- Email: [email protected]
On Edwards’ theory of powders
Physica A 339 (2004) 1 – 6
www.elsevier.com/locate/physa
On Edwards’ theory of powders A. Coniglio∗ , A. de Candia, A. Fierro, M. Nicodemi, M. Pica Ciamarra, M. Tarzia Dipartimento di Fisica, Universita degli studi di Napoli “Federico II”, INFM, Unita di Napoli, Comoplesso Universitario Monte Sant’Angelo, Via Cinthia, I-80126 Napoli, Italy
Abstract We brie.y review a Statistical Mechanics approach for the description of granular materials following the original ideas developed by Edwards. We show that Edwards’ assumptions works very well for a class of lattice models and we discuss some recent analytic results obtained in such a framework on their phase diagram, jamming transition and mixing/segregation properties. c 2004 Published by Elsevier B.V. PACS: 64.70.Pf; 45.70−n; 75.10.Nr Keywords: Granular materials; Glasses; Mean :eld
The thermodynamics of macroscopic systems evolving at equilibrium is well described by Statistical Mechanics. But in nature there are many systems typically found in “frozen states”, which do not evolve at all. Granular materials [1] at rest are an important example of system frozen in mechanically stable microstates. Grains are “frozen” because, due to dissipation and their large masses [1], the thermal kinetic energy is negligible compared to the gravitational energy; thus the external bath temperature, Tbath , can be considered equal to zero. Following the original ideas by Edwards [2], it is indeed possible to develop a Statistical Mechanics description of granular materials. In such systems the dynamics, from one stable microstate to another, can be induced by sequences of “taps”, in which the energy is pumped into the system in pulses. Due to inelastic collisions the kinetic energy is totally dissipated after each tap, and the system is again frozen in one of its inherent states (see Fig. 1). Edwards proposed that, at stationarity, time averages ∗
Corresponding author. E-mail address: [email protected] (A. Coniglio).
c 2004 Published by Elsevier B.V. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.03.038
2
A. Coniglio et al. / Physica A 339 (2004) 1 – 6 qj
Phase Space
Shaking Amplitude
qi
τ0
time
Fig. 1. Time evolution in phase space of a system subjected to tap dynamics. At each tap energy is pumped into the system. Due to inelastic collisions the kinetic energy is totally dissipated after each tap, and the system is again frozen in a mechanically stable state (black-:lled circles). This let the system explore the space of inherent states.
of a system under tap dynamics coincide with suitable ensemble averages over its mechanically stable states. Under the assumption that these mechanically stable states have the same a priori probability to occur, the probability distribution, Pr , of :nding the system in the inherent state r of energy Er in the stationary state [4] can be found as the maximum of the entropy S = − r Pr ln Pr under the constraint that the system energy, E = r Er Pr , is given. This assumption leads to the Gibbs result Pr ˙ e− conf Er ;
(1)
where conf is a Lagrange multiplier characterizing the distribution, called inverse con:gurational temperature, enforcing the above constraint on the energy: conf =9Sconf =9E with Sconf = ln (E). Here (E) is the number of mechanically stable states with energy E ( conf can be related to the shaking amplitude T , for instance, from the equality between the time average of the energy, e(T K ), and the ensemble average, e(Tconf )). These assumptions settle a theoretical Statistical Mechanics framework to describe granular media. Typical values of the inverse eLective temperature conf for a real system are of order of 1=mgd, where m is the grain mass, g the gravitational acceleration and d the grain diameter. For example for a sand grain of mass m=10−6 Kg and diameter d = 10−3 m the inverse con:gurational temperature conf is of order of 108 J−1 . If we compare this value to the room temperature we obtain that room = conf ∼ 1012 . Edwards’ hypothesis has been tested on a schematic model [5]: a monodisperse system of hard spheres under gravity on a cubic lattice (see inset of Fig. 2). The Hamiltonian of the system is: H = Hhc ({ni }) + mg i ni zi , where zi is the height of the site i, g is the gravity acceleration, m is the grains mass, ni = 0; 1 is the usual occupancy variable (ni = 0 if site i is empty, 1 if it is :lled) and Hhc is a hard-core interaction term that prevents the overlapping of nearest-neighbor grains. The system undergoes a “tap dynamics”, i.e., a sequence of Monte Carlo taps: the system is kept in contact with a thermal bath at temperature T for a period of time of length 0 . After that the bath temperature is put equal to zero and grains can only move
A. Coniglio et al. / Physica A 339 (2004) 1 – 6
3
Fig. 2. Time averages of energy .uctuations plotted as a function of time averages of energy, obtained for diLerent tap dynamics. Empty circles refers to ensemble averages, obtained according to Eq. (1).
