Coupling Molecular Dynamics and Lattice Boltzmann to Simulate Brownian Motion with Hydrodynamic Interactions

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1 Couplng Molecular Dynamcs and Lattce Boltzmann to Smulate Brownan Moton wth Hydrodynamc Interactons Burkhard Dünweg Max Planck Insttute for Polymer Research Ackermannweg 10, Manz, Germany and Department of Chemcal Engneerng, Monash Unversty, Melbourne, VIC 3800, Australa E-mal: In soft matter systems where Brownan consttuents are mmersed n a solvent, both thermal fluctuatons and hydrodynamc nteractons are mportant. The artcle outlnes a general scheme to smulate such systems by couplng Molecular Dynamcs for the Brownan partcles to a lattce Boltzmann algorthm for the solvent. By the example of a polymer chan mmersed n solvent, t s explctly demonstrated that ths approach yelds (essentally) the same results as Brownan Dynamcs. 1 Introducton Remark: The present contrbuton ntends to just gve a very bref overvew over the subject matter. It s an updated verson of a smlar artcle 1 that the author has wrtten on occason of the 2009 NIC wnter school. For more detaled nformaton, the reader s referred to a longer revew artcle, Ref. 2. Many soft matter systems are comprsed of Brownan partcles mmersed n a solvent. Prototypcal examples are collodal dspersons and polymer solutons, where the latter, n contrast to the former, are characterzed by non trval nternal degrees of freedom (here: the many possble conformatons of the macromolecule). Fundamental for these systems s the separaton of length and tme scales between large and slow Brownan partcles, and small and fast solvent partcles. Mesoscopc smulatons focus on the range of length and tme scales whch are, on the one hand, too small to allow a descrpton just n terms of contnuum mechancs of the overall system, but, on the other hand, large enough to allow the replacement of the solvent by a hydrodynamc contnuum. Ths latter approxmaton s much less severe than one would assume at frst glance; detaled Molecular Dynamcs smulatons have shown that hydrodynamcs works as soon as the length scale exceeds a few partcle dameters, and the tme scale a few collson tmes. To smulate such systems consstently, one has to take nto account that the length and tme scales are so small that thermal fluctuatons cannot be neglected. The Boltzmann number Bo (a term nvented by us) s a useful parameter for quantfyng how mportant fluctuatons are. Gven a certan spatal resoluton b (for example, the lattce spacng of a grd whch s used to smulate the flud dynamcs), we may ask ourselves how many solvent partclesn p correspond to the scaleb. On average, ths s gven byn p = ρb 3 /m p, whereρs the mass densty andm p the mass of a solvent partcle (and we assume a three 1

2 dmensonal system). The relatve mportance of fluctuatons s then gven by ( ) 1/2 Bo = Np 1/2 mp = ρb 3. (1) It should be noted that for an deal gas, where the occupaton statstcs s Possonan,Bo s just the relatve statstcal naccuracy of the random varablen p. In soft matter systems,b s usually small enough such that Bo s no longer neglgble. Furthermore, hydrodynamc nteractons must be modeled. In essence, ths term refers to dynamc correlatons between the Brownan partcles, medated by fast momentum transport through the solvent. The separaton of tme scales can be quantfed n terms of the so called Schmdt number Sc = η kn D, (2) where η kn = η/ρ s the knematc vscosty (rato of dynamc shear vscosty η and mass densty ρ) of the flud, measurng how quckly momentum propagates dffusvely through the solvent, andd s the dffuson constant of the partcles. Typcally, n a dense fludsc for the solvent partcles, whle for large Brownan partcles Sc s even much larger. Fnally, we may also often assume