Jamming Phase Diagram Of One Dimensional Hard Rods

Introduction

Despite the simplicity and lack of certain thermody-namic properties, the availability of exact results for theone–dimensional hard rod makes this system a valuabletool for the analysis of phenomena observed in morecomplex systems. Prigogine introduced the generalderivation of the thermodynamic properties of a one–dimensional mixture and Percus obtained the analyticalfree energy functional for both pure and binary systems. The non–additive hard rod system has beena subject of studies from a number of different aspects. A detailed investigation of the inherent structureand glass transition was performed [7], but still some keypoints in relation to the jamming phase diagram and howthe fluid samples the landscape have yet to be explored. Formulating the inherent structure paradigm for a sys-tem of one dimensional hard rods on a line is straight for-ward because the particles are unable to pass each otherand they only interact with their immediate neighbors. Compressing the system will map a fluid configurationto its inherent structure where all the particles contacttheir two nearest neighbors to satisfy the local jammingcriteria. Since the particles cannot pass, there are alsono collective motions that can lead to unjamming, so thestructures are collectively jammed.

The partition func-tion for a glass is then given by all the configurations ofthe particles in a fixed order and the full partition func-tion of the fluid is formed summing over all the possiblearrangements of the particles. For a single componentsystem of additive hard rods, there is just a single in-herent structure with jammed occupied volume, ϕJ = 1. The thermodynamic properties are the result of free vol-ume only and can be calculated exactly [8]. For exam-ple, the isobaric heat capacity (Cp) of an additive onedimensional system is a constant value and has no Cpmaximum. The additive binary mixture has many dif-ferent inherent structures arising from the distinguish-able particle arrangements, but they all have the sameϕJ and the same free volume as a function of density,so the inherent structure landscape plays no role in thethermodynamics.

Distribution of Glasses

The number of jammed states, NJ, with the density ϕJcan be obtained by considering the number of differentways the particles can be arranged on the line such thatthere are xAB interactions [1, 7]. Then the entropy ofthe jammed states can be defined as SJ/NkB = lnNJ,which is given bySJ(ϕJ)NkB= − (1− xAB) ln (1− xAA) + xAB lnxAB. (3)Within the inherent structure paradigm the vibra-tional (free volume) entropy of a single glass, relativeto the ideal gas at the same temperature and density,∆igSg (ϕ, ϕJ), can be obtained from the partition func-tion of the rods constrained to remain in the same or-der on the line. A fluid can sample all possible inherentstructures but at a given ϕ, it will generally sample theset of basins with the ϕJ that maximize its total entropy,0. 5 0. 6 0. 7 0. 8 0. 9 1φJ00. 10. 20. 30. 40. 50. 6Sc /NkB∆ = 0. 1∆ = 0. 2∆ = 0. 3∆ = 0. 4∆ = 0. 5∆ =0. 6∆ = 0. 7∆ = 0. 8∆ = 0. 9∆ = 1. 0FIG. 3.

The distribution of glasses with jammed density ϕJfor different values of the non–additivity parameter, ∆. ∆igSf (ϕ). This gives,∆igSf (ϕ) = SJ (ϕJ) + ∆igSg (ϕ, ϕJ), (4)where the equilibrium ϕJ is found using the condition(∂∆igSf (ϕ)/∂ϕJ)ϕ = 0 and the configurational entropyof the fluid Sc (ϕ) = SJ (ϕJ). Figure 3 shows the distribution of inherent structuresin terms of SJ (ϕJ) for different values of the non–additivity parameter. The width of the distribution asa function of ∆ decreases and in the limit of ∆ → 0the system becomes additive with a single jammed state. Figure 4 shows how the equilibrium liquid samples thebasins in the landscape. It can be seen that the sys-tem samples the deeper basins as the fluid quenches fromhigher densities. By considering Figs. 3 and 4 together, it reveals someinsight about the entire landscape of the system. Fig-ure 5 combines these two figures and shows the connec-tion between them more explicitly. There is only oneconfiguration at ϕJ max (SJ = 0) and at ϕJ min, while thedistribution goes through a maximum at an intermedi-ate density, ϕ∗J, when xAB = 0. 5 (Fig. 5a). There areinherent structures all the way down to ϕJ min, but itbe can seen from Fig. 5b that the equilibrium fluid onlysamples inherent structure basins above ϕ∗J. This impliesthat any compression of an ideal gas configuration, thatdoes not allow the system to escape its local basin, wouldend up jamming at ϕ∗J.

