- Email: [email protected]
Dynamically arrested states of matter
Advances in Colloid and Interface Science 122 (2006) 35 – 38 www.elsevier.com/locate/cis
Dynamically arrested states of matter Kenneth A. Dawson ⁎, Paolo De Gregorio, Aonghus Lawlor Irish Centre for Colloid Science and Biomaterials, Department of Chemistry, University College Dublin, Belfield, Dublin 4, Ireland Available online 8 August 2006
Abstract We outline current developments in our understanding of dynamical arrest, that phenomenon in which many particles stop moving in a collective manner. However in addition to the question of true dynamical arrest itself, we emphasize the development of new tools that can describe relatively sharp changes in the way that ergodic systems may be explored. We discuss the concept of new order parameters (dynamically available volume), and indicate how they may be applied to understand dramatic slowing phenomena present in particle systems, and other arenas of soft and complex matter. © 2006 Published by Elsevier B.V.
In the last decades, condensed matter science has advanced in many directions, both conceptually and practically, and is arguably one of the most fertile and vibrant arenas of modern scientific research. Many arenas involving new condensed state phenomena have been, or are being, successfully developed, sufficiently so that they have a major impact our every day lives. Yet there remain certain heritage problems of the condensed state of matter where efforts over many decades have not yet lead to a clear understanding of the phenomena, in some cases perhaps not even a clear appreciation of the issues involved. In some of these arenas scientific interest has moved on, perhaps in the belief that no very simple paradigms will emerge. However in others, amongst them the arena of dynamical arrest and glass-formation, attempts to master the phenomena persist. At the simplest level, the observations are well known from everyday life, as well as the laboratory. Many systems, from the familiar inorganic “window glass” [1,2] to soft- [3] and biomatter [4,5], upon changing some experimental conditions, exhibit a sharp, and entirely reproducible change of state at which, for many practical purposes, the fluid appears to “solidify”. The modern view tends to see this as a highly collective “transition” at which the system, without minimizing the free energy, falls out of equilibrium, particles stop moving, and the system remains trapped for large periods of time in limited regions of phase space. There are some indications that ergodicity might be fully broken under more extreme conditions, a phase transition then accompanying final arrest [6,7], but this is ⁎ Corresponding author. E-mail address: [email protected] (K.A. Dawson). 0001-8686/$ - see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.cis.2006.06.007
not the situation usually of importance or interest in this field. Rather, we seek to understand the nature of this collective slowing phenomenon. The central problem, of course, is that the early thinkers, such as Boltzmann and Gibbs, who reduced Newtonian mechanics to the much simpler statistical treatment (soon to be known as the statistical mechanics) found it possible to deal with simple distributions of phase space in which the system is either uniformly distributed in energy (the microcanonical ensemble) and related probabilities such as the canonical ensemble where the temperature is held fixed. Further development allowed treatment of broken ergodicity systems (phase-transitions) that involve a (statistically defined) complete confinement of the system to a given part of phase space. Thus, the infinite-time distributions in phase-space change sharply, as expressed by discontinuities in the first and second derivatives of quantities (such as the partition function and free-energy) related to the distribution. Simple and elegant models [8,9] (and infinite-time order-parameter formulations of them [10]) have evolved, leading to considerable understanding of phase transitions. The result has been some of the outstanding achievements of condensed matter science [8,11–14]. However, for many systems, this confinement phenomenon may reduce but not eliminate access to extensive parts of phasespace, again in a statistical sense. Then if we wait long enough, ergodicity will be restored, though the time required may be so long as to be experimentally unrealizable (window glass is a good example). In such cases (we refer to these as “dynamical arrest”) experiments involving averages carried out at usual timescales
36
K.A. Dawson et al. / Advances in Colloid and Interface Science 122 (2006) 35–38