that the solvent dynamcs s n the creepng flow regme,. e. that the Reynolds number Re = ul η kn, (3) where u denotes the velocty of the flow and l ts typcal sze, s small. Ths s certanly true as long as the system s not drven strongly out of thermal equlbrum. These consderatons lead to the natural (but, n our opnon, not always correct) concluson that the method of choce to smulate such systems s Brownan Dynamcs (BD) 3. Here the Brownan partcles are dsplaced under the nfluence of partcle partcle forces, hydrodynamc drag forces (calculated from the partcle postons), and stochastc forces representng the thermal nose. However, the techncal problems to do ths effcently for a large number N of Brownan partcles are substantal. The calculaton of the drag forces nvolves the evaluaton of the hydrodynamc Green s functon, whch depends on the boundary condtons, and has an ntrnscally long range nature (such that all partcles nteract wth each other). Furthermore, these drag terms also determne the correlatons n the stochastc dsplacements, such that the generaton of the stochastc terms nvolves the calculaton of the matrx square root of a 3N 3N matrx. Recently, there has been substantal progress n the development of fast algorthms 4 ; however, currently there are only few groups who master these advanced and complcated technques. Apart from ths, the applcablty s somewhat lmted, snce the Green s functon must be re calculated for each new boundary condton, and ts valdty s questonable f the system s put under strong nonequlbrum condtons lke, e. g., a turbulent flow t should be noted that the Green s functon s calculated for low Re hydrodynamcs. Therefore, many soft matter researchers have rather chosen the alternatve approach, whch s to smulate the system ncludng the solvent degrees of freedom, wth explct momentum transport. The advantage of ths s a smple algorthm, whch scales lnearly wth the number of Brownan partcles, and s easly parallelzable, due to ts localty. The dsadvantage, however, s that one needs to smulate many more degrees of freedom than 2

3 those n whch one s genunely nterested and to do ths on the short nertal tme scales n whch one s not nterested ether. It s clear that such an approach nvolves essentally Molecular Dynamcs (MD) for the Brownan partcles. Many ways are possble how to smulate the solvent degrees of freedom, and how to couple them to the MD part. It s just the unversalty of hydrodynamcs that allows us to nvent many models whch all wll produce the correct physcs. The requrements are rather weak the solvent model has to just be compatble wth Naver Stokes hydrodynamcs on the macroscopc scale. Partcle methods nclude Dsspatve Partcle Dynamcs (DPD) and Mult Partcle Collson Dynamcs (MPCD) 5, whle lattce methods nvolve the drect soluton of the Naver Stokes equaton on a lattce, or lattce Boltzmann (LB). The latter s a method wth whch we have made qute good experence, both n terms of effcency and versatlty. The effcency comes from the nherent ease of memory management for a lattce model, combned wth ease of parallelzaton, whch comes from the hgh degree of localty: Essentally an LB algorthm just shfts populatons on a lattce, combned wth collsons, whch however only happen locally on a sngle lattce ste. The couplng to the Brownan partcles (smulated va MD) can ether be done va boundary condtons, or va an nterpolaton functon that ntroduces a dsspatve couplng between partcles and flud. In ths artcle, we wll focus on the latter method. 