Shaking a box of sand I – a simple lattice model

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Shaking a box of sand I – a simple lattice model

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7 Shaking a box of sandI–asimplelattice model 7.1 Introduction Vibrating sand results in very varied dynamics, ranging from glassy [132, 172, 173] to fluidised [193–197]. In much of this book, we will focus on the former, while a subsequent chapter will contain a review of the latter regime. Of course, it is important to have a theoretical understanding of how one regime gives way to the other; it is for this reason that the model discussed in this chapter is a simple model of a vibrated sandbox, which interpolates between the glassy and fluidised regimes through an extremely interesting intermediate regime, whose properties are not fully understood to date, and are the subject of current investigations. The model [174] is based on the generalisation of an earlier cellular automaton (CA) model [22, 75, 83, 165] of an avalanching sandpile. This version of the model contains only near-neighbour interactions, with grains being in one of two orienta- tional states. It shows both fast and slow dynamics in the appropriate regimes and, in its simplest form, reduces to an exactly solvable model in the frozen or jammed regime. Of course, in order to replicate the truly glassy behaviour of the jammed regime, one needs to introduce true long-range interactions. This will be done in the next chapter, where in particular the effect of grain shapes will be probed using an extension of this model [34, 35]. 7.2 Definition of the model As mentioned above, a principal motivation for this model is the addition of ori- entational rearrangement to the normal flow mechanism of CA models. In this spirit, consider a rectangular lattice of height H and width W with N ≤ HW grains located at its lattice points. Each ‘grain’ is a rectangle with sides 1 and a ≤ 1, respectively, so that a grain which lies on its long(short) side is said to be Granular Physics, ed. Anita Mehta. Published by Cambridge University Press. C  A. Mehta 2007. 94 7.2 Definition of the model 95 ∆ h ∆H Fig. 7.1 A vertical grain needs to be tilted through the height h to reach the unstable equilibrium position and flop to the horizontal, while a horizontal grain needs to be tilted through an additional height H to reach the vertical. horizontal(vertical). Let this box be shaken with vibration intensity . The most intuitively reasonable rules for its dynamics should involve (a) grains falling under gravity to a void below (without thresholds) or diagonally adjacent (provided appro- priate height thresholds are met, as in usual CA models [65]); (b) grains flying directly or diagonally above themselves, provided they are given sufficient energy by the vibration intensity ; (c) grains flipping easily to their horizontal orientation, and flipping back less easily to their vertical one (Fig. 7.1); and (d) vertical grains being more unstable, they should add to the height of the column, and thus increase the grain’s propensity to fall to sites which are diagonally below them, i.e. ‘down the pile’. We write below the mathematical formulation of these rules; consider a grain (i, j)inrowi, column j whose height at any given time is given by h ij = n ij− + an ij+ , with n ij− the number of vertical grains and n ij+ the number of horizontal grains below (i, j). We give in the following a prescription [174] for the dynamics of this grain under shaking: r If lattice sites (i + 1, j − 1), (i + 1, j), or (i + 1, j + 1) are empty, grain (i, j ) moves there with a probability exp(−1/), in units such that the acceleration due to gravity, the mass of a grain, and the height of a lattice cell all equal unity. r If the lattice site (i − 1, j) below the grain is empty, it will fall down. r If lattice sites (i − 1, j ± 1) are empty, the grain at height h ij will fall to either lower neighbour, provided the height difference h ij − h i−1, j±1 ≥ 2. r The grain flips from horizontal to vertical with probability exp(−m ij (H + h)/), where m ij is the mass of the pile (consisting of grains of unit mass) above grain (i, j). For a rectangular grain, H = 1 − a is the height difference between the initial horizontal and the final vertical state of the grain. Similarly, the activation energy for a flip reads h = b − 1, where b = √ 1 + a 2 is the diagonal length of a grain. r The grain flips from vertical to horizontal with probability exp(−m ij h/). In this form, although the interactions are nearest, or at most next-nearest neigh- bour, the model is not exactly solvable, because of the presence of voids in the system. The solutions in much of this chapter will therefore be largely numerical, with an analytical solution obtained only when voids are strictly absent. 