Athermal Jamming vs. glassy dynamics for particles with exponentially decaying repulsive pair interaction potentials with a cutoff
aa r X i v : . [ c ond - m a t . s o f t ] S e p Athermal Jamming vs. glassy dynamics for particles with ex-ponentially decaying repulsive pair interaction potentials with acutoff
Nicolas Wohlleben and Michael Schmiedeberg Institut f¨ur Theoretische Physik I, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg (FAU), Staudtstraße 7, 91058Erlangen, Germany
PACS – Colloids
PACS – Glasses
PACS – Liquids
Abstract –We study athermal jamming as well as the thermal glassy dynamics in systems com-posed of spheres that interact according to repulsive interactions that exponentially decay as afunction of distance. As usual, a cutoff is employed in the simulations. While the athermaljamming transition that is determined by trying to remove overlaps is found to depend on thearbitrary and therefore unphysical choice of the cutoff, we do not find any athermal jamming tran-sition or crossover that only relies on the physical decay length. In contrast, the glassy dynamicsmainly depends on the decay length. Our findings constitute another demonstration of the factthat the athermal jamming transition is not related to thermal glassy dynamics. In addition, weargue that interactions without sharp physical cutoff should be considered more often as a modelsystem in jamming. By exploring how widely-used theoretical approaches or methods of analysisin the field of jamming have to be changed in order to not depend on unphysical cutoffs will leadto deeper insights into the nature of athermal and thermal jamming.
Introduction. –
Jamming in athermal systems thatare composed of spheres with finite-ranged repulsive in-teractions can be described and studied by a energy land-scape exploration method introduced by O’Hern et al.[1, 2]. The spheres are initially placed at random posi-tions and then one tries to remove the overlaps by min-imizing the energy without crossing energy barriers. Incase all overlaps can be removed, the system is called un-jammed, while in case remaining overlaps the system istermed jammed. In the large system limit a sharp tran-sition is found from unjammed to jammed packings ofspheres [1, 2] for increasing packing fraction. Close to thisathermal jamming transition critical power law scaling ofvarious quantities can be observed (cf. [3] as a review).In thermal systems jamming can be defined in variousways. In many particulate systems one can observe a dra-matic slowdown of the dynamics if the density is increased,the temperature is decreased, or external perturbationslike shear forces are decreased [4]. Here we mainly focus atthe slowdown of the dynamics as a function of the packingfraction at small temperatures. In case typical timescales of rearrangements in the systems become larger than thetime that is accessible in experiments or simulations suchthat the system becomes effectively non-ergodic, one of-ten speaks of the (dynamical) glass transition (see, e.g.,[5]). Note that the dynamical glass transition usually dif-fers from a possible structural or ideal glass transition thatmight be defined by the configurational entropy [5–8].While it has been conjectured initially that the ather-mal jamming transition might be the end point of the dy-namical glass transition as some kind of thermal jammingtransition in the limit of small temperatures [2, 4], theo-retically they can occur in the same mean field approachesbut as distinct phenomena [6–8]. In simulations the differ-ences between athermal and thermal jamming have beendiscussed, though they usually occur close together (see,e.g., [9]).In this work we study athermal jamming and glassy dy-namics for a system with repulsive interactions that decayexponentially with the distance but ideally do not possesa natural cutoff. This model interaction can be seen assimplification of DLVO-interactions that occur in charge-p-1. Wohlleben et al. stabilized colloidal systems [10, 11] or for dust particles inplasmas [12], where the exponential decay is due to screen-ing characterized by the so-called screening length. In sim-ulations the interactions are usually implemented with acutoff. Here we demonstrate that athermal jamming thatis determined with the usual protocol is dominated by the(unphysical) choice of the cutoff, while the glassy dynam-ics in thermal systems mainly depends on the screeninglength. While these results might not be considered to beunexpected, we want to stress the importance of studyingathermal and thermal jamming in systems with interac-tions that do not possess a sharp, unique cutoff as manydefinitions and protocol depend on such a cutoff. We arguethat the study of interactions without natural cutoff mighthelp to reveal the true nature of jamming that should notdepend on an arbitrary, unphysical cutoff.