The equilibrium properties of thefluid are determined by the competition between free vol-ume entropy and configurational entropy. Basins withdenser inherent structures have more free volume, butthere are fewer of them, so the equilibrium fluid samplesdeeper basins on the landscape as it is compressed un-til the system eventually becomes unavoidably jammedas ϕ → ϕJ max. At densities below ϕJ min, there are noconfigurations of the particles that are jammed. 30 0. 2 0. 4 0. 6 0. 8 1Density (φ)0. 60. 70. 80. 91φ J ∆ = 0. 1∆ = 0. 2∆ = 0. 3∆ = 0. 4∆ = 0. 5∆ = 0. 6∆ = 0. 7∆= 0. 8∆ = 0. 9∆ = 1. 0FIG. 4. The jammed state densities that the equilibrium fluidsamples as a function of density, for different values of thenon–additivity parameter, ∆. 00. 20. 40. 6SJ/NkB0. 5 0. 6 0. 7 0. 8 0. 9 1φJ00. 20. 40. 60. 8φφJ maxφJ*φJ minAccessible PackingsInaccessible Packings(a)(b)FIG. 5. Distribution of inherent structures (a) and how theequilibrium fluid samples the inherent structures (b) for thesystem with ∆ = 1. Figure 6 shows Sc as a function of ϕ for the systemswith different values of non–additivity. Here, it can beseen that the ideal gas samples the inherent structuresat the maximum of the distribution, ϕ∗J, then the fluidmoves to basins with a higher ϕJ with increasing density. The basins with ϕJ < ϕ∗J are never sampled by the equi-librium fluid. At low ϕ, the configurational entropy of thefluid decreases slowly before it begins a rapid decrease atintermediate occupied volume fractions. An extrapola-tion of the Sc to higher ϕ, based on its behavior in thisintermediate regime, would suggest the system exhibits0 0. 2 0. 4 0. 6 0. 8 1Density(φ)00. 10. 20. 30. 40. 50. 60. 7Sc /NkB∆ = 0. 1∆ = 0. 2∆ = 0. 3∆ = 0. 4∆ = 0. 5∆ = 0. 6∆ = 0. 7∆ = 0. 8∆ = 0. 9FIG. 6. Sc as a function of ϕ for the systems with differentvalues of non–additivitya Kauzmann catastrophe where the configurational en-tropy goes to zero at a ϕ well below ϕJ max. However, Scplateaus at high ϕ and only approaches zero in the limitϕ → 1. As a result, there is no ideal glass transition inthis system.

Transfer Matrix Method

Transfer matrix method (TMM) is used to solve va-riety of one dimensional models with nearest neighbourinteractions. The hard rod systems with positive nonadditivity also allow the first neighbour contact. There-for, in the following the TMM implication to obtain en-semble of inherent structures will be discussed. For abinary system of non–additive hard rod system, thereare four different types of interactions, two interactionsfrom particles A and B interacting with their own typeand two interactions from interacting with different types(AB and BA). These configurations can be combined tocreate locally jammed configuration of N particles thatcan be described by an ordered list of the particle types. The jammed states in one dimensional systems are alsocollectively jammed structures, because the particles areunable to pass each other. The interaction lengths for the system are introducedin II as σAA, σBB and σAB. Here, σAA and σBB are con-sidered to be the same length. For fixed N, the volumeof the system will fluctuate depending on the number oftype AB and BA interactions and also the magnitudeof non additivity ∆ in the configuration. The transfer4matrix then takes the form: M =[eβPσAA eβPσABeβPσBA eβPσBB](5)where β is kT.

The exponential term is the Gibbs mea-sure appropriate for the N,P, T ensemble. The matrixM has two eigenvalues and in the thermodynamic limit,the partition function will be dominated with the largereigenvalue. The partition function for the system in thethermodynamic limit is given by∆(N,P, T ) = NkT ln (λ+), (6)where λ+ is the largest eigenvalue of M. The jammingdensity, ϕJ, is then given by,ϕJ =NLJ= − 2∂ (lnλ+) /∂P, (7)where LJ is the length of the system in the jammed state. The entropy of jammed states SJ = −k lnNJ, where NJis the number of jammed configurations with ϕJ, can bewritten asSJ/Nk = lnλ+ + T∂ (lnλ+) /∂T. (8)The resulting eigenvalues are necessarily functions ofN,P and T. The factors associated with N are dealtwith by considering the system in the thermodynamiclimit and calculating quantities on a per particle basis. T plays no direct role in the hard particle system, exceptto provide the velocity distribution of the particles. Herewe are only dealing with jammed structures where thereis no free volume and the particles are unable to move,which implies that T = 0. In the absence of tempera-ture, there is no internal pressure caused by the collisionbetween particles. However, it is still necessary for thesystem to do work against the pressure P when it expandsso the equation of state for the ensemble of jammed con-figurations results from the connection between the workrequired to expand the volume of the system and SJ. We obtain the full distribution of states by varying thepressure from −∞ → ∞. The results obtained using thismethod are identical to those discussed above and Fig. 3shows that the distribution of jammed states.