downwards. The properties of the system are measured when it reaches a blocked state (i.e., no grains can move downwards without violating the hard core repulsion). In Fig. 2 are plotted time averages of energy .uctuations PE 2 as a function of time average of energy EK for diLerent “tap dynamics” and ensemble averages obtained according to Eq. (1). The collapse of the data obtained with diLerent dynamics shows that the stationary states of the system are characterized by a single thermodynamic parameter. Finally, the agreement with the ensemble averages with distribution of Eq. (1) shows the success of Edwards’ approach to describe the system macroscopic properties. Thus, the above schematic model for granular media is well described by Edwards’ assumption we can treat it analytically. We have :rst investigated the phenomenon of the “jamming transition”. Experiments on granular media exhibit a strong form of “jamming” [3,6,7], i.e., an exceedingly slow dynamics, which shows deep connections to “freezing” phenomena observed in many thermal systems such as glass formers [8,9]. We have solved at a mean :eld level in the Bethe approximation [10–12] the partition function aQ la Edwards of such a model: Z= exp{−Hhc ({ni }) − conf mg ni zi }r ; (2) r∈{microstates}
i
where r is a projector on the space of the mechanically stable states (i.e., r = 1 if the state r is mechanically stable else r = 0). The mean :eld phase diagram of the system is shown in Fig. 3. At low Ns (grains surface density, Ns = NTOT =L2 ) or high Tconf a .uid-like phase is found, characterized by horizontal translational invariance. For a given Ns , by lowering Tconf a phase transition to a crystal phase is found at Tm . The .uid phase still exist below Tm as a metastable phase, corresponding to a supercooled .uid found when crystallization is avoided. This supercooled .uid has a thermodynamic-phase transition, at a point TK , to a Replica Symmetry Breaking “glassy” phase [11,12] with the same structure found in mean :eld theory of glass
4
A. Coniglio et al. / Physica A 339 (2004) 1 – 6
Fig. 3. The system mean :eld-phase diagram is plotted in the plane of its two control parameters (Tconf ; Ns ): Tconf is Edwards’ “con:gurational temperature” and Ns the average number of grains per unit surface in the box. At low Ns or high Tconf , the system is found in a .uid phase. The .uid forms a crystal below a melting transition line Tm (Ns ). When crystallization is avoided, the “supercooled” (i.e., metastable) .uid has a thermodynamic-phase transition, at a point TK (Ns ), to a Replica Symmetry Breaking “glassy”. In between Tm (Ns ) and TK (Ns ) a dynamical freezing point, TD (Ns ), is located.
