Capillary attraction induced collapse of colloidal monolayers at fluid interfaces

Capillary attraction induced collapse of colloidal monolayers at fluid interfaces J. Bleibel,* ab A. Dom´ ınguez, c M. Oettel a and S. Dietrich bd We investigate the evolution of a system of colloidal particles, trapped at a fluid interface and interacting via capillary attraction, as a function of the range of capillary interactions and temperature. We address the collapse of an initially homogeneous particle distribution and of a radially symmetric (disk-shaped) distribution of finite size, both theoretically by using a perturbative approach inspired by cosmological models and numerically by means of Brownian dynamics (BD) and dynamical density functional theory (DDFT). The results are summarized in a “dynamical phase diagram”, describing a smooth crossover from a collective (gravitational-like) collapse to local (spinodal-like) clustering. In this crossover region, the evolution exhibits a peculiar shock wave behavior at the outer rim of the contracting, disk-shaped distribution. 1 Introduction Partially wetted colloidal particles can be strongly trapped at uid interfaces. There they form colloidal monolayers, which constitute systems of low spatial dimensionality exhibiting a rich and interesting phenomenology. This is in part due to the effective capillary force acting between the particles, which is tied to the presence of a deformable uid interface. Driven by basic research as well as application perspectives, capillary interactions have recently received renewed interest both in theory and in experiment, see, e.g., ref. 1–3 and references therein. If the colloidal particles deform the interface as a consequence of an external force acting on them, such as their weight or an externally imposed electric eld among several actual experimental realizations, the dominant contribution to the capillary interaction is the so-called “capillary monopole”.† In the (experimentally realistic) limit of small interfacial deformations, this force is always attractive and has a range given by the capillary length l¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g=ðgDrÞ p, where gis the surface tension of the interface, gis the acceleration of gravity, and Dris the mass density difference between the uids on both sides of the interface. In typical experimental congurations the interparticle separation (micron) is much smaller than l(millimeter), so that any given particle interacts simultaneously with a huge number of neighbors. Furthermore, at separations smaller than l, the capillary attraction varies logarithmically and is formally analogous to two-dimensional (2D) Newtonian gravity. In these circumstances, the capillary interaction is equivalent to “screened 2D Newtonian gravity”, with a screening length lmuch larger than any microscopic length scale. Therefore we call this interaction “long-ranged”.In this context the tunability of such a system is of special interest. The range of the interaction may in principle be varied by changing the capillary length, which is accessible via changes of the surface tension of the interface, e.g., due to the addition of surfactants. Additionally, the relevant ratios of the interaction range to other length scales of the system (most notably, the interparticle separation and the system size) can be easily altered also by changing these other lengths in experimental setups.‡Likewise, the amplitude of the capillary monopole and, correspondingly, the strength of the interaction are controllable by means of the external force. Therefore, for micron-sized colloidal particles, such a monolayer can serve as a 2D model system for studying the inuence of a “long-ranged”interaction as a function of its parametric characteristics. long-ranged interactions are a subject of active research in various branches of physics (see, e.g., ref. 6 and references therein), involving rather distinct scales. Ranging from cosmology down to chemotaxis in bacterial populations, a common feature of these systems is their temperature a Institut f¨ ur Angewandte Physik, Auf der Morgenstelle 10, Eberhard Karls Universit¨ at, 72076 T¨ ubingen, Germany. E-mail: johann[email protected] b Max-Planck-Institut f¨ ur Intelligente Systeme, Heisenbergstr. 3, 70569 Stuttgart, Germany c F´ ısica Te´ orica, Universidad de Sevilla, Apdo. 1065, 41080 Sevilla, Spain d IV Institut f¨ ur Theoretische Physik, Universit¨ at Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany †In the absence of an external force, the dominant contribution is the “capillary quadrupole”, which is related to deviations from the spherical symmetry of the particles. See, e.g., ref. 4 for a recent review. Cite this: Soft Matter,2014,10,4091 Received 10th December 2013 Accepted 18th February 2014 DOI: 10.1039/c3sm53070a www.rsc.org/softmatter ‡In this regard, we briey remark on the possibilities offered by the technique of optical tweezers. They provide effective walls for the colloidal particles (see, e.g., ref. 5) but do not alter the interface, so that the system size is controllable without affecting the capillary forces acting on the particles. This is unlike the more direct method of connement by means of physical walls, which may induce an undesired interfacial curvature or “external”capillary forces. This journal is © The Royal Society of Chemistry 2014 Soft Matter,2014,10,4091–4109 | 4091 Soft Matter PAPER Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online View Journal | View Issue dependent instability with respect to clustering. 7–10 In recent studies, we investigated this clustering instability in monolayers of micron sized particles driven by a monopolar capillary interaction, 10–12 paying special attention to the inuence of the ratio between the capillary length and the system size. In the formal limit l/N, corresponding to 2D Newtonian gravity, cosmology offers a useful analogy for tackling the problem within the so-called “cold collapse”approximation, which is widely used to study clustering of collisionless matter: one neglects any force other than gravity (including diffusion by thermal motion). Within this approximation, the dynamical equations of our model are amenable to an analytical solution, which provides the basis for perturbative calculations when lis nite. Numerical solutions and N-body simulations allow for going beyond the simplifying assumptions of the analytical approach. Here our goal is to provide a complete characterization of the evolution induced by the clustering instability as a function of the relevant parameters of the system. In this respect, in ref. 10 and 12, a homogeneous particle distribution (either macroscopically extended or with periodic boundary conditions) was studied, while ref. 11 has addressed a radially symmetric (disk-shaped) distribution with a nite radius, which is more realistic from an experimental point of view. There a “dynamical phase diagram”was proposed in order to summarize the ndings (see Fig. 1 in ref. 11): there is a region of stability at high temperatures, and a region of instability at low temperatures. In the latter region, the evolution proceeds as in the case of gravitational cold collapse if lis larger than the system size, and as in the case of spinodal decomposition if lis much smaller than the system size. There is no sharp borderline between these two limiting cases but instead a crossover region which exhibits characteristic features of both regimes. In a disk-shaped particle distribution, the presence of the explicit outer boundary brings into focus this crossover region as in the course of the collapse a shock wave-like structure is formed. First