2 Couplng Scheme As long as we vew LB as just a solver for the Naver Stokes equaton, we may wrte down the equatons of moton for the coupled system as follows: d dt r = 1 p, m (4) d dt p = F c + F d + F f, (5) t ρ+ α j α = 0, (6) t j α + β π E αβ = β η αβγδ γ u δ +f h α + β σ f αβ. (7) Here, r, p and m are the postons, momenta, and masses of the Brownan partcles, respectvely. The forces F actng on the partcles are conservatve (c,. e. comng from the nterpartcle potental), dsspatve (d), and fluctuatng (f). The equatons of moton for the flud have been wrtten n tensor notaton, where Greek ndexes denote Cartesan components, and the Ensten summaton conventon s used. The frst equaton descrbes mass conservaton; the mass flux ρ u, where u s the flow velocty, s dentcal to the momentum densty j. The last equaton descrbes the tme evoluton of the flud momentum densty. In the absence of partcles, the flud momentum s conserved. Ths part s descrbed va the stress tensor, whch n turn s decomposed nto the conservatve Euler stress π E αβ, the dsspatve stress η αβγδ γ u δ, and the fluctuatng stress σ f αβ. The nfluence of the partcles s descrbed va an external force densty f h. The couplng to a partcles ntroduced va an nterpolaton procedure where frst the flow veloctes from the surroundng stes are averaged over to yeld the flow velocty rght 3

4 at the poston of. In the contnuum lmt, ths s wrtten as u u( r ) = d 3 r ( r, r ) u( r), (8) where ( r, r ) s a weght functon wth compact support, satsfyng d 3 r ( r, r ) = 1. (9) Secondly, each partcle s assgned a phenomenologcal frcton coeffcent Γ, and ths allows us to calculate the frcton force on partcle : ( ) F d p = Γ u. (10) m A Langevn nose term F f s added to the partcle equaton of moton, n order to compensate the dsspatve losses that come from F d. F f satsfes the standard fluctuaton dsspaton relaton F f α = 0, (11) F f α (t)ff jβ (t ) = 2k B TΓ δ j δ αβ δ(t t ), (12) where T s s the absolute temperature and k B the Boltzmann constant. Whle the conservatve forces F c conserve the total momentum of the partcle system, as a result of Newton s thrd law, the dsspatve and fluctuatng terms ( F d and F f ) do not. The assocated momentum transfer must therefore have come from the flud. The overall momentum must be conserved, however. Ths means that the force term enterng the Naver Stokes equaton must just balance these forces. One easly sees that the choce f h ( r) = ( F d + ) F f ( r, r ) (13) satsfes ths crteron. It should be noted that we use the same weght functon to nterpolate the forces back onto the flud; ths s necessary to satsfy the fluctuaton dsspaton theorem for the overall system,. e. to smulate a well defned constant temperature ensemble. The detaled proof of the thermodynamc consstency of the procedure can be found n Ref. 2. We stll need to specfy the remanng terms n the Naver Stokes equaton. The vscosty tensor η αβγδ descrbes an sotropc Newtonan flud: η αβγδ = η (δ αγ δ βδ +δ αδ δ βγ 23 ) δ αβδ γδ +η b δ αβ δ γδ, (14) wth shear and bulk vscostes η and η b. Ths tensor also appears n the covarance matrx of the fluctuatng (Langevn) stressσ f αβ : σ f αβ = 0, (15) σ f αβ ( r,t)σf γδ ( r,t ) = 2k B Tη αβγδ δ( r r )δ(t t ). (16) 4

5 Fnally, the Euler stress π E αβ = pδ αβ +ρu α u β (17) descrbes the equaton of state of the flud (p s the thermodynamc pressure), and convectve momentum transport. 3 Low Mach Number Physcs At ths pont an mportant smplfcaton can be made. The equaton of state only matters for flow veloctes u that are comparable wth the speed of sound c s,. e. for whch the Mach number Ma = u c s (18) s large. In the low Mach number regme, the flow may be consdered as effectvely ncompressble (although no ncompressblty constrant s mposed n the algorthm). The Mach number should not be confused wth the Reynolds number Re, whch rather measures whether nertal effects are mportant. Now t turns out that essentally all soft matter applcatons lve n the low Ma regme. Furthermore, largema s anyway naccessble to the LB algorthm, snce t