96 Shaking a box of sandI–asimplelattice model 123456789 ln t −1.0 −0.5 0.0 0.5 φ−φ 00 0.1 0.2 0.5 1.0 2.0 3.0 5.0 Fig. 7.2 Plot of φ − φ ∞ versus ln t, for different values of , indicated on the curves. Note that φ ∞ decreases with increasing , and is thus distinct for each curve. 7.3 Results I: on the packing fraction In this section, we examine the behaviour of the packing fraction of the model, as a function of the vibration intensity .LetN − and N + be the numbers of vertical and horizontal grains in the box. The packing fraction φ is: φ = N + − aN − N + + aN − , (7.1) which we use as an order parameter. The vertical orientation of a grain thus wastes space proportional to 1 − a, relative to the horizontal one. We examine the response of the packing fraction for typical parameter values (H = 0.3, h = 0.05) to shaking at varying intensities in Fig. 7.2. Clearly, the asymptotic values of the packing fraction φ ∞ will differ for each intensity; in fact, as was shown in earlier chapters, it decreases with increasing intensity [61, 62, 130]. The asymptotic packing fraction was determined for each intensity, and the difference φ − φ ∞ plotted as a (logarithmic) function of time T in Fig. 7.2. Note that the initial packing fraction was the same in each case. The dynamical response of the shaken sandbox [174] includes three distinct regions, each illustrated by representative curves in the figure. We note first a fluidised region (for   1), where we observe an initial increase (caused by a nonequilibrium and transient ‘ordering’ of grains in the boundary layer) of the packing fraction that quickly relaxes to the equilibrium values φ ∞ in each case. This overshooting effect in Fig. 7.2 increases with , since grains ever deeper in the sandbox can now overcome their activation energy to relax to the horizontal. This inhomogeneous relaxation has been mentioned in earlier chapters, and observed in computer simulations of shaken spheres [131]. Very strikingly, the overshoot has also been observed in the context of a totally different model, the random graphs model of granular compaction [152, 153]. To date there is no satisfactory 7.4 Results II: on annealed cooling 97 quantitative explanation for this overshoot, which seems firmly established in view of its model-independence. Next, note an intermediate region (for  ≈ 1), where the packing fraction remains approximately constant in the bulk, while the surface equilibrates via the fast dynamics of single-particle relaxation. The specific φ ∞ at which this occurs (0.917 here), is the single-particle relaxation threshold density (SPRT) observed in Ref. [152, 153]; nonequilibrium, non-ergodic, fast dynamics allows single particles locally to find their equilibrium configurations at this density. Analogous effects have been observed in recent experiments on colloids [124], where the correlated dynamics of fast particles was seen to be responsible for most relaxational behaviour before the onset of the glass transition. Once again, the non-ergodic dynamics in the vicinity of the SPRT is very poorly understood, at the time of writing this book. Additionally, this intermediate regime, with properties in between the jammed and fluidised states, with its balance of individual and collective dynamics, remains one of the most fascinating mysteries of granular matter. Lastly, we see a frozen region (for   1), where the slow dynamics of the system results in a logarithmic growth of packing fraction with time: φ − φ ∞ = b()lnt + a, (7.2) where b() increases with , in good agreement with experiment [172, 173]. The slow dynamics has been identified in a previous chapter with a cascade process, where the free volume released by the relaxation of one or more grains allows for the ongoing relaxation of other grains in an extended neighbourhood. As  decreases, the corresponding φ ∞ increases asymptotically towards the jamming limit φ jam , identified with a dynamical phase transition in a previous chapter [152, 153]. 