(say in everyday life) will exhibit just as remarkable “transitions” in their behavior as those possessing discontinuous behavior in the free-energy. Also, for these cases infinite-time averages such as density, energy, and other common order parameters are expected to show no true discontinuous behavior, and it is unlikely that any infinite-time “equilibrium” order parameter will characterize the system in the manner so familiar to equilibrium statistical mechanics. Still, the phenomena are just as real. The sense that many have of such arrest phenomena having another (less clear) status in the repertoire of condensed matter science is more a reflection on the power of our tools and language, than on the importance that Nature confers to them. In this discussion we explore whether it is possible to formulate a new statistical treatment of phase space, analogous to that of Boltzmann and Gibbs, that might describe dynamical arrest phenomena, thereby framing them into as coherent and organized body of knowledge as the phase-transitions. We emphasize our desire to avoid a mere reformulation of Newton's equations (in which time would be retained as a central quantity), wishing instead to preserve the ease of infinite-time order parameters, familiar from equilibrium statistical mechanics. To do so it is clear that we must develop some dynamically defined criterion to decide which parts of each configuration belong to the “order” being captured in the order parameter. Otherwise, as we have noted above, there can be no real distinction in the order parameter prior to and after the “transition” we wish to study. Ideally we would like to have a description close to that familiar from order parameter Landau–Ginzburg–Wilson (LGW) Hamiltonian. Of course, the practicability of such a formulation would have to be established afterwards as we must avoid replacing one hard problem with a new equally hard one. Again the analogy is to the Landau Ginzburg Wilson approach in which the dependence on the order parameter is determined on very general grounds using arguments based on symmetry, dimension and other general aspects of the order. Perhaps at this point, to clarify the intention of our program we should give a simple example. Consider the Kob–Anderson type models [15] in which dynamical arrest is expressed as a consequence of local constraints on the dynamics. Thus, to each site of a regular lattice in D spatial dimensions we define a lattice-gas type variable ρ = 0, 1 indicating (respectively) absence (or presence) of a particle. Then, any particle adjacent to an empty site that is surrounded by more than a fixed number of particles (c) (or if, upon moving to that vacancy, it there has more than c neighbors) is forbidden to move. These constraints may be considered as a simple description of the local effects of caging, but for our present purpose the models are considered illustrative of the ideas. Empty sites that are blocked by constraints are named vacancies, whereas those into which one can move at least one neighboring particle are named holes (density νt) [16,17]. Such models typically have “cages” being (perhaps extended) perimeters filled with particles that mutually block movement of each other. Holes inside such cages (density νd) are disconnected from rest of the configuration unless the cage is broken by particle movement at the exterior of the cage. Holes not being confined by such cages may, with some sequence of movements, eventually be used to mobilize every particle in the system. They are named
connected holes (density νc = νt − νd) because they are the agencies by which many configurations may be connected to each other using only legitimate movements of the particles. Note that the density of disconnected and connected holes are well-defined infinite-time averages (though they have small fluctuations from one configuration to another). That is, while the particle (ρ) and vacancy (v = 1 − ρ) densities are held fixed, holes constitute a type of Grand Ensemble with a well-defined infinite time average. We wish to highlight once more that, though the hole densities (connected and disconnected) possess many qualities of equilibrium order parameters, their identity is a consequence of their dynamical role. It is because they contain this information about dynamics that we consider the connected hole density to be an example of a static “order parameter” that would readily recognize detailed changes in the systems exploration of phase, and thereby represent near-arrest phenomena. In fact, as particle density increases, and matter becomes more highly coupled, hole density vanishes (for it is the absence of connected holes that leads to the loss of sustained motion) and thereby arrest occurs. It is of interest to note that in the near-arrest limit, connected holes are widely separated, and systems can relax only via motions that commence from a connected hole. Thus each connected hole has a region of influence of typical size ξd within which only highly co-operative and constrained motions can occur. Systems of size less than this are not expected to have a connected hole, and are therefore dynamically arrested. This length is also termed the dynamical correlation length, and is related to the connected hole density via νd = ξd−D where D is the dimensionality of space. We also note that it would be natural