provdes only a fnte set of lattce veloctes and these essentally determne the value of c s. In other words, the LB algorthm smply cannot realstcally represent flows whose velocty s not small compared to c s. For ths reason, the detals of the equaton of state do not matter, and therefore one chooses the system that s by far the easest the deal gas. Here the equaton of state for a system at temperaturet may be wrtten as k B T = m p c 2 s. (19) In the D3Q19 model (the most popular standard LB model n three dmensons, usng nneteen lattce veloctes, see below) t turns out that the speed of sound s gven by b 2 c 2 s = 1 3h2, (20) where b s the lattce spacng and h the tme step. Therefore the Boltzmann number can also be wrtten as ( ) 1/2 ( mp 3kB Th 2 ) 1/2 Bo = ρb 3 = ρb 5. (21) 4 Lattce Boltzmann 1: Statstcal Mechancs The lattce Boltzmann algorthm starts from a regular grd wth stes r and lattce spacng b, plus a tme step h. We then ntroduce a small set of veloctes c such that c h connects two nearby lattce stes on the grd. In the D3Q19 model, the lattce s smple cubc, and the nneteen veloctes correspond to the sx nearest and twelve next nearest neghbors, plus a zero velocty. On each lattce ste r at tmet, there are nneteen populatonsn ( r,t). 5

6 Each populaton s nterpreted as the mass densty correspondng to velocty c. The total mass and momentum densty are therefore gven by ρ( r,t) = j( r,t) = n ( r,t), (22) n ( r,t) c, (23) such that the flow velocty s obtaned va u = j/ρ. The number of lattce Boltzmann partcles whch correspond ton s gven by ν = n b 3 m p n µ, (24) wherem p s the mass of a lattce Boltzmann partcle, andµthe correspondng mass densty. It should be noted that µ s a measure of the thermal fluctuatons n the system, snce, accordng to Eq. 21, one has Bo 2 = µ/ρ. If we now assume a velocty bn to be n thermal contact wth a large reservor of partcles, the probablty densty for ν s Possonan. Furthermore, f we assume that the velocty bns are statstcally ndependent, but take nto account that mass and momentum densty are fxed (these varables are conserved quanttes durng an LB collson step and should therefore be handled lke conserved quanttes n a mcrocanoncal ensemble), we fnd ( ) ( ν ν P ({ν }) e ν δ µ ) ( ν ρ δ µ ) ν c j. (25) ν! for the probablty densty of the varables ν. Ths must be vewed as the statstcs whch descrbes the local (sngle ste) equlbrum under the condton of fxed values of the hydrodynamc varables ρ and j. The parameter ν s the mean occupaton mposed by the reservor, and we assume that t s gven by ν = a c ρ µ, (26) where a c > 0 s a weght factor correspondng to the neghbor shell wth speed c. From normalzaton and cubc symmetry we know that the low order velocty moments of the weghts must have the form a c = 1, (27) a c c α = 0, (28) a c c α c β = σ 2 δ αβ, (29) a c c α c β c γ = 0, (30) a c c α c β c γ c δ = κ 4 δ αβγδ +σ 4 (δ αβ δ γδ +δ αγ δ βδ +δ αδ δ βγ ), (31) 6

7 where σ 2, σ 4, κ 4 are yet undetermned constants, whle δ αβγδ s unty f all four ndexes are the same and zero otherwse. Employng Strlng s formula for the factoral, t s straghtforward to fnd the set of populatons n eq whch maxmzes P under the constrants of gven ρ and j. Up to second order nu(low Mach number!) the soluton s gven by ( ) n eq = ρa c 1+ u c + ( u c ) 2 σ 2 2σ2 2 u2. (32) 2σ 2 The low order moments of the equlbrum populatons are then gven by n eq = ρ, (33) n eq c α = j α, (34) n eq c αc β = ρc 2 sδ αβ +ρu α u β. (35) The frst two equatons are just the mposed constrants, whle the last one (meanng that the second moment s just the hydrodynamc Euler stress) follows from mposng two addtonal condtons, whch s to choose the weghts a c such that they satsfy κ 4 = 0 and σ 4 = σ 2 2(= c 4 s). From the Chapman Enskog analyss of