The dynamics of colloids in a narrow channel driven by a non-uniform force
aa r X i v : . [ c ond - m a t . s o f t ] O c t The dynamics of colloids in a narrow channel driven by a non-uniform force
D. V. Tkachenko , V. R. Misko , and F. M. Peeters , Department of Physics, University of Antwerpen,Groenenborgerlaan 171, B-2020 Antwerpen, Belgium and Departamento de F´ısica, Universidade Federal do Cear´a, 60455-900 Fortaleza, Cear´a, Brazil (Dated: December 5, 2018)Using Brownian dynamics simulations, we investigate the dynamics of colloids confined in two-dimensional narrow channels driven by a non-uniform force F dr ( y ). We considered linear-gradient,parabolic and delta-like driving-force profiles. This driving force induces melting of the colloidalsolid (i.e., shear-induced melting), and the colloidal motion experiences a transition from elastic toplastic regime with increasing F dr . For intermediate F dr (i.e., in the transition region) the responseof the system, i.e., the distribution of the velocities of the colloidal chains υ i ( y ), in general doesnot coincide with the profile of the driving force F dr ( y ), and depends on the magnitude of F dr ,the width of the channel and the density of colloids. For example, we show that the onset ofplasticity is first observed near the boundaries while the motion in the central region is elastic. Thisis explained by: i) (in)commensurability between the chains due to the larger density of colloidsnear the boundaries, and ii) the gradient in F dr . Our study provides a deeper understanding ofthe dynamics of colloids in channels and could be accessed in experiments on colloids (or in dustyplasma) with, e.g., asymmetric channels or in the presence of a gradient potential field. PACS numbers: 82.70.Dd, 64.60.Cn, 64.70.Rh
I. INTRODUCTION
During the last decade there has been a growing in-terest in the research of physical properties of colloidalsystems. This interest is by part due to perspectives ofpractical use of colloids, e.g., in biology, medicine, meteo-rology, food production, etc. On the other hand, colloidsserve as a convenient model system for studying, e.g.,phase transitions [1, 2, 3], diffusion [4], or commensurate-incommensurate transitions [5]. Typical dimensions ofcolloids are in the micrometer regime and their dynamicsis governed by a time scale in the microsecond regime andtherefore is suitable for direct observation in real spaceand time. Furthemore, it has been possible to tune theinter-particle interaction potential which opens thereby awide area for research of fundamental properties of clas-sical systems.In Refs. [6, 7, 8] structural properties of mag-netic mono-colloidal mixtures confined in narrow two-dimensional (2D) channels were studied under equilib-rium conditions. Structural deviations from an infinite2D crystal and related to it oscillations in structuralproperties were predicted. Transport properties of para-magnetic mono-colloidal mixtures in narrow channels be-ing in non-equilibrium but under stationary and homo-geneous external conditions (in the gravity field) wereinvestigated in Refs. [9, 10]. It was shown that gravita-tion action results in the occurrence of a density gradientalong the channel, that in its turn leads to a gradient inthe number of chains. The latter was predicted for adriven colloidal system in the presence of a constrictionin the channel [11]. Besides, authors of Refs. [9, 10] alsostudied the relation between velocities of colloidal mo-tion, diffusive behavior and the self-organized order inthe system. The transition from a hexagonal to a chain-like orderedphase was studied in Ref. [12] for Yukawa particles (appli-cable for charged colloids and dusty plasma [13, 14, 15])confined in a two-dimensional parabolic channel. It wasshown that the system crystallized in a number of chainssimilar to a colloidal system confined in a narrow chan-nel. The authors of Ref. [12] analyzed the structuraltransitions between the ground state configurations andshowed that such system exhibited a rich phase diagramat zero temperature under continuous and discontinuousstructural transitions.The purpose of this study is to achieve a deeper un-derstanding of the influence of structural properties onthe motion of paramagnetic colloids in narrow channelsunder non-uniform distributed driving force. In particu-lar, we consider: (i) a linear increasing driving force fromone to the other boundary of the channel, i.e., a linear-gradient driving force, and (ii) a parabolic-profile drivingforce, i.e., equal to zero at the boundaries and maximumat the center of the channel. Although the experimen-tal realization of such profiles is not as obvious as for thesimplest case of a constant driving force (created, e.g., bygravity in an inclined channel), they can serve as a modelof a very important case of a channel modulated in a non-uniform manner, e.g., by a non-uniform distributed dis-order (“pinning”) or surface roughness. The combinationof a uniform force applied to colloids with a non-uniformmodulation in such a channel will result in an effectivenon-uniform driving, i.e., F dr,nu = F dr,u − f m,nu . Thespatial period of the modulation should be chosen, obvi-ously, essentially smaller than the colloidal radius. Forexample, if the modulation depth of a channel linearlygrows from one edge of the channel to another, the effec-tive driving force distribution is described by a linearlygrowing function. Moreover, the application of a non-uniform driving force to a discrete system of interactingparticles confined to a quasi-1D channel is expected toresult in a rich dynamics interesting from the point ofview of fundamental research, and it could be useful forunderstanding the dynamics of other interacting systemsmoving in narrow channels.The paper is organized as follows. The model is de-scribed in Sec. II. In Sec. III, we present the results of ournumerical calculations of the response of the colloidal sys-tem to a non-uniform, i.e., linear-gradient and parabolic,driving force. In Sec. IV, we study the mobility of stripesand dynamical phases of their motion. The conclusionsare presented in Sec. V. II. THE MODEL AND METHOD
