Stripes, Zigzags, and Slow Dynamics in Buckled Hard Spheres
aa r X i v : . [ c ond - m a t . s o f t ] D ec Stripes, Zigzags, and Slow Dynamics in Buckled Hard Spheres
Yair Shokef ∗ and Tom C. Lubensky Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA (Dated: October 29, 2018)We study the analogy between buckled colloidal monolayers and the triangular-lattice Ising an-tiferromagnet. We calculate free volume-induced Ising interactions, show how lattice deformationsfavor zigzag stripes that partially remove the Ising model ground-state degeneracy, and identify theMartensitic mechanism prohibiting perfect stripes. Slowly inflating the spheres yields jamming aswell as logarithmically slow relaxation reminiscent of the glassy dynamics observed experimentally.
PACS numbers: 82.70.Dd,75.50.Ee,75.10.Hk,81.30.Kf
Geometric frustration, manifested for example in theanti-ferromagnetic (AF) Ising model on a triangular lat-tice [1], occurs whenever local interaction energies cannotbe simultaneously minimized. It gives rise to highly de-generate ground states, unusual phases of matter [2], andpossibly slow or glassy dynamics [3], whose propertieseven after decades of research are not fully understood.Here we present a theoretical study of a colloidal system,one of a class of artificial frustrated systems in which thestate of each constituent can be directly visualized [4],that provides new insight into the microstructure of frus-trated systems and its connection with their dynamics.We study hard spheres confined between parallelplates. For plate separation slightly greater than thesphere diameter and at sufficiently high sphere density,the spheres buckle upward or downward [5, 6, 7] fromtheir lower-density positions on a hexagonal lattice. Thisbuckling gives rise to a choice of two states for eachsphere, analogous to the two states of Ising-model spins[5, 8]. The tendency of spheres to maximize free vol-ume introduces an effective repulsive interaction that fa-vors configurations with neighboring spheres in oppositestates just as the AF Ising interaction favors oppositestates of neighboring spins. As in the triangular-latticeAF Ising model, frustration arises because it is impossibleto arrange the three particles on any triangular plaquettesuch that all pairs of neighboring particles are in oppositestates. In the AF Ising model on a rigid lattice, there isan extensive number of ground-state configurations (im-plying an extensive ground-state entropy) in which neigh-boring spins on two of the three bonds on each plaquetteare in opposite states. Given the analogy just drawn be-tween our colloidal system and the AF Ising model, it isreasonable to conjecture that the colloidal system mightexhibit ground-state degeneracies and dynamics similarto those of the rigid-lattice AF Ising model. Recent ex-periments in diameter-tunable-microgel systems revealedsub-extensive ground-state entropy and glassy dynamics[9]. Our theoretical study will address the differencesbetween the colloidal system and the rigid lattice Isingmodel and make some conjectures about a likely closeranalogy between the colloidal system and the AF Isingmodel on a compressible lattice [10]. We show that the short-ranged AF behavior in thishard-sphere system may be explained by a simple geo-metrical model relating it to the nearest-neighbor Isingmodel. However, the out-of-plane buckling induces localin-plane lattice distortions that, as in the elastic Isingmodel, partially remove the Ising ground-state degen-eracy and select configurations with zigzagging stripesof up and down spheres. This ‘ground-state’ lacks thelocal zero-energy modes found in the Ising model on arigid triangular lattice, and as a result, the colloidal sys-tem exhibits dynamics that are qualitatively slower thanthose of the Ising model. Moreover, stripes require globaldeformations that are incompatible with the system’sboundary conditions; consequently they break up into aMartensite [11] and form a new partially disordered andhighly degenerate ‘ground-state’.The free energy of our hard-sphere system is dictatedby its phase-space volume, which is a collective functionof all the particles in the system; hence this system is notexactly equivalent to an Ising model with pairwise addi-tive interactions. Nonetheless, we may compare our sys-tem to the nearest-neighbor Ising model on the triangularlattice and ask what is the strength of AF interactionsthat best describes the hard-sphere system.A ‘microscopic’ state is specified by the positions { x i , y i , z i } of all particles 1 ≤ i ≤ N . We coarse-grainthese