formers. In between Tm and TK a dynamical freezing point, TD , is located, where the system characteristic time scales diverge. In the same framework we studied the phenomenon of segregation as well. Granular mixtures subjected to shaking can mix or, under diLerent conditions, surprisingly spontaneously segregate their components. The criteria underlying these behaviors, although of deep practical and conceptual relevance, are still largely unknown [1]. We have shown that Edwards’ approach is well grounded to describe a schematic lattice model of a binary hard spheres mixture too [13], with Hamiltonian H = Hhc + m1 gH1 + m2 gH2 , where H1 and H2 are the heights of the two species and Hhc is the hard core potential. In particular, we have shown [13] that the stationary states of such a system are characterized by two control parameter such as the gravitational energy of the :rst and of the second species. More speci:cally the weight of a given state r is: exp{−Hhc − 1 m1 gH1 − 2 m2 gH2 } · r , where 1 and 2 are two thermodynamic parameters (inverse con:gurational temperatures) conjugated to the gravitational energies of the two species. We have solved analytically at a mean :eld level the partition function aQ la Edwards of such a model and derived its phase diagram and mixing/segregation properties [14] (see, Fig. 4). For low values of N2s the system is found in a .uid-like phase. By increasing N2s the system has a discontinuous melting transition to a crystal phase. The presence of .uid–crystal-phase transitions in the system drives segregation as a form of phase separation. Within a given phase, gravity can also induce a kind of “vertical” segregation, not associated to phase transitions. The crossover points from BNE to RBNE within a given phase depend on the ratio m2 =m1 and also, as shown in Fig. 4, on the surface densities of the two species Ns1 and Ns2 . In particular, we :nd that BNE is favored for m2 . 2m1 and for N1s ¿ N2s , while RBNE is instead favored
A. Coniglio et al. / Physica A 339 (2004) 1 – 6
5
Fig. 4. Phase diagram, from Bethe approximation, of the hard sphere binary mixture model under gravity in the plane (N1s ; N2s ) for m1 1 = 0:8 and m2 2 = 1:25 and m1 1 = 1:25 and m2 2 = 0:8 (inset). In both cases a .uid and a crystal phase are found, divided by a discontinuous (but for N1s ∼ 0) melting transition. For the :rst case we have also indicated the regions where the system is found in a BNE (small particles on the top and big particles at the bottom), RBNE (big particles on the top and small ones at the bottom) or mixed state. The phase “C+F” in the phase diagram in the inset refers to a phase characterized by a bed of small .uid particles beneath a crystal formed by the big ones.
if m2 & 2m1 and/or N2s ¿ N1s . As shown in the inset of Fig. 4, for some values of the control parameters we also :nd a phase characterized by a .uid bed of small particle beneath a crystal formed by the big ones. In conclusion, even though the general validity of Edwards approach to granular systems has just begun to be assessed, it turns out that a :rst reference framework is emerging to understand the physics of such materials. In particular in this framework it is possible to extend Statistical Mechanics concepts to non thermal systems and to :nd analytical results and predictions that can be compared with experiments. For example the density pro:le, the compaction curve, the correlations functions can be computed by means of analytical tools as much as for any thermal system. Here, for instance, we have studied analytically the jamming transition and the mixing/segregation phenomena. Work supported by MURST-PRIN 2002, MUIR-FIRB 2002, CRdC-AMRA, INFM-PCI. References [1] H.M. Jaeger, S.R. Nagel, R.P. Behringer, Rev. Mod. Phys. 68 (1996) 1259; J.M. Ottino, D.V. Khakhar, Ann. Rev. Fluid Mech. 32 (2000) 55; A. Coniglio, A. Fierro, H.J. Herrmann, M. Nicodemi (Eds.), Unifying Concepts in Granular Media and Glasses, Elsevier, Amsterdam, in press. [2] S.F. Edwards, R.B.S. Oakeshott, Physica A 157 (1989) 1080. [3] J.B. Knight, et al., Phys. Rev. E 51 (1995) 3957. [4] A. Coniglio, M. Nicodemi, Physica A 296 (2001) 451. [5] A. Coniglio, A. Fierro, M. Nicodemi, Europhys. Lett. 59 (2002) 642. [6] G. D’Anna, G. Gremaud, Nature 413 (2001) 407.
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A. Coniglio et al. / Physica A 339 (2004) 1 – 6
[7] P. Philippe, D. Bideau, Europhys. Lett. 60 (2002) 677. [8] M. Nicodemi, A. Coniglio, H.J. Herrmann, Phys. Rev. E 55 (1997) 3962; M. Nicodemi, A. Coniglio, Phys. Rev. Lett. 82 (1999) 916. [9] A.J. Liu, S.R. Nagel, Nature 396 (1998) 21. [10] A. Coniglio, A. de Candia, A. Fierro, M. Nicodemi, M. Tarzia, Europhys. Lett. 66 (2004), in press. [11] G. Biroli, M. MQezard, Phys. Rev. Lett. 88 (2002) 025501. [12] M. MQezard, G. Parisi, Eur. Phys. J. B 20 (2001) 217. [13] A. Coniglio, A. Fierro, M. Nicodemi, Europhys. Lett. 60 (2002) 684. [14] A. Coniglio, A. Fierro, M. Nicodemi, M. Tarzia, to be published.