to represent transport coefficients as a function of connected hole density [16,17], or equivalently in terms of the dynamical length [18–20] rather than in the conventional particle density or temperature representations. There are therefore two steps to be accomplished in our program. The first is to relate transport coefficients to connected hole density (or dynamical length), the second is to relate that connected hole density to the particle density (or more generally temperature, pressure and other experimentally determined parameters). The connection between dynamical length and transport coefficients is at present only argued phenomenologically, and we will not discuss it here. In the latter step, connection between the dynamical length or connected hole density, progress is beginning to occur rapidly, including the development of the first exact solutions [17,21]. Here we emphasize the physical picture derived from such models, instead of the detail, which may be found in the original literature [22]. The purpose of this paper is to point to a broader range of understandings from more varied systems that may arise from these ideas. We have outlined how the connected hole density is expected to be the order parameter of extended motion in near-arrested systems. This being so, it is interesting that even the simplest models (such as those of KA discussed above) already exhibit striking changes in νc well before final arrest. For example, it has been shown [22] for a number of different models in different dimensions there is quite a sharp change in the derivative of the connected hole density with respect to total holes (equation here). Note that one may view this derivative as a type of “chemical
K.A. Dawson et al. / Advances in Colloid and Interface Science 122 (2006) 35–38
potential” relation that tells us how easy it is to insert a new piece of usable empty space that the particles can use to move on long length scales. This is essentially a generalization to dynamical phenomena of the well known insertion formula of equilibrium statistical mechanics [11]. It has been shown [22] (see also Fig. 1) that, at lower particle density (higher connected hole density) the derivative is almost unity, while beyond some threshold particle density this derivative becomes close to, but not yet precisely, zero. We have identified this as a geometrical “transition” such that, at lower particle density, the connected holes associate with each other to form a network, that de-percolates at higher particle density. Thus, beyond this threshold the easily available space is no longer extended, becoming “droplet-like” regions in the vicinity of each connected hole. Adding a new piece of empty space in the percolating regime leads it naturally to be connected, whereas in the de-percolated regime, the empty space is rarely usable to generate new dynamical pathways. We believe that this reorganization transition in which the order parameter (connected hole density) depercolates is quite a general phenomenon, and argue that in the past it may well have been identified as true dynamical arrest (sometimes itself loosely called the “glass transition”). For example at packing fractions of 0.58 in hard sphere dispersions there is often considered to be a dynamical arrest transition, a contention not shared by all [23–25]. If our interpretation is correct, this is merely a de-percolation transition of the useful empty space, and true arrest would not occur in these systems until much higher density, close to what is loosely described as “close packing” fraction. This interpretation has many consequences. For example, motion may be so dramatically reduced at such thresholds as to be often mis-identified as the “glass transition” itself, whereas it is only a geometrical change, rather than vanishing of the order parameter. This in itself would render the whole conceptual structure of these order parameter ideas of importance. We may pursue this line a little further, for particles with shortranged attractions [3,26] where particle (physical) gellation has begun to be interpreted as a type of “glass transition” or dynamical arrest. In Fig. 2 we see the typical phase diagram exhibited by particles with a repulsive core, and short ranged attraction. The specific example discussed here [27] is hard core repulsion, and a
Fig. 1. ∂νc / ∂νt for 2D Kob–Andersen model c = 2. The derivatives with respect to density ρ are also shown.
37
Fig. 2. Phase diagram for the Yukawa fluid with screening parameter b = 30 [27]. The crosses (×) represent the fluid–solid phase transition, the open circles (ο) represent the glass transition line as evaluated by Mode Coupling Theory, the continuous line is the binodal and the dashed one is the spinodal. The filled circle is the critical point. The labels I, II and III are chosen after Muschol and Rosenberger [28] and refer to regimes suitable for crystallisation (I), two-liquid coexistence (II) and the “arrested” region (III), as described in the text.