Motion of the system of paramagnetic colloids in atwo-dimensional (2D) narrow channel under the actionof a non-uniform distributed driving force F dr ( y ) is in-vestigated using the Langevin equations of motion in theoverdamped regime, i.e., the Brownian dynamics (BD)method. It is supposed that a magnetic field is ap-plied perpendicularly to the channel plane (i.e., perpen-dicular to the xy − plane). Following Ref. [9], we choose B = 1 mT . In this case the magnetic field induces mag-netic moments in the colloids M = χB ( χ = 3 · − [9]) and, therefore, the interparticle interaction is de-scribed by the dipolar repulsive potential V ( r ij ) (for r ij ≥ r col , and by hard core interaction for r ij < r col ): V ( r ij ) = ( µ π M r ij , if r ij ≥ r col , ∞ , if r ij < r col , (1)where µ is a magnetic constant, ~r ij is the distance be-tween i th and j th particles. Following Ref. [9], we havechosen the colloidal radius r col = 2 . µm . The diame-ter of a colloidal particle is denoted as σ col = 2 r col .The use of the BD method assumes that colloids areconsidered in the limit of large viscosity. The system ofoverdamped equations of motion in this case is given by η d r i dt = − X j,j = i ∇ V ( r ij ) + F dr ( y i ) + F Ti , (2)where the friction constant η = 1 . · − is Stokesfriction for a spherical particle of radius r col [9], r i is theposition vector of i th colloid. An external non-uniformstationary force F dr ( y i ) is applied along the channel anddepends only on the transverse coordinate y . The ther-mal stochastic term F Ti entering Eq. (2) obeys the fol-lowing conditions: h F Ti ( t ) i = 0 (3)and h F Ti ( t ) F Tj ( t ′ ) i = 2 ηk B T δ ij δ ( t − t ′ ) . (4) To find the ground state (or an initial state with thelowest free energy close to the ground state) of the sys-tem, we first solve Eq. (2), for nonzero temperature inthe absence of driving, thus simulating annealing processused for obtaining the ground state of, e.g., colloids [16]or vortices in a superconductor (see, e.g., [17, 18, 19]).Then we set temperature equal to zero and solve Eq. (2)for colloids driven by the external force F dr ( y i ). Notethat inclusion of temperature fluctuation would lead tothe degradation of the colloidal stripe structure and, asa result, to smearing of the velocity profile of differentcolloidal chains. We focus on the features of the veloc-ity profiles related to the structure of the chains drivenby a non-uniform force, and we neglected temperature-induced fluctuations in our calculations. In our numericalsimulations, we use a simulation cell with sizes equal tothe channel width in the y -direction and typically 140 r col along the channel, i.e., in the x -direction. We use peri-odic boundary condition along the x -direction. Interac-tion of colloids with lateral walls is hard-wall. The systemof overdamped equations (2) is solved using a variabletime step, calculated from the condition max { d r i } ≤ δ ,where dr i is the displacement of i th colloid, and δ wastypically chosen r col /
200 or r col / r col /
20 was sufficient.
III. RESPONSE OF THE SYSTEM TONON-UNIFORM DRIVING FORCEA. The ground state configurations
As was shown in Refs. [6, 8, 11], in the absence of ex-ternal forces interacting colloids in narrow 2D channelsare ordered in stripe structures. Therefore, one mightexpect that appliyng an external force along the channelwill result in an ordered drift of colloidal stripes in thechannel. (This obviously applies to the case of ratherweak driving force or its gradient and very narrow chan-nels. Strong inhomogeneous driving in a wide enoughchannel could lead to the appearance of instabilities inthe transverse direction to the applied driving and thusto the migration of colloids between stripes.) Thus theinvestigation of the motion of colloids in narrow channelscan be reduced to the analysis of the drift of colloidalstripes. Note that in this case the order of individualcolloids in a stripe does not change with time, and themotion of colloids in stripes resembles single-file diffusion(see, e.g., [20, 21, 22, 23, 24]).For studying the colloidal motion in channels we choosesuch ground state configurations which are characterizedby strongly pronounced stripe structure and thus by ahigh value of stripe order parameter [9]Ψ n l = 1 N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j =1 e i π ( n l − y i /L y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5)The order parameter Ψ n l = 1 for particles distributed FIG. 1: (Color online) A typical view of a highly-orderedground-state (Ψ n l = 0 .
