Rectified motion induced by ac forces in periodic structures

P/2;.ç 1 Fiani / (1994) 4 e 1551-1561 1994, OCTOBER 1551 PAGE Cla,,ification Ab.~fi.fi<'1< Phi'.~i<.~ 05.61) ()~.40 87.10 Rectified motion Ajdari ('. Armand i') Laboratoire 05. Ceiiex i~) i~) by induced ), David ('. ~), Mukamel Phy,ico-Chimie de forces ac (~.~) Peliti Luca Théorique periodic in ESPCI, ). 10 structures Jacques and Vauquelii. rue (') Prost 752,~l Pari, France ot Coiiplex Phy,ique et Departmeiit Section de Sy,tem,. Weizmann The Chimie, Curie, In;titut ot'Science, In,titute 13 P. rue Rehovot, Curie, M, et 76100, Pari; 75231 Israél Cedex 05, France Dipartmiciito di dl Napoli, l~) Scieiiée Univer,ità (Re(.eii'ed analyze periodic potential in a potential is asymmetric of the re;ulting complex ;impie ,awtooth potential. feature, frameworL of ,eparation method, a analy,1, and of be can ,et ,pace behaviour niacro,copic force ac into or ai allow, Fi,ica dalla Materii the and ?ero an a;yil~metric low teniperature, of force ac tinte. We within the in experiil~ents ;ugge;t to u, by motion i, proil~oting a;,eil~blie,. protein mater di Italy /994) in Thi, Napoli, 8()1?5 5./u/). fil?a/ foi-ni il? Nazioiialc In,tituto 19, Pad. if the value mean a(.(.epled /994, Uniià, and d'oltremare, particle A Abstract~ zero Aprt/ II Fi~ichc Mo,tra Introduction. On symmetry application way could the or lead tu partiale <t force of grounds, of any force other (3 (. As at zero low Reynolds j j ( ~) J?ifii/ u-r-a- : such « as brutal valves picture » is armand@iurner.e~pci.fr. CNRS asymmetric potentials without that dissipation is forced one aimed at understanding how investigate in this article the we bath a pinning potential and an pumps, submitted tu and diods hâve the sense in provided by force of a time-periodic ac the temporal asymmetry in or in effort selective number~ obtained (1-3 (, provided motion gener~il more ~i of be may of this concept behaviour of externat ac value. average setter A influence potential current direction of part elaboration Asymmetric units They are time. the particle the the long dir/ction. a in 138~. the of ac that been known they essentially particle in of evolution zero signal mean results a value, in tu when a act rectifiers as forbid for transport periodic potential spatial asymmetry a rectified average quite in a une under of the macroscopic JOURNAL 1552 drift. This from clear is overdamped particle the in the inspection of absence noise of PHYSIQUE DE the I describing equation generic the motion N° 10 of an jl'=-é(~(-i)+y4(1) where and é. y spatial and U(.i + and ~b = in the have and.i / U(.;) 1) symmetric amplitudes the measure units (/ + 1) rescaled (/). ~b = potential of the been and temporal The respectively. The temporal periodicity conditions read: force ac that su (1) the of average ~b is If zero. the is system that sense u(~ u(.;) ;)t (2) and ~b shown (3) impose of the ~inalysis The which Conversely symmetries (2, 3) does resulting velocity. the of present is ~b (/ considerations a more or alternating force provide a paradigm the The (3) lt noise. 1/2). + initial However, conditions. average de drift can of this article with differs the however (1.e. (3) does time in -.;(/ the subject resemblance some structured ~isymmetric be to net a hold. net bear less (1), ~b = must velocity. average zero a two 1/2) + corresponds a solution be independent of solution.;(/) (I), to any below, the velocity average from then (t two be (2) and when ~iny quantitative the of way~ second hold), net obtained is work recent in be wiII as Thus (4( reference first we we do in for allow try net to protein assemblies (3-6], but rather mater to want of rectifying which take advantage the describe. promote experiments processes we describing the ~iverage drift velocity as a function of The ~in~ilysis of the « phase di~igrams the amplitudes of the pinning potential and of the ~ic force, reveals complexity surprising a of of including the Devil and already observed behaviour existence st~iircases resonant s an further rectification (4]. We show that the effects of spatial the and temporal competing in cancel the rectifying in asymmetry interesting process can ways. of an asymmetric sawtooth potential and a square We first an~ilyze the symmetric case ~ic for the function of » « force with no (or very rectifying » little) noise. This allows to us extract