Yukawa attractions, but the details are not important to our current purpose. The point is that, for sufficiently short ranged attraction (in this example b = 30), the traditional methods (mode coupling theory) that describe dynamical arrest predicts that the phenomenon that appears at 0.58 volume fraction for hard spheres moves to much lower volume fractions, and is there identified as “gellation” of the particles. Now, with the more simple lattice models where we can look in more detail at the dynamical processes, and are not restricted to the mode coupling limit, we can see that particle gellation not a true arrest, or glass transition, but a dramatic restructuring of the dynamically available volume. We note that dynamically available volume need does not require infinite interactions or barriers to be defined, and it is possible to apply much the same ideas with finite energies and barriers to movement. In future it will become increasingly possible to describe many of these near-arrest phenomena in these more subtle terms, not as a true loss of ergodicity, but as precipitous changes in the type of ergodic motion of the system. At present the signs are that many phenomena of slowing, near-solidification, or arrest that occur in particulate or soft matter systems will be describable in terms of relatively few classes, each of which is governed by quite general and simple rules. Underlying this is a simple LGW-type description of the systems in which this dynamically available volume plays the central role. Finally, we note that as some confidence in these simple descriptions begin to grow for more complicated systems, we are increasingly becoming more confident that we will be able to treat these order parameter ideas in a much deeper, in some cases exact manner. Thus, recently, it has been possible to develop an approach that allows us to systematically add up the consequences of dynamical blockages or traps met by particles in nearly arrested systems in an formally exact manner. We have shown that for one example in two spatial dimensions it is even possible to calculate the dynamical correlation length exactly [17,21]. This approach is not unlike renormalization group methods for the critical point, a
38
K.A. Dawson et al. / Advances in Colloid and Interface Science 122 (2006) 35–38
method that adds up the effects of (static) correlations on all length scale. The indications are that we are beginning to see a new conceptual framework within which it will be possible to describe many natural phenomena that may reasonably be named a “change of state”, but that do not involve any singular aspect to the derivatives of the free energy. There is a great deal to be done yet, particularly in building a microscopic continuum approach in the arena. However, based on the simple models, our feeling is that the outcome will lead us increasingly to see phase-transitions as only one extreme of a range of such phenomena. Then the whole machinery of equilibrium statistical mechanics will also be seen as a particular case of a more general picture in which many of the same elegant classifications of “states” and “transitions” obtain. For those that have grown from the great schools of equilibrium statistical mechanics, with the tastes and sensibilities that accompany this background, such an outcome would be particularly satisfying. We acknowledge interactions at various stages with our colleagues in the an EU funded research program (see www. arrestedmatter.net), contract number MRTN-CT- 2003-504712. References [1] Angell CA, Ngai KL, McKenna GB, McMillan PF, Martin W. J Appl Phys 2000;88:3113. [2] Mézard M, Parisi G. J Phys Condens Matter 2000;12:6655. [3] Dawson K. Curr Opin Coll Int Sci 2002;7:218. [4] McManus JJ, Rdler JO, Dawson KA. J Am Chem Soc 2004;126:15966. [5] Stradner A, Sedgwick H, Cardinaux F, Poon WCK, Egelhaaf SU, Schurtenberger P. Nature 2004;432:492. [6] Kauzmann W. Chem Rev 1948;9:219.
[7] Huang D, Simon SL, McKenna GB. J Chem Phys 2003;119:3590. [8] Domb C, Green MS, editors. Phase Transitions and Critical Phenomena, vol. 5a. New York: Academic; 1978. [9] Dawson KA, Lipkin MD, Widom B. J Chem Phys 1988;88:5149. [10] Lifshitz, EM and Landau, LD. Statistical Physics: Volume 5 (Course of Theoretical Physics. Volume 5) (Butterworth-Heinemann, 1980). [11] Widom B. J Chem Phys 1965;43:3892. [12] Kadanoff L. Physics 1966;2:263. [13] Wilson KG, Fisher ME. Phys Rev Lett 1972;28:240. [14] Wilson K. Rev Mod Phys 1983;55:583. [15] Kob W, Andersen HC. Phys Rev E 1993;48:4364. [16] Lawlor A, Reagan D, McCullagh GD, De Gregorio P, Tartaglia P, Dawson KA. Phys Rev Lett 2002;89:245503, URL http://link.aps.org/abstract/ PRL/v89/e245503. [17] De Gregorio P, Lawlor A, Bradley P, Dawson KA. Phys Rev Lett 2004;93:025501, URL http://link. aps.org/abstract/PRL/v93/e025501. [18] Ertel W, Frobröse K, Jäckle J. J Chem Phys 1988;88:5027. [19] Jäckle J, Krönig A. J Phys Condens Matter 1994;6:7633. [20] Toninelli C, Biroli G, Fisher D. Phys Rev Lett 2004;92:185504. [21] Gregorio PD, Lawlor