96) colloidal distribution in a channel,for the total density ρ = 0 . σ − col . equidistantly in n l chains across the channel, and Ψ n l ≪ L y is the channel width, y i is the y coordinate of i th colloid and N is the number ofparticles in the system. Such a choice of the initial con-figurations predetermined a well-defined stripe structureand allowed us to focus on the study of stripe dynam-ics rather than on the dynamics of a low-ordered phase(i.e., for Ψ n l ≪ F dr ( y ) is dominating. Contrarily, in case of weak drive,the motion of colloidal stripes is mainly determined bythe structural distribution of colloids. In intermediatecases the character of colloidal motion results from a com-plex competition between these two factors.The distribution of the density of colloids in the trans-verse direction ρ chain ( y ) is essentially non-uniform, i.e.,presence of the channel boundaries leads to a relativelyhigher colloidal density on the periphery and lower in thecentral chains [6, 8, 9]. The distribution of colloidal den-sity becomes more homogeneous with increasing channelwidth L y , and in the limit of a wide channel, L y ≫ σ col ,the colloidal structure corresponds to a regular hexagonallattice.Another essential feature of the colloidal distributionin a channel consists in oscillating dependence of the de-fect concentration on the channel width L y [6]. Thus in-crease of the channel width is periodically accompaniedby a qualitative rearrangement of the colloidal structure,namely, by the occurrence of new chains. The valuesof the channel width, for which the colloidal structuresrearrange, correspond to low-ordered states (Ψ n l ≪ n l with increasing width of the channel. It isworth to note however that the fact that Ψ n l is large doesnot mean that the defect concentration is low. Actuallyit means only that stripes are well pronounced and alldefects are mainly topological. B. The integral of motion
Before discussing results of our numerical calculations,we note a useful property that directly follows from sys-tem of equations (2), namely, the integral of motion: η X i d~r i dt = X i ~F dr ( y i ) . (6)Separating the longitudinal component we obtain, interms of average stripe velocities, η X n ¯ υ xn = X n F dr ( y n ) , (7)where n denotes the n th chain, ¯ υ xn is the average velocityof n th stripe. This expression has a clear meaning: thesum of all average chain velocities is defined by the mo-mentum of the external force transferred to the system.From Eq. (7) it follows that the colloidal system is in thestatic state, i.e., does not move, only in case of: (i) zerodriving force; (ii) sign-alternating driving force satisfyingthe condition P n F dr ( y n ) = 0 and max | F dr ( y n ) | < F c ,where F c is some critical force.This result is remarkable due to the fact that it differsfrom the case of particles moving under the action of anexternal force in any stationary periodic potential. In thelatter case, for a commensurate configuration (i.e., whenthe number of colloids per unit length coincides with thenumber of potential minima), there is always a thresholdvalue of the force related to the “static friction”. Thusthe system of particles moves if the driving force exceedsthe threshold value. In our case, the system of colloidsmoves always when the sum of forces in (7) is larger thanzero.Note that the conservation law (7) is fulfilled for anydistribution of the driving force F dr ( y ) and it is not validif temperature is non zero. C. Stripe velocity versus non-uniform driving force
In this subsection, we study a response of the colloidalsystem confined in a channel to an external non-uniformdriving force. Clearly, this case is more general than theearlier studied case of a uniform driving created, e.g., bythe gravity in inclined channels [9]. A gradient drivingforce produces a shear stress that allows us to examineelastic properties of the colloid “lattice” and to revealthe onset of plastic motion. The transition from elasticto plastic motion leads to a number of different dynamicalregimes.Without loss of generality, we consider two typical pro-files of the non-uniform distributed driving force: (i) alinear distributed force given by F dr ( y ) = F max (cid:18)
12 + yL y (cid:19) , (8) FIG. 2: (Color online) The profile of average chain veloc-ities h υ x i as a function of the transverse coordinate y for aparabolic ((a) to (c)) and a linear ((d) to (f)) profiles of theapplied driving force for various values of α : α = 0 .