Heat Capacity

This model also can be used to study how featuresof the landscape influence the thermodynamic proper-ties of the system, such as the heat capacity and theequation of state (EOS). The isobaric heat capacity,Cp = (∂H/∂T )p, where for the hard rods model, the en-thalpy is H = (1/2)NkBT+PL, with the compressibilityfactor PL/NkBT = 1 + ϕ/ (ϕJ − ϕ). Note that here, Pis the 1d pressure and the equivalent to the pressure tothe bulk systems and ϕJ is the jammed density that theequilibrium fluid samples and not the most dense config-uration’s density, ϕJ max. Therefore, the heat capacity isgiven byCp/NkB = 1/2+(PL/NkBT )2(PL/NkBT + (PL/NkBT − 1) {1 + (PL/NkBT − 1) (1− dϕJ/dϕ)}). (9)Figure 7 represents the inverse of compressibility factoras a function of density for different non–additivities. Figure 8 shows that Cp as a function of (ϕPL/NkBT )−1goes through a maximum as a result of the maximum inthe term dϕJ/dϕ (See Fig. 5). The peak also sharpensand moves to lower temperatures as ∆ → 0, in a man-ner that is somewhat similar to a system approaching acritical point. This coincides with the narrowing of theinherent structure distribution which collapses to a sin-gle state at ∆ = 0. It is also interesting to note that thedensity of the fluid at the CP maximum is equal to ϕ∗J asthis seems to connect the properties of the fluid at highdensities to the ideal gas, through the inherent structurelandscape. 0 0. 2 0. 4 0. 6 0. 8 1Density (φ)00. 20. 40. 60. 81NkBT / PL∆ = 0. 0∆ = 0. 1∆ = 0. 2∆ = 0. 3∆ = 0. 4∆ = 0. 5∆ = 0. 6∆ = 0. 7∆ = 0. 8∆ = 0. 9∆ = 1. 0FIG. 7. EOS for different non–additivity parameter as a func-tion of density(ϕ).

Inherent Structure Pressure

Shell et. al. [9] showed that the properties of the EOSof a fluid could be related to the inherent structure land-scape by separating the equilibrium pressure into contri-butions from the inherent structure pressure, PIS, andvibrational pressure, Pvib, so that,P = (PIS + Pvib). (10)50 0. 5 1 1. 5 2(φPL/NkBT)-11. 51. 61. 71. 81. 92Cp /NkB∆ = 0. 1∆ = 0. 2∆ = 0. 3∆ = 0. 4∆ = 0. 5∆ = 0. 6∆ = 0. 7∆ = 0. 8∆ = 0. 9∆ = 1. 0FIG. 8. CP /NkB for different non–additivity parameter as afunction of (ϕPL/NkBT )−1. Making use of the general relation P = T (∂S/∂V )U, theinherent structure pressure can be calculated asβPIS = −ϕ2(∂Sc/NkB∂ϕ)U, (11)and then obtain Pvib from Eq. 10. Both contributions tothe pressure are shown in Figs. 9 and 10. The vibrationalpressure increases monotonically as a function of ϕ whilePIS exhibits a maximum at densities that are slightlyhigher than where the Cp maximum appears. 0 0. 2 0. 4 0. 6 0. 8 1Density (φ)0246810βPISσ∆ = 0. 1∆ = 0. 2∆ = 0. 3∆ = 0. 4∆ = 0. 5∆ = 0. 6∆ = 0. 7∆ = 0. 8∆ = 0. 9∆ = 1. 0FIG. 9. Inherent structure pressure as function of density fordifferent values of non–additivity, ∆. 0 0. 2 0. 4 0. 6 0. 8 1Density (φ)02468βPvibσ∆ = 0. 1∆ = 0. 2∆ = 0. 3∆ = 0. 4∆ = 0. 5∆ = 0. 6∆ = 0. 7∆ = 0. 8∆ = 0. 9∆ = 1. 0FIG. 10. Vibrational pressure as function of density for dif-ferent values of non–additivity, ∆. FIG. 11. The jamming phase diagram for the non–additivebinary mixture of hard rods model, including ϕJ max, ϕJ min,and ϕ∗J, as a function of ∆.