A, Bradley P, Dawson KA. Proc Natl Acad Sci 2005;102:5669, URL http://www.pnas.org/cgi/content/abstract/102/16/ 5669. [22] Lawlor A, De Gregorio P, Bradley P, Sellitto M, Dawson KA. cond-mat/ 0503089 (2005), URL http://arxiv.org/abs/cond-mat/0503089. [23] Pusey PN, van Megen W. Phys Rev Lett 1987;59:2083. [24] Zhu J, Li M, Rogers R, Meyer W, Ottewill RH. STS-73 Space Shuttle Crew. Nature 1997;387:883. [25] Simeonova NB, Kegel WK. Phys Rev Lett 2004;93:035701. [26] Dawson K, Foffi G, Fuchs M, Götze W, Sciortino F, Sperl M, et al. Phys Rev E 2001;63:011401. [27] Foffi G, McCullagh G, Lawlor A, Zaccarelli E, Dawson K, Sciortino F, et al. Phys Rev E 2001;65:031407, URL http://link.aps.org/abstract/PRE/v65/ e031407. [28] Muschol M, Rosenberger F. J Chem Phys 1995;103:10424.
Dynamically arrested states of matter Kenneth A. Dawson ⁎, Paolo De Gregorio, Aonghus Lawlor Irish Centre for Colloid Science and Biomaterials, Department of Chemistry, University College Dublin, Belfield, Dublin 4, Ireland Available online 8 August 2006
Abstract We outline current developments in our understanding of dynamical arrest, that phenomenon in which many particles stop moving in a collective manner. However in addition to the question of true dynamical arrest itself, we emphasize the development of new tools that can describe relatively sharp changes in the way that ergodic systems may be explored. We discuss the concept of new order parameters (dynamically available volume), and indicate how they may be applied to understand dramatic slowing phenomena present in particle systems, and other arenas of soft and complex matter. © 2006 Published by Elsevier B.V.
In the last decades, condensed matter science has advanced in many directions, both conceptually and practically, and is arguably one of the most fertile and vibrant arenas of modern scientific research. Many arenas involving new condensed state phenomena have been, or are being, successfully developed, sufficiently so that they have a major impact our every day lives. Yet there remain certain heritage problems of the condensed state of matter where efforts over many decades have not yet lead to a clear understanding of the phenomena, in some cases perhaps not even a clear appreciation of the issues involved. In some of these arenas scientific interest has moved on, perhaps in the belief that no very simple paradigms will emerge. However in others, amongst them the arena of dynamical arrest and glass-formation, attempts to master the phenomena persist. At the simplest level, the observations are well known from everyday life, as well as the laboratory. Many systems, from the familiar inorganic “window glass” [1,2] to soft- [3] and biomatter [4,5], upon changing some experimental conditions, exhibit a sharp, and entirely reproducible change of state at which, for many practical purposes, the fluid appears to “solidify”. The modern view tends to see this as a highly collective “transition” at which the system, without minimizing the free energy, falls out of equilibrium, particles stop moving, and the system remains trapped for large periods of time in limited regions of phase space. There are some indications that ergodicity might be fully broken under more extreme conditions, a phase transition then accompanying final arrest [6,7], but this is ⁎ Corresponding author. E-mail address: [email protected] (K.A. Dawson). 0001-8686/$ - see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.cis.2006.06.007
not the situation usually of importance or interest in this field. Rather, we seek to understand the nature of this collective slowing phenomenon. The central problem, of course, is that the early thinkers, such as Boltzmann and Gibbs, who reduced Newtonian mechanics to the much simpler statistical treatment (soon to be known as the statistical mechanics) found it possible to deal with simple distributions of phase space in which the system is either uniformly distributed in energy (the microcanonical ensemble) and related probabilities such as the canonical ensemble where the temperature is held fixed. Further development allowed treatment of broken ergodicity systems (phase-transitions) that involve a (statistically defined) complete confinement of the system to a given part of phase space. Thus, the infinite-time distributions in phase-space change sharply, as expressed by discontinuities in the first and second derivatives of quantities (such as the partition function and free-energy) related to the distribution. Simple and elegant models [8,9] (and infinite-time order-parameter formulations of them [10]) have evolved, leading to considerable understanding of phase transitions. The result has been some of the outstanding achievements of condensed matter science [8,11–14]. However, for many systems, this confinement phenomenon may reduce but not eliminate access to extensive parts of phasespace, again in a statistical sense. Then if we wait long enough, ergodicity will be restored, though the time required may be so long as to be experimentally unrealizable (window glass is a good example). In such cases (we refer to these as “dynamical arrest”) experiments involving averages carried out at usual timescales