01 to 0.08((a), (d)), α = 0 . α = 1 to 8 ((c),(f)). The width of the channel is L y /σ col = 4 . ρ = 0 . σ − col . The symbols connected bysolid lines are average velocities of chains while dashed linesshow velocities of (non-interacting) chains corresponding tothe distribution of the applied force. and (ii) a parabolic distributed driving F dr ( y ) = F max − (cid:18) yL y (cid:19) ! , (9)where F max = mg sin( α ) is the maximum value of thedriving force which we express, for convenience, via theprojection of the gravity force that acts on a free colloidalparticle being on an inclined plane at angle α ; m is themass of the colloidal particle for the density of colloids ρ col = 1600 kg/m [9].The average velocity of different chains ¯ υ x as a functionof their transverse coordinate y n is presented in Fig. 2for values of α varying in the range 0 . ≤ α ≤
8. Theprofiles of the calculated average stripe velocity ¯ υ x ( y n )are shown by different symbols (corresponding to differ-ent values of driving force expressed in terms of angle α ,– see the figure captions) connected by solid lines. Forcomparison, the profiles of stripe velocities in the absenceof interaction between the stripes (i.e., renormalized ap-plied external force F dr ( y ) /η ) are shown by dashed lines,for different magnitudes of the driving force. With in-creasing α and, accordingly, the total amplitude F max ofthe driving force the role of the shape of the distributionof applied force on ¯ υ x ( y n ) increases, while the influence of the structure of colloids and the density of individ-ual chains, ρ chain ( y ), on the contrary, decreases. In thelimit of very large driving force F dr (1 . α ≤ F dr ( y ), as one can ex-pect in the plastic limit (see Figs. 2(c),(f), for parabolicand linear-gradient driving, correspondingly). On thecontrary, for small amplitudes F max of the driving force(0 . ≤ α . . υ xn from the profile ofthe driving force F dr ( y ) are essential (Fig. 2(a),(d)). Thislimit corresponds to the elastic or quasi-elastic regime.In the intermediate region of F dr ( y ), 0 . . α .
1, thefunction ¯ υ x ( y n ) is essentially influenced by both factors, F dr ( y ) and ρ chain ( y ).The above behavior is rather general and is typical, inprinciple, for any profile of the external driving. At thesame time, the shape of the driving-force profile is im-portant, especially, in the region of intermediate drivingforces. In other words, for the same set of parameters(i.e., the total density, ρ , the channel width, L y , and α )the difference between F dr and ¯ υ x varies for parabolicand linear profiles of the external driving force.In order to introduce a measure of conformity of theprofiles of the calculated average velocity ¯ υ x ( y ) and theexternal force F dr , we define a normalized standard de-viation ∆ in the form:∆ = vuut X chain (cid:18) F dr ( y chain ) /η − v chain F max (cid:19) . (10)Low values of ∆ correspond to small difference between¯ υ xn ( y ) and F dr ( y ). For a parabolic driving-force profile,the gradient of the applied force and thus the shear stressbetween inner chains is much less than between those atthe periphery, while for a linear profile the gradient isconstant.The results of our calculation of the standard deviation∆ are shown in Fig. 3 versus driving force (a) and density(b,c), for linear and parabolic profiles of the driving force.Note that for small driving forces, i.e., for elastic mo-tion, the deviation of the velocity profile ¯ υ xn ( y ) from thatof the applied driving F dr ( y ) is less in case of a lineardriving-force profile. This means that the elastic defor-mation of the colloidal “solid” related to a linear drivingis stronger than that for a parabolic driving. Contrary,for overall large driving (i.e., α ), linear-gradient-drivenstripes follow the profile of the driving force to a lesserextent than in case of parabolic driving (see Fig. 3(a)).This means that the average dynamical friction betweenchains is larger in case of a linear-driving applied force.This conclusion seems to be quite surprising keeping inmind the large gradient of the driving force (experiencedby peripheral chains) in case of a parabolic driving. Aswe show below, it can be understood in terms of incom-mensurability between peripheral chains (i.e., there thedriving gradient is maximum) related to different densi-ties of colloids in different chains. The function ∆( ρ ) ver-sus the overall colloidal density is presented in Figs. 3(b) FIG. 3: (Color online) The normalized standard deviation ∆as a function of α , calculated from data presented in Fig. 2(i.e., for a four-chain structure calculated for ρ = 0 . σ − col ).Squares (connected by a dot-dashed line) correspond to alinear profile of the driving force; circles correspond to aparabolic profile (a). The function ∆ vs. the total colloidaldensity ρ for: a parabolic driving force (b) and a linear-gradient driving force (c), for different values of α = 0 . and 3(c), respectively, for parabolic and linear-gradientdriving, for different α , and L y /σ col = 4 .