states into Ising-like configurations specified by { s i } with s i = sign( z i ) ( x, y are the coordinates in the planeof the confining walls, z is perpendicular to the walls,and z = 0 is at the middle of the cell). The probabilityof a particular configuration { s i } is equal to the 3 N -dimensional integral V ( { s i } ) over all states belonging tothat configuration, divided by the total phase-space vol-ume V tot of all configurations: P HS ( { s i } ) = V ( { s i } ) /V tot . (1)We would like to equate this to the probability of findingthe corresponding configuration in the Ising model, P I ( { s i } ) = exp (cid:16) βJ X s i s j (cid:17) /Z, (2)where β = 1 /k B T is the inverse temperature, J is the in-teraction strength, the sum runs over all nearest neighborpairs, and Z is the canonical partition function. V + V - d/2d/2d 2L h- - -V - E - =-2J - + -V + E + =2J(A)(B) (C) FIG. 1: (Color online) Free volume model: A) Top view. Con-tributions to free volume originate from motion along axes toeach of the neighbors. B) Side view. Up particle surroundedby down particles has more free volume (lower energy) than adown particle. C) Surrounding neighbors touch wall and areseparated by 2 L , central particle is confined to the verticalplane. Large circles are volumes excluded by neighbors, hor-izontal bands are volumes excluded by the walls, remainingwhite region is the central particle’s free volume, divided atthe middle of the cell height (dotted line) into V + and V − . Unlike the commonly used cell-model, which approxi-mates the phase-space volume of a system as a productof single-particle free volumes, our model equates P HS and P I by assuming that V ( { s i } ) is a product of con-tributions from all nearest-neighbor ‘bonds’: V ( { s i } ) = Q v ( s i s j ; h, d, L ) , where the pair contribution v dependson s i s j and on the wall separation h , the sphere diameter d , and the in-plane number density, which we character-ize by the spacing L of the underlying triangular lattice.We evaluate v ( s i s j ) in a quasi-one-dimensional approxi-mation by allowing particles i and j to move only in thevertical plane passing through the axis connecting theirlattice positions (see Fig. 1A). We consider a particle,which we call the central particle, and its two neighborsalong one of the principal lattice directions (see Fig. 1B).If the two neighbors are in opposite Ising states, the freevolumes resulting from the central one being up or downare equal by symmetry. When the two neighbors are inthe same state (down, without loss of generality), thecentral particle has more free volume ( V + ) when it is upthan when it is down ( V − ). We calculate V + and V − fromthe geometrical setting of Fig. 1C and equate the ratioof the probabilities of finding the two configurations inFig. 1B for hard spheres to that of the Ising model: V + V − = exp( − βE + )exp( − βE − ) = exp( − βJ ) . (3)From this,we deduce that the hard-sphere system corre-sponds to an Ising model with an effective AF interaction, βJ eff ( d, h, L ) = − ln ( V + /V − ) / < d , h , L , we evalu-ate V + and V − and determine the effective interactionstrength βJ eff . We then use the exact solution of theIsing model [1] to calculate the average number h N f i offrustrated neighbors per particle (we refer to s i s j = − s i s j = 1 as frustrated), which pro- < N f > h/L=1.1h/L=1.3h/L=1.5 FIG. 2: (Color online) Average number of frustrated bondsper particle vs sphere diameter and cell height. Free-volumemodel (lines) agrees with Monte-Carlo simulations (symbols). vides a measure of short-range AF order. h N f i = 2 isthe value in the ground state, and h N f i = 3 correspondsto a random configuration. Figure 2 shows the agreementbetween our simple model and three-dimensional Monte-Carlo (MC) simulations. Simulations included N = 1600spheres with steps consisting of small displacements ofsingle spheres as well as area-preserving box deforma-tions, in which the angle or aspect ratio of the simu-lated parallelogram was allowed to change [7]. To probecases with d > L , we start with striped configurationsthat can accommodate the maximal sphere diameter bylattice deformation (see below), wait 4 × steps, andthen average h N f i over additional 4 × steps. We plotresults only of cases for which d was large enough forthe system to have long-range six-fold orientational or-der Ψ ≡ h exp( i θ jk ) i > . θ jk is the angle the bondbetween j and k forms with an arbitrary axis, and theaverage is over all nearest-neighbors pairs [12]). Smallspheres ( d < L ) in a wide cell ( h/L = 1 .
