Using Brownian dynamics simulations, density functional theory, and
analytical perturbation theory we study the collapse of a patch of
interfacially trapped, micrometer-sized colloidal particles, driven by
long-ranged capillary attraction. This attraction {is formally analogous} to
two--dimensional (2D) screened Newtonian gravity with the capillary length
\hat{\lambda} as the screening length. Whereas the limit \hat{\lambda} \to
\infty corresponds to the global collapse of a self--gravitating fluid, for
finite \hat{\lambda} we predict theoretically and observe in simulations a
ringlike density peak at the outer rim of a disclike patch, moving as an
inbound shock wave. Possible experimental realizations are discussed.Comment: 5 pages, 3 figures, revised version with new Refs. added, matches
  version accepted for publication in PR

Bleibel, J.

Dietrich, S.

Dominguez, A.

Oettel, M.

English

arXiv

J. Bleibel

S. Dietrich

A. Domínguez

M. Oettel

Crossref

Physical Review Letters

Shock Waves in Capillary Collapse of Colloids: A Model System for Two-Dimensional Screened Newtonian Gravity

Max Planck Society eDoc Server

Shock waves in capillary collapse of colloids: a model system for two-dimensional screened Newtonian gravity

Using Brownian dynamics simulations, density functional theory, and analytical perturbation theory we study the collapse of a patch of interfacially trapped, micrometer-sized colloidal particles, driven by long-ranged capillary attraction. This attraction is formally analogous to two-dimensional (2D) screened Newtonian gravity with the capillary length ˆλ as the screening length. Whereas the limit ˆλ→∞ corresponds to the global collapse of a self-gravitating fluid, for finite ˆλ we predict theoretically and observe in simulations a ringlike density peak at the outer rim of a disclike patch, moving as an inbound shock wave. Possible experimental realizations are discussed.DFG SFB-TR6Gobierno de España FIS2008-01339 AIB2010DE-0026

