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Filamentous networks in phase-separatingtwo-dimensional gels

O. Peleg, M. Kroger, I. Hecht and Y. Rabin

EPL, 77 (2007) 58007

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March 2007

EPL, 77 (2007) 58007 www.epljournal.org

doi: 10.1209/0295-5075/77/58007

Filamentous networks in phase-separating two-dimensional gels

O. Peleg1, M. Kroger2, I. Hecht1 and Y. Rabin1

1Department of Physics, Bar–Ilan University - Ramat-Gan 52900, Israel2 Polymer Physics, ETH Zurich, Department of Materials - CH-8093 Zurich, Switzerland

received 20 October 2006; accepted in final form 19 January 2007published online 22 February 2007

PACS 82.70.Gg – Gels and solsPACS 64.70.Kb – Solid-solid transitionsPACS 68.55.Ac – Nucleation and growth: microscopic aspect

Abstract – We introduce a toy model that contains the basic features of microphase separationin polymer gels: a stretched elastic network of Lennard-Jones particles, studied in two dimensions.When temperature is lowered below some value T ∗, attraction between particles dominates overboth thermal motion and elastic forces, and the network separates into dense domains of filamentsconnected by three-fold vertices, surrounded by low-density domains in which the network ishomogeneously stretched. The length of the filaments decreases and the number of domainsincreases with decreasing temperature. The system exhibits hysteresis characteristic of first-orderphase transitions: pre-formed filaments thin upon heating and eventually melt at a temperatureT ∗∗ (>T ∗). Although details may vary, the above general features are independent of networktopology (square or hexagonal), system size, distribution of spring constants, and perturbationsof initial conditions.

Copyright c© EPLA, 2007

Introduction. – Phase transitions in polymernetworks (gels) involve two coupled yet distinct processes:volume transition and phase separation [1]. During avolume transition the gel undergoes a uniform changeof volume by expelling some of the solvent containedwithin it [2–6]. Since this process takes place by (slow)cooperative diffusion, it is well-separated in time fromthe fast local reorganization of the gel which leads toits separation into domains of high and low polymerconcentration (at constant total volume of the gel). Whilethe volume transition is well understood, most of thework on phase separation in polymer gels focused onsurface instabilities [7] and the question of what happensin the bulk of the gel remained largely unresolved. Thisquestion is of considerable interest from the fundamentalperspective because, unlike binary liquids in which twomacroscopic phases are formed in the process of phaseseparation, a gel is a connected network that cannotundergo phase separation on macroscopic scales. While itis obvious that only local reorganization of the polymerconcentration profile can take place in such a system,little is known about the details of this process. Inparticular, it is not clear whether the characteristic lengthscale of microphase separation is of the order of the meshsize or whether cooperative behavior that results fromlong-range elastic interactions can lead to the formation

of much larger domains. The former scenario accordingto which the wavelength of microphase separation isdetermined by molecular length scales, takes place indiblock copolymer mesophases. However, previous theo-retical [8] and experimental [9] investigations of phaseseparation in gels suggest the presence of much largerstructures (spongelike domains in ref. [8] vs. filamentoushoneycomb-shaped networks in ref. [9]).The goal of the present work is to show that the

appearance of such mesoscopic structures is a quite generalfeature of phase separation in a connected network andto obtain some insight about their properties and themechanism of their formation.

The simulation model. – Since microphase sepa-ration in gels arises as the result of interplay betweenattractive forces which promote the appearance of apolymer-rich phase and elastic network forces thatoppose phase separation, we introduce a “minimal”two-dimensional model in which these two (and onlythese two) aspects of real gels are represented, albeit inan idealized fashion.We begin with a perfect square network (see inset to

fig. 1) subjected to periodic boundary conditions. Ateach node in the network we place a particle such thateach of the particles is permanently connected through

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O. Peleg et al.