3. Note thatin both cases ∆( ρ ) has a minimum at about ρ = 0 . σ − col corresponding to a perfect chained structure and incom-mensurate number of colloids in different chains. As aresult, the chains easily slide with respect to each other,and the velocity profile reproduces that of the drivingforce (if the driving is strong enough, see Figs. 3(b) and(c)). The minimum in ∆( ρ ) at ρ = 0 . σ − col is followed bya sharp increase of ∆( ρ ) at about ρ = 0 . σ − col explainedby the appearance of defects in the chained structure.The defects lock the motion of different chains, similarlyto the case of low densities (i.e., ρ ∼ . σ − col ), and thesystem displays a rigid-body(RB)-like motion. With in-creasing ρ , the maximum in ∆ alternates with a grad-ual decrease indicating increasing ordering of colloids in stripes. Note that the behavior of ∆ is in general similarfor the linear and parabolic driving force (i.e., for highdensities ρ ), the absolute values and dispersion of ∆ fordifferent α are lower in case of the linear-gradient driving,in agreement with the result shown in Fig. 3(a). D. The role of colloidal density in stripes
In order to study the influence of two competing fac-tors, the structural factor ρ chain ( y ) (i.e., the transversecolloidal density) and the distribution of the externalforce F dr ( y ), we performed simulations of colloidal mo-tion for various values of the total density ρ in the inter-mediate range of values of α , 0 . ≤ α ≤ .
8. The averagestripe velocity ¯ υ x as a function of the transverse coordi-nate y at various values of α and different total colloidaldensity ρ is presented in Figs. 4 and 5, respectively, forparabolic and linear-gradient profiles of the external force F dr ( y ). The corresponding distributions of the (normal-ized) cross-section chain density ρ chain ( y ) are shown inthe bottom panels of Fig. 4.As seen from Fig. 4(a), for the density ρ = 0 . σ − col and α < . y = 0 then the critical value α of the RB-to-plastic mo-tion increases up to 0 .
5. Such an increase of the criticalvalue of α is explained by the simultaneous locking ofmotion of the central stripe by two adjacent stripes. Thetransition from the RB mode to the plastic mode occursonly for high enough magnitudes of F max when colloidsof the central stripe are capable to overcome the poten-tial barriers created by peripheral stripes. However, incase of an asymmetric (with respect to the channel axis y = 0) distribution ρ chain ( y ), the motion of the centralstripe is assisted by defects. The potential-energy profileasymmetry, together with the transverse degree of free-dom, leads to such type of motion of the central stripewhen it avoids obstacles and moves along the potential-energy minimum lines, i.e., along serpentine-like trajec-tories. This kind of motion provides a lower, as comparedto the symmetric case, critical value of α of the transitionfrom RB to plastic mode.For density ρ = 0 .
41, the RB mode is observed only forthe lowest value α = 0 . α =0.1 to 0.8 (Fig. 4(c)-(f)) but it can be realized for evensmaller values of α . Note, however, that for ρ = 0 . σ − col (Fig. 4(d)) (the least dense state of a four-chain phase) at α = 0 . F dr ( y ) this modeis observed for values α ≤ . ρ =0 . σ − col (i.e., the lowest- ρ three-chain phase), that meansa lower threshold magnitude F max of the transition from FIG. 4: (Color online) Top panels: The profile of averagechain velocities h υ x i as a function of the transverse coordinate y for a parabolic profile of the applied driving force for variousvalues of α ( α = 0 . L y /σ col = 3 .
7. Bottom panels: The correspondingdistributions of the chain densities ρ chain ( y ) shown for com-parison (i.e., normalized on ρ max , where ρ max is the maximumdensity of the chain in the colloidal n l -chain configuration).The results are shown for the total density ρ : ρ = 0 . σ − col (a), ρ = 0 . σ − col (a), ρ = 0 . σ − col (c), ρ = 0 . σ − col (d), ρ = 0 . σ − col (e), and ρ = 0 . σ − col (f). As in Fig. 2, symbolsconnected by solid lines show average velocities of chains, anddashed lines show velocities of (non-interacting) chains. elastic to plastic mode of colloidal motion (cp. Figs. 4(a)and 5(a)). At the same time, for the lowest- ρ four-chainphase, this threshold value is higher ( α ≤ .