5) are weaklyconfined, and the approximation that the surroundingspheres in Fig. 1C touch the walls fails, giving rise tosmall differences between the model and simulations.For large spheres ( d > L ), simulations remain jammedin striped configurations and do not increase the valueof h N f i beyond the initial value of 2, even though theyare expected to do so from the free volume considera-tions incorporated in the model. To further explore thisjamming, we conducted MC simulations that started ata disordered configuration with d = L and then orderedas the sphere diameter was gradually increased to somelarger value d . The spheres were initially on a triangu-lar lattice in the xy plane with each sphere randomlytouching either the top or bottom wall. To speed thesimulations, we considered random jumps in the z direc-tion between touching either walls, while keeping the xy displacements continuous. During the swelling process,once every MC step the diameter of all spheres was in-creased to the maximal value allowable without overlaps.Figure 3 shows results of simulations with wall separation h/L = 1 .
3. For d/L = 1 . h N f i = 2 .
12, and the simulation indeed slowly equi-librates to that value by ∼ MC steps. For d/L = 1 . d / L (A)10 MC steps < N f > (B) d/L=1.005d/L=1.01d/L=1.015d/L=1.01 rigid 22.53 Ising −2 −1 FIG. 3: (Color online) A) Sphere diameter, and, B) averagenumber of frustrated bonds per particles following swelling todifferent sphere diameters. Normalized cell height h/L = 1 . T = 0. the system’s relaxation to the value of h N f i = 2 pre-dicted by the model includes a logarithmically slow de-cay to h N f i ≈ . MC steps,followed by a sharp jump to the equilibrium state. For d/L = 1 . h N f i = 2, it gets jammed during the swellingprocess at a state with h N f i = 2 .
35 and does not leaveit over the time scales investigated here. Note that inFig. 3 we plot a single realization for each case, howeverwe observed similar behavior when repeating the simu-lations with multiple realizations. Neither jamming norlogarithmically slow relaxation occur in the Ising modeleven when quenched to zero temperature (see inset).Densely-packed spheres exhibit slower dynamics thanlow-temperature Ising spins on a rigid lattice because themorphology of the maximally-packed hard-sphere config-urations differ from those of the Ising ground-state. Un-like the highly disordered Ising ground-state [1], the hard-sphere ‘ground-state’ consists of parallel zigzag stripes(Fig. 4A). In the Ising model, each triangular plaquttehas one bond frustrated and two satisfied. Although onethird of the bonds in the system are frustrated, and theaverage number of frustrated neighbors per particle is h N f i = 2, not all particles have exactly two frustratedneighbors. By considering the six triangles surrounding acertain spin in the lattice, Fig. 5A shows the five possibleways (up to rotations and spin inversions) to align themsuch that each triangle will have a single frustrated bond.The central spin may have N f = 0, 1, 2, or 3 frustratedneighbors. This leads not only to disorder but also tofast relaxation dynamics since spins with N f = 3 are freeto flip without an energetic cost. For close-packed buck-led spheres, each triplet of spheres in contact defines anequilateral triangle with sides d . As in the Ising model,one of the three spheres is up (or down) and two down(or up), thus tilting this equilateral triangle with respectto the horizontal plane. When projected onto the plane, (A)(B) (C) FIG. 4: (Color online) Final configurations following swellingto d/L = 1 .
01 at h/L = 1 .