the LB dynamcs (see below) t follows that the asymptotc behavor n the lmt of large length and tme scales s compatble wth the Naver Stokes equaton only f Eq. 35 holds, and ths n turn s only possble f the abovementoned sotropy condtons are satsfed. Together wth the normalzaton condton, we thus obtan a set of three equatons for the a c. Therefore at least three neghbor shells are needed to satsfy these condtons, and ths s the reason for choosng a nneteen velocty model. For D3Q19, one thus obtans a c = 1/3 for the zero velocty, 1/18 for the nearest neghbors, and 1/36 for the next nearest neghbors. Furthermore, one fnds c 2 s = σ 2 = (1/3)b 2 /h 2. For the fluctuatons around the most probable populatons n eq, n neq = n n eq, (36) we employ a saddle pont approxmaton and approxmate u by zero. Ths yelds ( P ({n neq }) exp ) ( ) ( ) (n neq ) 2 δ n neq 2µρa c δ c n neq. (37) We now ntroduce normalzed fluctuatons va ˆn neq = nneq µρa c (38) and transform to normalzed modes (symmetry adapted lnear combnatons of the n, see Ref. 2) ˆm neq k va an orthonormal transformaton ê k : ˆm neq k = ê kˆn neq, (39) 7

8 k = 0,...,18, and obtan P ({m k }) exp 1 2 k 4 m 2 k. (40) It should be noted that the modes number zero to three have been excluded; they are just the conserved mass and momentum denstes. 5 Lattce Boltzmann 2: Stochastc Collsons A collson step conssts of re-arrangng the set of n on a gven lattce ste such that both mass and momentum are conserved. Snce the algorthm should smulate thermal fluctuatons, ths should be done n a way that s () stochastc and () consstent wth the developed statstcal mechancal model. Ths s straghtforwardly mposed by requrng that the collson s nothng but a Monte Carlo procedure, where a Monte Carlo step transforms the pre collsonal set of populatons,n, to the post collsonal one,n. Consstency wth statstcal mechancs can be acheved by requrng that the Monte Carlo update satsfes the condton of detaled balance. Most easly ths s done n terms of the normalzed modes ˆm k, whch we update accordng to the rule (k 4) ˆm k = γ k ˆm k + 1 γk 2r k. (41) Here the γ k are relaxaton parameters wth 1 < γ k < 1, and the r k are statstcally ndependent Gaussan random numbers wth zero mean and unt varance. Mass and momentum are automatcally conserved snce the correspondng modes are not updated. Comparson wth Eq. 40 shows that the procedure ndeed does satsfy detaled balance. The parameters γ k can n prncple be chosen at wll; however, they should be compatble wth symmetry. For example, mode number four corresponds to the bulk stress, wth a relaxaton parameter γ b, whle modes number fve to nne correspond to the fve shear stresses, whch form a symmetry multplett. Therefore one must choose γ 5 =... = γ 9 = γ s. For the remanng knetc modes one often usesγ k = 0 for smplcty, but ths s not necessary. 6 Lattce Boltzmann 3: Chapman Enskog Expanson The actual LB algorthm now conssts of alternatng collson and streamng steps, as summarzed n the LB equaton (LBE): n ( r + c h,t+h) = n ( r,t) = n ( r,t)+ {n ( r,t)}. (42) The populatons are frst re arranged on the lattce ste; ths s descrbed by the so called collson operator. The resultng post collsonal populatonsn are then propagated to the neghborng stes, as expressed by the left hand sde of the equaton. After that, the next collson step s done, etc.. The collson step may nclude momentum transfer as a result of external forces (for detals, see Ref. 2); apart from that, t s just gven by the update procedure outlned n the prevous secton. 8