Dietrich, Siegfried

Domínguez Álvarez, Álvaro

Oettel, Martin

idUS. Depósito de Investigación Universidad de Sevilla

Shock Waves in Capillary Collapse of Colloids: A Model Systemfor Two-Dimensional Screened Newtonian GravityJ. Bleibel,1,4 S. Dietrich,1,2 A. Domı́nguez,3 and M. Oettel41Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, 70569 Stuttgart, Germany2Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany3Fı́sica Teórica, Universidad de Sevilla, Apdo. 1065, 41080 Sevilla, Spain4Institut für Physik, WA 331, Johannes Gutenberg Universität Mainz, 55099 Mainz, Germany(Received 6 May 2011; published 16 September 2011)Using Brownian dynamics simulations, density functional theory, and analytical perturbation theory westudy the collapse of a patch of interfacially trapped, micrometer-sized colloidal particles, driven by long-ranged capillary attraction. This attraction is formally analogous to two-dimensional (2D) screenedNewtonian gravity with the capillary length ̂ as the screening length. Whereas the limit ̂ ! 1corresponds to the global collapse of a self-gravitating fluid, for finite ̂ we predict theoretically andobserve in simulations a ringlike density peak at the outer rim of a disclike patch, moving as an inboundshock wave. Possible experimental realizations are discussed.DOI: 10.1103/PhysRevLett.107.128302 PACS numbers: 82.70.Dd, 47.11.Mn, 47.40.xThe dynamics of matter under the influence of long-ranged attractions is studied intensively in several branchesof physics [1], in particular, with respect to inherent insta-bilities. Most prominently, the structure formation in theuniverse is understood as the consequence of an instabilityin self-gravitating matter, and cosmological theories inconjunction with numerical simulations have been success-fully applied to explain the dynamical formation of clus-ters, galaxies, or dark matter halos on large scales [2].More recently, other systems with gravitational-like attrac-tions have been investigated, including seemingly unre-lated phenomena like bacterial chemotaxis [3,4] orcapillary-driven clustering in colloids trapped at fluid in-terfaces [5,6]. In these systems the interaction is effectivelycut off beyond a finite range, albeit much larger than themean interparticle separation.In a self-gravitating fluid any homogeneous mass distri-bution is unstable with respect to small fluctuations onsufficiently large scales [7] (Jeans’s instability). In systemswith a cutoff gravitational-like attraction, this instabilityonly occurs below a critical temperature [4,5]. As the rangeof the interaction is scaled [6] from infinity down to amicroscopic length like the size of the particles, the dy-namical evolution of the instability crosses over fromgravitational collapse to spinodal decomposition. A stan-dard theoretical approach to the gravitational collapse incosmology is the so-called cold collapse approximation(see, e.g., Ref. [8]), within which any force other thangravity (in particular the thermal pressure of the fluid) isneglected altogether. The view on applications to otherphysical situations raises the natural question of how thephenomenology of this scenario is affected by a nonvan-ishing thermal pressure and a large but finite range ofattractions and specifically how the crossover to the spino-dal decomposition scenario occurs.Colloidal particles with radii in the micrometer range,which are trapped at a fluid interface, lend themselves tostudy this issue. Their weight results in a force f on eachparticle perpendicular to the interface which deforms it andgives rise to long-ranged capillary interactions between theparticles. For large colloid center-to-center distances d, theleading interaction term (dominating the collective col-lapse dynamics) is a pair interaction with the potential[9] VðdÞ ¼ ½f2=ð2ÞK0ðd=̂Þ with the modified Besselfunction K0, the capillary length ̂ (OðmmÞ), and theinterfacial tension . For d < ̂ this reduces to 2DNewtonian gravity, V0ðdÞ  ln d, whereas for d > ̂ itdecreases exponentially. This soft matter system is ofparticular interest because the range ̂ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi=ðgmÞp istunable via the dependence of  on temperature T̂ and theconcentration of surfactants or through the mass densitydifference m between the two fluid phases forming theinterface; g is Earth’s gravitational acceleration.Main results.—We study the time evolution of the 2Dparticle number density %̂ðr̂ ¼ fx; yg; t̂Þ of an initially cir-cular patch of radius L with particles of radius R on the flatinterface with a homogenous density %̂0. The range ismeasured by the dimensionless parameter  ¼ ̂=L andwe introduce the effective, dimensionless temperatureT ¼ kBT̂f2%̂0L2: (1)For ̂ ! 