3. A) Deformable box. B-C) Rigidbox. System has N = 1600(A-B), 6400(C) spheres. Spherestouching top/bottom wall are dark/bright. Simulation boxesare deformed parallelograms with periodic boundary condi-tions. For ease of presentation we copy the simulated regionand plot a rectangular region of the periodic system. N f =0 N f =1 N f =3N f =2(B)(A) FIG. 5: (Color online) Tiling with: A) equilateral trianglesfor the Ising ground-state, and, B) isosceles triangles for close-packed buckled hard spheres. The large angle β is blackened. the tilted equilateral triangle is deformed to an isoscelestriangle with one long side d along the frustrated bondand two shorter sides x = p d − ( h − d ) < d alongthe satisfied bonds. Each of these isosceles triangles hastwo small angles α = cos − (cid:0) d x (cid:1) < π and a large an-gle β = π − α > π . Now, close-packed configurationsfor the buckled spheres are equivalent to tiling the planewith these isosceles triangles. To completely cover theplane, the angles of the six triangles meeting at each ver-tex must sum to 2 π . Figure 5B demonstrates that for N f = 0 , , β > π , 2 α +4 β > π , and6 α < π , respectively, and thus that the triangles can-not fit together: for the two configurations with N f = 2the angles sum to 4 α + 2 β = 2 π , enabling a perfect tilingcorresponding to the maximal-density close-packed state.The slow dynamics observed for large sphere diametersresult from the lower degeneracy of these zigzagged stripeconfigurations compared to the Ising ground-state. Moreimportantly, the close-packed states with N f ≡ N f = 3 that are crucial forthe low-temperature dynamics in the Ising model [13].Here, many spheres need to cooperatively rearrange inorder for the system to find configurations that maximizethe free volume. Spheres swollen to a very large diame-ter ( d/L = 1 .
015 here) hardly move vertically to changetheir Ising-configuration because the neighboring spheresdo not have enough room to rearrange in the horizontaldirections and to accommodate the lattice deformationsrequired to achieve optimal packing.When the spheres swell slowly enough, they find aconfiguration that maximizes free-volume by each spherehaving exactly two frustrated neighbors. Such configu-rations consist of parallel zigzagging stripes (Fig. 4A).Stripes run only along two of the three principal lat-tice directions, hence the local distortions are non-isotropic and require a macroscopic deformation of thesystem. This is possible in the simulations describedabove in which the shape of the simulation’s bounding-box changes dynamically [7]. However, experimentallythe spheres form crystalline domains separated by grainboundaries [12], which may be better described theoret-ically by rigid boundary conditions. Then, the local ten-dency for zigzag stripes is incompatible with the rigidboundary conditions. The tiling rules for the isosce-les triangles induce local deformations along two of thethree principal lattice directions, and for the systemto be globally isotropic it must break up into domainswith stripes running along different directions. We sus-pect that this is the mechanism leading to the bro-ken stripes seen experimentally [9] and we indeed ob-served such Martensitic states [11] when repeating theswelling simulations without allowing the simulation boxto deform [14]. For instance, the case of h/L = 1 . d/L = 1 .
01 relaxes in the deformable box simulationsto the zigzagged striped state with N f ≡
2, whereas ina rigid box, the average number of frustrated neighborsrelaxes to h N f i = 2 .