7.4 Results II: on annealed cooling, and the onset of jamming We next investigate the analogue of ‘annealed cooling’, where  is increased and decreased cyclically, and the response of the packing fraction observed [172, 173]. The results obtained here are similar to those [131] seen using more realistic mod- els of shaken spheres, but the simplicity of the present model allows for greater transparency. Starting with the sand in a fluidised state, as in experiment [172, 173], the sandbox is submitted to taps at a given intensity  for a time t tap and the intensity is increased in steps of δ; at a certain point, the cycle is reversed, to go from higher to lower intensities. The entire process is then iterated twice. Figure 7.3 shows the resulting behaviour of the volume fraction φ as a function of , where an ‘irreversible’ branch and a ‘reversible’ branch of the compaction curve are seen, which meet at 98 Shaking a box of sandI–asimplelattice model 0246810 Γ −0.5 0.0 0.5 1.0 φ 1 −> 2 <− 3 −> 4 <− 5 −> 6 <− 1234 5 Fig. 7.3 Hysteresis curves. Left:  = 0.1, t tap = 2000 time units. Right:  = 0.001, t tap = 10 5 time units. Note the approach of the irreversibility point  ∗ to the ‘shoulder’  jam , as the ramp rate δ/t tap is lowered. the ‘irreversibility point’  ∗ [172, 173]. The left- and right-hand sides of Fig. 7.3 correspond respectively to high and low values of the ‘ramp rate’ δ/t tap [172, 173]. As the ramp rate is lowered, we note that the width of the hysteresis loop in the so-called reversible branch decreases. The ‘reversible’ branch is thus not reversible at all; hard-sphere simulations of shaken spheres [61, 62, 130] confirm the first- order, irreversible nature of the transition. As mentioned in an earlier chapter, the density may attain values that are substantially higher than random close packing, and quite close to the crystalline limit [131, 132]. An analogous transition has also been observed experimentally in the compaction of rods [133]. Note also that the ‘irreversibility point’  ∗ (the shaking intensity at which the irreversible branch and the reversible branch meet) approaches  jam (the shaking intensity at which the jamming limit φ jam is approached), in agreement with results on other discrete models [190]. Of course, as mentioned in an earlier chapter, this feature is an idealisation; in reality the transition to the crystalline limit is not approached quite so smoothly as predicted here, since force networks act as mechanical impediments. However, it is interesting to see how the simple use of phenomenology in this model can get some of the results that are more cumbersomely obtained from hard-sphere simulations. We next use the simplicity of the model to explore the onset of jamming, probing in between the regimes where fast and slow dynamics respectively predominate. This is explored via a configurational overlap function χ(t ref ,t) = 1 N  i, j [B i, j (t ref ), B i, j (t ref + t)]. (7.3) Here B i, j (t) can take three distinct values depending on whether the lattice site (i, j) at time t is (a) empty, (b) occupied by a + grain, or (c) occupied by a − grain; [X, Y ] = 1 − δ X,Y ; i.e., [X, Y ] = 0ifX = Y ; and t is the time lag. The func- tion χ(t ref ,t) is therefore 0 when configurations are identical at t ref and t ref + t 7.4 Results II: on annealed cooling 99 012345 0 0.02 0.04 0.06 0.08 0.10 χ 012345 log ∆t 0 0.1 0.2 0.3 0.4 012345 0 0.2 0.4 0.6 0.8 Γ = 0.1 Γ = 0.7 Γ = 5.0 Fig. 7.4 Overlap functions χ (t ref ,t), Eq. (7.3), for  = 0.1, 0.7, 5.0. Line styles distinguish five reference times from t ref = 1 (full line) to t ref = 10 4 (dotted line). The time unit is defined as HW attempted Monte Carlo moves. and takes larger values depending on the differences of the configurations at those times. Figure 7.4 shows results for different values of , for  = 0.1, 0.7, 5.0. The left-hand panel ( = 0.1) shows the logarithmic behaviour characteristic of ageing; here the leftmost (full) line shows the behaviour at t ref = 1, while the dotted lines to the right of it represent increasing values of t ref . We see clearly that ‘older’ systems change more slowly (as in life!); in other words, with increasing waiting times, χ (t ref ,t) increases much more slowly from zero. Clearly also, with an appropriate rescaling of time, an older system can be made to look like a younger one. This, and the logarithmic increase of χ(t ref ,t) with t, are classic symptoms of glassy dynamics, a point to which we will return later. The right-hand panel ( = 5) shows the quick equilibration virtually indepen- dent of waiting times, which characterises the fluidised regime. Notice that for all waiting times t ref , configurations begin to change rapidly; there is no ageing, no slow dynamics and every system finds its equilibrium quickly. The middle panel ( = 0.7) exemplifies the behaviour characteristic of the tran- sition between the two regimes, the intermediate phase referred to in the earlier section: this has features of both ‘glassy’ and fluidised regimes. On the one hand, younger systems find their equilibrium quickly, as in the fluidised case (χ rises quickly from zero); and on the other hand, there is an age-dependence, whereby older systems seemingly equilibrate to metastable configurations that are not very different from their starting positions, depending on the waiting