characteristics some of these complementary situation of a resonant processes as potential with an asymmetrtc ac force is then shown to produce similar features. In a symmetric the of potential force third stage, illustrate how and produce competition asymmetric we ac can drifts of variable sign. We show that situations (zero velocity at antiresonance average such the conditions) can by suggesting we separation devices are intermediate end noise new 1. then behaviour. be proposed, which in well as commenting these of influence could processes of investigation as the on rectifying thermal of use assemblies. protein mater be Model. particle motion of a massless periodic pinning potential W(x ) simple Langevin equation f where the Einstein friction coefficient relation fis (F(t) in immersed The an After obtained. experiments The and a a fluid homogeneous ~~ ~~ = related to r(t')) 2 = (x the + medium and submitted to externat force F (t be F (t + t'). constant a described r(t We by a (4) function ~iutocorrelation fkTô (t can bath write of the white W(x AU = noise (x/p) rby and N° AND FORCES Ac f0 1553 MOTION RECTIFIED U lac ' X a , ÎR(Î C) Fig. Shape l, and de~cribing parameter, F (t) B~b (t/r to B) of the potential exhibit externat (Î () with x/p, .; / = = kTz/fp~. a t/r, (1) F. = Ar/fp~, F potential wwtooth U the and periodicities (p and z), shapes (U and force. Equation (4) can then be the = and the Br/fp, y = y4(/) + + (0 à. force ac amplitudes (A ç§) and rewritten and : (5) 0(/) (/ 0 a « (t')) 2 -1') à (/ a = where = Equation (1) corresponds drift velocity, V aver~ige the to = i particle. This being particle at First, . of iii the . Second, grants that the periodicity (/,.;jj any in w.t(/~.tj + initial w ~b~ the n that noise One y. ~, depend not the on that show can position initial knowledge of 1(/,.ijj). A~ a implies that 1(/,.;o) w.i(/,.ij potential and the homogeneity pinning for differential in of the 1(/,.io) = does the of the = entirely 0 so )Il, 0 a order / cross, .i and time c~innot Hence, sh~ipes U first ~i ;(0 ~o > for seen (1) trajectories 1(/, easily is negligible of ltmit ;(/ Lim + the equ~ition, define~ the trajectory location the consequence~ time / any of the extern~il <.ij force n with.ijj w.ij w.;jj + satisfying condition.ij will lead to ~i trajectory therefore velocity 1(/,.;n) + 1, of the than.i~j. For given saine ~iverage velocity V is thus only a function of yand é, but not of the initial average conditions. will We where 2. We of a, focus start .; ~ / ~ h with results and asymmetric the in det~iil some j0, c/ [0, cl a = Spatially the with U (.i ~b ; .i ~ la, 1/2 (. t ~ IL., than 1. = smaller are As U (.i ~b a force, and (t) (1 = illustrated 1)Il> 1/2(1 we (6) (7) (.) = convention figure in take ai h. potential. analysis of this .ila = (t) following potential the on case is the action summartzed of a in temporally figure 2. symmetric force Continuous fines c 1/2. = delineate The essence isovelocity JOURNAL 1554 PHYSIQUE DE I N° 10 12 c 8 / , , ,' , ° / 1') Î / ~/,Î ,' ' ' 8 O ~ 16 a) 32 24 y 2 c i b~ o Fig. a) 2. Continuou, only the theoretical ,how; j2ù0 2 y to x ~ù() 1/6. 3 >16 diagram Pha;e « fine; de;cribe threshold~ to de;cription what 6 5 4 the attain » in the lower (y. > ot contour 1/2, 3 1/3, 1/2, = given 1/3, 3 plane for an 1;ovelocity 1/4, velocities by equation j8), wiggling a,pect of the extent Only are b) Blown-up view. as 1/4, 3 1/5, 3 The the curve, and 3. potential a;ymmetric for,ake demain, 1, 2, 3,.,, due i; V to 2 The,traight ot Da;hed fine,, the;e to compati;on repre;ented 1/6 8 ~ the 1/2. fine; valid finite 2 0.75, (a 1/~ ). readability we di,play fine, carre;pond to the in the number 1/3. 2 repre,ent y region. of,imulation, upper 1/4, >16 = 1/5, 2 2 h and N° 1555 MOTION RECTIFIED AND FORCES Ac 10 2 o 20 10 30 7 Fig. -Velocity ~. 1/2, ( >. temperature They are obtained two ~ uphill in Clearly will we call backward the the integer by average number of the y it direction. next Regime the force reversai. velocity in sign this regime n = This Regime the next 2 the time y~ pinning regimes potential é. é-16= the extract to + é./h. has a particle from the leads to may the in integer an to,ij 0+ does with not a little have time an half-period puise), first (reverse forwards move as the encountered well~ n ~ y backward exactly moves during the half-penod will it be velocity lé-16 ~ go obtatned. be can a in cannot and The there stays threshold to until obtatn the this or = Î ~é.la travels second bottom next average it : backwards, then particle the slope~ smallest whereas drift average or » however it uphill the Fig. I), penod, particle During the ii. + fait to time in ), the situations Two right time one drive to climb can the necessary not tends corresponds parttcle period unable rectification « during if (see Fig. y . particle following (to (n ~~ starts and different The constant = is the is ori gin = This y cases. 