www.elsevier.com/locate/physa
On Edwards’ theory of powders A. Coniglio∗ , A. de Candia, A. Fierro, M. Nicodemi, M. Pica Ciamarra, M. Tarzia Dipartimento di Fisica, Universita degli studi di Napoli “Federico II”, INFM, Unita di Napoli, Comoplesso Universitario Monte Sant’Angelo, Via Cinthia, I-80126 Napoli, Italy
Abstract We brie.y review a Statistical Mechanics approach for the description of granular materials following the original ideas developed by Edwards. We show that Edwards’ assumptions works very well for a class of lattice models and we discuss some recent analytic results obtained in such a framework on their phase diagram, jamming transition and mixing/segregation properties. c 2004 Published by Elsevier B.V. PACS: 64.70.Pf; 45.70−n; 75.10.Nr Keywords: Granular materials; Glasses; Mean :eld
The thermodynamics of macroscopic systems evolving at equilibrium is well described by Statistical Mechanics. But in nature there are many systems typically found in “frozen states”, which do not evolve at all. Granular materials [1] at rest are an important example of system frozen in mechanically stable microstates. Grains are “frozen” because, due to dissipation and their large masses [1], the thermal kinetic energy is negligible compared to the gravitational energy; thus the external bath temperature, Tbath , can be considered equal to zero. Following the original ideas by Edwards [2], it is indeed possible to develop a Statistical Mechanics description of granular materials. In such systems the dynamics, from one stable microstate to another, can be induced by sequences of “taps”, in which the energy is pumped into the system in pulses. Due to inelastic collisions the kinetic energy is totally dissipated after each tap, and the system is again frozen in one of its inherent states (see Fig. 1). Edwards proposed that, at stationarity, time averages ∗
Corresponding author. E-mail address: [email protected] (A. Coniglio).
c 2004 Published by Elsevier B.V. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.03.038
2
A. Coniglio et al. / Physica A 339 (2004) 1 – 6 qj
Phase Space
Shaking Amplitude
qi
τ0
time
Fig. 1. Time evolution in phase space of a system subjected to tap dynamics. At each tap energy is pumped into the system. Due to inelastic collisions the kinetic energy is totally dissipated after each tap, and the system is again frozen in a mechanically stable state (black-:lled circles). This let the system explore the space of inherent states.
of a system under tap dynamics coincide with suitable ensemble averages over its mechanically stable states. Under the assumption that these mechanically stable states have the same a priori probability to occur, the probability distribution, Pr , of :nding the system in the inherent state r of energy Er in the stationary state [4] can be found as the maximum of the entropy S = − r Pr ln Pr under the constraint that the system energy, E = r Er Pr , is given. This assumption leads to the Gibbs result Pr ˙ e− conf Er ;
(1)
where conf is a Lagrange multiplier characterizing the distribution, called inverse con:gurational temperature, enforcing the above constraint on the energy: conf =9Sconf =9E with Sconf = ln (E). Here (E) is the number of mechanically stable states with energy E ( conf can be related to the shaking amplitude T , for instance, from the equality between the time average of the energy, e(T K ), and the ensemble average, e(Tconf )). These assumptions settle a theoretical Statistical Mechanics framework to describe granular media. Typical values of the inverse eLective temperature conf for a real system are of order of 1=mgd, where m is the grain mass, g the gravitational acceleration and d the grain diameter. For example for a sand grain of mass m=10−6 Kg and diameter d = 10−3 m the inverse con:gurational temperature conf is of order of 108 J−1 . If we compare this value to the room temperature we obtain that room = conf ∼ 1012 . Edwards’ hypothesis has been tested on a schematic model [5]: a monodisperse system of hard spheres under gravity on a cubic lattice (see inset of Fig. 2). The Hamiltonian of the system is: H = Hhc ({ni }) + mg i ni zi , where zi is the height of the site i, g is the gravity acceleration, m is the grains mass, ni = 0; 1 is the usual occupancy variable (ni = 0 if site i is empty, 1 if it is :lled) and Hhc is a hard-core interaction term that prevents the overlapping of nearest-neighbor grains. The system undergoes a “tap dynamics”, i.e., a sequence of Monte Carlo taps: the system is kept in contact with a thermal bath at temperature T for a period of time of length 0 . After that the bath temperature is put equal to zero and grains can only move
A. Coniglio et al. / Physica A 339 (2004) 1 – 6
3
Fig. 2. Time averages of energy .uctuations plotted as a function of time averages of energy, obtained for diLerent tap dynamics. Empty circles refers to ensemble averages, obtained according to Eq. (1).