time t ref . It should be borne in mind that these extremely interesting features come out of a very simple and physical model, and have at least a qualitative relationship with the phenomena that they are seeking to describe. However, care should be taken not to over-interpret these results. The ageing we see cannot be the result of real glassy dynamics, as there are no long-range interactions in the system, which is the reason that we have referred to its glassiness within quotes. Why it manifests 100 Shaking a box of sandI–asimplelattice model ageing, and many other glassy phenomena, is still somewhat mysterious, although ongoing work at the time of writing this book is shedding some light on this issue. 7.5 Results III: when the sandbox is frozen When there are no free voids within the sandbox, we refer to it as frozen or ‘glassy’ (taking care to use quotes, since, as mentioned above, there are no long-range inter- actions, and hence no real glassy behaviour in this model). Here, the model reduces [174] to an exactly solvable model of W independent columns of H noninteracting ‘grains’ σ n (t) =±1, with σ =+1 denoting a horizontal grain, and σ =−1 denot- ing a vertical grain. The orientation of the grain at depth n, measured from the top of the system, evolves according to a Markov dynamics with depth-dependent rates  w(−1 →+1) = exp(−nh/), w(+1 →−1) = exp(−n(H + h)/), (7.4) as m ij = n = H + 1 − i . This indicates clearly that the deeper a grain is in the sandbox, the less free it is to move, and also that the horizontal to vertical transition is more hindered than its reverse, both of which rules make sense. Other than their depth-dependence, where each grain carries the mass of the grains above it, there are no interactions between the grains. We emphasise this issue here, in order to contrast it with the situation of the next chapter, where long-range interactions involving grain orientations will be introduced into the model. The order parameter describing the mean orientation, which we hereafter refer to as ‘orientedness’, M(t) = (1/H )  H n=1 M n (t), with M n (t) =σ n (t), is related to the packing fraction of Eq. (7.1) as M = (1 + a)φ − (1 − a) 1 + a − (1 − a)φ . (7.5) At equilibrium, the orientedness profile is given by M n,eq = tanh  n/(2ξ eq )  , (7.6) while the local equilibration time diverges exponentially with depth n as τ n,eq ≈ exp(n/ξ dyn ). (7.7) In other words, the equilibration rate of an n-dependent order parameter will be ∼ ξ dyn ln t (cf. the overlap function χ defined in the earlier subsection or Eqs. 7.12, 7.13 to follow). These expressions involve two characteristic lengths of the model, the equilibrium length ξ eq and the dynamical length ξ dyn , which read ξ eq =  H ,ξ dyn =  h . (7.8) 7.5 Results III: when the sandbox is frozen 101 In the scaling regime where the height H and both lengths ξ eq and ξ dyn are large, the mean orientedness is M eq ≈ (2ξ eq /H ) ln cosh(H/(2ξ eq )). (7.9) For H  ξ eq ,wehave M eq ≈ H/(4ξ eq )  1. (7.10) In this case, the system is very weakly ordered, even at equilibrium. For H  ξ eq , we have M eq ≈ 1 − (2 ln 2)ξ eq /H. (7.11) Now, the system is strongly ordered at equilibrium, except for its top skin layer, whose depth is of order ξ eq . The length ξ eq is therefore the length up to which disorder persists in the granular material when it has attained equilibrium. As the equilibration time diverges exponentially with the depth, orientational order propagates down the system logarithmically slowly. More specifically, for a large but finite time t, only a top layer up to an ‘ordering length’ (t) has equilibrated, with (t) ≈ ξ dyn ln t. (7.12) We have M n (t) ≈ M n,eq for n  (t), whereas M n (t) ≈ 0 for n  (t). The most ordered grains are situated at a depth comparable to (t); the length ξ dyn therefore determines the length to which order has propagated in the granular material in the ‘glassy’ regime. Grains at depth (t) have a maximum orientedness M max (t) ≈ tanh  (ω/2) ln t  , (7.13) where [174] ω = ξ dyn ξ eq = H h = 1 − a b − 1 (7.14) is the ratio of both characteristic lengths. For grains where there are nearly equiv- alent orientations (a ∼ 1), even at equilibrium, there will be a large number of ‘disordered’ configurations in the top layer, since these will be almost equivalent to the strictly ordered one. Both lengths ξ eq and ξ dyn have, in experimental terms, the interpretation of the depth of the boundary layer in a vibrated granular system; in the first case, this description applies when equilibrium has been reached, while in the second case, this applies to the nonequilibrium evolution of a vibrated granular bed. 102 Shaking a box of sandI–asimplelattice model Table 7.1 Two different