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Simulation of olloidal hain movements under a magneti (cid:28)eld Mareike Grünzel Institut für Theoretis he Physik, Universität zu Köln, Zülpi her Straÿe 77, D-50937 Köln, Germany (Dated: February 2, 2008) Short olloidal hains are simulated by the slithering-snake-algorithm on a simple ubi latti e. The dipole hara ter of the olloidal parti les leads to a long range dipole-dipole intera tion. The 5 solvent is simulated by the nearest neighbor Ising model. The aligning of the dipoles under a 0 magneti (cid:28)eldgives rise to the hains to align on their partwith the(cid:28)eld dire tion. 0 2 n Keywords: Larson model, Monte-Carlo, dipole Initially, the hains are aligned in the x-dire tion in a every simulation. The magneti (cid:28)eld is oriented in z- J dire tion. The omputation of the energy is summed up 8 I. INTRODUCTION 1 5 ] The initial idea of this paper is based on the arti le of ft Goubault et al. [1℄, whose resear hpartlydealt with the 4 o aligning of olloidal hains and their urvature behavior nd-mat.s bhTeausnylehatiddentnregwoebspiaaignshtshhe(cid:29)iosboruboroemitr nfo psttlpeahoedii lenlyubmaplIyensresiodnar,gngwar ahhmenmaxie oitnuhdeisstrearnlawara[eln2lito ℄smh.teimhnaTaetguhernlhreaepytstmrdeuio drorgorv(cid:28)wrpoeaehuimlmtdinhle.idn ttehthdhsaeetobanfysdeit,amohriae--l Average of chain orientations23 o hains are simulated by the slithering-snake algorithm. 1 c This program was reated on the basis of Larson's PhD [ thesis [3℄. 1 0 0 20000 40000 60000 80000 100000 v time 4 II. ALIGNING OF THE CHAINS 2 FIG.1: Averageof hain10a×lig1n0i×ng10of5 hainsoflength6( on- 4 ◦ 1 TheIsingmodelandtheslithering-snakealgorithmare en3t.r0aJti/oknBof0.03)ina latti eatthetemperatSure of (100 runs). (+): x-aligning; ( ): y-aligning; ( ): 0 taken over to the new program. The (cid:28)rst aim of the z-aligning. 5 work is to add the long-range dipole-dipole intera tion 0 a ording to the olloidal hara ter of the hains to the / for one hain intera ting with the entire system due to t program. Themagneti dipolein~µtera tion~µenergyfortwo a i j the long range of the intera tion. This is repeated for m tinratenrsafo trimngedmtoagsn eatlai rsmpoinmveanrtiaablesasnid=±1(eso.g.tha[t4℄t)hiiss every sha=in−. 1Thespinsofthe hainsmo=le0 ulesrangefrom i i - leads to the Hamiltonian [5℄ value s =for1the tail, over to for the nseu=tra2l d i i 1 enter, and forthe se ondmole ule overto n H=− J s s i,j i j for the head. o 2 with c Xi,j Inthis ase,the magneti (cid:28)eld issimulatedindire tly by : aligning the dipoles in the dire tion of the (cid:28)eld. The v rXi Ji,j =(−J(−g/(rgi/jr)3ij·)31·(cid:0)−13−·3(r·iz(jr/izrji/jr)2ij)2(cid:1) ei,lsje,neighbors, ainh haFiineigvs.eda1lia,gfwntehri1en0arep×tphtrh1eoe0x(cid:28)×iamevl1dea0rtaldygier3e0o 0rtii0eo0nnttaiumtnioetnilssteeoqpfus5itlio bhrbaieuinmsseeoinsf a g = µ2/a3 (cid:0) (cid:1) rz length 6 in a latti e is shown. where is t~hre d=ip(orlxe,srtyre,nrgzt)h, ij is the om- The(cid:28)rst methodto al ulatethealigningisthedeter- ponent of the ve tor ij ij ij ij between two re- mination of the distan es of the head and the tail of the r ij gardeddipolesandi isthedistan Le3betweLen×tLhe×mL. The hains. This method