Fig. 1: (Colour online) Particles and connecting linear springsfor a part of the 100× 100 (periodic) network at T = 0.44. Theinitial “high temperature” square grid topology is depicted inthe inset. Springs connecting isolated particles are shown inblue; those connecting clusters to particles and to other clustersare orange and those connecting particles in the same clusterare green.

identical stretched harmonic springs of zero equilibriumlength and of spring constant k, to four other particles,precisely those which were its nearest neighbors in theinitial configuration. Similarly to the phantom chainmodel of polymer networks [10], the springs are notendowed with any physical attributes such as mass orexcluded volume; nevertheless, since the harmonic springsare stretched (because of periodic boundary conditions,the network is wrapped around a torus), their presencemakes the network behave as an elastic solid. The nextstep is to introduce attractive interactions by havingall the particles interact via the Lennard-Jones (LJ)potential which is known to lead to condensation of gas ofparticles at sufficiently low temperatures [11]. The systemis studied by molecular dynamics simulations in the(N,V, T ) ensemble (N is the total number of particles,V the total area and T the temperature).The system size is chosen large enough to avoid the rela-

tive displacement of any pair of particles above half thebox size, because one cannot uniquely introduce a near-est image convention for springs compatible with thenearest image convention for the LJ particles. This modelfeature poses no difficulty in practice, because its validitycan be monitored during the simulation. The choice of theLJ potential is dictated by convenience: it is the simplestmodel potential that combines both attraction andexcluded-volume effects. Throughout this paper all quan-tities (length, times, energies etc.) are made dimensionlessby expressing them in LJ units [11]. We truncate the LJpotential at an interparticle distance of rcut = 3× 21/6,

and use a velocity Verlet algorithm [12] with integrationtime step ∆t= 0.004 to integrate Newton’s equationof motion; temperature is kept constant by rescalingvelocities [11,12]. We here choose the initial state to bea simple square (or alternatively, a hexagonal) 100× 100grid with grid spacing l0 = 3.5, where four (six) nearestneighbors are assigned to each particle (inset to fig. 1for the square grid). Even though the present model ismotivated by polymer gels which possess entropic elasti-city, in this work we neglect any temperature dependenceof the spring constant k. With the present choices ofk= 1/10 (square grid) and N = 104, the systems can beequilibrated within accessible simulation times (kineticbarriers to structural reorganization increase with springstiffness), for sufficiently broad range of temperatures inthe two-phase region. The above choice of the value of thespring constant is quite arbitrary —the generic behaviorreported in this work occurs for spring constants in therange kmin < k < kmax, where kmin vanishes in the limit ofinfinite system size and kmax is estimated by equating theLJ energy gain for bringing two adjacent lines of particlesparallel to the y-axis close together (each line of lengthL contains

√N particles), to the elastic energy loss due

to uniform stretching of the rest of the network along thex -axis (the lowest-energy elastic-deformation mode).

Percolating high-density clusters. – When such asystem, initially placed to the simple grid, is brought toa temperature higher than some value T ∗ (T ∗ = 0.435±0.01), only isolated particles and small clusters (denseaggregates of particles with nearest neighbors separatedby distance � 1.5) of size n� 15 that break up and reformcontinuously, are observed. A typical snapshot at T = 0.44(started off from the square grid) is shown in fig. 1. Theprobability that a particle belongs to a (small) clusteris slightly above the value obtained for an ideal gas ofnon-interacting particles at the same concentration andwill be quantified below.In order to study phase separation at lower tempera-

tures, we begin with the corresponding ideal (square orhexagonal) configuration and perform a series of temper-ature quenches in the range T > 0.3 (at T < 0.3, steadystate was not reached even after 108 time steps!). Foreach temperature above T = 0.3, a steady state is reachedin the sense that all monitored parameters in the system(e.g., the total number of particles and the total energyof each of the phases) do not change in time, apart fromsmall fluctuations about their mean values. In figs. 2a-b wepresent two typical steady-state patterns observed after55× 106 time steps, at T = 0.41, i.e., slightly below T ∗.The two percolating high-density clusters (PHDCs) can beobtained by using slightly different distortions of the initialperiodic lattice or different choices of seeds for the randomnumber generator of the initial velocities of the particles.Even though the shapes of the two PHDCs in figs. 2a-bappear to be quite different, in both cases the observedshape is generated by a similar mechanism: nucleation