0. obvious An force ~ the il is the particle yi ~ in (forward . y ~ forward Suppose that starting at puise) a distance though the externat even potential gradient wins y V at 0.75, = = (1). equation é-la yj « (a Iine) (dotted 0.005 = yj velocity periods ii of force potential asymmetric an temperature resolution perturbation externat for y weak progressively the clearly scales are initially. w~is e-ç, fine) and numerical a increasing force important a) Small_fi)ic.e.ç, y potential well in which b) fiileintecliale_fi)t.< direction a intensity force ac 0 (fui( by understood The of function a zero best (Fig. 3). as ; = domains. are 4) V y to fait shift. ail the Upon way ~~~ ~é.la down repetition to the of the bottom, time it periods thus the JOURNAL 1556 V PHYSIQUE DE I N° " 15 " x x x ~ x * ~~~~-- - - x x ~ ~ « x ~ f Î %~ « ~. Î ~ ~ ~~ » ~~ ~É 05 )2 16 20 7 a) 2 V X i 8 $ l 6 ~-~ [ , ~~=~ ~ l.4 ~~ 12.5 12.7 12.9 z b) Fig. --Velocity 4. 1'=1/2, F staircme » potential with 4, uructure of 5, 0.5 = yn + sin ày force ac intensity magnification ((a) parabolic ;hape of (d) curve. ni y of Succe,,ive the (2 (cas > function a a~ 0)1 « U(.t) parameters V (4 yji ni " ))/2 w, and 10,65566975). for y to jc)) the force ;cale the «,tairca;e à(r) potential asymmetric an of » cm exhibit; obtained (2 WI) when are (a the « the used 0.6, Devil, ~mooth together 10 10 N° FORCES Ac AND 1557 MOTION RECTIFIED 33 V ~ h t 32 ~ l l 3 l " 29 t2.988 C) 1.12 1-1 1.08 ~_ -- l.06 5med ~-_ .04 1 1 '- 98 -2.0 Ai d) forward subsequent particle il Î ~ Fig. so shifts and step returns = n y ~~ Îé/fi 2 curve of egime + /é./h equires e Î 1). e, )-th I its bottom. average y~ is threshold The y Î Ii after Thus velocity m + I periods being h. /h to this obtain the e n - hat he 2 xact ntinuation the of = n velocity />.ll> y ~ 2 - its to similar y of (n the in position 1/(m - ~ e faits a to threshold of regime 1 indeed 2 reads JOURNAL 1558 both in 2a, + = threshold is puise. forward the é-la y the cases during with direct a PHYSIQUE DE N° I determined by the fact that a Upon increasing y, non zero jump to V =1 if é ~2 ah, 1) (ii distance velocitie~ ~tarting or + is fi 10 travelled are obtained by rational when values otherwise. Lai,qe Ji)1<e.ç, c) backwards. Clearly, allowance increasing of long y return point, that for 3. Temporally As explained drift. We (a one for ~i) = of that to drives, r~itional becomes but and and velocity the puises reverse forward inferred and one con the particle suit describe motion, backward cascade of a but the uphill climb can of most the data. feature main although decreases average on ;traight the asymmetric ~b n-th The successive singular blow than first, diagonal, velocity ii from defining a needed if or translated switch velocities, rational « up of the V y those described Devil curve in singularity for the square-root slope) di;played (Fig~. 4b, c). a (finite fine of the the intersects less (Fig. 4d) analysis obtained. is through force a an map return ii/m shown as from velocity evident U expect~ in;tead of ,tairca;e bath may be m-th through potential a generically the a now which structure the involved more the is some local of the m-th observed. behaviour occurs ii favours Whenever map. fixed stable during now becomes backward be general The y,, ~ situation can increase y The (6) defining a to velocity staircase s », (Fig. 4). and Note (7), enveloppe one of the force. breaking either (2) or (3) leads to the exi~tence of an average brietly consider the sole breaking of the temporal keeping the spatial symmetry, similar h 1/2). A typical phase diagram is shown in figure 5, The structure is fairly figure ? introduction~ in the here = y ~ 4 the >.