downwards. The properties of the system are measured when it reaches a blocked state (i.e., no grains can move downwards without violating the hard core repulsion). In Fig. 2 are plotted time averages of energy .uctuations PE 2 as a function of time average of energy EK for diLerent “tap dynamics” and ensemble averages obtained according to Eq. (1). The collapse of the data obtained with diLerent dynamics shows that the stationary states of the system are characterized by a single thermodynamic parameter. Finally, the agreement with the ensemble averages with distribution of Eq. (1) shows the success of Edwards’ approach to describe the system macroscopic properties. Thus, the above schematic model for granular media is well described by Edwards’ assumption we can treat it analytically. We have :rst investigated the phenomenon of the “jamming transition”. Experiments on granular media exhibit a strong form of “jamming” [3,6,7], i.e., an exceedingly slow dynamics, which shows deep connections to “freezing” phenomena observed in many thermal systems such as glass formers [8,9]. We have solved at a mean :eld level in the Bethe approximation [10–12] the partition function aQ la Edwards of such a model: Z= exp{−Hhc ({ni }) − conf mg ni zi }r ; (2) r∈{microstates}
i
where r is a projector on the space of the mechanically stable states (i.e., r = 1 if the state r is mechanically stable else r = 0). The mean :eld phase diagram of the system is shown in Fig. 3. At low Ns (grains surface density, Ns = NTOT =L2 ) or high Tconf a .uid-like phase is found, characterized by horizontal translational invariance. For a given Ns , by lowering Tconf a phase transition to a crystal phase is found at Tm . The .uid phase still exist below Tm as a metastable phase, corresponding to a supercooled .uid found when crystallization is avoided. This supercooled .uid has a thermodynamic-phase transition, at a point TK , to a Replica Symmetry Breaking “glassy” phase [11,12] with the same structure found in mean :eld theory of glass
4
A. Coniglio et al. / Physica A 339 (2004) 1 – 6
Fig. 3. The system mean :eld-phase diagram is plotted in the plane of its two control parameters (Tconf ; Ns ): Tconf is Edwards’ “con:gurational temperature” and Ns the average number of grains per unit surface in the box. At low Ns or high Tconf , the system is found in a .uid phase. The .uid forms a crystal below a melting transition line Tm (Ns ). When crystallization is avoided, the “supercooled” (i.e., metastable) .uid has a thermodynamic-phase transition, at a point TK (Ns ), to a Replica Symmetry Breaking “glassy”. In between Tm (Ns ) and TK (Ns ) a dynamical freezing point, TD (Ns ), is located.