4) in caseof a linear driving than for a parabolic, as can be seenfrom Figs. 4(d) and 5(d). This means that the RB modemanifests itself for: (i) a parabolic driving force F dr ( y )in the case of the lowest- ρ three-chain phase and (ii) alinear F dr ( y ) in case of the lowest- ρ four-chain phase.The observed behavior is explained as follows. In caseof a parabolic driving of the lowest- ρ three-chain struc-ture, the driving is applied to the central stripe. Themotion of this stripe is locked by the potential createdby two adjacent stripes. The most stable is a symmet-ric configuration, i.e., when two peripheral stripes aremirror-symmetric with respect to the axis of the channel FIG. 5: (Color online) The profile of average chain veloc-ities h υ x i as a function of the transverse coordinate y fora liner-gradient profile of the applied driving force for vari-ous values of α ( α =0.1 to 0.8) and for relative width of thechannel L y /σ col = 3 .
7. The results are shown for the totaldensity ρ : ρ = 0 . σ − col (a), ρ = 0 . σ − col (a), ρ = 0 . σ − col (c), ρ = 0 . σ − col (d), ρ = 0 . σ − col (e), and ρ = 0 . σ − col (f).As in Fig. 2, symbols connected by solid lines show averagevelocities of chains, and dashed lines show velocities of (non-interacting) chains. y = 0, and thus the potential created by them is doublethe potential created by a single stripe. Thus, the sym-metric configuration favors the RB mode. For asymmet-ric configurations, the RB-to-plastic mode threshold islower. In this case the motion of the central stripe (i.e.,for driving higher than a critical value) is serpentine-like, i.e., the central stripe deforms in the y -directionfollowing the minimum potential-energy path. A typicaldispersion of the trajectory of the central stripe in thetransverse direction is ∆ y ≈ . y ≈ − to10 − for the symmetric case). As a result, the lowest- ρ three-chain colloidal structure is characterized by a rela-tively high threshold of the RB-to-plastic transition, un-der a parabolic-profile driving (see Fig. 4(a)). In case of alinear-gradient driving of the lowest- ρ three-chain phase(Fig. 5(a)), the maximum driving force is applied to oneof the peripheral chains. In the RB mode, its motionis locked only by the adjacent central chain. When un-locked, the peripheral chain also provides an additionaldriving applied to the central chain. The central chainadjusts itself to follow the minimum potential-energypath between the two peripheral chains. This motionis characterized by a rather large transverse dispersion,∆ y ≈ . .
3, that facilitates an easy sliding of thethree chains with respect to each other. (a)(b) (c)(d) r s / - r s / - <> v x <> v x FIG. 6: (Color online) Average stripe velocities h υ x i as afunction of the total colloidal density ρ for a linear ((a), (b))and a parabolic ((c), (d)) profiles of the applied driving force,for the channel width L y /σ col = 5, and for different driving: α = 0 . α = 0 . h υ x i of different chains markedfor clarity by numbers 1, 2, 3, 4, and 5 (the numbers areshown by the same color as the corresponding curves). The situation is quite different in case of the lowest- ρ four-layer phase shown in Figs. 4(d) and 5(d). In theground-state low- ρ four-chain configurations, two centralchains have lower density than the peripheral chains (seebottom panels in Figs. 5(d), (e), and (f)). Thus, thepotential profile created by the central chains is deeper,and they survive a rather strong shear stress (similar tothe case of low-density 1D chains in a harmonic potentialin the Frenkel-Kontorova model [25]). Such shear stressis negligible for a parabolic driving (it appears as a re-sult of transverse instabilities). Thus the central partof the four-chain structure remain in the RB mode forvery large values of driving, and it also does in case ofa linear-gradient driving (see Figs. 4(d), (e), and (f)).However, the higher- ρ peripheral chains produce shal-lower potential-energy profiles, and thus the friction be-tween them and the central chains is weak [25]. It is clearthan that in case of a parabolic driving the peripheralstripes can easily slide with respect to the central stripes,and thus the RB-to-plastic mode threshold is rather low(Fig. 4(d)). In contrast, a relatively weak (i.e., for thesame F max ) gradient of the driving force in case of alinear-gradient driving provides a high threshold of theRB-to-plastic mode transition (Fig. 5(d)).As we discussed above, the colloidal distribution in achannel depends on the width of the channel and the to-tal colloidal density. For certain “matching” sets of theseparameters, colloids form well-defined stripe structurescharacterized by a maximum value of the order param-eter Ψ n l = 1, while small values