corresponding results are communicated in ref. 11. Here we elaborate and extend the theoretical model and the perturbative calculations underlying this study, including the proposed “dynamical phase diagram”. Resorting to Brownian dynamics (BD) simulations and dynamic density functional theory (DDFT) we investigate this diagram concerning the dynamic evolution of a disk-shaped nite distribution of particles. Both the BD and the DDFT results support the picture captured by the “dynamical phase diagram”. Thepaperisorganizedsuchthatinthenextsectionwe shall present the theoretical model. In Subsection 2.1 we shall introduce the denitions and the terminology used throughout the paper. We rst recall briey the theory for colloidal systems with monopolar capillary attractions and the model for the dynamical evolution. Next, in Subsection 2.2 we shall discuss the linear stability analysis of the homogeneous system, based on which we identify various dynamical regimes in the “dynamical phase diagram”as a function of the screening length land a rescaled, effective temperature. This will be followed by a formal denition of the cold collapse model, which serves as the baseline for the formulation of an analytical perturbation theory for the capillary collapse of colloids trapped at a uid interface. The main results are summarized in Subsection 2.3 and certain derivations are giveninAppendixA.InSection3weshalldiscussthe numerical methods we have applied in order to investigate the capillary collapse. In Section 4 we shall report on simulations for various setups so as to explore the “dynamical phase diagram”also for the case of nite-sized circular patches of particles, as a function of an effective temperature and of the range of the interaction. In order to render a comparison with future experiments feasible, in Section 5 we shall briey discuss how a change of the interface tension affects the relevant variables for the setup of a homogeneous system or for that of a collapsing disk. Finally, in Section 6 we shall summarize and present our conclusions. 2 Theoretical model 2.1 Basic features and description of the dynamic evolution We consider a two-dimensional set of interacting particles (Fig. 1). For the system under study, we map a distribution of spherical colloidal particles of radius R, which are trapped at a uid interface, onto a two-dimensional distribution of disks in a at reference plane. For small deformations of the interface as considered here, this renders a two-dimensional distribution of circular disks with radius R 0 . 1,13,14 (The relationship between Rand R 0 depends on the contact angle of the interface at the particle surface.) Their dynamical model as used here is introduced in ref. 10 and is brieyrecalled here. The particles are subject to capillary forces. These are generated by the interfacial deformation upon the action of an external force fon the particles (which in turn is, e.g., caused by the weight of the particles and their buoyancy at the interface) in the direction perpendicular to the reference plane. Within mean-eld theory, the linearized Young–Laplace equation governing the interfacial deformation U(^ r), ^ r˛R 2 , is given by 1,13 ^ V2UU l2¼f gr;(1) where r(^ r) is the 2D particle number density eld in the reference plane and gis the surface tension. (The interfacial deformation for micron-sized particles is in the nm range, which justies the linearized approximation.) This interfacial deformation gives rise to an in-plane capillary force described by the areal force density Fcap ¼fr^ VU:(2) The capillary force between two particles can be cast into a form such that a corresponding pair potential is given by 1 VcapðdÞ¼ f2 2pgK0d l(3) in terms of the modied Bessel function K 0 , as obtained from Green's function for eqn (1), and the center-to-center distance d between the particles. The capillary length lacts as a cutofffor 4092 |Soft Matter,2014,10,4091–4109 This journal is © The Royal Society of Chemistry 2014 Soft Matter Paper Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online the interaction range because K0ðd=l/NÞ¼ 2 p d l  1=2 ed=l. In the opposite limit, dl, this potential coincides with the Newtonian gravitational interaction in d¼2, K 0 (d/l/0)  ln(l/d), which is nonintegrable in the sense of statistical mechanics, i.e., it leads to a nonadditive (hyperextensive) internal energy. In view of this property and due to the wide separation of scales in colloidal systems, l¼O(mm) and R 0 ¼ O(mm), for the present purposes we call V cap (d)a“long-ranged interaction”, as opposed to other interactions which will be called “short-ranged”. Concerning the dynamics, we assume an overdamped motion for the particles adsorbed at the interface. For reasons of simplicity, as a rst step we neglect hydrodynamic interactions§so that the in-plane number density current of the particles is proportional to the driving force, rv¼G(F short +F cap )(4) with an effective single-particle mobility G.F short is the sum of the areal thermodynamic force densities associated with thermal motion and with any other (short-ranged) interaction between the particles. Assuming local equilibrium, one can write Fshort ¼^ Vp, where the 2D pressure eld p¼p(r)is given by an appropriate equilibrium equation of state describing the macroscopic manifestation of the short-ranged forces. Therefore, the continuity equation for the particle number density is vr v^ t¼^ V$ðrvÞ¼G^ V$^ Vpfr^ VU  :(5) We introduce a characteristic particle density r 0 and the length scale L 0 associated with the size of the system. (For the homogeneous systems considered below, L 0 is the side length of the periodically replicated square box and for the collapsing disk scenario, it is the initial radius of the particle-covered disk.) We also introduce the time scale T:¼g/(Gf 2 r 0 ), which is the so-called Jeans' time associated with the initial homogeneous density r 0 . 10 This allows us to form the following dimensionless variables and quantities: r:¼^ r L0 ;t:¼ ^ t T;k:¼L0 l;nðr;tÞ:¼r^ r;^ tðÞ r0 ;(6a) wðr;tÞ:¼g fr0L02U^ r;^ tðÞ;PðnÞ:¼g f2r02L02pðrÞ:(6b) In these terms, eqn (1) and (5) take the form: V 2 wk 2 w¼n,(7a) vn vt¼V$½nVwVP:(7b) It is useful to introduce the (dimensionless) chemical potential m¼^ mg/(f 2 r 0 L 02 ) which is associated with the pressure via the Gibbs–Duhem relationship:{ vm vP ¼1 n 0mðn;TÞ¼ðn n* dn0 n0 dP dn0þmðn*;TÞ;(8) with a certain constant n*so that eqn (7b) can be also written as vn vt¼V$½nVðmwÞ:(9) Alternatively, one can introduce a free energy functional F[n,w] (see ref. 10), such that dF dn¼mw;(10) so that eqn (7a) and (7b) can be written as Fig. 1 Sketch of the system under consideration. Colloids are trapped at the interface between two fluids with surface tension g. Around each colloid a dimple is formed because the external force facting on each colloid pushes the colloids into the lower fluid (often water). The dimple depth is proportional to f/g. Thus for micron-sized colloids with fcaused by their own weight (or buoyancy) the depth is in the range of a few nm. Therefore the system is effectively two-dimensional, and the colloids interact via the attractive, long-ranged capillary pair potential V cap (d) (see eqn (3)) in addition to short-ranged, repulsive interactions of different natures. This pair potential constitutes the dominant term of the colloidal interaction within the multipolar expansion which amounts to an expansion in terms of the inverse interparticle separation 1/d. 2,13,14 The actual three-dimensional (3D) configuration is mapped onto a two-dimensional (2D) configuration of disks in a reference plane. §This can be justied for dilute systems (see ref. 10 and 12). Nevertheless, recent results 11,15 provide evidence that the main effectofthelong-ranged hydrodynamic interaction on the dynamics of the capillary collapse consists of a reduction of the time scales, without altering the qualitative picture which we present here. {The additive function of only temperature is irrelevant for our present purposes, given that the system evolves under isothermal conditions. This journal is © The Royal Society of Chemistry 2014 Soft Matter,2014,10,4091–4109 | 4093 Paper Soft Matter Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online dF dw¼0;vn vt¼V$nVdF dn;(11) respectively. This establishes a formal analogy between eqn (7) and the so-called dynamic density functional theory (DDFT) 16 for the ensemble-averaged density. The dynamics described by these equations has been already the subject of previous studies. 