Brownian-Dynamics Simulation of Colloidal Suspensions with Kob-Andersen Type Lennard-Jones Potentials 1

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1 Brownan-Dynamcs Smulaton of Collodal Suspensons wth Kob-Andersen Type Lennard-Jones Potentals 1 Yuto KIMURA 2 and Mcho TOKUYAMA 3 Summary Extensve Brownan-dynamcs smulatons of bnary collodal suspenton wth Kob- Andersen Lennard-Jones potental are performed n order to study the statc and the dynamc propertes of supercooled lqud and also to elucdate the mechansm of glass transton. The numercal results of several physcal quanttes, such as an effectve pressure, a radal dstrbuton functon, a mean square dsplacement, a non-gaussan parameter, and a self-ntermedate scatterng functon, are then nvestgated. Thus, t s shown that as the temperature s decreased, a crossover from a lqud state to a supercooled state occurs smoothly wthout any frst-order phase transtons. The slow relaxaton processes at lower temperatures are also dscussed from a unfed pont of vew. 1. Introducton The glass s a famlar and useful materal to the human beng from ancent tmes. The tradtonal method to obtan the glass s to cool the vscous lquds rapdly from a hgher temperature to a lower temperature. Wth a suffcently rapd coolng rate, the system can avod the crystallzaton at ts freezng pont and become a supercooled lqud. If one cools the system further, the system fnally goes nto a glass state. The transton from the supercooled lqud to the glass s called the glass transton. From the physcal pont of vew, however, t s not fully understood yet [1]. Recently, the slow dynamcs near the glass transton has attracted remarkable attenton [2]. About two decades ago, the appearance of the mode-couplng theory (MCT) trggered that ths feld was warmed up [3, 4]. However, near the glass transton, MCT fals to descrbe the whole dynamcs of the system snce t assumes the sngular behavor for physcal quanttes near the glass transton. Recently, Tokuyama has proposed the mean-feld theory (MFT) [5, 6], whch was obtaned from the nonlnear stochastc dffuson equaton for collodal suspensons [7, 8], and shown that MFT well descrbes the unversal behavor near the glass transtons. Computer smulatons have also provded us wth much helpful nformaton for deeper understandng about those phenomena. For a typcal example, Kob and Andersen have smulated the model of N 80 P 20, whch s known as one of metallc glass former, and compared ther results wth mode-couplng theory [9, 10, 11]. Today, ther model, the so-called Kob-Andersen model, s well known as a good glass-formng system on computer smulatons. Recently, Flenner et. al. have performed the Brownan-dynamcs smulatons on Kob-Andersen model wth 1000 partcles [12, 13] and recovered the supercooled state at temperatures lower than those n the orgnal Kob-Andersen model system. In the present paper, we perform extensve Brownan-dynamcs smulatons of bnary collodal suspensons wth Kob-Andersen type Lennard-Jones potentals wth partcles and explore the statc and the dynamc propertes of the model system from a unfed pont of vew. 1. Manuscrpt receved on Aprl 23, Graduate Student, Department of Nanomechancs, Graduate School of Tohoku Unversty. 3. Professor, Insttute Flud Scence, Tohoku Unversty.

2 86 KIMURA and TOKUYAMA / Rep. Inst. Flud Scence, Vol.19 (2007) 2. Startng Equatons We consder the three-dmensonal system whch conssts of N partcles wth mass m and radus a n the equlbrum solvent wth temperature T and vscosty η. The whole system s gven by a cubc box of sze L(= V 1/3 ), where V s the total volume of the system. There are two types of partcles, N A partcles of type A and N B partcles of type B. We assume that the radus a s of order m. Hence they undergo a Brownan moton. Let r α (t) be a poston vector of th partcle of type α at tme t. Then, the partcles obey the Langevn equatons m d2 dt 2 rα (t) = γ d dt rα (t) + A,B N β β j (t) + Rα (t), (1) where γ(= 6πηa) denotes the frcton constant and j (t) the force between th partcle of type α and jth partcle of type β. Here R α (t) denotes the Gaussan, Markov random force and satsfes < R α (t) > = 0, (2) < R α (t)r β j (t ) > = 2γk B T δ(t t )δ j δ αβ 1, (3) where the brackets < > denote the equlbrum ensemble average. Eq. (1) holds on the tme scale of order t B, where t B (= m/γ) s a Brownan relaxaton tme. In the followng, we neglect the hydrodynamc nteractons between partcles. Ths s because they are manly descrbed by the Oseen tensor and show long-range nteractons whch are dffcult to deal wth numercally. We are nterested only n the dffuson process on the tme scale of order t D, where t D (= a 2 /D 0 ) s a structural relaxaton tme. Here D 0 (= k B T/γ) s a sngle-partcle dffuson constant. Snce t D >> t B, on the tme scale of order t D, we have d 2 r α(t)/dt2 0. Then, Eq. (1) reduces to d dt rα (t) = 1 γ A,B N β β j (t) + f α (t), (4) where f α (t) denotes the random velocty gven by f α(t) = Rα (t)/γ and satsfes < f α (t)f β j (t ) >= 2D 0 δ(t t )δ j δ αβ 1. (5) Ths s a startng equaton to solve numercally. We dscuss ths next. Fg.1 The effectve pressure versus temperature. 3. Smulatons We choose N A = 8781 and N B = 2195 here, where N = and N A /N 0.8. The force between the partcles j s gven by the Kob-Andersen type Lennard-Jones potental j = U αβ (r αβ j ), (6) [ (σαβ ) 12 ( σαβ ) ] 6 U αβ (r) = 4ɛ αβ, (7) r r where ɛ αβ and σ αβ are chosen as n Ref. [9]; ɛ AA = 1.0, σ AA = 1.0, ɛ AB = 1.5, σ AB = 0.8, ɛ BB = 0.5, and σ BB = In ths model, both the crystallzaton and the phase separaton are known not to occur. The length of force cutoff r c s set on 2.5σ AA. The system sze L s fxed on 20.89σ AA. Length, tme, energy, and temperature are scaled by σ AA, t D, ɛ AA, and ɛ AA /k B, respectvely. Perodc boundary condtons are used. We ntegrate Eq. (4) at dfferent temperatures by usng Euler method wth tme step dt = The ntal confguratons are created by the random packng method [14]. Before we take the data, we wat for long tmes to equlbrate the system. 4. Smulaton Results We frst dscuss the effectve pressure P of the system. One calculates t by usng the vral theorem