Jamming Phase Diagram

The key elements of this model can be summarized inthe form of the jamming phase diagram (see Fig. 11). ϕJ max = 1 for all ∆. The jammed packings above ϕ∗Jare all accessible in the sense they can be reached bycompressing the fluid from an equilibrium configurationat the appropriate density. The jammed states belowϕ∗J are inaccessible and there are no jammed configura-tions below ϕJ min. The figure also identifies the densityof fluid at the maxima in Cp and PIS, highlighting theconnection between the inherent structure landscape andthe thermodynamic properties of the fluid. VIII. THE FLUIDHeat capacity in 1D is calculated from: Z =11− η, (12)CP = 1/2 +Z1 + ηZ∂Z∂η, (13)η/Z = η (1− η), (14)∂Z∂η=1(1− η)2, (15)6CP = 1/2 +1/ (1− η)1 + η (1− η)× 1(1−η)2,= 1/2 +1/ (1− η)1 + η(1−η)= 1/2 + 1= 1. 5(16)The isobaric heat capacity is Cp = (∂H/∂T )P. Forhard rods the enthalpy is H = (1/2)NkT + PL andCp/Nk = 1/2 + Z/ (1 + d ln {Z} /d ln {ϕ}), (17)where Z = PL/NkT is the compressibility, ϕ =N (xAσAA + xAσAA) /L and N = NA + NB is the to-tal number of particles, and L is the length of the line. In the case of the of equimolar binary hard rod system,Z = 1+ ϕ/(ϕJ − ϕ), and the heat capacity can be calcu-lated as,Cp/Nk = 1/2+Z2/ (Z + (Z − 1) {1 + (Z − 1) (1− dϕJ/dϕ)}). (18)

As the system is compressed, there is competition be-tween free volume entropy and configurational entropy. Basins with denser inherent structures have more freevolume, but there are fewer of them. The equilibriumstate is determined by the competition between thesefactors, so that as the density of the fluid increases, itsamples deeper basins on the landscape. Fig. 8 showsthe heat capacity for different nonadditivity parameterswith different distribution of inherent structures. All ofthe systems show the characteristic maximum in heat ca-pacity of associated with a binomial distribution of states.

In general the pressure of the system is given byP = T(∂S∂V)U. (19)The equilibrium pressure of a liquid can be separated intocontributions from the inherent structure pressure, PIS,and vibrational pressure, Pvib, so that [? ]P = (PIS + Pvib). (20)The inherent structure pressure, PIS, can be calculatefrom configurational entropy which counts the numberof inherent structures. In this system, V = L and combing and simplifyingEqs. 19 and 20 will lead to,βPIS = −ϕ2(∂Sc/Nk∂ϕ)U. (21)Pvib calculated by using equation 20 and subtractingPIS from P. Fig. shows the vibrational and inherentstructure part of the pressure as a function of density. PIS goes through a maximum while Pvib is always anincreasing function.

The equilibrium assumption aboutthe arrangements of the rods on a line leads the systemto samples the basins that maximize the total entropy,so movement between basins results from competitionbetween vibrational entropy and configurational entropy. From PIS and Pvib, this is the PIS that goes through amaximum which could be the reason for the heat capacitybehaviour. Fig. 11 depicts that the maximum of inherent struc-ture pressure as a function of density along with ϕJ minand the CP maximum as a function of density for differ-ent nonadditivity parameters. As it can be seen fromthis figure, isobaric heat capacity get’s it’s maximumvalue as the configurational entropy reaches it’s maxi-mum. As discussed above, ϕJmax is the phase separatedcase and for all non-additivity vales is equal to 1, while,ϕJmin is the case with alternate hard rods. The idealgas configurations under the fast quench condition mapsto the jammed states with ϕJig which associated withxAB = 0. 5. The jammed configurations with densitiesless than ϕJig are not accessible by the equilibrium flu-ids. Despite the many advantages regarding hard rod sys-tem, still the dynamics connection with the jamminglandscape is missing. Therefore in the following, we willdescribe a system of hard disks confined into a narrowchannel that has the all advantages from the hard rodmodel also, the dynamics of the system can be studied.

Concluding Remarks

The properties of liquids, glasses and jamming phe-nomena have always been an interesting and challengingtopics in the field of soft and condensed matter, and af-ter decades of research, many of their features are stillnot well understood. The potential energy landscape, orits hard particle equivalent, was introduced to provide aframework for describing the properties of these systems. However, the complexity of landscape and the challengesassociated with mapping configurations to their local in-herent structures make it difficult to determine exactlyhow the thermodynamics and dynamics are related to thefeatures of the landscape.