of the order parameterΨ n l ≪ ρ = 0 . σ − col , the colloids form four stripes(marked by numbers 1, 2, 3, and 4 of the same color asthe corresponding lines/symbols), and for all values of α (i.e., α = 0 . ρ chain and are incommensurate with the central chains)slide with respect to the central chains. With increas-ing ρ , e.g., ρ = 0 . σ − col for α = 0 . α = 0 . ρ = 0 . σ − col and they unlock ata higher value of ρ : ρ = 0 . σ − col . For even higher den-sity, ρ = 0 . σ − col , the inter-shell defects lock the motionof the central chains 2 and 3 (and partially of all thechains), and the whole stripe structure collapses. If westill increase the colloid density, a new stripe (5) origi-nates from the “disordered” phase, and for larger ρ (i.e., ρ = 0 . σ − col ) the motion of all five stripes unlock, andthey move with individual velocities. Note that the tran-sition of the system from the state with n l stripes to thestate with n l + 1 stripes occurs always through the dis-ordered phase (i.e., when all or some of the stripes losetheir identity, and colloids of these “stripes” move withthe same velocity as, e.g., stripes 2, 3, and 4 in Fig. 6(a)and (b) at ρ = 0 . σ − col ). Thus the disordered phase ofthe motion serves as a “bifurcation point” that gives riseto a new state with n l + 1 velocity branches. The evolu-tion of stripe velocities for a parabolic driving occurs in asimilar way (Fig. 6(c) and (d)), although the degeneratevelocities of the central stripes, 2 and 3 (and those of theperipheral stripes, 1 and 4, which are separated from thevelocities of the central stripes by a wide gap), unlockonly for rather high density.It is also interesting to note an oscillating behavior ofthe velocity of motion of the peripheral stripes drivenonly by the interaction with adjacent stripes. The os-cillation in velocity reflects the oscillation of the frictionforce between the stripes with increasing total colloidaldensity. IV. MOBILITY OF STRIPES ANDDYNAMICAL PHASES OF THEIR MOTION
As shown above, the response of the system of col-loidal stripes (i.e., the profile of individual velocities ofdifferent stripes) to the external non-uniform driving is,in general, nonlinear: the velocity profile of the stripesis not scalable with the driving (i.e., for all values of thedriving, although there are regions of scalability whichwill be discussed below).In order to systematically examine the response of thecolloidal system to an external driving, we analyze themobility of different chains under the action of a delta-like driving force applied to one of the chains: F dr ( y ) = F drk ( y ) = F max δ ( y − y k,i ) , (11)where δ ( y − y k,i ) = 1, if y = y k,i , and δ ( y − y k,i ) = 0,if y = y k,i , y k,i is the coordinate of i th particle of k thstripe [27] and F max is the driving force as defined inEq. (9). We simulated the colloidal motion in a chan-nel of width L y /σ col = 4 for various colloidal densities ρ .For such a channel and considered densities ρ , the groundstate of the system corresponds to a three-stripe struc-ture characterized by a high value of the order parameterΨ ≈
1. Fig. 7 shows the results of calculation for themobilities versus driving force F drk for different stripes B n,k = ¯ υ xn /F drk , where ¯ υ xn is the average velocity of n th stripe, and F drk is the force applied to k th stripe.(Here we show only the case when driving is applied toone peripheral stripe. Applying driving to other stripesresults in a similar behavior.) Note that the mobility isdefined by B k,k = ¯ υ xk /F drk , while B n,k for k = n is amobile response of the stripes.As seen from Fig. 7, the mobility as a function of theapplied force exhibits a variety of modes of motion. First,we stress the most general features of the behavior. Astripe mobility B k,k tends to unity in the limit of large ap-plied force F drk , and mobile responses B n,k , correspond-ingly, vanish, as seen from Fig. 7. This limit correspondsto the plastic mode of motion. Another general conclu-sion refers to the RB-to-plastic transition: this transitionshifts towards lower drivings with increasing the totalcolloidal density (cp. Figs. 7(a) and (b)). As we dis-cussed above, a growth of the colloidal density results ina hardening of the rigidity of the elastic stripes [25] anda simultaneous weakening of the inter-stripe friction.However, apart from these general features, the mo-bilities display other important peculiarities. To discussthese, let us define dynamical phases corresponding todifferent regimes of colloidal motion in channels. Wedenote the first regime as “RB” (“rigid body”, or elas-tic motion). This regime is most pronounced for lowcolloidal density, ρ = 0 . σ − col (Fig. 7(a)), where it ex-tends to F dr ≈ .