favors (cid:28)nite-size e(cid:27)e ts in smaller spa e oordinate varies from 1 to for lat- systems, be ause the hains an spread over the latti e. ti es. The al ulation of the distan e is being done by - That is why only the se ond method has been used. It the al ulation of the oordinates of the two regarded al ulatesthedi(cid:27)eren ebetweentwo hainlinkssituated dipoles with the help of the modulo-fun tion. The x- next to ea hother. Then it isbeing de ided in whi hdi- oordinates,for example, arethe mLodulo-fun tion of the re ±ti1on they are standing to ea h other. If the di(cid:27)ere±nL e positions and the system width . The di(cid:27)eren e be- is ,thex-aligningisraisedby1;±ifLth2edi(cid:27)eren eis , tweenthex-,y-andz- oordinatesgivesthedistan eve - they-aligningisraisedandif it is it on ernsto the ~r ij tor of the dipoles. z-aligning. The averageis al ulated for every hainand 2 every run of the program at every timestep to get the dipolestorotateinthepositivez-dire tion,be ausethey ourseof hainaligningindependen eonthetime. Tak- aree(cid:27)e tiveagainstthemagneti (cid:28)eld. Simulationswith ingtheexampleofa hainwiththelengthof6,it anbe a positive magneti (cid:28)eld result in a suppression of the arrangedat a maximum of 5 in one dire tion. aligninginthedire tionofthe(cid:28)eld. Instead,simulations Simulations with a variation of the on entration of the with anegativemagneti (cid:28)eld showthe expe ted behav- hains result in a faster rise of the average alignments iorofthedipolesunderamagneti (cid:28)eldwithvarious(cid:28)eld towards the (cid:28)eld dire tion for small on entrations. strengths (Fig. 2). In Fig. 2 the average value of ea h omponent is omputed for 70 timesteps from 30 to 100 1 for every magneti (cid:28)eld strength. This shows the equal arrangementof the dipoles without a magneti (cid:28)eld in a 0.8 ni e way, be ause every dipole omponent has the equal 0.5 value . Simulations result in a onstant value after Average of dipole aligning 00..46 3rSbxo0-iemdrhstiaurimvoela fieotttsiirooht.ennepssesfwo arirvtehteahrtateihgneiegndviataiipasluuolel(cid:30)e so aanielrdinegittntiiaoonvongersrnameogsatuelllrvtaatnoilnudebot.mehTevbhissueiatbmelirene-. 5 0.2 4 0 0 5 10 15 20 25 30 35 40 45 50 (oFx f-Io G3on..m 0e2Jpn:/otkrnAaBetvnie(ot1r;na0(g00◦e.0)r3:uo)nfyis-dn fioaopmro1lp0eea×o na1hle0ing×m-tnB;1ain(0gSgnla)eo:tttfiz i- 5 (cid:28)eoe amlhtdaptishontnesreetnneomtfg.tlphee)nr.gat(th+ur)6e: Average of chain aligning23 1 III. MODIFIED MODEL 0 0 5 10 15 20 25 30 35 40 45 -B The program has a modi(cid:28)ed version, in whi h the dipoles are three-dimensional and rotate randomly. The FIG. 3: Average of the 1h0a×in1a0l×ig1n0ing of 5 hains of length 6 rotationtakes pla e with the method of Barker& Watts ( o3n. 0eJn/tkraBtio◦n0.03)ina latti eatthetemperature in their simulation of water (1969) [6℄. Mole ules ro- of (100 runs for ea Sh magneti (cid:28)eld strength). (+): x-aligning; ( ): y-aligning; ( ):z-aligning. tate with a random angle betwπeen zero and a spe i(cid:28)ed 2 maximum angle (in this ase ) to all sides about one spa e-(cid:28)xed axes hosen at random. The Hamiltonian is Fig. 3 shows the hain aligning of 5 hains depend- now hanged as follows: ing on the magneti (cid:28)eld strength. The average value is 1 H=− J − B~ ·~s omputed for 850timesteps from 150to 1000. As Fig. 1 i,j i 2 with shows,thisdurationisnotenoughto ompletethealign- i,j i X X ing, but its tenden y is shownin Fig. 3. For the interval B = 0 B = 3 from to the aligning in the dire tion of J = J −(g/ri3j) ~si~sj −3(~si·~rij)(~sj ·~rij)/ri2j the (cid:28)eldTd=oe3s.0nJot/ktBake pla e due to the high tempera- i,j (−(g/ri3j) ~si(cid:0)~sj −3(~si·~rij)(~sj ·~rij)/ri2j (cid:1) ttuhreereoifs a saturation .ofFtohrehmigahgnmeatig n(cid:28)eteil d,(cid:28)seoldthstartenagrtihses (cid:0) (cid:1) of the aligning is no longer possible. Instead, the mid- After the al ulation of the energy to align the hains dle interval of Fig. 3 shows a linear pro ess of aligning thereisanother al ulationforeverydipoleofthe hains in dire tion of the (cid:28)eld. The gradient of the aligning in relativeoftherotationofthedipoles. Thedipolestrength x- and y-dire tion is nearly the same. The line point- has the value 2 for the head of every hain and 1 for the inginto the x-dire tion, however,is situateda bit higher rest in all modi(cid:28)ed simulations. The magneti (cid:28)eld now than the one pointing into the y-dire tion, be ause the in(cid:29)uen es dire tly the rotation of the dipoles and there- hains have not arranged yet due to the short time; in fore indire tly the aligning of the hains in the dire tion the beginning they were arranged into the x-dire tion. of the (cid:28)eld. The magneti (cid:28)eld is negative in all simu- Despite the small time for the average value, this is a lations due to the fa t, that a negative (cid:28)eld a(cid:27)e ts the lear tenden y. 3 π/2 S−iπm/u2lations with a dis rete (random angle of or penden eonthetemperatureorthe hainlengthand on- ) aligning result in the same behavior but not as entration an be examined. The results from Goubault exa t as it is in the ontinuous aligning. The dis rete et. al. [1℄ an likewise be resear hed with little e(cid:27)ort, model shows various weak points in ontrast to the on- above all the urvature of the hains. The easiest way tinuous ase. Among other things, the dipole aligning to examine this is taking the (cid:28)rst method, whi h has takes pla emu h fasterto the value 1. In relationto the been mentioned in se tion II. For this, bigger systems hain aligning, there is no visible linear dependen e on are ne essary. The linear dependen e of the aligning on the magneti (cid:28)eld. themagneti (cid:28)eld shouldbeveri(cid:28)edforbiggersystems. IV. CONCLUSION The result is a good working program o(cid:27)ering lots of possibilities to work with. Among other things, the de- [1℄ C. Goubault, P. Jop, M. Fermigier, J. Baudry, E. tion, John Wiley &Sons, In .(1975). Bertrand, J. Bibette, Phys. RevLett. 91, 260802 (2003). [5℄ A. Mi helsen, Diploma Thesis: "Monte-Carlo- [2℄ D. Stau(cid:27)er, N. Jan, R. B. Pandey, Physi a A 198, 401 Simulationen mit langrei hweitigen Dipol- (1993). We hselwirkungen", Cologne (2000). [3℄ R. G. Larson, L. E. S riven, H. I. Davis, J. Chem. Phys. [6℄ M. P. Allan, D. J. Tildesley : "Computer Simulation of 83, 2411 (1985). Liquids",Chapter4.7, ClarendonPress-London(1990). [4℄ J. D. Ja kson: "Classi al Ele trodynami s", Se ond Edi-