<., externat force is large enough never tu the move p~irticle 16 / 12 c 8 4 o O Fig. 5. jfi 1/2, = velocitie; « Pha;e IN). v IN, diagram » Continuou, 1/3, 16 8 1/~, in the fine, 2, (Y, > de;cribe 3,. plane the for a lower potential ,ymmetric contour 32 24 y of and isovelocity an demain, asymmetric ac force carre,ponding to 1() N° b) FORCES Ac for y 4 ~ macroscopic if y é<., motion is and case. For y (regime 1), appearing ~it asymmetry p~irticles the + é-t- absent, If 4 0. V V is increasing an threshold a + function é(1-1)- ~4 l~ ~iround y ~ of only by : given MOTION oscillate é(. = right the to move 4 ~ still RECTIFIED AND 4 ~ The y. integer y é(1 <.) (regime 2). Note for whereas appear 4 y + = ><. c) Upon 4 1, but motion Combining 4. in for <. é é (1 ), Stans staircases devil encountered are ii ~~~~ >.<. velocities of form the -1/(m n velocity jumps directly fiom values. with rational velocity larger will decrease start to as be may 1) + 0 to the velocity ), the and é motion case (1 y rational <.) the to but only spatial -~4 ~ >.<. Il (4 similar is V = y l/2 ~ opposite yabove allowed, is ~4 y ~ that the increasing backw~ird -1 minimum energy the particle can <. é situation velocities ~ é.(. given a ~4 -~4 ~~~ 1559 and ai larger encountered. asymmetries. both spatial effects the rectifying either add asymmetry can this limit large potential barriers I. In the w up or seen é. compared the spatial period and the general during be large time to excursions one can of the velocity trends be understood without paying attention to the exact rational nature con which of values : V can be estimated V the number spatial periods (n,) in are as ni iii n~ furthermore visited during one forward (backward) é./h~ get the step. Focusing we on y function of <. (we take a 2 h) corresponding ~inalytical for V as a variation In gener~il the both of case This subtract. is most temporal easily period ~ind for = = ~ ~~~~ ~ <. from 0~ m Heavyside o is the where varied 0 to for other 1, <. ones ~ ~/3 2 m in The sole pinning det~iiled 5. the Influence When a motion. of nature « its (é ~la ~isymmetry energies. excursions When of the small (- é order la ). con the If of one gets or a few clearly for c. ii = As for temporal and e~ich oppose with is c. é./2 h : h))"~) le~iding to velocity provides cancellation limiting periods, decre~ises V increases, spatial they the whereas cancels The regime. /2 ~ m has one and values of changes a give but the estimate scale keep to course sign, direct the of track the of the discontinuous visible. noise. temperature the exp of are particle neighbours, é, in c. ~~ ~~ allowed). is values positive direction~ Il a(a ? + motion from For 1~ m equation (h~ + backward no decreases (forw~ird) asymmetry this Again as c. velocity changes a (b temporal of the measure spatial the l/2~ ~ c. same given by velocity ~° of <. for a the ~Î(1 ~Î.ÎÎ~~~ÎIa)Î ~~(Î monotonously contribute processes l/2. The ~ ~ (when function step velocity the negative to rectification ~/)Î ~~Î ~Îl Î~Î~ÎI~~~ is small occasionaly typical an extern~il but finite, make a hopping force noise thermal rate charactenzed has to be taken hop from one well following a by y is ~ipplied, into of the will bias to one of formula Kramers-like it For account. potential this hopping : 1560 JOURNAL motion, that so for Indeed present. " is which « a ~ The we ~, macroscopic Ta find of case Î Î~ N° velocity is obtained if ~ ~ ÎÎÎ Î.)~ ~~ 10 asymmetry is behaviour asymmetry owing tu was also finite at ~ ~~~~ ~ 8 spatial (temporal) a frequency low I : 1~(1 ~.) in the non-zero r-h-s- spatial y of values j~~P Î~ ~ of the low at even PHYSIQUE DE temperature (second) term in [4] for a first the analyzed asymmetry. Note that (a-h) experimental an i'ic/ study in point, cancellation the regime this and lead can measure a both to of of measure a pinning the the anisotropy comp~ired energy to LT. generally, More singularities the of the V (>., diagram y progressively are washed ont as a (Fig. 3). is increased 6. Concluding remarks. Although rectification have been for a long time, known show in this analysis processes we they take interesting characteristics the periodic consider in this article. structures we in They could lead to a new generation of separation techniques, and to new tools to probe and analyze asymmetric such as assemblies. protein motor systems Indeed, separation techniques up to now rest upon the use of an externat continuous most induces field, that the migration of particles at a speed which characterizes