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Filamentous networks in phase-separating two-dimensional gels

(b) (c)(a)

Fig. 2: Typical snapshots of PHDC steady state patterns(relaxed from the initial square grid): (a) single fila-ment, (b) two connected 3-fold vertices (both at T = 0.41),(c) network with eight nodes at T = 0.3.

of a high-density filament that elongates by absorbingsmall clusters at its ends. A linear filament forms whenthis growth takes place along the x- or y-axis (the peri-odic directions) and the ends meet upon traversing theperiodic lattice, thus forming a circle along one of theprincipal axes of the torus. When the growth takes placeat any other direction, the ends wind around the torus,as can be seen in the accompanying movie1, until theychange direction due to thermal fluctuations and collidewith previously formed portions of the filament. A 3-foldvertex forms at each of the collision points of the two ends,resulting in the observed double-vertex shape (>-<) ofthe percolating cluster. Upon closer inspection we noticethat the total area occupied by the PHDC is smaller inthe linear than in the double-vertex case and that moresmall clusters are present in the former case than in thelatter (compare figs. 2a and b). A more quantitative analy-sis shows that the total energy of the system is signifi-cantly lower in the case where the double vertex PHDCis observed, than in the linear PHDC case. Since entropyis not measured in our simulation, this does not tell uswhich of the two states corresponds to a lower free energyof the system. Nevertheless, the analogy with late stagegrowth in phase-separating binary liquids, where surface-energy–driven droplet coalescence processes continue longafter the final equilibrium composition was reached withinindividual droplets of the daughter phase, suggests thatthe state with linear PHDC is metastable with respectto coalescence of the remaining small spherical clusterswith the linear cluster, and the formation of the doublevertex PHDC. Indeed, at T = 0.4 we observed a sequenceof events whereby a linear PHDC was first formed and,later on, a new linear cluster nucleated and grew untilboth its ends collided with the original linear filament,giving rise to a double-vertex structure (not shown).As temperature is further lowered, multiple nucleation

events take place in the simulation box, followed bythe growth of linear filaments and their termination bycollisions. This leads to the formation of a “super network”of dense filaments connected by 3-fold vertices, embedded

1Movies and additional material permanently available athttp://www.complexfluids.ethz.ch/gels (login name and pass-word: EPL).

(a) (b) (c)

Fig. 3: Snapshots of transient patterns (square grid) followingquench to T = 0.2: (a) absorption stage, (b) small clustersorganized along a line, (c) shortly later —a linear filament isformed.

in a dilute phase of isolated particles connected by stronglystretched springs. A typical snapshot of the steady stateof the system at T = 0.3 is shown in fig. 2c where bothhexagons and squares are observed. Interestingly, theshapes resemble those of two-dimensional foams [13], eventhough the physics is very different —interplay of elasticityand attractions in our case, vs. interplay of surface tensionand gas pressure in foams.

Nucleation, growth and hysteresis. – In orderto examine the sequence of events that precede thebirth of a filament, we decreased the temperature toT = 0.2 and followed the dynamics of the system in time(see movie B). Following the quench, numerous compactclusters of up to about n= 10 particles appear in thesimulation box. Each of the particles in such a clusteris typically connected by several springs to the “outside”and therefore there are about n springs that connectneighboring clusters (figs. 3a and b). Since the pullingforce on a cluster due to n parallel springs is n timeslarger than the force exerted by a similarly stretched singlespring (between the cluster and an isolated neighboringparticle), in mechanical equilibrium this force has to bebalanced by a force equal in magnitude and acting alongthe opposite direction. The same argument can be appliedto the neighboring cluster as well, and clusters arrangethemselves along lines of high stress. This breaks radialsymmetry and the resulting critical nucleus has the shapeof a linear filament (fig. 3c), reminiscent of the string-like arrangement of magnetic or electric dipoles. Theelectric/magnetic analogy is not accidental since such acluster can be described as a force dipole [14]. In fig. 4we plot the probability that a particle belongs i) to thePHDC, ii) to a “small” cluster, or that it is iii) an isolatedparticle, as a function of temperature. As T decreases,the probability for a particle to belong to a small clusterdrops sharply from a value exceeding 0.5 to less than 0.1at T ∗ and approaches zero at lower T . A much smallerdrop at T ∗ is observed for the probability to observe anisolated particle. We conclude that the formation of thelarge cluster occurs mainly at the expense of small clustersthat are absorbed by it, reminiscent of late stage growthof droplets in a phase-separating binary liquid (in themetastable region of its phase diagram).