9 A convenent way to fnd the dynamc behavor of the algorthm on large length and tme scales s a mult tme scale analyss. One ntroduces a coarse graned ruler by transformng from the orgnal coordnates r to new coordnates r 1 va r 1 = ǫ r, (43) where ǫ s a dmensonless parameter wth 0 < ǫ 1. The ratonale behnd ths s the fact that any reasonable value for the scale r 1 wll automatcally force r to be large. In other words: By consderng the lmtǫ 0 we automatcally focus our attenton on large length scales. The same s done for the tme; however, here we ntroduce two scales va and t 1 = ǫt (44) t 2 = ǫ 2 t. (45) The reason for ths s that one needs to consder both wave lke phenomena, whch happen on thet 1 tme scale (. e. the real tme s moderately large), and dffusve processes (where the real tme s very large). We now wrte the LB varables as a functon of r 1,t 1,t 2 nstead of r,t. Snce changng ǫ at fxed r 1 changes r and thus n, we must take nto account that the LB varables depend on ǫ: The same s true for the collson operator: n = n (0) +ǫn (1) +ǫ 2 n (2) +O(ǫ 3 ). (46) = (0) +ǫ (1) +ǫ 2 (2) +O(ǫ 3 ). (47) In terms of the new varables, the LBE s wrtten as n ( r 1 +ǫ c h,t 1 +ǫh,t 2 +ǫ 2 h) n ( r 1,t 1,t 2 ) =. (48) Now, one systematcally Taylor expands the equaton up to order ǫ 2. Sortng by order yelds a herarchy of LBEs of whch one takes the zeroth, frst, and second velocty moment. Systematc analyss of ths set of moment equatons (for detals, see Ref. 2) shows that the LB procedure, as t has been developed n the prevous sectons, ndeed yelds the fluctuatng Naver Stokes equatons n the asymptotc ǫ 0 lmt however only for low Mach numbers; n the hgh Mach number regme, where terms of order u 3 /c 3 s can no longer be neglected, the dynamcs defntely devates from Naver Stokes. In partcular, ths analyss shows that the zeroth order populatons must be dentfed wth n eq, and that t s necessary that ths encodes the Euler stress va sutably chosen weghts a c. Furthermore, one fnds explct expressons for the vscostes: η = hρc2 s 2 η b = hρc2 s 3 1+γ s 1 γ s, (49) 1+γ b 1 γ b. (50) 9

10 7 A Polymer Chan n Solvent In Ref. 6 we explctly amed at a comparson between BD and coupled LB MD for the same system. We chose a well studed standard benchmark system, a sngle bead sprng polymer chan ofn monomers n good solvent n thermal equlbrum. The BD algorthm s realzed va r α (t+h) = r α (t)+(k B T) 1 D jαβ F jβ h+ 2hB jαβ W jβ, = 1,2,...,N. (51) Here r s the coordnate of the th partcle, h s the BD tme step, D j s the dffuson tensor couplng partclesandj, and F j denotes the determnstc force on partclej (here sprng force and excluded volume force). We assume summaton conventon wth respect to both Cartesan and partcle ndexes. The tensor B j s the matrx square root of D j, D jαβ = B kαγ B jkβγ, (52) whle W s a dscretzed Wener process, W α = 0 and W α W jβ = δ j δ αβ. For the dffuson tensor we used the Rotne Prager tensor. The computatonally most demandng part s the calculaton of the matrx square root. The exact numercal soluton of ths problem va Cholesky decomposton has a computatonal complexty O(N 3 ). We therefore rather used Fxman s trck 7, 8 to speed up the calculatons. Ths s based on the observaton that the square root functon, f vewed as a functon actng on real numbers, needs to be evaluated only wthn a fnte nterval, spannng from the smallest to the largest egenvalue. Snce ths nterval does not contan the sngularty at zero, a truncated (Chebyshev) polynomal expanson approxmates the functon qute well. The same expanson can then also be used to evaluate the matrx square root. The number of terms needed s emprcally found to scale as O(N 0.25 ). Furthermore, for each term one needs to do a matrx tmes vector operaton, whch scales as O(N 2 ), such that the algorthm n total has a computatonal complexty O(N 2.25 ). We dd not employ an FFT based superfast BD algorthm 4 ; ths would have been qute complcated, and also requred to assume a smulaton box of sze L 3 wth perodc bondary