8,10–12,17 The main goal of the following analysis is the investigation of the dynamics as a function of the capillary length if the initial condition is a circular patch of colloidal particles, i.e., the solution of these equations with the property that w(r,t) is regular and with the boundary conditions lim r/Nwðr;tÞ¼0;(12) and a so-called “top-hat”prole as the initial condition n0ðrÞ:¼nðr;t¼0Þ¼1;r\1 0;1\r:(13) This initial conguration complements and is closer to experimental realizations than the idealized case of an innitely extended homogeneous distribution. Compared with the cases of an innitely extended system or a system in a box with periodic boundary conditions (as addressed in ref. 8, 10, 12 and 17), the presence of an explicit boundary brings into focus the effect of changing the range of the capillary attraction relative to the system size, leading to a peculiar phenomenology. 11 For the following discussion we introduce the useful notions of the “cold limit”(P/0 in eqn (7b) or m/0 in eqn (9)) and of the “Newtonian limit”(k/0 in eqn (7a)). The rst notion refers to the interpretation of vanishing pressure as a sort of athermal limit of the uid; the second notion refers to the case that the capillary attraction (eqn (3)) becomes identical to Newtonian gravitational attraction in 2D (see, e.g., ref. 10 and 18). Regarding this latter notion, we briey remark on the opposite limit k/N,i.e., a very short-ranged capillary attraction. From the explicit integral of eqn (7a), wðrÞ¼ 1 2pðℝ2 d2r0nðr0ÞK0ðk|rr0|Þ;(14) one deduces wzn k2;k[1;(15) in regions where the density eld is smooth, and eqn (7b) becomes vn vtzV2P;PðnÞ¼PðnÞn2 2k2;(16) with V(nVw)zk 2 V(nVn)¼(k 2 /2)V 2 (n 2 ). This is the diffusion equation for a uid the equation of state P(n) of which accounts for the attractive force through the van der Waals-like contribution n 2 /(2k 2 ). In view of that, one should note that, motivated by the wide scale separation mentioned above (l[R 0 ), we are interested in the comparison of the range of the interaction with a characteristic system size, as quantied by the parameter k. This is distinct from the more usual approach, in which the range would be compared with a microscopic length, e.g., the particle radius. 2.2 Stability with periodic boundary conditions We rst review briey the linear stability analysis of a homogeneous distribution in a nite square box with side length L 0 and periodic boundary conditions. The ndings can be summarized in a “dynamical phase diagram”, which is also useful for understanding qualitatively the mathematically more difficult case of a nite disk. Consider the equilibrium, homogeneous state n eq (r)¼1 with r˛[0, 1] [0, 1] and periodic boundary conditions. In order to account for a locally perturbed density n(r,t)¼1+d(r,t) one introduces the Fourier transform of the density deviations: ~ dkðtÞ:¼ð½0;12 d2rdðr;tÞeik$r(17) with the wavevectors (in units of 1/L 0 ) k¼2p(m x ,m y ), m x ,m y ˛Z.(18) Eqn (7) can be linearized in dresulting in ~ d k (t)¼~ d k (0)e t/s(k) ,(19) where in units of Tthe inverse characteristic time for each mode kis given by 1 sðkÞ¼k21 k2þk21 K2;(20) in terms of Jeans' length K 1 (in units of L 0 )dened as K2:¼dP dnn¼1 hP0ðn¼1Þ:(21) (This quantity is actually the inverse of the isothermal compressibility of the uid due to thermal motion and shortranged forces, i.e., in the absence of capillary attraction.) Fig. 2 depicts 1/s(k) for two qualitatively distinct cases characterized by the values of Kand k. In the cold (K/N) Newtonian (k¼0) limit all modes grow at the same rate, i.e.,s(k)h1 (horizontal blue line in Fig. 2(a)). (Actually, in this limit the growth of perturbations can be computed exactly beyond the linear regime; see the discussion in Subsection 2.3). A nonzero pressure (K<N) causes a slowing down of the growth of the small-scale (k/N) perturbations or even leads to damping (1/s(k) < 0) due to the nite compressibility (red line in Fig. 2(a)). The effect of a nite range of the attraction (k> 0) is to slow down the growth of the large-scale (k/0) perturbations (green line in Fig. 2(a)). If K/k< 1, all perturbations are damped (red curve in Fig. 2(b)); if K/k> 1, the amplitude of all Fourier modes below a critical wavenumber grow in time (blue curve in Fig. 2(b)). The analysis is facilitated by the introduction of a dimensionless parameter which we call “effective temperature”: Teff :¼ð2pÞ2þk2 K2:(22) 4094 |Soft Matter,2014,10,4091–4109 This journal is © The Royal Society of Chemistry 2014 Soft Matter Paper Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online This parameter quanties the ratio between the energy associated with thermal motion and short-ranged forces and with the potential energy of the capillary attraction. Indeed, if for simplicity one considers an ideal gas, p(r)¼ rk B T, in physical units eqn (22) turns into Teff ¼gkBT f2Nneigh ;(23) where the capillary potential energy of a pair of particles is f 2 /g(see eqn (3)) and Nneigh ¼r0"2p L02 þ1 l2#1 (24) gives the approximate average number of neighbors with which any given particle interacts via capillary attraction. This quantity allows one to quantify the notions of a “large system”, for which N neigh r 0 l 2 if k¼L 0 /l[1, and of a “small system”, with N neigh r 0 L 02 if k¼L 0 /l1. In terms of these parameters T eff (eqn (22)) and k, for a general system eqn (20) becomes 1 sðkÞh1 sk;k;Teff ¼k2 ð2pÞ2þk2"ð2pÞ2þk2 k2þk2Teff #;(25) so that the homogenous state is linearly stable (s(k)<0forall allowed values of k)ifandonlyifT eff >1.AsT eff decreases, more and more modes at smaller length scales become unstable.ThisisvisualizedinFig.3,whichfacilitatesthe discussion of the evolution of the density eld in real space as afunctionofthetwoparametersT eff and kcharacterizing the initial state. We have found that two properties are relevant in order to identify different “regimes of instability”:rst, the number of unstable Fourier modes, as determined by the position of the zero in Fig. 2, and second, the kind of mode exhibiting the fastest growth, determined by the position of the maximum in Fig. 2. (1) The limit of stability given in Fig. 3 of a given mode k¼2p(m x ,m y ) is determined using the conditions 1 sk;k;Teff ¼00Teff mx;my;k¼1þðk=ð2pÞÞ2 mx2þmy2þðk=ð2pÞÞ2: (26) In particular, for the fundamental mode (k¼2p) this denes the line T eff ¼1 of absolute stability (“line S”in Fig. 3). (2) The loci in Fig. 3 where a given mode k¼2p(m x ,m y ) exhibits the fastest possible growth is determined using the condition d dk 1 sk;k;Teff ¼00Teff mx;my;k ¼k 2p  21þðk=ð2pÞÞ2 hmx2þmy2þðk=ð2pÞÞ2i2:(27) In particular, for the fundamental mode (k¼2p) this denes the line T eff ¼[1 + (2p/k) 2 ] 1 (red “line F”in Fig. 3) above which the fundamental mode is the fastest growing one.k It also proves to be useful to dene the following loci: (1) The