formers. In between Tm and TK a dynamical freezing point, TD , is located, where the system characteristic time scales diverge. In the same framework we studied the phenomenon of segregation as well. Granular mixtures subjected to shaking can mix or, under diLerent conditions, surprisingly spontaneously segregate their components. The criteria underlying these behaviors, although of deep practical and conceptual relevance, are still largely unknown [1]. We have shown that Edwards’ approach is well grounded to describe a schematic lattice model of a binary hard spheres mixture too [13], with Hamiltonian H = Hhc + m1 gH1 + m2 gH2 , where H1 and H2 are the heights of the two species and Hhc is the hard core potential. In particular, we have shown [13] that the stationary states of such a system are characterized by two control parameter such as the gravitational energy of the :rst and of the second species. More speci:cally the weight of a given state r is: exp{−Hhc − 1 m1 gH1 − 2 m2 gH2 } · r , where 1 and 2 are two thermodynamic parameters (inverse con:gurational temperatures) conjugated to the gravitational energies of the two species. We have solved analytically at a mean :eld level the partition function aQ la Edwards of such a model and derived its phase diagram and mixing/segregation properties [14] (see, Fig. 4). For low values of N2s the system is found in a .uid-like phase. By increasing N2s the system has a discontinuous melting transition to a crystal phase. The presence of .uid–crystal-phase transitions in the system drives segregation as a form of phase separation. Within a given phase, gravity can also induce a kind of “vertical” segregation, not associated to phase transitions. The crossover points from BNE to RBNE within a given phase depend on the ratio m2 =m1 and also, as shown in Fig. 4, on the surface densities of the two species Ns1 and Ns2 . In particular, we :nd that BNE is favored for m2 . 2m1 and for N1s ¿ N2s , while RBNE is instead favored
A. Coniglio et al. / Physica A 339 (2004) 1 – 6
5
Fig. 4. Phase diagram, from Bethe approximation, of the hard sphere binary mixture model under gravity in the plane (N1s ; N2s ) for m1 1 = 0:8 and m2 2 = 1:25 and m1 1 = 1:25 and m2 2 = 0:8 (inset). In both cases a .uid and a crystal phase are found, divided by a discontinuous (but for N1s ∼ 0) melting transition. For the :rst case we have also indicated the regions where the system is found in a BNE (small particles on the top and big particles at the bottom), RBNE (big particles on the top and small ones at the bottom) or mixed state. The phase “C+F” in the phase diagram in the inset refers to a phase characterized by a bed of small .uid particles beneath a crystal formed by the big ones.
if m2 & 2m1 and/or N2s ¿ N1s . As shown in the inset of Fig. 4, for some values of the control parameters we also :nd a phase characterized by a .uid bed of small particle beneath a crystal formed by the big ones. In conclusion, even though the general validity of Edwards approach to granular systems has just begun to be assessed, it turns out that a :rst reference framework is emerging to understand the physics of such materials. In particular in this framework it is possible to extend Statistical Mechanics concepts to non thermal systems and to :nd analytical results and predictions that can be compared with experiments. For example the density pro:le, the compaction curve, the correlations functions can be computed by means of analytical tools as much as for any thermal system. Here, for instance, we have studied analytically the jamming transition and the mixing/segregation phenomena. Work supported by MURST-PRIN 2002, MUIR-FIRB 2002, CRdC-AMRA, INFM-PCI. References [1] H.M. Jaeger, S.R. Nagel, R.P. Behringer, Rev. Mod. Phys. 68 (1996) 1259; J.M. Ottino, D.V. Khakhar, Ann. Rev. Fluid Mech. 32 (2000) 55; A. Coniglio, A. Fierro, H.J. Herrmann, M. Nicodemi (Eds.), Unifying Concepts in Granular Media and Glasses, Elsevier, Amsterdam, in press. [2] S.F. Edwards, R.B.S. Oakeshott, Physica A 157 (1989) 1080. [3] J.B. Knight, et al., Phys. Rev. E 51 (1995) 3957. [4] A. Coniglio, M. Nicodemi, Physica A 296 (2001) 451. [5] A. Coniglio, A. Fierro, M. Nicodemi, Europhys. Lett. 59 (2002) 642. [6] G. D’Anna, G. Gremaud, Nature 413 (2001) 407.
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A. Coniglio et al. / Physica A 339 (2004) 1 – 6
[7] P. Philippe, D. Bideau, Europhys. Lett. 60 (2002) 677. [8] M. Nicodemi, A. Coniglio, H.J. Herrmann, Phys. Rev. E 55 (1997) 3962; M. Nicodemi, A. Coniglio, Phys. Rev. Lett. 82 (1999) 916. [9] A.J. Liu, S.R. Nagel, Nature 396 (1998) 21. [10] A. Coniglio, A. de Candia, A. Fierro, M. Nicodemi, M. Tarzia, Europhys. Lett. 66 (2004), in press. [11] G. Biroli, M. MQezard, Phys. Rev. Lett. 88 (2002) 025501. [12] M. MQezard, G. Parisi, Eur. Phys. J. B 20 (2001) 217. [13] A. Coniglio, A. Fierro, M. Nicodemi, Europhys. Lett. 60 (2002) 684. [14] A. Coniglio, A. Fierro, M. Nicodemi, M. Tarzia, to be published.