Method. –
A monodisperse system of N spheres inthree spatial dimensions is considered. The interactionbetween the spheres is purely repulsive and given by thepair potential V ( r ) = ( ǫ exp ( − r/l s ) − V c ( r ) , r < l c , , r ≥ l c , (1)where r is the distance between the particles, ǫ sets theenergy scale, l s corresponds to the screening lengths, i.e.,the physically motivated length scale of the interaction po-tential, and l c is a cutoff length that is usually introducedas a purely technical parameter in order to simplify sim-ulations. The function V c ( r ) is a linear function that ischosen such that the interaction potential and in case ofathermal jamming in addition its derivative do not possessany discontinuity at r = l c .To determine the jamming behavior at zero tempera-ture, we employ the same energy landscape explorationmethod as in [1, 2]: The particles are initially placedat random positions into a simulation box with periodicboundary conditions and volume V and with a given num-ber density ρ = N/V . Here N = 700 particles are used.Note that depending on what length one considers as typ-ical size of the particle, one can define a packing frac-tion φ s = πρl / l s or φ c = πρl / l c . By em-ploying a conjugate gradient method we then minimizethe energy, i.e., we determine the local minimum of theenergy without crossing any energy barriers. As in [1, 2]the system is considered to be (athermally) unjammed, ifall overlaps can be removed by the minimization. If over-laps prevail, the system is called jammed. To avoid thelengthy final removal of small overlaps we consider config-urations to be unjammed if the overlap energy E becomessmaller than 10 − ǫ . As a consequence, for counting thenumber of contacts, overlaps are only counted if largerthan 10 − l c . We checked that the results do not dependon these choices and that for harmonic interactions theresults from [1, 2] can be reproduced. In order to explore the dynamics of thermal systems, weperform local Monte Carlo simulations, i.e., Monte Carlosimulations with only local moves such that the numberof Monte Carlo sweeps can be used as a approximate mea-sure of the time [13]. We determine the relaxation time(number of Monte Carlo sweeps) τ for which the self-intermediate scattering function for k = 2 πφ / /l decaysto 1 /e of its initial value. For the analysis where the cutoffis taken as length scale, φ = φ c and l = l c . If the screen-ing length is taken in the analysis, φ = φ s and l = l s .For the dynamical simulations, we employ a bidispersesystem, where half of the N = 1000 particles are largerby a factor 1 .
4, i.e., the same interaction potential as forathermal jamming is employed, but all lengths are scaledby a factor 1 . . φ c or φ s = 0 .
05 and later increasethe packing fraction in intervals of 0 .
05. At each density,the relaxation time is measured after an initial relaxationof 10 Monte Carlo sweeps with adaptive maximum dis-placements per step. Afterwards the relaxation time ismeasured with a constant maximum size of displacementsper Monte Carlo step of l/φ / /
20. Different temperaturesfrom k B T /ǫ = 0 . k B T /ǫ = 0 . Results. –
Athermal Jamming.
In fig. 1 the overlap energy thatis reached after a minimization towards a local minimum,i.e., without crossing energy barriers is shown. In fig. 1(a)it is plotted as a function of φ c . In fig. 1(b) the same datais shown as a function of φ s . Therefore, in fig. 1(a) thecutoff length l c determines the packing fraction while infig. 1(b) the data is organized by the screening length l s .In fig. 1(a) the well known behavior with no overlapsbelow a transition packing fraction φ J and an increasingoverlap energy above the transition is observed. The tran-sition occurs at the same φ J for all screening lengths l s and is in agreement with the transition packing fractionof spheres with harmonic interactions as reported in [1,2].The inset of fig. 1(a) shows the difference of the numberof per particle contacts Z c , i.e., neighbors that are closerthan the cutoff length l c , and the isostatic contact number6 as a function of the difference of the packing fraction φ c and the transition packing fraction φ J . For large packingfractions φ c the data depends slightly