maximum value 1=sm¼max kð1=sðkÞÞ is given by 1 sm¼"1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Teff 1þð2p=kÞ2 s#2 ;(28) which renders a line T eff (k,s m ) with a parametric dependence on s m . This time s m is typically of order 1, i.e., Jeans' time, and it can become signicantly larger only close to the instability threshold, T eff /1, and for a “large system”,k/N; see in Fig. 3 the blue dashed line corresponding to s m ¼10. (2) The line along which the characteristic time s(k¼2p)of the fundamental mode has a given value s f is given by 1 sf¼1Teff 1þðk=2pÞ2:(29) Fig. 2 Examples of the dependence on kof the growth rate 1/s(k)defined in eqn (20) for various values of the parameters Kand k(see the main text). In (a), K 1 ¼1 for the red line and k¼0.2 for the green line; in (b), K 1 ¼1 and k¼5 for the red line, and K 1 ¼1 and k¼0.2 for the blue line. kThe fundamental mode can actually still be the fastest growing one in the region even below this line but above the line corresponding to the next mode, k¼2pffiffiffi 2 p. However, this is irrelevant for our purposes because the borders separating the different “regimes of instability”are not sharp anyhow. This journal is © The Royal Society of Chemistry 2014 Soft Matter,2014,10,4091–4109 | 4095 Paper Soft Matter Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online As illustrated in Fig. 3, these lines run parallel to the line S and then become vertical at a sharp value (k 0 /2p) 2 ¼s f 1. In this sense, the notion of a “small vs. large system”is also quantied by the value of s f 1. We can now summarize the overall picture in Fig. 3 as follows: In the region below the line S (stability limit) and above the line F (the fundamental mode is the fastest growing one) there are in general many unstable modes (thin blue lines), but the fastest growing one is the fundamental mode (i.e., with the smallest k), so that the evolution of the density eld is dominated by features at the largest scales in real space (“collective collapse”). Furthermore, in this region the effect of a nite range of the capillary attraction i.e.,ks0isirrel- evantandtherelativechangeinthegrowthratebetween different modes is due to the effect of pressure (eqn (20)). This is the region of gravitational-like collapse, which proceeds on a time scale of the order of Jeans' time (because most part of this region lies well below the line s m ¼10 and in the “small system”region, s f T1). In the region well below the red line F and in the “large system”region, i.e., far to the right of the lines s f T1, there are many unstable modes, but the fastest growing ones correspond to perturbations on small scales (compared to the size of the system), which evolve on a time scale substantially faster than the fundamental mode. Therefore, the evolution of the density eld is dominated by the growth of small-scale perturbations of a certain preferred size (“local collapse”). This is the region of spinodal-like instability, which also proceeds on a time scale of the order of Jeans' time, except in the corner region k/N, T eff 1 small (note the location of the line s m ¼10). In this corner the evolution appears to be substantially slowed down on the scale of Jeans' time and it corresponds precisely to the usual scenario of spinodal decomposition in uids aer quenching to a temperature below but not too far from the critical point (note that the spinodal line is actually the line S). Fig. 3 Diagram showing the different “regimes of instability”for an initially homogeneous distribution with periodic boundary conditions in terms of T eff (eqn (22)) and k¼L 0 /l. Blue lines, depending on the squared modulus m 2 ¼m x2 +m y2 of the mode indices, indicate the limit of stability for those modes (eqn (26)). The thick blue line S of absolute stability corresponds to m 2 ¼1; subsequent lines correspond to m 2 ¼2, m 2 ¼4, and m 2 ¼5. Red lines indicate that a mode with squared modulus m 2 is the fastest growing one (eqn (27)). The thick red line F corresponds to m 2 ¼1. The thick black line is an iso-Kline T eff (k)¼K 2 [(2p) 2 +k 2 ] (eqn (22)), containing the state points of the Brownian dynamics simulations presented in ref. 12 with a Jeans' length of K 1 ¼8.75 10 3 (filled squares). Finally, the lines s m ¼10 (i.e., the fastest mode has the characteristic time 10T) and s f ¼(1.01, 1.45, 41.5) (from left to right; the fundamental mode has the characteristic time 1.01T, 1.45T, and 41.5T, respectively) are given by eqn (28) and (29), respectively. The lines for given values of s f T1 indicate the transition region. The black symbols (symbols of different shapes correspond to different Jeans' lengths) indicate the systems consisting of a finite circular initial patch of particles investigated via simulations and are discussed in Section 4. For the points (a)–(c), (e), and (f), the corresponding inverse characteristic times 1/s(k) (as shown in Fig. 2, see eqn (20)) are shown in the side panels on the right with the discrete modes indicated by blue vertical lines. Note that the horizontal scales for kvary whereas the vertical scale always ranges from 0.1 < 1/s< 1. The green horizontal line corresponds to 1/s¼0 and the vertical black lines indicate the maxima of 1/s(k). 4096 |Soft Matter,2014,10,4091–4109 This journal is © The Royal Society of Chemistry 2014 Soft Matter Paper Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online The transition between the two regimes discussed above is smooth and corresponds roughly to the region below the line F but in the “small system”region, i.e., to the leof or around the lines s f T1. There the evolution consists of the simultaneous clustering at a rate of the order of Jeans' time of widely differing length scales, including both very small scales and the largest possible ones. This regime thus shares characteristic features of both the “spinodal instability”and the “gravitational instability”. Which feature dominates the structure formation (global or local collapse) depends critically on the distribution of the initial amplitudes of the density perturbations. This picture is exemplied in ref. 12 with N-body simulations in a nite box with periodic boundary conditions corresponding to an initial dilute (ideal-gas) density with 1/Kz 8.75 10 3 xed and values of kranging from 0.1 up to 100; in Fig. 3 the corresponding states are represented by lled red squares lying on the iso-K(thick black) line T eff (k)¼ K 2 [(2p) 2 +k 2 ]. In these simulations all initial state points were chosen below the stability line S, and therefore the inherent mode instabilities led to clustering. For the two lowest state points (I and II) with k<1,perfectscalingofthe time evolution of various geometric cluster measures with Jeans' time was observed, indicating the gravitational-like “collective collapse”. The third state point (III) with k¼1.85 is located in the transition region and deviations from the above time scaling are noticeable. For all other state points (IV–VII) with k$5, the dynamics became increasingly slow and the scaling with Jeans' time is completely lost. Together with the spatial information inferred from various snapshots (see Fig. 3 in ref. 12), these state points are characteristic of the spinodal regime. 