1, T is the ratio of the thermal to the (mean-field)attractive inner energy of the patch because each particleinteracts with N  %̂0L2 other particles. For ̂ < L, thenumber reduces to N  %̂0̂2 and this ratio approximatelyequals ð1þ 2ÞT. Figure 1 summarizes qualitatively thedynamic phases we have found. If T is large enough (abovethe black line), the capillary attraction cannot confine thePRL 107, 128302 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending16 SEPTEMBER 20110031-9007=11=107(12)=128302(4) 128302-1  2011 American Physical Societyparticles to the circular patch, so that it becomes more andmore diluted as time progresses. If T is small (dashed line)the patch collapses, but how this proceeds depends on . Inthe limit  ! 1 (gravitational collapse), the fastest grow-ing modes span the patch and the evolution is dominated bythe collapse of the structure as a whole. In the oppositelimit  ! R (spinodal decomposition), the fastest growingmodes have a characteristic length scale well below L andthe evolution is dominated by the coarsening of thesedomains. We have explored the transition region betweenthese limits using perturbation theory, Brownian dynamics(BD) simulations, and dynamic density functional theory(DDFT). Accordingly, the most prominent feature of thistransition is a density peak forming at the outer rim of thecollapsing circular patch which exhibits properties of ashock wave. The latter is defined by the crossing of thecharacteristic curves of a differential equation. For Eq. (2)below, in the presence of radial symmetry, these character-istics are the trajectories of rings of particles. Rings withinitially larger radii travel faster than those with smallerradii and, upon crossing, form a density singularity asT ! 0 [Fig. 2(a)]. This feature becomes more pronouncedwith  becoming smaller, because the time for building upthe shock wave diminishes relative to the collapse time ofthe whole patch (Fig. 3). On the other hand, for smaller additional small clusters form in the interior of the patchand the shock wave amplitude is reduced [Fig. 2(b)].Theory.—We consider a 2D fluid within a mean-fieldapproximation appropriate for the long-ranged, capillaryinteractions and take into account nonzero temperature andshort-ranged interactions through a pressure equation ofstate p̂ð%̂; T̂Þ [5], containing the effects of short-rangedinteractions such as hard or soft cores and subleading termsin the capillary forces. The particle dynamics is assumed tobe in the overdamped regime with  as the associatedinterfacial mobility of the particles and hydrodynamicinteractions are neglected. The relevant time scale of col-lapse is Jeans’s time T ¼ =ðf2%̂0Þ. We note that thetime it takes a particle to diffuse a distance L by Brownianmotion alone is L2=ðkBT̂Þ ¼ T =T. Dimensionlessvariables are introduced as % ¼ %̂=%̂0, p ¼ p̂=ðkBT̂%̂0Þ,r ¼ r̂=L, and t ¼ t̂=T together with Eq. (1). Mass conser-vation reads [5]@%@t¼ r  ð%rU½%  TrpÞ; (2)with the dimensionless potential of capillary interac-tion U½%ðrÞ ¼ ½1=ð2ÞR dr0%ðr0ÞK0ðjr r0j=Þ. Foridealized point particles the pressure is pid ¼ %; wehave also considered a fluid of hard discs of radius R,described by [10] phdð%Þ ¼ %ð%c þ %Þ=ð%c  %Þ, where%c ¼ ð2ffiffiffi3p%̂0R2Þ1 is the dimensionless density of closeFIG. 2 (color). Evolution of the radial density profile for T ¼ 3:1 104 (for further parameters, see the main text). Panel (a): ¼ 0:25, comparison between 2D DDFT (colored lines), and BD (symbols). Panel (b):  ¼ 0:075, only BD (DDFT results omittedfor clarity). Unlike as in (a), small transient peaks distinct from the one at the outer rim are observed due to clustering near the center ofthe collapsing disc.FIG. 1 (color). Proposed sketch of dynamical regimes for acircular patch of radius L with particles of radius R as a functionof the range  ¼ ̂=L of the interaction and a rescaled effectivetemperature Tð1þ 2Þ [see Eq. (1) and below]. The rescalingfactor for T leads to a horizontal border separating the collapseand dilution regimes. Neglecting a possible temperature depen-dence of ̂, isotherms are parallel to the dashed black line. Thetransition region is bounded approximately by the line for whichlinear stability theory for an infinite homogeneous distribution[5,6] predicts that the fastest growing density mode has a wavenumber 2=L.PRL 107, 128302 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending16 SEPTEMBER 2011128302-2packing. Equation (2) can be viewed as a simple DDFT forthis system [5] which, given the long range of the attrac-tions, is expected to hold at least for scales much largerthan R and to describe correctly the collective aspects ofthe dynamics. Deviations are likely to occur for smallerscales and are mainly attributable to the local densityapproximation, i.e., the