36
K.A. Dawson et al. / Advances in Colloid and Interface Science 122 (2006) 35–38
(say in everyday life) will exhibit just as remarkable “transitions” in their behavior as those possessing discontinuous behavior in the free-energy. Also, for these cases infinite-time averages such as density, energy, and other common order parameters are expected to show no true discontinuous behavior, and it is unlikely that any infinite-time “equilibrium” order parameter will characterize the system in the manner so familiar to equilibrium statistical mechanics. Still, the phenomena are just as real. The sense that many have of such arrest phenomena having another (less clear) status in the repertoire of condensed matter science is more a reflection on the power of our tools and language, than on the importance that Nature confers to them. In this discussion we explore whether it is possible to formulate a new statistical treatment of phase space, analogous to that of Boltzmann and Gibbs, that might describe dynamical arrest phenomena, thereby framing them into as coherent and organized body of knowledge as the phase-transitions. We emphasize our desire to avoid a mere reformulation of Newton's equations (in which time would be retained as a central quantity), wishing instead to preserve the ease of infinite-time order parameters, familiar from equilibrium statistical mechanics. To do so it is clear that we must develop some dynamically defined criterion to decide which parts of each configuration belong to the “order” being captured in the order parameter. Otherwise, as we have noted above, there can be no real distinction in the order parameter prior to and after the “transition” we wish to study. Ideally we would like to have a description close to that familiar from order parameter Landau–Ginzburg–Wilson (LGW) Hamiltonian. Of course, the practicability of such a formulation would have to be established afterwards as we must avoid replacing one hard problem with a new equally hard one. Again the analogy is to the Landau Ginzburg Wilson approach in which the dependence on the order parameter is determined on very general grounds using arguments based on symmetry, dimension and other general aspects of the order. Perhaps at this point, to clarify the intention of our program we should give a simple example. Consider the Kob–Anderson type models [15] in which dynamical arrest is expressed as a consequence of local constraints on the dynamics. Thus, to each site of a regular lattice in D spatial dimensions we define a lattice-gas type variable ρ = 0, 1 indicating (respectively) absence (or presence) of a particle. Then, any particle adjacent to an empty site that is surrounded by more than a fixed number of particles (c) (or if, upon moving to that vacancy, it there has more than c neighbors) is forbidden to move. These constraints may be considered as a simple description of the local effects of caging, but for our present purpose the models are considered illustrative of the ideas. Empty sites that are blocked by constraints are named vacancies, whereas those into which one can move at least one neighboring particle are named holes (density νt) [16,17]. Such models typically have “cages” being (perhaps extended) perimeters filled with particles that mutually block movement of each other. Holes inside such cages (density νd) are disconnected from rest of the configuration unless the cage is broken by particle movement at the exterior of the cage. Holes not being confined by such cages may, with some sequence of movements, eventually be used to mobilize every particle in the system. They are named
connected holes (density νc = νt − νd) because they are the agencies by which many configurations may be connected to each other using only legitimate movements of the particles. Note that the density of disconnected and connected holes are well-defined infinite-time averages (though they have small fluctuations from one configuration to another). That is, while the particle (ρ) and vacancy (v = 1 − ρ) densities are held fixed, holes constitute a type of Grand Ensemble with a well-defined infinite time average. We wish to highlight once more that, though the hole densities (connected and disconnected) possess many qualities of equilibrium order parameters, their identity is a consequence of their dynamical role. It is because they contain this information about dynamics that we consider the connected hole density to be an example of a static “order parameter” that would readily recognize detailed changes in the systems exploration of phase, and thereby represent near-arrest phenomena. In fact, as particle density increases, and matter becomes more highly coupled, hole density vanishes (for it is the absence of connected holes that leads to the loss of sustained motion) and thereby arrest occurs. It is of interest to note that in the