2.3 Collapse of a nite-sized, radially symmetric distribution As compared to the case discussed in the previous subsection, the theoretical analysis of a collapsing nite-sized disk is complicated by the fact that the unperturbed state is neither stationary nor spatially homogeneous. The theoretical description of such a system is facilitated by switching to Lagrangian coordinates. The details of this treatment are presented in Appendix A. Here we summarize and discuss the main results obtained from these calculations. The Lagrangian trajectory eld of volume elements is the mapping x1r¼r L (x,t)(30) such that ris dened physically as the position at time tof that volume element which was at position xinitially (i.e., at time t¼ 0); xis called the Lagrangian coordinate of the volume element. The trajectory eld allows one to express any other eld as a function of the Lagrangian coordinates x. In particular, the Lagrangian density eld is given by: n L (x,t):¼n(r¼r L (x,t), t). (31) Following the denitions and steps in Appendix A.1, the evolution equations for nand w(eqn (7a) and (7b)) can be put into their Lagrangian form (see eqn (41d), (41e), and (47)). Mass conservation is a built-in property of the Lagrangian scheme, so that it is not violated regardless of the approximations done in the Lagrangian equations. The subsequent discussion is divided into two parts: the radially symmetric evolution of the initial prole given by eqn (13) and the (linearized) evolution of perturbations to this initial prole. Radially symmetric evolution. It is described by Lagrangian trajectories of the form r L (x,t)¼a(x,t)x. The formal solution for a(x,t) is derived in Appendix A.2. The theoretical analysis has been divided into steps of increasing difficulty. (1) Newtonian (k¼0), cold (P¼0) limit. The evolution equation for n L (eqn (47)) can be solved exactly and the solution can be used to obtain a(x,t) from eqn (54). In this manner one nds (see Appendix A.2.1) aðx;tÞ¼ ffiffiffiffiffiffiffiffiffiffi 1t p;x#1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1t x2 r;1\x 8 > < > : (32a) and nðr;tÞ¼1=a2ðtÞ;jrj\aðtÞ 0;aðtÞ\jrj:(32b) This is the so-called cold collapse solution, for which in the course of the evolution the disk shrinks and the density inside the disk does not depend on the radial position, so that all particles reach the center simultaneously. This is the 2D analogue of the cosmological “big crunch”solution (i.e., a time-reversed “big bang”solution), where aplays the role of the expansion factor and da/dtthat of the Hubble function. (2) Finite screening (ks0), cold (P¼0) limit. Analytical results can be obtained using a perturbative approach in the two limiting cases of “small systems”(k1) and of “large systems”(k[1). This amounts to approximately solving eqn (47) for n L by replacing the term proportional to k 2 by one corresponding to a reference solution. In the case k1, this is the cold-collapse solution given by eqn (32). The perturbative solutions for a(x,t)andn L (x,t) (eqn (63) and (64)) result in a deformation of the density proleduringthecollapse(see Fig. 4(a)). In particular, at the rim of the disk a peak forms which becomes singular before all matter collapses in the center. To the lowest order in k,theradialpositionr sw and the time t sw for the occurrence of this shock wave are given by (eqn (72) and (73)): tswz1þk2 81 22geln k2 4 rsw zk 4 9 > > > = > > > ; ;k1:(33) In the opposite limit of large system sizes, k[1, the collapse of the disk is slowed down substantially because initially the density eld is homogeneous almost everywhere (see eqn (16)). Thus one can use the initial condition as a reference solution as if it was stationary in order to obtain an approximate solution of eqn (61). At the beginning, the dynamics is driven solely by the This journal is © The Royal Society of Chemistry 2014 Soft Matter,2014,10,4091–4109 | 4097 Paper Soft Matter Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online net capillary attraction near the density inhomogeneity at the rim and again a density singularity appears there (see Fig. 4(b)). To the lowest order in k 1 , this occurs at a time and a position given by (see the text following eqn (77) and (78)) tsw z21 k rsw z11 k 9 > > = > > ; ;k[1:(34) Aer this, the position of the singularity is shied by the collapse of the disk, which occurs on a much longer time scale which is at least T1/k(see eqn (78)). Thus, the formation of a singularity is a generic feature of the cold limit. The time evolution is formally undened beyond its appearance, but actually the singularity is regularized by the effect of a nonzero compressibility caused by corrections to the cold limit. (3) Effects of nonzero P. At nonzero temperatures, the pressure term in the evolution equation for n(eqn (7b) or, equivalently, the Lagrangian form in eqn (47)) can become important. If Pis sufficiently large (formally the limit of high temperature, T/N), pressure is the dominant term in the evolution equation and the initial density distribution ends up with two-dimensional evaporation instead of a collapse. This is the analogue of the region T eff > 1 shown in Fig. 3. We are more interested, however, in the case that the disk collapses and how a nonzero but still sufficiently low temperature Taffects the evolution. In Appendix A.2.3 we show that a nonzero pressure quickly smoothens any nite steps in the initial density distribution. The analytical study of the effect of pressure on the density singularity, which forms as a divergence for P¼0 and k s0, is more difficult and we do not have explicit analytical results for this behavior, but it can be expected that the pressure will regularize also this singularity. This is clearly visible in our numerical solutions and in the simulated proles (see Section 4 below). Thus the collapse will proceed beyond the time t sw and the regularized peak at the rim of the disk is shied to the center in the manner of a shock wave. Perturbations of radial symmetry. This issue has been studied in Appendix A.3. For the Newtonian, cold limit one can solve the evolution equation for the Lagrangian density eld exactly (see eqn (84)). There is enhanced clustering localized at initially overdense regions and depletion at underdense regions. In astrophysics this is called the “fragmentation instability”.For an initially small local overdensity, 0 < dn 0 1, the local time of collapse (signaled as a density singularity) is still of the order of Jeans' time, t s z1dn 0 .Again,itisdifficult to treat the case of nonzero Tand k.However,fordensityperturbationslocalized near the center of the disk such that the effectoftheboundarycan be neglected, the calculations in Appendix A.3 show that the linear stability analysis of a time-independent, homogeneous state as discussed in Subsection 2.2 is useful also for this kind of perturbation. Particularly relevant observations are that the shortest temporal scale for the growth of perturbations is in general Jeans' time (see the line denoted as s m shown in Fig. 3) and that during disk collapse further fragmentation of the inhomogeneities is counteracted by a nonzero pressure. Although Fig. 3 derives from the linear stability analysis of a stationary and homogeneous state, it is also useful in order to summarize and rationalize the picture which emerges from the theoretical analysis of the collapsing disk, with T eff and kreferring to the initial disk state. Thus, the disk collapses if T eff is well below 1. A generic feature of the disk-collapse is the formation of a singularity, eventually regularized by pressure, at the outer rim at a time slightly larger than Jeans' time. As an inbound shock wave this feature can dominate the evolution only in an intermediate transition region where kis of the order of one and the time scale of the overall collapse is comparable to Jeans' time (see the lines denoted as s f in Fig. 3). If kis very small, the gravitational-like global collapse is so fast that the formation of the singularity is practically unobservable. If kis very large, the evolution is dominated by spinodal-like growth, on Jeans' time scale, of density perturbations with large wavenumbers inside the disk, while the collapse of the disk remains frozen and with it the singularity at the outer rim. This picture is conrmed by numerical calculations, as explained in the subsequent sections. Fig. 4 In the cold limit time evolution of the radial density profile for a disk computed perturbatively (see eqn (63) and (64)). (a) Small systems: for k¼2/3 the shock wave singularity appears at the radius r sw z0.17 and at the time t sw z1.086 (see eqn (33)). (b) Large systems: for k¼10 the shock wave singularity appears at the radius r sw z0.9 and at the time t sw z1.9 (see eqn (34)). 