condtons, such that an extrapolaton L would have been necessary. Such a fnte box sze, combned wth an extrapolaton, s however precsely what s needed for LB MD. We therefore ran these smulatons for at least three dfferent values of L n order to allow for meanngful extrapolatons (and used the total tme for these three systems to estmate our CPU effort). The typcal box szes that are needed are gven by the requrement that the polymer chan should ft ncely nto the box, wthout much back foldng. Snce n a good solvent the polymer radus R scales as R N ν, where ν 0.59 s the Flory exponent, we fnd L 3 N 3ν. Furthermore, the computatonal cost s completely domnated by the operatons requred to run the solvent, and hence the computatonal complexty s O(N 3ν ) = O(N 1.8 ). We see that ths s slghtly better than BD; however, the prefactor s much smaller for BD. In practce, we fnd that BD s roughly two orders of magntude faster than LB MD, for the typcal chan lengths used n smulatons, see Fg. 1. The stuaton s expected to be qute dfferent when one studes a semdlute soluton, where the monomer concentraton s stll qute low (such that the LB MD CPU effort s stll domnated by the solvent), but the chans are so long that they strongly overlap. For 10

11 Fgure 1. Comparson of the CPU tme needed by the LB MD and BD systems for the equvalent of 1000 LB MD tme steps for varous chan lengthsn. From Ref. 6. example, Ref. 9 studed 50 chans of length N = 1000, beng well n the semdlute regme. Whle the addtonal chans for the LB MD system pose essentally no computatonal burden at all (rather on the contrary: Flory screenng makes the chans shrnk, such that one can afford to run the smulaton n a somewhat smaller box), the BD effort (for our algorthm) s expected to ncrease by a factor of ,. e. more than three orders of magntude or even more, snce one needs a more complcated scheme to evaluate the hydrodynamc nteractons for a perodc system. In other words: For such a system, BD can at best be compettve f the superfast 4 verson s mplemented and to our knowledge, ths has not yet been tested. In order to allow a meanngful comparson, both systems have to be run for the same system and the same parameters. Ths mples, frstly, dentcal nteracton potentals between the beads, and the same temperature. From ths one concludes (and numercally verfes) that the statc propertes lke gyraton radus, statc structure factor, etc., must all be dentcal. For the dynamcs, t s mportant that both smulatons are run wth the same value for the shear vscosty η, whch s easy to acheve, plus wth the same value for the monomerc frcton coeffcent. At ths pont, one has to take nto account that the frcton coeffcent ζ that appears n the BD algorthm (on the dagonal of the dffuson tensor) s a long tme frcton coeffcent, whch descrbes the asymptotc statonary velocty v of a partcle that s dragged through the flud wth a force F, F = ζ v, whle the frcton coeffcent Γ that appears n the LB MD algorthm va the couplng prescrpton F = Γ( v u) (see Eq. 10) s a correspondng short tme coeffcent that does not yet take the backflow effects nto account. Indeed, for an experment n whch a partcle s dragged through the 11

12 Fgure 2. The dmensonless long tme dffuson constant for the center of mass at varous box lengths L. From Ref. 6 LB flud, t s clear that the flow velocty u wll be nonzero, and typcally slghtly smaller than v. Hence, F = ζ v = Γ( v u),. e. ζ s smaller thanγ. Snce hydrodynamcs allows us to estmate u up to a numercal prefactorg va a Stokes lke formula, F = gηa u, where a s the range of the nterpolaton scheme, one fnds (see also Ref. 2) ζ 1 = Γ 1 +(gηa) 1. (53) For nearest neghbor lnear nterpolaton, one fnds g 25 f a s dentfed wth the LB lattce spacng. One hence needs to choose the