperature if s_ ! 1, _ Tg , the glass temperature for a ®nite cooling rate s, and the glass temperature for s_ 0 TG . To derive such a diagram, we make what is probably far too crude an approximation but one that leads to quite reasonably looking curves. We argue that when the slow con®gurational and fast modes are in equilibrium
Fig. 5. The expected time dependence of the Z X _ given for decreasing values of the cooling rate s.
Fig. 4. The inverse viscosity g 1 as a function of temperature T given for dierent values of the cooling rate asymptotically approaching in®nity.
Fig. 6. Dependence of the function Z t on the ratio of temperature dierence and the cooling rate.
oP 1 oT V
282
S.F. Edwards / Journal of Non-Crystalline Solids 293±295 (2001) 279±282
4. Conclusion Crude as this formula (Eq. (8)) is, it does show some of the features of real glasses and suggests that it is worthwhile to produce a better developed theory based on the relaxation of con®gurational entropy as compared to a thermal system. Acknowledgements I wish to acknowledge the ®nancial support of Leverhulme Foundation and thank Dr D.V. Grinev for helpful discussions. References [1] E.R. Nowak, J.B. Knight, M.L. Povinelli, H.M. Jaeger, S.R. Nagel, Powder Technol. 94 (1997) 79.
[2] E.R. Nowak, J.B. Knight, E. Ben-Naim, H.M. Jaeger, S.R. Nagel, Phys. Rev. E 57 (1998) 1971. [3] S.F. Edwards, Ann. NY Acad. Sci. 371 (1981) 210. [4] S.F. Edwards, J. Phys.: Condens. Matter 2 (1990) SA63. [5] S.F. Edwards, in: Disorder in Condensed Matter Physics, Oxford, 1991. [6] S.F. Edwards, R.B.S. Oakeshott, Physica A 157 (1989) 1080. [7] A. Coniglio, M. Nicodemi, Physica A 296 (2001) 451. [8] A. Barrat, J. Kurchan, V. Loreto, M. Sellitto, Phys. Rev. Lett. 85 (2000) 5034. [9] A. Barrat, J. Kurchan, V. Loreto, M. Sellitto, Phys. Rev. E 63 (2001) 051301. [10] S.F. Edwards, D.V. Grinev, Phys. Rev. Lett. 82 (1999) 5397. [11] S.F. Edwards, D.V. Grinev, Physica A 294 (2001) 451. [12] R.C. Ball, R. Blumenfeld, ArXiv cond-matt/0008127. [13] C.C. Moun®eld, S.F. Edwards, Physica A 210 (1994) 279.