05. For ρ = 0 . σ − col (Fig. 7(b)) thethreshold value reduced by a factor of five, F dr ≈ . F dr . The RB regimeis followed by a broad transition region (“Tr”) character-ized by a monotonic increase of the mobility B k,k (andmonotonic decrease of the mobile responses B n,k ). Whilethis growth is close to linear in Fig. 7(a), for higher den-sity it displays oscillations in Fig. 7(b). Remarkably, forhigher colloidal densities these oscillations turn to regionsof decreasing colloid mobility B k,k versus driving force.Note that this regime (denoted as “S”) is similar to thedynamically-ordered phase of vortex motion in supercon- FIG. 7: (Color online) The mobility B k,k and mobile response B n,k ( n = k ) of different stripes as a function of driving forceapplied to k th stripe ( k = 1) at various values of the totaldensity: ρ = 0 . σ − col (a), ρ = 0 . σ − col (b), ρ = 0 . σ − col (c), and ρ = 0 . σ − col (d). The mobility (mobile response)of different stripes are shown by different symbols: B , ( (cid:3) ), B , ( (cid:13) ), and B , ( △ ). ductors with arrays of regular pinning sites [28, 29]. Thisordered phase follows a disordered phase with a higherdensity of average vortex flow and thus leads to the ap-pearance of a negative differential resistivity (NDR) partof N-type in the VI-curve [29]. In our case of colloidalmotion in a narrow channel, we observe a similar effectcharacterized by a slowing down of the motion of thestripe driven by an external force (with a simultaneousincrease of the velocity of the adjacent stripes) when in-creasing the applied driving force. In this region, thesystem of colloids displays a partial “reentrant” behav-ior (cp. Refs. [30, 31]) when, being melted by increasingshear stress, it starts evolving towards a solid phase withfurther increasing driving. Note that this dynamically-induced “solidification” has common features with the re-cently discovered transition “freezing by heating” [32, 33]where by increasing temperature it leads to the crystal-lization of moving repulsive interacting particles whichare in a molten state.The mechanism of the observed colloid solidificationcan be understood as follows. As we discussed above,the longitudinal motion of colloidal stripes is accompa-nied by transverse oscillations of the stripes’ trajectories(i.e., serpentine-like motion) related to the asymmetryof the potential-energy profiles created by the adjacentstripes. For low drivings (but larger than the criticaldriving force of the RB-to-plastic transition), the stripedriven by an external force moves along some serpentine-like trajectory. When moving, the stripe itself deformsadjusting to the potential-energy landscape. In its turn,it elastically deforms other stripes due to the interactionwith colloids in the adjacent stripes. At low velocities,colloids in adjacent stripes relax to the initial state thusproviding low friction between the stripes. Increasingthe velocity of motion of the driven stripe (at high col-loid density) leads to increasing rate of the inter-stripecollisions and to the development of instabilities in thetransverse direction. This results in an increase of thedynamical friction between stripes which explains the ob-served decrease of the mobility B k,k (and increase of themobile response B n,k ), i.e., the appearance of “S” phase.Very large driving, however, straightens the trajectoriesof motion of the driven stripe. In the limit of F dr → ∞ ,the driven stripe moves along unrelaxed in the y -directionstraight trajectory, and the mobility B k,k → B n,k → V. CONCLUSIONS
We have studied the dynamics of colloids driven byan external non-uniform force in a narrow channel. Wefocused on colloidal densities which provide well-definedcolloidal stripe structures. In the limit of a very lowdriving force the system moves as a rigid body (elasticmotion) independent of the profile of the driving force.In the opposite limit of a very strong driving force, theprofile of the average velocities of colloidal stripes fol-low the profile of the applied driving (plastic regime).While these limiting-case results are easily understand-able, the dynamics of colloidal stripes in the general caseis rather complex. It is governed by the interplay of sev-eral factors, i.e.: (i) the magnitude and profile of driv-ing force; (ii) the total density of the colloidal systemand the distribution of the colloidal density in stripes,and, as a consequence, (iii) commensurability effects be-tween the adjacent chains. We have shown that depend- ing on the density and number of stripes, the transitionfrom elastic to plastic motion occurs at different values ofthe driving force. For example, the lowest-density three-stripe colloidal configuration is shown to be more ro-bust with respect to a parabolic driving than to a linear-gradient driving, while for the lowest-density four-stripeconfiguration the situation is opposite. This result is ex-plained by the colloidal density distribution over the cen-tral (lower density) and the peripheral (higher density)stripes and commensurability effects. We have analyzedthe mobility of colloidal stripes and have identified thedynamical phases of their motion. In particular, it hasbeen shown that the transition from elastic (rigid-body-motion phase) to plastic mode, depending on the densityof colloids, could be either monotonic (i.e., characterizedby a gradual increase of the mobility), or it could containan NDR-type part (i.e., characterized by a drop of mo-bility versus driving force). This unusual “solidification”is related to the dynamically-induced increase of the fric-tion in the colloidal system and is similar to the recentlydiscovered “freezing by heating” transition. The resultsof our study, with corresponding changes, can also be ap-plied to other systems of interacting particles driven bya non-uniform force in narrow channels, e.g., in physicsor biology.
VI. ACKNOWLEDGMENTS
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