4098 |Soft Matter,2014,10,4091–4109 This journal is © The Royal Society of Chemistry 2014 Soft Matter Paper Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online 3 Brownian dynamics and particle based DDFT The Brownian dynamics simulation introduced in ref. 12 relies on the particle-mesh method known from cosmological simulations. We rst briey recall the basic concept of this method, which provides the foundation of an easily applicable extension corresponding to a solution of a two-dimensional (2D) dynamic density functional theory (DDFT). Within the simulation, the straightforward way to implement colloidal dynamics based on a pairwise additive potential such as the capillary potential of eqn (3) consists of summing all forces from all possible pairs of colloids. However, as explained aer eqn (3), the potential in eqn (3) is long-ranged and nonintegrable in 2D in a system of size L<l. In using periodic boundary conditions, colloidal particles from periodic images will therefore contribute signicantly to the total force acting on a single colloid. In order to obtain the correct value of the force experienced by a particle, the sum of the forces has to be extended over many periodic images. Alternatively, here we use the particle-mesh method, in order to exploit the underlying mean eld character of eqn (1). It connects the average interfacial deformation U(r) with the average mean number density r(r). Provided one has access to the mean density, e.g., discretized on a grid or mesh, one can solve eqn (1) via Fourier transformation. The resulting mean interfacial deformation on the grid enables one to compute the capillary forces acting on the colloidal particles by means of interpolation, whereby the effect of periodic images is taken into account properly by the Fourier transform. In order to simulate colloidal particles trapped at a uid interface, one also has to consider the effect of the thermal heat bath provided by the uid, as well as the nonzero size of the particles. In continuum theory, both aspects are taken into account by the presence of the (local) term fVp(r) in eqn (5). Within BD simulations, the nonzero particle size is accounted for by short-ranged repulsive forces between the colloids. We use the cutoffand shied repulsive part of the Lennard–Jonespotential (the so-called soWCA potential, as used in ref. 12). Temperature enters by adding a stochastic noise term to the equations of motion of the particles. Hence, within BD one integrates the corresponding overdamped Langevin equation. Thus, our BD simulations incorporate two different methods for the computation of the forces: the particle-mesh method for the long-ranged capillary forces and the direct summation of short-ranged repulsive forces. It would be even more convenient, if the particle-mesh method could be extended towards the short-ranged forces. Such an approach has been used in ref. 19 for describing the very dilute intergalactic medium. The same idea can be employed for eqn (9) because the forces fV(mw) can easily be calculated for the discretized density grid. Since the uctuations, giving rise to the noise present in BD simulations, are now also incorporated within a mean eld approximation, this method describes the dynamical evolution of the discretized density according to an initial condition for the density eld set by the distribution of particles at time zero. The evolution of the ensemble-averaged density eld is obtained by averaging over initial conditions, i.e., the initial distributions of particles. In that manner, the obtained averaged evolution corresponds to the solution of the 2D DDFT dened by the evolution equation (11). We call it particle based, because in every time step one discretizes the density, i.e., one assigns all particles to points on the grid and calculates the forces V(mw) on the grid. Then, instead of integrating eqn (9) one interpolates these forces back to the positions of the particles which they occupied before they were assigned to grid points, and integrates their deterministic equations of motion. The next step, which is to discretize the density at time t+Dt, completes the evolution of the density as the basic quantity of our analysis. As has been pointed out in ref. 19, in comparison with the standard particle-mesh method, this discretization and integration scheme comes at the cost of an increased numerical noise at small scales of the order of the grid-spacing. However, for any physical observable noise we have studied so far, this increased numerical noise turned out to be not relevant. 4 Results from simulations Using BD simulations and 2D DDFT as described in the previous section, we explore the “dynamical phase diagram” (Fig. 3) for the initial condition of a nite-sized patch of particles, arranged in a circular disk.** In order to locate in the T eff –k plane of Fig. 3 the kind of evolution of a certain conguration, the value of the effective temperature (eqn (22)) is determined by the initial conditions inside the disk. To this end we recall the central result from the linear stability analysis in Subsection 2.2 and the perturbative analysis for the disk evolution in Subsection 2.3: when following an isothermal line†† from large capillary lengths l(i.e., small cutoffparameters k¼L 0 /l) down to l¼2R,i.e., the value at which the range of the interaction equals the particle diameter, one rst observes a collective collapse, followed by a regime where shock waves become visible and a spinodal decomposition stage for rather shortranged attractions, i.e.,l/L 0 1ork[1. We x the effective temperature of the system by choosing a disk of radius L 0 ¼180Rand setting the number of particles with radius Rto N¼1804 so that r 0 x574/L 02 . The associated Jeans' length‡‡ in units of L 0 is K 1 ¼0.018, corresponding to ** In order to minimize the effect due to periodic images of the disk within the particle-mesh method, we use a large simulation box with side length L¼800R z4.4L 0 , such that the disk is surrounded by empty space and therefore is substantially separated from its images. For the largest capillary length under consideration (l¼1.5L 0 ), the distance between the rim of the disk and its nearest periodic image is given by 3.4L 0 z2.2l. †† We recall that the physical temperature Tenters Jeans' length via K 1 ¼ (dP/dn) 1/2 . Thus for the repulsive interaction taken to be of the hard core type, one has K 1 fT 1/2 . Since the effective temperature T eff used in Fig. 3 is given by T eff ¼[(2p) 2 +k 2 ]/K 2 (eqn (22)), in the double-logarithmic T eff –kplane of Fig. 3 an isotherm is a line which is horizontal for small kand crosses over to a line with slope 2 at large kdue to T eff fk 2 . ‡‡ As discussed in ref. 12, we employ an ideal gas equation of state for the initial condition to determine Jeans' length according to K1zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gkBT=ðf2r0Þ pwith f 2 /(2pg)¼0.89k B T(see eqn (3); this numerical value corresponds to the choice 1 R¼10 mm) so that K 1 z0.42r 01/2 . This journal is © The Royal Society of Chemistry 