nonequilibrium regimes Regime I Regime II ξ eq  (t)  H (t)  ξ eq , H (ω ln t  1) (ω ln t  1) M(t) (t)/H [ (2t) ] 2 /(4Hξ eq ) S(t, s)1− [(t) − (s)]/H 1 − [ (2(t − s)) ] /H C(t, s)1− [2(t) − (t + s)]/H 1 − [ (2(t − s)) ] /H 7.6 Results IV: two nonequilibrium regimes Let us recall that (t) is the equilibration length of this model; clearly then, in order for it to exhibit interesting nonequilibrium or ageing effects, the system size must be much less than this length, i.e. one must have (t)  H. The two-time quantities we investigate to explore ageing are the full two-time correlation function, S n (t, s) =σ n (t)σ n (s), (7.15) and the connected one, C n (t, s) = S n (t, s) − M n (t)M n (s), (7.16) with 0 ≤ s (waiting time) ≤ t (observation time). In terms of the overlap function of Eq. (7.3), these can be written as S(t ref + t, t ref ) = 1 − 2χ(t ref ,t). (7.17) We are led to consider two different non-equilibrium regimes, which result from different ratios of the two characteristic lengths of the system. In each case, the mean observables can be expressed in terms of these lengths alone (see Table 7.1). In Regime I, the maximal ordering is very close to perfect, as 1 − M max (t) ∼ t −ω  1. This is the conventional frozen regime (to which the data in Fig. 7.4 corre- spond). The top layer of the system is strongly ordered, most of the grains are flat, and likely to stay that way: the ageing phenomenon corresponds to the slow order- ing attempts of grains deeper in the bulk, quantified by the logarithmic growth of the ordering length (t). Table 7.1 shows that the mean orientedness is nothing but the fraction (t)/H of the system that has equilibrated. The two-time correlations are non-stationary, and they involve (s), (t) and (t + s). In Regime II, the maximal ordering is very weak, as M max (t) ≈ (ω/2) ln t  1. This regime exists only for ω  1, i.e., a → 1 in the geometrical model. It corresponds to an even slower dynamics, since now any attempts at ordering 7.7 Discussion 103 are hindered additionally by a strong probability that a horizontal grain will flip to the vertical orientation. Table 7.1 shows that the mean orientedness involves the square of the ordering length, while the two-time correlations do not exhibit any non-stationary features characteristic of ageing, at least to leading order, in this scaling regime. The physical difference between the two scenarios is comprehensible in terms of orientational modes of shaped grains. Recall that in Regime I, ξ eq  ξ dyn so that ω  1; from Eq. (7.14), this implies that one side of the rectangular grain is much larger than the other, so there is a strong tendency to prefer one of the two orientations on stability grounds. In a spirit of generalisation, we say therefore that when grains are very asymmetrically shaped, and there is a strong preferred orientation, the nonequilibrium regime of granular dynamics will carry all the usual characteristics of ageing. Now recall that that in Regime II, ξ eq  ξ dyn so that ω  1; from Eq. (7.14), this implies that the grain is nearly square (a ∼ 1), and any grain flip is easily reversed by a corresponding flop! Generalising once again, we suggest that where grains are symmetrically shaped and the restoring ‘force’ to get to a particular orientation is weak, the signatures of ageing will be hard to detect even in a highly nonequilibrium dynamical regime. Even bearing in mind that the present model is an extremely simplified repre- sentation of a granular medium, it would be interesting to test these speculations experimentally: would ageing experiments carried out on differently shaped grains of identical materials give different results? 7.7 Discussion In conclusion, the simplicity of the sandbox model makes it a useful conceptual tool for probing the dynamical responses of vibrated sand, from the fluidised to the frozen regimes. In the latter case, the model is exactly solvable, which allows one to describe the by now well-established picture of logarithmic compaction, in terms of two characteristic lengths, as well as providing an interesting insight into shape-dependent ageing. The improvement of this necessarily qualitative picture by the addition of more realistic and complex interactions, while still retaining the overall conceptual simplicity of the model, constitutes the subject matter of the next chapter. . Shaking a box of sandI–asimplelattice model 7.1 Introduction Vibrating sand results in very varied dynamics, ranging from glassy [132, 172, 173] to fluidised. mechanism of CA models. In this spirit, consider a rectangular lattice of height H and width W with N ≤ HW grains located at its lattice points. Each ‘grain’ is

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