crude form of the term / rp forthe short-ranged interactions. Here, improvement could beachieved using more sophisticated expressions from DFT.The cold collapse (CC) scenario [formally T ¼ 0 inEq. (2)] is best studied using Lagrangian coordinates: thecharacteristic curves of Eq. (2) are the radial trajectories ofinfinitesimally thin rings of particles, assuming that theinitial radial symmetry is preserved by the mean-fieldevolution. If the initial radius is r0, the trajectory is de-scribed by a time dependent mapping r ¼ rLaðr0; tÞ. TheJacobian of this mapping provides the density field inLagrangian coordinates:1%Laðr0; tÞ ¼rLar0@rLa@r0; ðr0  1Þ: (3)The mapping is well defined and invertible as long as tworings do not cross. If two rings cross, the Jacobian vanishesand the density field exhibits a singularity. For  ! 1, theCC approximation yields [5,11] rLa ¼ r0ffiffiffiffiffiffiffiffiffiffiffi1 tp (t  1)and %Laðr0; tÞ ¼ 1=ð1 tÞ. Thus, the initial homogeneityinside the patch is preserved and a singularity arises att ¼ 1 (i.e., at Jeans’s time), when all the rings reach thecenter simultaneously. For finite  we have applied pertur-bation theory in terms of 1= leading to [6]rLar02 ¼ 1 tþ 2 4Euler  r20  lnð42Þð4Þ2 tð2 tÞ 2 lnð42Þð4Þ2 ð1 tÞ2 lnj1 tj; (4)1%Laðr0; tÞ ¼rLar02  tð2 tÞr042: (5)This result predicts the formation of an overdensity forthe outermost ring (r0 ¼ 1) which becomes singular(1=%La ¼ 0) at a time ts  1þ lnð1:44Þ=ð2Þ2 when itreaches a radius rs ¼ rLað1; tsÞ  1=ð4Þ. (The singularityis a consequence of the CC approximation; it is actuallyregularized into a density peak moving like a shock waveby the term / rp.) Thus, with decreasing  the collapsesingularity occurs later and at larger radii. Intuitively, fordecreasing  the range of the interaction is confined tosmaller regions around any point of the disc. In addition theouter rim of the disc experiences no balancing pull fromthe outside, so that there the first overdensity builds up,which then attracts more and more particles.Numerical methods and discussion.—In order to test thistheoretical analysis we have performed simulations andsolved Eq. (2) numerically. Simulation parameters werechosen to reflect the conditions in an actual experimentalrealization [5] (patch size L ¼ 1:83 mm, particleradius R ¼ 10 m, capillary potential depth f2=ð2Þ ¼0:89kBT̂, particle mobility  ¼ 3waterR with waterbeing the water viscosity at room temperature, and theparticle hard core realized by the repulsive part of theLennard-Jones potential [12]). The simulations were car-ried out with N ¼ 1804 particles using BD [6,13] and theradial density profile was obtained through angular andensemble (120 runs) averages. Numerical solutions wereobtained either through a numerical integration of Eq. (2)with enforced radial symmetry or through a particle-based(Lagrangian) integration scheme of the full 2D equation(labeled as 2D DDFT) [6,14]. In this latter case, the densityfield is probed by a discrete number of (fictitious) particleswhich follow the characteristic curves of Eq. (2) and theFIG. 3 (color). Snapshots from BD simulations for temperature T ¼ 3:1 104 and  ¼ 1:5 (upper row) or  ¼ 0:25 (lower row).Particle distributions in each column have the same radial extent. Clusters (i.e., particles with at least 3 neighbors within a distance of3:25R) are depicted in red. For  ¼ 1:5 the global collapse appears to be faster than the formation of individual smaller clusters. For ¼ 0:25, small clusters predominantly form at the outer rim and collectively move towards the center.PRL 107, 128302 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending16 SEPTEMBER 2011128302-3density profile is obtained like the profiles from BDsimulations.We find that the value  ¼ 1:5 (i.e., ̂  2:7 mm,which is the capillary length of the air-water interface atambient conditions) is at the limit of applicability of theperturbation theory: the DDFT solutions for T ¼ 0 confirmthe occurrence of the singularity at the predicted valuesrs  0:16, ts  1:1 and that the formation of the singular-ity indeed slows down and occurs at larger values of rs withdecreasing  (rs  0:69, ts  1:7 for  ¼ 0:25). TheDDFT solutions at nonzero T ¼ Oð103–105Þ showthat, as for T ¼ 0, an overdensity peak forms at the rimwhile traveling inwards with increasing amplitude, but thedevelopment into a singularity is inhibited.The results from the BD simulations confirm this andprovide further details about the dynamical evolution, seeFigs. 