2014 Soft Matter,2014,10,4091–4109 | 4099 Paper Soft Matter Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online time t sw z21/k(compare with eqn (72)). Nevertheless, in both cases the radial displacement given by eqn (64) is very small, aðx;tÞz1t ktsðxÞ;t tsðxÞ #1;(78) so that the physically meaningful singularity at the rim has a position r sw ¼a(1, t sw )z11/k.Fig.4(b)illustrates the evolution of the density prole according to this perturbative calculation. Formally speaking, for points with x<1 the approximation in eqn (76) is a short-time expansion with respect to the time scale of evolution (see eqn (77)). Since this scale is much larger than Jeans' time, the approximation is also valid for describing the evolution on Jeans' time scale. In summary, a nonzero value of kinduces a deformation of the initial top-hat prole by forming an enhanced density peak at the outer rim. For any value of kthe time scale for forming the singularity is of the order of Jeans' time, but the radial position of the peak depends strongly on the value of k. The singularity eventually evolves into a shock wave when the effect of pressure becomes relevant locally. A.2.3 “Hot”effects (Ps0). Here we add a few remarks concerning the effect of the pressure Pin eqn (7b) on the radial evolution. Generically, the cold limit P/0 is singular because Pis associated with the highest order of the spatial derivatives in eqn (7b), so that one cannot neglect the term VPuniformly throughout the disk. There are two particular aspects for which the effect of Pis relevant, that is the jump of the initial density at the outer rim of the disk and the formation of the shock wave, because both features are associated with the formal behavior VP/N. The initial discontinuity at r¼1 (see eqn (13)) implies that locally the early stages of the evolution will be dominated by the regularizing effect of the pressure, no matter how small Pis. In order to be specic we consider the ideal gas approximation, P ¼3n, in which the small parameter 3/0 can be identied with a dimensionless temperature (see eqn (6b)). The very early evolution of the density discontinuity can be described by neglecting locally any other force but pressure, so that in Eulerian coordinates vn vtzV2P¼31 r v vrrvn vr;rz1:(79) In order to analyze the effectofpressureweintroducea new independent variable z:¼ðr1Þ=ffiffi3 p(centered at the initial density jump and rescaled in proportion to the, as it will turn out, thickness of the regularized discontinuity). This simplies the equation as 3/0atnite values of z(i.e.,oneeffectively neglects the curvature of the regularized discontinuity and approximates it locally by a straight line): vn vtzv2n vz2;nðz;t¼0Þ¼1;z\0 0;0\z:(80) Expressed in terms of the original variables, the solution is given by the error function: nðr;tÞz1 21erfr1 ffiffiffiffiffiffi 43t p¼1 21erfz ffiffiffiffi 4t p:(81) This represents a regularized jump discontinuity or kink of thickness ffiffiffiffi 3t p. One can use this solution in order to determine self-consistently the time t p beyond which the contribution of the pressure term no longer dominates eqn (7b) even near r¼1 compared with the term V$(nVw), which is initially of order unity in terms of dimensionless quantities: maxr 31 r v vrrvn vr zmaxz v2n vz2 (10tpTð8peÞ1=2z0:12:(82) The fact that this time is independent of 3means that the regularization of the initial discontinuity takes a small but nite fraction of the total time of collapse in the Newtonian limit, which is of order 1 (see Subsection A.2.1 and Fig. 4). However, this effect is spatially localized because the thickness of the resulting kink (eqn (81)) vanishes as 3/0. In the same manner, when a shock wave appears the density gradients can become so large that the effect of a small pressure eventually dominates the evolution in the localized regions of large gradients. This is the hallmark of the formation of a moving boundary layer. The involved mathematical analysis of this phenomenon is beyond the scope of the present study; here we conne our effort to the insight that this mechanism provides a qualitative explanation of the regularization of shock waves as observed in simulations. A.3 Non-radially symmetric perturbations. We now explore the robustness of the radially symmetric evolution against small perturbations of the initial radial symmetry. In this respect the ultimate goal is a generalization of the stability analysis as already carried out for the static (i.e., time-independent) homogeneous background (see Subsection 2.2). However, the mathematical problem is substantially more complicated, given that the unperturbed state, i.e., the radially symmetric density prole studied in the previous sections, is neither static nor homogeneous. Lagrangian perturbation theory has proved to be successful in the study of cosmological structure formation in an expanding homogeneous universe 20,22 and, as it will turn out below, it is also helpful for the present problem. But, compared to applications in cosmology, progress is limited due to two difficulties: (i) the presence of the initial inhomogeneity at the boundary of the disk, and (ii) corrections to the Newtonian limit. Therefore, here we can address the full stability analysis only partially. A.3.1 Newtonian, cold limit. We rst consider the exact solution in eqn (55) specialized to an initial density distribution of the form n0ðxÞ¼1þdn0ðxÞ;x\1 dn0ðxÞ;1\x;(83) i.e., a perturbation of the top-hat prole given by eqn (13). One can distinguish two cases. (1) For perturbations inside the disk (x< 1) one has nLðx;tÞ¼ 1þdn0ðxÞ 1t½1þdn0ðxÞ;(84) 4106 |Soft Matter,2014,10,4091–4109 This journal is © The Royal Society of Chemistry 2014 Soft Matter Paper Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online and a density singularity arises at the time t s (x)¼[1 + dn 0 (x)] 1 . Therefore any local overdensity (dn 0 > 0) will grow faster and collapse earlier than the disk as a whole. This is the counterpart of the so-called “fragmentation instability”of a collapsing spherical cloud in astrophysics. 23,24 (2) For perturbations outside the disk (x> 1), representing in particular deviations from the circular shape of the initial disk, one has nLðx;tÞ¼ dn0ðxÞ 1tdn0ðxÞ;(85) and a density singularity arises at the time t s (x)¼1/dn 0 (x). For small perturbations (0 < dn 0 1), this time is larger than the time of collapse of the disk. We emphasize that these results are exact, which is a peculiarity of the overdamped dynamics. In the astrophysical counterpartofthisproblem,therelevance of inertia prevents one from nding a closed equation for the density eld analogous to eqn (55) and thus it is unavoidable to resort to perturbation theory. Nevertheless, the problem is not fully solved until the Lagrangian-to-Eulerian map r L (x,t) has been computed. Since this cannot be done exactly it is useful to apply Lagrangian perturbation theory. To this end, one denes a perturbation dr L (r,t):¼r L (r,t)^ r L (x,t), (86) of the trajectory, where the hat over a symbol denotes the reference, unperturbed evolution. The initial condition dr L (x,t¼0) ¼0(87) holds by denition of the Lagrangian map (see eqn (39)). Likewise, one denes perturbations of the Lagrangian elds as dn L (x,t):¼n L (x,t)^ n L (x,t), (88) and similarly for dM,dg L ,dw L , and dm L . Eqn (41) can be linearized with respect to the small perturbations:kk dMz^ M$ðVxdrLÞ$^ M;(89a) ddrL dtzdgLð^ M$VxÞdmLðdM$VxÞ^mL;(89b) dnLzdet ^ Mhdn0þ^ n0tr^ M1 $dMi;(89c) dmLz^m0 LdnL¼ ^ P0 L ^ nL dnL;(89d) dgLz^ M$VxÞdwLþðdM$VxÞ^ wL; ð(89e) ð^ M$VxÞ$dgLþðdM$VxÞ$^ gLzdnLþk2dwL;(89f) where ^m 0 L :¼dm/d^ n L ,^ P 0 L :¼dP/d^ n L ,dn 0 (x):¼dn L (x, 0) represents the