on l s , but otherwise l s hardly affects the overlap energy.If the overlap energy is plotted as a function of the pack-ing fraction φ s that is given by the screening length l s , allcurves with φ c ≤ φ J are zero everywhere, and the curvesfor φ c > φ J are slightly increasing with increasing φ s . Oth-erwise, there is no indication for any transition or crossoverthat could explain a connection between the rapid slow-down of the dynamics that we present in the next subsec-tion and the inherent structures whose overlap energy isshown here.The inset of fig. 1(b) shows the number of contacts Z s given by the number of neighbors that are closer than thep-2thermal Jamming vs. glassy dynamics for exponentially decaying interactions E / ε φ c E / ε φ s Z c - φ c - φ J Z s φ s Fig. 1: Overlap energy per particle after the minimization forvarious cutoff lengths and screening lengths that are denotedby the corresponding packing fractions φ c and φ s , respectively.In (a) the overlap energy is shown as a function of the packingfraction φ c based on the cutoff length l c , while in (b) the samedata is plotted as a function of φ s based on the screening length l s . In all figures the color denotes φ c as can be read of in (a)and in addition is shown in the legend of fig. 2(b) while thesymbols mark φ s as visible in (b) and in the legend of fig. 2(a).The grey line in the main figure of (a) is a quadratic fit functionthat starts from zero at a transition packing fraction φ J . Fromthe fit, we find φ J = 0 . Z c per particle, defined of overlaps concerning thecutoff length l c , in excess of the isostatic contact number 6 as afunction of the packing fraction φ c above the transition packingfraction φ J . In the inset in (b) Z s , which gives the number ofneighbors per particle that are closer than the screening length l s , is plotted as a function of φ s . screening length l s as a function of φ s . A sharp increaseonly occurs in case φ s ≈ φ c ≈ φ J . For φ c < φ J the numberof contacts Z s start to deviate from 0 below φ J while for φ c > φ J there is a continuous increase that roughly startsat φ J . As a consequence, no universal transition behaviorcan be found if l s is used for the analysis of contacts. φ c =0.66 g (r) r/l c φ s =0.30 φ s =0.40 φ s =0.50 φ s =0.60 φ s =0.62 φ s =0.63 φ s =0.64 φ s =0.65 φ s =0.66 φ s =0.70 φ s =0.80 φ s =0.90 1 10 0 0.5 1 1.5 2 2.5 3 3.5 4(b) φ s =0.66 g (r) r/l c φ c =0.30 φ c =0.40 φ c =0.50 φ c =0.60 φ c =0.62 φ c =0.63 φ c =0.64 φ c =0.65 φ c =0.66 φ c =0.70 φ c =0.80 φ c =0.90 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 3.5 4(c) φ s =0.66 g (r) r/l s Fig. 2: Pair distribution function of the structures obtainedafter minimizing the overlap energy without crossing energybarriers. The length is plotted in units of (a,b) the cutoff length l c or (c) the screening length l c . (a) All curves are for the same φ c = 0 .
66 and various φ s . (b,c) φ s is kept constant at 0.66and φ c is varied. The colors and symbols, as also shown in thelegend, are the same as in fig. 1. In fig 2 we present pair distribution functions g ( r ) forthe structures obtained after the minimization process. Infig. 2(a) the length is plotted in units of the cutoff length l c and curves slightly above the athermal jamming tran-sition, i.e., for constant φ c = 0 .
66, are shown for various φ s . All curves collapse onto each other within the sizeof the symbols. Therefore, close to athermal jamming, g ( r ) is almost independent of the screening length l s . Asa consequence, the inherent structures close to athermaljamming depend on the cutoff length l c but not on thescreening length l s .p-3. Wohlleben et al. In fig. 2(b,c) g ( r ) for a constant φ s = 0 .
66 and vari-ous φ c are shown. Therefore, cases further away from theathermal jamming transition are included. If the lengthis plotted in units of the cutoff length l c as depicted infig. 2(b), one finds that the first peak of g ( r ) occurs at l c but otherwise the other features of g ( r ) are not deter-mined by l c . If the length is shown in units of l s (seefig. 2(c)), there are still significant differences between thecurves. However, for sufficiently large density ( φ c ≥ . Glassy dynamics of thermal systems.