05 (Fig. 3B), and the final configura-tion (Fig. 4B) consists of broken stripes. We saw similarstructures (Fig. 4C) and relaxation to the same value of h N f i for N = 100 , , , h N f i slowlygoes to 2 in the large- N limit.The maximal sphere diameter possible in a zigzag con-figuration is equal to that of straight stripes, and thefree-volume-cell approximation does not distinguish be-tween the two N f = 2 configurations corresponding to astraight segment of a stripe and to a bend in the stripes.However simulations and experiments seem to indicate apossible preference for straight stripes over zigzags. It isunclear if the observed zigzag patterns represent equiva-lence between straight and zigzagged stripes, or whetherthe system falls into zigzagged configurations due to ki-netic reasons. It would be interesting to go beyondthe mean-field description, as was done when comparingthe face-centred cubic and the random hexagonal-close-packed structure of hard spheres in three dimensions [15].The relief of frustration by lattice deformation resem-bles the elastic Ising model [10], which when analyzed ex-actly at the microscopic level yields by our isosceles tilingscheme a zigzag-stripe ground-state. It would be interest-ing to further investigate the finite temperature behavior of that model, as well as other models with zigzag-stripeground-states [16].We thank Yilong Han, Matt Lohr, Arjun Yodh, andPeter Yunker for involving us in their experimentalstudy of this topic, and Bulbul Chakraborty, RandyKamien, Andrea Liu, Carl Modes, Yehuda Snir, and An-ton Souslov for helpful discussions. This work is sup-ported by NSF MRSEC grant DMR-0520020. ∗ Present Address: Physics of Complex Systems, Weiz-mann Institute of Science, Rehovot 76100, Israel.[1] G.H. Wannier, Phys. Rev. , 357 (1950); Phys. Rev.B , 5017 (1973). R.M.F. Houtappel, Physica , 391(1950); Physica , 425 (1950).[2] R. Moessner and A.R. Ramirez, Phys. Today , 24(2006).[3] G. Tarjus, S. A. Kivelson, Z. Nussinov, and P. Viot, J.Phys.: Condens. Matter R1143 (2005).[4] D. Davidovi´c et al. , Phys. Rev. Lett. , 815 (1996);Phys. Rev. B , 6518 (1997). H. Hilgenkamp et al. , Na-ture , 50 (2003). R.F. Wang et al. , Nature , 303(2006). Y. Qi, T. Brintlinger, and J. Cumings, Phys. Rev.B , 094418 (2008). A. Lib´al, C. Reichhardt, and C.J.Olson Reichhardt, Phys. Rev. Lett. , 228302 (2006).[5] P. Pieranski, L. Strzelecki, and B. Pansu, Phys. Rev.Lett. , 900 (1983).[6] B. Pansu, Pi. Pieranski, and Pa. Pieranski, J. Physique , 331 (1984). D.H. Van Winkle and C.A. Murray,Phys. Rev. A , 562 (1986). T. Chou and D.R. Nelson,Phys. Rev. E , 4611 (1993). P. Melby et al. , J. Phys.Cond. Mat. , S2689 (2005). N. Osterman, D. Babiˇc, I.Poberaj, J. Dobnikar, and P. Ziherl, Phys. Rev. Lett. ,248301 (2007).[7] M. Schmidt and H. L¨owen, Phys. Rev. Lett. , 4552(1996); Phys. Rev. E , 7228 (1997). R. Zangi and S.A.Rice, Phys. Rev. E , 7529 (1998); , 660 (2000).[8] T. Ogawa, J. Phys. Soc. Jpn. Suppl. , 167 (1983).[9] Y. Han et al. , Nature , 898 (2008).[10] Z.Y. Chen and M. Kardar, J. Phys. C: Solid State Phys. , 6825 (1986). L. Gu, B. Chakraborty, P.L. Garrido,M. Phani, and J.L. Lebowitz, Phys. Rev. B , 11985(1996).[11] K. Bhattacharya, Microstructure of Matersite (OxfordUniversity Press, New-York, 2003).[12] Y. Han, N.Y. Ha, A.M. Alsayed, and A.G. Yodh, Phys.Rev. E , 066127(2003).[14] Note that this equilateral-to-isosceles deformation differsfrom the square-to-triangle Martensitic transition stud-ied in: J.A. Weiss, D.W. Oxtoby, D.G. Grier, and C.A.Murray, J. Chem. Phys. , 1180 (1995).[15] W.G. Rudd, Z.W. Salsburg, A.P. Yu, and F.H. Stillinger,J. Chem. Phys. , 4857 (1968). P.N. Pusey et al. , Phys.Rev. Lett. , 2753 (1989). S.C. Mau and D.A. Huse,Phys. Rev. E , 4396 (1999). C. Radin and L. Sadun,Phys. Rev. Lett. , 015502 (2005). H. Koch, C. Radin,and L. Sadun, Phys. Rev. E , 016708 (2005).[16] Z. Nussinov, Phys. Rev. B69