near-arrest limit, connected holes are widely separated, and systems can relax only via motions that commence from a connected hole. Thus each connected hole has a region of influence of typical size ξd within which only highly co-operative and constrained motions can occur. Systems of size less than this are not expected to have a connected hole, and are therefore dynamically arrested. This length is also termed the dynamical correlation length, and is related to the connected hole density via νd = ξd−D where D is the dimensionality of space. We also note that it would be natural to represent transport coefficients as a function of connected hole density [16,17], or equivalently in terms of the dynamical length [18–20] rather than in the conventional particle density or temperature representations. There are therefore two steps to be accomplished in our program. The first is to relate transport coefficients to connected hole density (or dynamical length), the second is to relate that connected hole density to the particle density (or more generally temperature, pressure and other experimentally determined parameters). The connection between dynamical length and transport coefficients is at present only argued phenomenologically, and we will not discuss it here. In the latter step, connection between the dynamical length or connected hole density, progress is beginning to occur rapidly, including the development of the first exact solutions [17,21]. Here we emphasize the physical picture derived from such models, instead of the detail, which may be found in the original literature [22]. The purpose of this paper is to point to a broader range of understandings from more varied systems that may arise from these ideas. We have outlined how the connected hole density is expected to be the order parameter of extended motion in near-arrested systems. This being so, it is interesting that even the simplest models (such as those of KA discussed above) already exhibit striking changes in νc well before final arrest. For example, it has been shown [22] for a number of different models in different dimensions there is quite a sharp change in the derivative of the connected hole density with respect to total holes (equation here). Note that one may view this derivative as a type of “chemical
K.A. Dawson et al. / Advances in Colloid and Interface Science 122 (2006) 35–38
potential” relation that tells us how easy it is to insert a new piece of usable empty space that the particles can use to move on long length scales. This is essentially a generalization to dynamical phenomena of the well known insertion formula of equilibrium statistical mechanics [11]. It has been shown [22] (see also Fig. 1) that, at lower particle density (higher connected hole density) the derivative is almost unity, while beyond some threshold particle density this derivative becomes close to, but not yet precisely, zero. We have identified this as a geometrical “transition” such that, at lower particle density, the connected holes associate with each other to form a network, that de-percolates at higher particle density. Thus, beyond this threshold the easily available space is no longer extended, becoming “droplet-like” regions in the vicinity of each connected hole. Adding a new piece of empty space in the percolating regime leads it naturally to be connected, whereas in the de-percolated regime, the empty space is rarely usable to generate new dynamical pathways. We believe that this reorganization transition in which the order parameter (connected hole density) depercolates is quite a general phenomenon, and argue that in the past it may well have been identified as true dynamical arrest (sometimes itself loosely called the “glass transition”). For example at packing fractions of 0.58 in hard sphere dispersions there is often considered to be a dynamical arrest transition, a contention not shared by all [23–25]. If our interpretation is correct, this is merely a de-percolation transition of the useful empty space, and true arrest would not occur in these systems until much higher density, close to what is loosely described as “close packing” fraction. This interpretation has many consequences. For example, motion may be so dramatically reduced at such thresholds as to be often mis-identified as the “glass transition” itself, whereas it is only a geometrical change, rather than vanishing of the order parameter. This in itself would render the whole conceptual structure of these order parameter ideas of importance. We may pursue this line a little further, for particles with shortranged attractions [3,26] where particle (physical) gellation has begun to be interpreted as a type of “glass transition” or dynamical arrest. In Fig. 2 we see the typical phase diagram exhibited by particles with a repulsive core, and short ranged attraction. The specific example discussed here [27] is hard core repulsion, and a
Fig. 1. ∂νc / ∂νt for 2D Kob–Andersen model c = 2. The derivatives with respect to density ρ are also shown.