Γ value n the LB MD smulatons n such a way that t reproduces the BD ζ value. The dffuson constant of the LB MD chan depends on the box sze, as a result of the hydrodynamc nteracton wth the perodc mages. Snce the latter decays lke r 1, one concludes an L 1 fnte sze effect, whch s ncely borne out by the data of Fg. 2. From these data one sees also that for an accurate descrpton of the dynamcs t s necessary to not only thermalze the stress modes n the LB algorthm (only these matter n the strct hydrodynamc lmt), but also the knetc modes, as suggested by the more mcroscopc theory outlned above. Takng the fnte sze effect and the proper thermalzaton nto account, the remanng devaton between BD and LB MD s only a few percent. The nternal Rouse modes of the chan are defned as (p = 1,2,...,N 1) X p = 1 N N [ pπ r n cos N n=1 ( n 1 2 )]. (54) Fgure 3 shows the decay of the normalzed mode autocorrelaton functon up to p = 5. Obvously the agreement wth BD s qute good,. e. the fnte sze effect s qute weak. The reason s the followng: The dffuson constant corresponds to the frcton of the chan 12

13 Fgure 3. Normalzed autocorrelaton functon of the frst 5 Rouse modes X p for LB MD smulatons at fxed L = 25 and BD smulatons at L. From Ref. 6. Fgure 4. The autocorrelaton functon for the frst Rouse mode X 1 at a fnte tme value of t = 700 for LB MD smulatons at varous box lengths L and BD smulatons at L. From Ref. 6. as a whole,. e. to an experment where the chan s beng dragged through the flud wth a constant force. Ths gves rse to a flow feld that decays lker 1, and thus anl 1 fnte sze effect. Ths total force may also be vewed as the monopole moment of a dstrbuton of forces actng on the polymer. The Rouse modes however study the nternal moton of 13

14 the chan,. e. n the center of mass system. Therefore, the monopole contrbuton of the forces has been subtracted, and only hghr order multpole moments reman. The dpole contrbuton vanshes for symmetry reasons,. e. the frst hgher order multpole s the quadrupole (ths may be vaguely understood by recallng that the mass dstrbuton has a monopole and a quadrupole moment, but not a dpole moment). The quadrupolar flow feld decays lke r 3, and hence one expects an L 3 fnte sze effect. For a more detaled dervaton, see Ref. 10. Ths fnte sze effect s ndeed observed, see Fg. 4, demonstratng that on the one hand the system s theoretcally qute well understood, and that on the other hand such smulatons are nowadays so accurate that even rather subtle effects can be analyzed unambguously. References 1. B. Dünweg, Computer smulatons of systems wth hydrodynamc nteractons: The coupled Molecular Dynamcs - lattce Boltzmann approach, n: Multscale Smulaton Methods n Molecular Scences, J. Grotendorst, N. Attg, S. Blügel, and D. Marx, (Eds.). Forschungszentrum Jülch, Jülch, B. Dünweg and A. J. C. Ladd, Lattce Boltzmann smulatons of soft matter systems, Advances n Polymer Scence, 221, 89, G. Nägele, Brownan dynamcs smulatons, n: Computatonal Condensed Matter Physcs, S. Blügel, G. Gompper, E. Koch, H. Müller-Krumbhaar, R. Spatschek, and R. G. Wnkler, (Eds.). Forschungszentrum Jülch, Jülch, A. J. Bancho and J. F. Brady, Journal of Chemcal Physcs, 118, 10323, M. Rpoll, Mesoscale hydrodynamcs smulatons, n: Computatonal Condensed Matter Physcs, S. Blügel, G. Gompper, E. Koch, H. Müller-Krumbhaar, R. Spatschek, and R. G. Wnkler, (Eds.). Forschungszentrum Jülch, Jülch, Tr T. Pham, Ulf D. Schller, J. Rav Prakash, and B. Dünweg, Journal of Chemcal Physcs, 131, , M. Fxman, Macromolecules, 19, 1204, R. M. Jendrejack, M. D. Graham, and J. J. De Pablo, Journal of Chemcal Physcs, 113, 2894, P. Ahlrchs, R. Everaers, and B. Dünweg, Physcal Revew E, 64, , P. Ahlrchs and B. Dünweg, Journal of Chemcal Physcs, 111, 8225,