The thermaljamming of particles that interact according to the DLVO-potential has been studied, e.g., in [14]. Here we want toexplore how the two length scales, i.e., the screening length l s and the cutoff l c , affect the glassy dynamics.In fig. 3 the relaxation time obtained by dynamicalMonte-Carlo simulations as described in sec. is plottedas function of the packing fraction. If plotted as functionof φ c and for an analysis based on the cutoff length l c (seefig. 3(a)), the slowdown of the dynamics for different ratiosof l s /l c occurs at very different packing fractions. Obvi-ously, the cutoff length is not a good way to characterizethe dynamics.As a function of φ s and with an evaluation also other-wise based on the screening length l s (see fig. 3(b)), thecurves arrange in groups where the slowdown for a giventemperature (as indicated by the color) is at least similareven for different l s /l c . As a consequence, the dynamicsof the system mainly depends on the screening length l s .The cutoff length still has an significant influence on thedynamics but in contrast to the athermal jamming cannotbe used as control parameter alone.These results probably are not surprising: Concerningthe dynamics in thermal systems, one usually does notexpect that the thermal dynamics strongly depends on acutoff length. For example, Yukawa-like particles in plas-mas can be studied by using the isomorph invariance [15]that relies on the screening lengths. Furthermore, thereare various way to approximately map the dynamics ofsoft particle onto the dynamics of hard spheres [16–18]and for the employed methods most cutoffs are negligible:An effective diameter that is determined according to theBarker-Handersen [19] or the Andersen-Weeks-Chandlermethod [20] is hardly affected by a change of the cut-off if the cutoff length is sufficiently large. As a conse-quence, from the mapping-approaches one indeed expectthat glassy dynamics does not strongly depend on the cut-off.Note that in fig. 3, short cutoff lengths were chosen to
100 1000 10000 0.1 0.5 1.0 5.0(a) τ φ c l s /l c =0.5l s /l c =0.6l s /l c =0.7l s /l c =0.8l s /l c =0.9l s /l c =1.0k B T/ ε =0.1k B T/ ε =0.2k B T/ ε =0.3k B T/ ε =0.4k B T/ ε =0.5 100 1000 10000 0.1 0.3 0.5 1.0 3.0 5.0(b) τ φ s
100 1000 10000 0.01 0.1 1.0(c) τ f BH3 φ s Fig. 3: Relaxation time from dynamical Monte Carlo simula-tions as function of (a) φ c , (b) φ s , and (c) φ s rescaled witha factor based on the Barker-Hendersen approximation [19] asexplained in the text. The temperature is denoted by the colorwhile the symbols indicate the ratios of the screening length l s and the cutoff length l c . demonstrate their (weak) influence. We can correct thepacking fraction φ s by employing a factor f BH that is givenp-4thermal Jamming vs. glassy dynamics for exponentially decaying interactionsas ratio of the Barker-Hendersen length [19] of the interac-tion potential without cutoff and the one of the potentialwith cutoff. In fig. 3(c) we show that this rescaling ofthe packing fraction reduces the dependence on the cut-off. Probably a further reduction of the cutoff-dependencewith improved rescalings as, e.g., in [16, 17, 20].Finally, we want to note that multiple reentrant glasstransitions are observed in ultrasoft systems for increas-ing overlaps [21–23]. Obviously this dynamics is not con-nected at all to the conventional definition of athermaljamming where only the difference between systems withoverlaps and systems without overlaps matters. Discussion and Conclusions. –
We studied theathermal jamming behavior for particles with exponen-tially decaying repulsive interaction potentials and a cut-off. We find that according to the conventional definitionof jamming the athermal jamming transition depends onthe cutoff length but not on the screening length. No othertransition or crossover is observed in athermal systems ifthe screening length is taken to define the packing fraction.In contrast, the slowdown of the thermal glassy dynamicsas a function of packing fraction mainly depends on thescreening lengths.Concerning the connection of structure and glassy dy-namics our results suggest that the dynamics mainly de-pends on the longer-ranged correlations while the struc-ture close to contact is less important. Note that in oursystem the repulsive forces close to contact are small andtherefore we are not close to the hard-sphere limit.On a first view the results might appear trivial. Theconventional definition of the athermal jamming transitionrelies on the range of the interaction potential. However,we want to point to important, non-trivial conclusions andquestions that arise due to our findings:Sometimes the athermal jamming transition has beendescribed as some kind of end point of a thermal jam-ming line that is related to the glass transition [2, 4]. Forthe system that we consider here such an interpretation isimpossible, because the athermal jamming transition de-pends on the cutoff while the glass transition is mainlycontrolled by the screening length. Athermal and