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O. Peleg et al.

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Fig. 4: Plots of probability that a particle belongs to a largecluster (�), a small cluster (©), or that it is an isolated particle(∗). Data for both hexagonal (shown) and square grids withN = 104 particles fall onto the same curves within statisticalcertainty and with the exception, that the high-T values forthe number of small clusters slightly differ corresponding idealgas values (0.438 and 0.487 for the square and hexagonal grids,respectively).

In order to gain further insight into the behavior of thesystem, we start from an equilibrated low temperature(T = 0.38) configuration which contains a PHDC, increaseT to some value larger than T ∗, and monitor the systemduring very long simulation runs (≫ 108 time steps).When T is increased to the range T ∗ <T <T ∗∗ (in whichno filaments are observed under cooling from an initialhigh-temperature state), filaments tend to thin by progres-sive “melting” at their surface and then break. In the 104

particles system the process continues until a single thinfilament remains which appears to be stable during thelongest simulations runs. Finally, at T ∗∗ no filaments areobserved. This behavior is summarized in fig. 5 where theratio of the number of particles in the PHDC to the totalnumber of particles in clusters of all sizes, is monitored.Under cooling from the homogeneous state, the systemfollows the lower branch and jumps to the inhomogeneousfilamentous state at T ∗; conversely, when the system isheated starting from the low-temperature inhomogeneousphase where PHDC are present, it follows the upperbranch up to T ∗∗ = 0.55± 0.02 (for N = 104) and thenundergoes a transition to the homogeneous phase. Suchhysteresis is familiar from the study of first-order phasetransitions (see, e.g., the isotropic-nematic transitionin liquid crystals [15]). The interpretation (based onmean field considerations) is that two free energy minimacorresponding to the two phases are present in the rangeT ∗ <T <T ∗∗. Even though only the lowest minimumcorresponds to the true equilibrium state, a system whichwas initially prepared in the other (metastable) state, willremain in it almost indefinitely if this local free energyminimum is deep enough (compared to thermal energy,T ). We therefore interpret T ∗ and T ∗∗ as the stability

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Fig. 5: Order parameter 0� S � 1 defined as S =L/C where Land C denote the number of particles in the largest cluster andin all clusters (isolated particles do not belong to any cluster),respectively. Data shown for both “cooling” and “heating”runs, results indistinguishable for both the hexagonal andsquare grids with N = 104 particles. States on both branchesare reached from all states within their branch upon heatingor cooling.

limits of the homogeneous and the inhomogeneous phases,respectively. Notice that unlike the true thermodynamictransition temperature which lies somewhere between T ∗

and T ∗∗ and which is strictly defined only in the limitof an infinite system, the latter temperatures have nothermodynamic significance and can depend on the sizeof the system.

Discussion. – All the data presented above wereobtained for a given set of parameters (lattice topol-ogy, system size, density, spring constant, initial config-uration). In order to test the generality of our resultswe performed a set of simulations in which these para-meters were varied. We find that when the square latticeis replaced by a hexagonal one (and the spring constant isreduced from k= 1/10 to k= 1/15 to compensate for theincreased coordination number), all steady-state resultsare little changed, including T ∗, the PHDC patterns andthe temperature ranges in which different patterns areobserved. We also tested the sensitivity of our results tomajor changes in the initial conditions: instead of plac-ing the particles on an ideal lattice, we started froman inhomogeneous “droplet” configuration consisting of adense block of particles (with grid spacing l0 = 1) centeredat the simulation cell, surrounded by a uniform low-density background. When this configuration is quenchedto T < T ∗, the droplet configuration breaks into severalfilamentous branches and a PHDC is formed which issimilar to that obtained by starting the simulation fromthe ideal lattice. Further, we made sure that N = 104

is large enough to not suffer from finite size effects bystudying systems being 5 times larger in each dimen-sion. While T ∗ appears to increase with system size, our