2 and 3. For  ¼ 0:25 at T ¼ 3:1 104, in contrastto the collapse at larger , a ring-shaped densified zoneforms which is composed of a number of smaller clusters.The 2D DDFT solutions agree well with the BD simulationdata for all times. This indicates that the features weare analyzing are not affected by the details of the micro-scopic correlations, the effect of which can be taken intoaccount through the macroscopic pressure term. Upondecreasing  further, the formation of even more individualclusters inside the disc is observed, as a prelude to thespinodal decomposition scenario. For the particular value ¼ 0:075 the averaged radial density profile still exhibitsthe collective behavior of the shock wave at the outer rimalbeit with a smaller amplitude. Additionally, transientpeak structures for smaller radii also become visible andare eventually absorbed by the shock wave travelinginwards.The phenomenology of local clustering due to smallinitial fluctuations is also visible in the azimuthal direction,both in the BD simulations and in the 2D DDFT solution.In both cases the initial conditions break the radial sym-metry, which is expected to be recovered by an ensembleaverage.Concerning a possible experimental realization, asuitable value of T can be arranged using colloids withR 10 m [5]. Observation of the ring-shaped densitybuildup and the ensuing shock wave requires =L & 0:25,corresponding to L * 10 mm for an air-water interface. Alikely relevant issue for experiments is the role of hydro-dynamic interactions. We have carried out simulationsincorporating these on the Rotne-Prager level (as formu-lated in Ref. [15]) for the specific case of particles with acontact angle close to zero (i.e., just touching the inter-face). Our results indicate that the qualitative features ofthe evolution discussed here are not affected and the col-lapse is simply accelerated. This is in line with the experi-mentally observed enhanced colloid self-diffusion due tohydrodynamic interactions in such a system [16].Summary and conclusions.—We have studied the col-lapse of a homogeneous, circular patch of colloidal parti-cles trapped at a fluid interface by means of analyticalperturbation theory, Brownian dynamics simulations, anddynamical density functional-like theories. The capillaryattraction is formally analogous to two-dimensional grav-ity with a tunable cutoff length . We find that a finite valueof  strongly influences the collapse features. While for ! 1 the evolution is dominated by the global collapseof the patch, a large but finite  induces the formation of aringlike overdensity which quickly becomes singular in thelimit of a vanishing pressure force (i.e., zero temperature orcold collapse). A nonvanishing pressure regularizes thesingularity into a collapsing shock wave. System parame-ters can be chosen such that these spatiotemporal structurescan be realized in experiments with micrometer-sized col-loids. Furthermore, this system appears to be ideally suitedto investigate the transition from Jeans’s gravitational in-stability ( ! 1) to a spinodal instability ( colloidradius).Financial support by the DFG through SFB-TR6‘‘Colloids in External Fields’’ (Project N01), by DAAD-PPP and by the Spanish Government through GrantsNo. FIS2008-01339 (partially financed by FEDER funds)and No. AIB2010DE-00263 is acknowledged.[1] A. Campa, T. Dauxois, and S. Ruffo, Phys. Rep. 480, 57(2009).[2] V. Springel et al. Nature (London) 435, 629 (2005).[3] E. F. Keller and L.A. Segel, J. Theor. Biol. 26, 399 (1970).[4] P. H. Chavanis and L. Delfini, Phys. Rev. E 81, 051103(2010).[5] A. Domı́nguez, M. Oettel, and S. Dietrich, Phys. Rev. E82, 011402 (2010).[6] J. Bleibel, A. Domı́nguez, M. Oettel, and S. Dietrich(unpublished).[7] J. H. Jeans, Phil. Trans. R. Soc. A 199, 1 (1902).[8] V. Sahni and P. Coles, Phys. Rep. 262, 1 (1995).[9] M. Oettel and S. Dietrich, Langmuir 24, 1425 (2008).[10] E. L. Grossman, T. Zhou, and E. Ben-Naim, Phys. Rev. E55, 4200 (1997).[11] P. H. Chavanis and C. Sire, Phys. Rev. E 83, 031131(2011).[12] J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem.Phys. 54, 5237 (1971).[13] M. P. Allen and D. J. Tildesley, Computer Simulation ofLiquids (Clarendon, New York, 1989).[14] N. Y. Gnedin and L. Hui, Mon. Not. R. Astron. Soc. 296,44 (1998).[15] B. Cichocki, M. L. Ekiel-Jezewska, G. Nägele, and E.Wajnryb, Europhys. Lett. 67, 383 (2004).[16] K. Zahn, J.M. Mendez-Alcaraz, and G. Maret, Phys. Rev.Lett. 79, 175 (1997).PRL 107, 128302 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending16 SEPTEMBER 2011128302-4

Shock waves in capillary collapse of colloids: A model system for two-dimensional screened Newtonian gravity

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Phys. Rev. Lett.

Shock waves in capillary collapse of colloids: a model system for
  two--dimensional screened Newtonian gravity

Shock waves in capillary collapse of colloids: a model system for two--dimensional screened Newtonian gravity

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