ther-mal jamming can occur at very different packing fractions.Therefore, by using the exponential interaction potentialwith a cutoff athermal and thermal jamming can be tunedand studied separately.Since the athermal jamming transition in our systemcan be seen as an artifact caused by basing the definitionon a cutoff, there is an imminent question: Is there anathermal jamming transition (or at least a crossover) atall for interaction potentials that are not finite-ranged?Obviously, mechanical properties like elastic modules of asuspension depend on the density. Our results suggest thatthe rigidification for increasing density is a smooth processas there is no obvious physical way to define whether thereis a contact between neighboring particles or not. As aconsequence, a term like isostaticity should not be used for particle where the interaction is not finite-ranged.In many theoretical works (see, e.g., [6–8]) the differ-ences between athermal jamming and the glass transitionare carefully considered. However, the jamming line thatoccurs as limiting line of the glassy behavior predicted bymean-field theory (cf., [7, 8]) probably does not exist atall without cutoff. As we have shown by using the expo-nential interactions with cutoff, the dynamics glass tran-sition and athermal jamming can be tuned almost inde-pendently. Therefore we suggest that it could be of greatinterest to use this model system to explore how other pre-dicted intermediate transitions behave, e.g., the Gardnertransition.Many simulation studies dealing with the thermal glasstransition assume or find that the thermal glass transitionoccurs close to athermal jamming (see, e.g., [5, 9, 24–27]).In addition, many works successfully employ an analysisthat is based on an athermal jamming-like definition withoverlaps [28–31] or deal with inherent structures (in thesense of local minima of the energy landscape) [27, 32–36]though without cutoff the energy landscape might consistof only one connected energy basin just as in systems withcutoff below athermal jamming [30]. However, in systemswhere athermal jamming is an artifact related to an ar-bitrary cutoff, why should the athermal jamming pointbe a valid starting point to explore glassy dynamics or tofind properties of the glass transition? Is there somethingspecial about the often employed harmonic or Hertzian in-teractions that athermal and thermal jamming seem to beclosely related in these systems?We believe that it is of great importance to find outwhich properties connected to glassy behaviors rely on acutoff lengths or not. The definitions, analysis, or theo-retical approaches should be changed such that they areindependent of unphysical cutoffs. The interaction poten-tial used in this letter is suitable to explore which proper-ties and phenomena depend on the screening length andtherefore can be related to the glassy dynamics and whichone rely on the cutoff.For example, in [30, 31, 37] we have found an directedpercolation transition in time that breaks ergodicity andtherefore corresponds to the dynamical glass transitionbetween ergodic states that can reach the ground stateand non-ergodic states where the ground state is inacces-sible [30,31]. In principle, such an ergodicity breaking canalso occur for particles without finite-ranged interactionsthough the recognition of ground states is more difficultbecause they no longer can be associated to configurationswithout overlaps. Therefore, in a future work, we wantto study ergodicity breaking without relying on athermaljamming or any definition based on overlaps.Furthermore, there are many protocols that investi-gate the energy or free energy landscape, e.g., to studyits hierarchical properties close to the glass transitionor the Gaardner transition [35, 38]. It is an open ques-tion whether the same results can be obtained in absenceof a nearby athermal jamming transition and thereforep-5. Wohlleben et al. whether these findings are really related to the glass tran-sition.Finally, in future works we want to find out how jam-ming in case of additional attractive interactions can beproperly defined. In case of short finite-ranged interac-tions with weak attractions, athermal jamming can bestudied in a similar way as for finite-ranged repulsive in-teractions [39]. However, in colloidal gel-forming systemsthe interactions and dynamics are more complex and usu-ally at longer distances there is a repulsive interaction asconsidered in this letter [40–43] and it is not expected thata cutoff length significantly affects the properties of a gel.However, it remains an open question how the mechani-cal solidification of a gel might be related to the athermaljamming, especially because gelation sometimes is asso-ciated to a rigidity percolation transition [43, 44] that isbased on whether strands of isostatic particles, as definedin athermal jamming with some cutoff, percolate or not. ∗ ∗ ∗
The project was supported by the Deutsche Forschungs-gemeinschaft (Grant No. Schm 2657/3-1).
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