initial density perturbation, and ^ n 0 (x):¼^ n L (x, 0) is the initial unperturbed density. These equations simplify signicantly when specialized to the perturbed evolution of a top-hat prole in the Newtonian, cold limit, which amounts to setting k/0, m L /0, and, restricting attention to the interior of the disk (x< 1), ^ nL¼1 ^ a2;^ M¼1 ^ aI;^ wL¼CðtÞ1 4x2;(90) with ^ aðtÞ:¼ffiffiffiffiffiffiffiffiffiffi 1t p(see eqn (50), (58) and (59)). Under these conditions one obtains dMz1 ^ a2VxdrL;(91a) ddrL dtzdgL;(91b) dnLz1 ^ a2"dn01 ^ aVx$drL#;(91c) dgLz1 ^ aVxdwLþ1 2^ a2ðVxdrLÞ$x;(91d) 1 ^ aVx$dgLþ1 2^ a3ðVx$drLÞzdnL:(91e) In order to proceed one introduces the auxiliary eld fL:¼dwLþ1 2^ adrL$x;(92) which can be viewed as a potential in Lagrangian coordinates: V x f L ¼ˆ adg L +dr L /(2ˆ a). Eqn (91b) and (91d) reduce to ddrL dtz1 ^ aVxfL1 2^ a2drL(93a) and Vx2fLzdn0þ1 ^ aVx$drL;(93b) from which one obtains*** d dtðVxdrLÞz1 2^ a2VxdrL;(94a) and d dtðVx$drLÞzdn0 ^ aþ1 2^ a2Vx$drL:(94b) These equations can be integrated with the initial condition given in eqn (87): V x dr L (x,t)¼ˆ a(t)V x dr L (x,t¼0) ¼0, (95a) kk Linearization of terms involving the matrix Mis achieved by applying the identities ln det M¼tr ln Mand M1¼X N n¼0ðIMÞn, where Iis the identity matrix and M 1 ¼V x r L (eqn (41a)). *** The vector eld dr L and V x consist of two components. Both can be extended to three-component vectors by adding a third (“vertical”) component taken to be zero. This way V x dr L is well dened and its only nonzero component is in the “vertical”direction. Accordingly, eqn (94a) reduces to a single equation for the “vertical”component. This journal is © The Royal Society of Chemistry 2014 Soft Matter,2014,10,4091–4109 | 4107 Paper Soft Matter Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online Vx$drLðx;tÞ¼1^ a2ðtÞ ^ aðtÞdn0ðxÞ:(95b) These equations indicate that the mass clusters at the initially overdense regions (dn 0 > 0; note that 1 ˆ a 2 $0) while, consistently, the underdense regions (dn 0 < 0) get depleted. Actually, the perturbations of the trajectories are parallel to the gravitational eld generated by the initial density perturbation. This follows from the form of eqn (95): the time dependence induced by ˆ a(t) can be factored out by a simple rescaling of the perturbation dr L , and the resulting equations are formally the eld equations for the Newtonian gravitational eld generated by a mass distribution given by dn 0 (x). This property is the counterpart of the so-called Zel'dovich approximation in the context of cosmological structure formation (see ref. 25 and references therein). A.3.2 Small-scale, central perturbations. The theoretical analysis beyond the cold, Newtonian limit is hampered by the fact that even the evolution of the reference, unperturbed state cannot be obtained in closed form. However, by focusing on perturbations localized close to the center of the disk (“central perturbations”), one can neglect the inuence of the boundary so that one can take advantage of the results presented in Subsection 2.2. The simplest case is the limit of large system sizes, i.e.,k[ 1. As argued in Subsection A.2.2, in this case the time scale of disk collapse is much larger than Jeans' time and, as far as the central perturbations are concerned, one can approximate the disk as a time-independent homogeneous distribution, namely the initial one, and the analysis presented in Subsection 2.2 holds. Therefore, according to Fig. 3, the fastest growing modes are the ones on small spatial scales and they are characterized by time scales of the order of Jeans' time (barring the exceptional case T eff /1), which is consistent with the underlying approximations. In the opposite limit of small system sizes, i.e.,k1, we can establish a connection with the results presented recently in ref. 17, which deals, in the present language, with the stability in the Newtonian limit (k¼0) of an innitely extended disk (i.e.,no boundary is considered). In this case one can argue that for the central perturbations the evolution of the reference state is indistinguishable from the cold, Newtonian limit during a certain initial period of time, because we have demonstrated before that the effect of the disk boundary is localized and that it takes some time for it to propagate into the interior of the disk. Thus, at early times eqn (89) can be evaluated in the region x1 by again using eqn (90): ddrL dtz1 ^ aVxfL1 2^ a2drLþP0^ nL ðÞ ^ a2VxðVx$drLÞ;(96a) Vx2fLzdn0þ1 ^ aVx$drLþ^ a2k2"fL1 2^ ax$drL#(96b) (compare with eqn (93)). Furthermore, in order to be consistent with this approximation one has to neglect the term fk 2 in eqn (96b) because only perturbations with length scales much shorter than k 1 [1 are to be addressed. This leads to d dtðVxdrLÞz1 2^ a2VxdrL(97a) and d dtðVx$drLÞzdn0 ^ aþ1 2^ a2Vx$drLþP0^ nL ðÞ ^ a2Vx2ðVx$drLÞ(97b) (compare with eqn (94)). Integration of eqn (97a) implies V x  dr L ¼0 due to the initial condition given in eqn (87) (compare with eqn (95a)), while eqn (97b) can be written in a more familiar form by introducing the so-called density contrast, dened as dLðx;tÞ:¼dnLðx;tÞ ^ nLðx;tÞz^ a2dnL¼dn01 ^ aVx$drL;(98) so that ^ a2ddL dtzdLþP0^ nL ðÞVx2dL:(99) By introducing the Fourier transform in Lagrangian coordinates, ~ d L,k :¼Ðd 2 xd L (x)e ik$x , one obtains ^ a2d~ dL;k dtz"1k KðtÞ2#~ dL;k(100) in terms of the time-dependent Jeans' length: K1ðtÞ:¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dP dnn¼^ a2ðtÞ r:(101) This is the same expression as the one obtained for the stability of the homogenous state (see eqn (19) and (20)), restricted to modes with k[1[k(i.e., valid only for perturbations of small length scales near the center of the disk). The only difference is the explicit time dependence of Kintroduced by the background density ^ n L (t). For realistic equations of state, e.g., for hard disks, Jeans' length will decrease in time and in Fig. 3 the effective state of the system for these “central perturbations”would describe a trajectory of steadily increasing T eff fK 2 (t). Thus, there is an increasing number of modes at small scales for which the effect of pressure will counteract the fragmentation instability. The solutions of eqn (100) have been studied recently by Chavanis 17 for the particular choice P(n)f n a as a function of the polytropic index a. The main conclusion follows from evaluating the time-dependent Jeans' length (see eqn (101)), K 2 (t)fˆ a 2(a1) , so that the amplitude of an initially unstable mode (k<K(t¼0)) will eventually die out aer a certain time (k>K(t)) only if a> 1 (corresponding to the critical index g 4/3 introduced in ref. 17 for the spatial dimension d¼2)†††. Acknowledgements J.B. thanks the German Research Foundation (DFG) for the nancial support through the Collaborative Research Center ††† As pointed out in ref. 17, this conclusion only holds provided the initial amplitude of the unstable mode is sufficiently small so that its growth and subsequent decay can be described by the linearized theory. If aer a certain time the amplitude enters the nonlinear regime, the conclusion concerning the asymptotic stability is no longer reliable. 4108 |Soft Matter,2014,10,4091–4109 This journal is © The Royal Society of Chemistry 2014 Soft Matter Paper Open Access Article. Published on 18 February 2014. Downloaded on 9/23/2022 1:19:05 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence. View Article Online (SFB-TR6) “Colloids in External Fields”Project N01. A.D. acknowledges support from the Spanish Government through Grants AIB2010DE-00263 and FIS2011-24460 (partially nanced by FEDER funds). 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