There them. are only a few scheme I?i Separation techniques such as Force Flux this general exceptions to Fractionation (FFF) (8] could be renewed by the use of alternating and structures asymmetric fields. versions of FFF, Let us just give here two examples of the brownian small in one non particles are set into motion, by either a hydrodynamic flow or a d-c- electric field, parallel to a confined homogeneous horizontal watt on which they are by gravity. Surfaces such as blazed and electric fields could allowing for the alternating be used, selection of gratings monodisperse particles. Thi~ technique would work well for particles in the 10-100 ~Lm range. second selection of polyelectrolytes of a given molar A example is adapted to the mass. where the confinement Currently, one of the processes FFF in ~i mode provided by a uses is gradient (Soret effect) and the driving field is of electric origin. Again, selection temperature would be greatly enhanced by the use of ~i bl~ized grating type of surface and ac electric fields. similar such zeolites~ driven by ~ippropriate In a symmetric structures ~isymmetric w~iy, as that electric field be fraction a compared pumps Let potential for the applied of the to should considered directed transport symmetry). a the inherent local propose to which cytoplasm tubulin voltage in along (kinesins a dyneins) of part of possesses walk « the along in such symmetry allow these local them the and of intrinsic an large differences. cytoskeleton Each be can Therefore precisely have filaments. » number voltages. in would structure, a concentration assemblies network which or sizeable maintaining are this reasonable difference the that such across charge the with even Note pumps. forced be times protein motor molecules proteins could 1/40 eV assemblies the motion pol~irity directed ~i efficient filaments is way when Different (ATP). models have been recently proposed to (3-6], making use of the f~ict that the adsorption » potential generation exerts a (broken Triphosph~ite filament use that These of Motor tubulin energy m efficient ionic selective as ion, given article. Adenosine with describe that in assembly line~ir fed this used alternating extremely point out be in a be kT thermal finally us could sequences, electrochemical « on a than asymmetry. Rather externat alternating dynein trying fields kinesin or in molecule provide clues motility assays to as [9, flat is to the loi on motor to average activity, investigate but we the with here main N° this field ac adsorption of the characteristics driving of FORCES Ac 1(1 should potential, allow also but about information get to about of friction the the frequency amplitude, molecules the 1561 MOTION amplitude, Varying potential. us RECTIFIED AND of the asymmetry and asymmetry shape filament. the on and Acknowledgments. Laboratoire de Physico-Chimie Théorique Physique Théorique and the Service de Etat initiated. Condensé this work their kind hospitality during a visit in which at CE Saclay for was of Science. Institute hospitality and financial help from the Weizmann A. Ajdari acknowledges figures. help wtth the for grateful J-F-BChauwin We to are D. Mukamel ESPCI at would like the and thank to members of the of the members Service the de References II [2] Ajdan Ajdari For Ph. A.. A., J., C.R. experimental an Chap. 7, The,i,, D. Prost Ai a</. realisation Univer,ité ,i(.i Rousselet see 6 Pari; (April 1992). (1992) 1635-1639. Salomé L., A jdari A., 315 Il Pfii1., J., Prost J., Naiii e 370 (1994) 446-448. [3] [4] [~[ [6[ J., Pro~t cl 7 R. C. S.. fil. Ed;. D.. meihods V., [Ll[ [10[ Svoboda Finer J. K.. T., G. F. P. Dondi B., 71 widely m S.. and Schmidt C. F., Simmon~ R. M., A., Lell. Osier G. u~ed Rel. P/t_1',v j1993) Roi. New-York, Met/to(/.~ Williams technique;, are Ajdari Lell. Phy.ç. M., jspringer, M, M.. Martin Bier L., Peliti Rel. P/t_i'.i Ermentrouni fields Pul,ed Oison [8] O,, M. A,tumian Pe,kin J.-F., Chauwin Magna,co 7211994) F., Cell j1994) 2652-2655, and Cellular Engineering, V. Mow 1994). for,eparaiion Guiochon Schnapp Spudich 72 1766-1769. Mechanic, of En=_i't~ir)/o,q_1 155 il 987) Theoretical advancement G. L~tl 1477-148I. Ed;. large in jKluwer B. J., BlocL J. A., Nfifiu(> S. DNA fragment;, see e-g- Carle G. F., 468-482. M.. 368 chromatography Acad. Nfifin( j1994) Pub[., 365 and L)92). j1993) 113-119. relaied 721-727. ~eparation