37
Fig. 2. Phase diagram for the Yukawa fluid with screening parameter b = 30 [27]. The crosses (×) represent the fluid–solid phase transition, the open circles (ο) represent the glass transition line as evaluated by Mode Coupling Theory, the continuous line is the binodal and the dashed one is the spinodal. The filled circle is the critical point. The labels I, II and III are chosen after Muschol and Rosenberger [28] and refer to regimes suitable for crystallisation (I), two-liquid coexistence (II) and the “arrested” region (III), as described in the text.
Yukawa attractions, but the details are not important to our current purpose. The point is that, for sufficiently short ranged attraction (in this example b = 30), the traditional methods (mode coupling theory) that describe dynamical arrest predicts that the phenomenon that appears at 0.58 volume fraction for hard spheres moves to much lower volume fractions, and is there identified as “gellation” of the particles. Now, with the more simple lattice models where we can look in more detail at the dynamical processes, and are not restricted to the mode coupling limit, we can see that particle gellation not a true arrest, or glass transition, but a dramatic restructuring of the dynamically available volume. We note that dynamically available volume need does not require infinite interactions or barriers to be defined, and it is possible to apply much the same ideas with finite energies and barriers to movement. In future it will become increasingly possible to describe many of these near-arrest phenomena in these more subtle terms, not as a true loss of ergodicity, but as precipitous changes in the type of ergodic motion of the system. At present the signs are that many phenomena of slowing, near-solidification, or arrest that occur in particulate or soft matter systems will be describable in terms of relatively few classes, each of which is governed by quite general and simple rules. Underlying this is a simple LGW-type description of the systems in which this dynamically available volume plays the central role. Finally, we note that as some confidence in these simple descriptions begin to grow for more complicated systems, we are increasingly becoming more confident that we will be able to treat these order parameter ideas in a much deeper, in some cases exact manner. Thus, recently, it has been possible to develop an approach that allows us to systematically add up the consequences of dynamical blockages or traps met by particles in nearly arrested systems in an formally exact manner. We have shown that for one example in two spatial dimensions it is even possible to calculate the dynamical correlation length exactly [17,21]. This approach is not unlike renormalization group methods for the critical point, a
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method that adds up the effects of (static) correlations on all length scale. The indications are that we are beginning to see a new conceptual framework within which it will be possible to describe many natural phenomena that may reasonably be named a “change of state”, but that do not involve any singular aspect to the derivatives of the free energy. There is a great deal to be done yet, particularly in building a microscopic continuum approach in the arena. However, based on the simple models, our feeling is that the outcome will lead us increasingly to see phase-transitions as only one extreme of a range of such phenomena. Then the whole machinery of equilibrium statistical mechanics will also be seen as a particular case of a more general picture in which many of the same elegant classifications of “states” and “transitions” obtain. For those that have grown from the great schools of equilibrium statistical mechanics, with the tastes and sensibilities that accompany this background, such an outcome would be particularly satisfying. We acknowledge interactions at various stages with our colleagues in the an EU funded research program (see www. arrestedmatter.net), contract number MRTN-CT- 2003-504712. References [1] Angell CA, Ngai KL, McKenna GB, McMillan PF, Martin W. J Appl Phys 2000;88:3113. [2] Mézard M, Parisi G. J Phys Condens Matter 2000;12:6655. [3] Dawson K. Curr Opin Coll Int Sci 2002;7:218. [4] McManus JJ, Rdler JO, Dawson KA. J Am Chem Soc 2004;126:15966. [5] Stradner A, Sedgwick H, Cardinaux F, Poon WCK, Egelhaaf SU, Schurtenberger P. Nature 2004;432:492. [6] Kauzmann W. Chem Rev 1948;9:219.
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