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Filamentous networks in phase-separating two-dimensional gels

100 x 100

500 x 500

hexagonal grid

Fig. 6: Snapshots of large N = 25×N4 and small (insert,N = 104 particles) system at same system parameters (springcoefficient k= 1/15, grid spacing l0 = 3.5, T = 0.3, after1.25× 108 time steps, at t= 50× 104 (initial configuration:hexagonal grid).

qualitative conclusions concerning the structure of thelow-temperature phase (the geometry and the kinetics offormation of the PHDC) remain unaffected (cf. fig. 6).What happens if a random network is employed? In orderto study this issue, we generated a random network ona square lattice (with N = 104 and l0 = 3.5), with springconstants sampled from a uniform random distribution ina symmetric interval about k= 1/10. For narrow intervalsof k values (0.08–0.12), the results are practically indistin-guishable from those for k= 0.1. For k values in the range0.01–0.19), the steady state PHDC patterns observed atlower temperatures remain the same as for the regularnetwork case but the point at which these clusters firstappear is shifted to some temperature above 0.5. The mostpronounced difference is in the temporal history: while inthe regular network case steady state is approached bysingle-step kinetics, the random network exhibits two-stepkinetics with a fast step in which the PHDC is formed,followed by a slow step involving dissolution of remainingsmall clusters and further growth of the PHDC.In conclusion, we would like to mention that the

microphase separation patterns reported in this workbear strong resemblance to those seen in sections ofelastin hydrogels observed by cryoscopic scanning electron

microscopy: a network of 7 nm thick and several hundrednm long filaments, the latter made of spherical beads (seefigs. 3a and b in ref. [9]). This provides support to ourbelief that, despite its simplicity, our model captures themain physical ingredients responsible for phase separa-tion in gels. Work on the effect of temperature on themechanical properties of our networks is in progress andpreliminary results indicate that the appearance of theHDPC is accompanied by a corresponding increase of theshear modulus [16].

∗ ∗ ∗

We would like to thank D.Rapaport for helpful discus-sions. YR’s work was supported by a grant from US-IsraelBSF, MK acknowledges support from the Europeancommunity, EU-NSF contract NMP3-CT-2005-016375.

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[3] Tanaka T., Phys. Rev. Lett., 40 (1978) 820.[4] Tanaka T. et al., Phys. Rev. Lett., 45 (1980) 1636.[5] Ilmain F., Tanaka T. and Kokufuta E., Nature, 349(1991) 400.

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[7] Tanaka T., Sun S. T., Hirokawa Y., Katayama S.,Kucera J., Hirose Y. and Amiya T., Nature, 325(1987) 796.

[8] Sekimoto K., Suematsu N. and Kawasaki K., Phys.Rev. A, 39 (1989) 4912.

[9] McMillan R. A., Caran K. L., Apkarian R. P. andConticello V. P., Macromolecules, 32 (1999) 9067.

[10] Flory P. J., Principles of Polymer Chemistry (CornellUniversity Press, Ithaca) 1953.

[11] Kroger M., Phys. Rep., 390 (2004) 453.[12] Rapaport D., The Art of Molecular Dynamics Simula-

tion, 2nd ed. (Cambridge University Press, Cambridge)2004.

[13] Avron J. E. and Levine D., Phys. Rev. Lett., 69 (1992)208.

[14] Bouchaud J.-P. and Pitard E., Eur. Phys. J. E, 6(2001) 231.

[15] Chaikin P. M. and Lubensky T. C., Principles ofCondensed Matter Physics (Cambridge University Press,Cambridge) 1995.

[16] Peleg O., Kroger M. and Rabin Y., in preparation.

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