3 Brownan-Dynamcs Smulaton of Collodal Suspensons wth Kob-Andersen Type Lennard-Jones Potentals 87 Fg.2 Radal dstrbuton functon of A-A correlaton for dfferent temperatures T = 0.45, 0.47, 0.50, 0.55, 0.60, 0.70, 0.80, 0.90, 1.0, 1.5, 2.0, 3.0 and 5.0 (from top to bottom). Fg.3 The log-log plot of the mean-square dsplacement of the partcles of type A for dfferent temperatures T =5.0, 3.0, 2.0, 1.5, 1.0, 0.9, 0.8, 0.7, 0.6, 0.55, 0.5, 0.47 and 0.45 (from left to rght). gven by P = Nk BT V 1 6V A,B N α N β α,β j r αβ j, (8) where r αβ j = r α r β j. In Fg. 1, the pressure P s shown versus temperature. As the temperature decreases, the pressure decreases smoothly. There exsts no jump n P over a whole temperature regon. Ths means that crystallzaton does not occur n the present system. Hence we dscuss how the structure of the system changes as the temperature decreases. Ths s done by calculatng the radal dstrbuton functon for A-A correlaton g AA (r) gven by NA 1 N A g AA (r) = δ(r rj AA ), (9) 2N A ρ A where ρ A (= N A /V ) s the number densty of the partcles of type A. In Fg. 2, g AA (r) s plotted for varous temperatures. At hgher temperatures, g AA (r) shows a structure observed n a lqud state. As T decreases, the splt generally appears on the second peak of g AA (r). Ths s well-known as one of characterstc propertes of supercooled lquds. Next, we dscuss the dynamcs of the system. It s convenent to ntroduce the mean square dsplacement M A 2 of the partcles of type A by M A 2 (t) = 1 N A N A < r A (t) r A (0) 2 >. (10) In Fg. 3, the smulaton result s shown for dfferent temperatures. In a short-tme regon, M2 A obeys a free dffuson process gven by M A 2 (t) = 6D 0 t. (11) In a long-tme regon, t obeys a long-tme dffuson process gven by M A 2 (t) = 6D L(A) s (T )t, (12) where D L(A) S (T ) denotes the long-tme self-dffuson coeffcent and depends on T. Thus, there s a crossover from a free dffuson process to a long-tme dffuson process. Ths crossover s ndependent of the temperature, although D L(A) S decreases drastcally as the temperature decreases. In an ntermedate-tme regon, however, the dynamcs of M2 A (t) strongly depends on the temperature. After short tmes, the many-body nteractons become mportant and prevent M2 A (t) to ncrease, leadng to plateaus at lower temperatures. Ths s the so-called cage effect. In order to nvestgate how the dynamcs at lower temperatures s dfferent from that at hgher temper-

4 88 KIMURA and TOKUYAMA / Rep. Inst. Flud Scence, Vol.19 (2007) Fg.4 The non-gaussan parameter of the partcles of type A versus tme for dfferent temperatures. The detals are the same as n Fg. 3. atures, t s also convenent to ntroduce the non- Gaussan parameter α2 A of the partcles of type A by α A 2 (t) = 3 5 M A 4 (t) (M A 2 where M A 4 (t) s the 4th moment gven by M A 4 (t) = 1 N A N A (t))2 1, (13) < r A (t) r A (0) 4 >. (14) In Fg.4, α2 A (t) s plotted versus tme for dfferent temperatures. As the temperature decreases, the peak poston τ α of α2 A ncreases and ts peak heght grows drastcally. Ths shows that the spatal heterogenetes play an mportant role at lower temperatures. The tme τ α s the so-called α-relaxaton tme. Fnally, we dscuss the self-ntermedate scatterng functon F S (k, t) of type A, whch s gven by F A S (k, t) = 1 N A N A =1 < e k [ra (t) ra (0)] >, (15) where k denotes the wave number and FS A (k, t = 0) = 1. The value of k s chosen as the frst-peak poston of the statc structure factor S AA (k), whch s gven by S AA (k) = 1 + 4πρ A 0 [g AA (r) 1] sn(kr) r 2 dr. kr (16) Fg.5 The statc structure factor of A-A correlaton versus wave number for dfferent temperatures. The detals are the same as n Fg. 2 In Fg. 5, S AA (k) s plotted versus wave number for dfferent temperatures. The frst-peak poston s found around k = In Fg. 6, FS A (k, t) s shown versus tme for dfferent temperatures at k For hgher temperatures, FS A (k, t) shows a sngle exponental lke decay. For lower temperatures, one can see the shoulders n the ntermedate-tme regon. Ths corresponds to the cage effect dscussed n the mean-square dsplacement M2 A (t). The long-tme decays around τ α are also dfferent from an exponental decay because the non-gaussan parameter α2 A (t) s large. Ths s easly seen f one can transform Eq. (15) nto a new form [ FS A (k, t) exp k 2 M 2 A (t) + 1 ( ) M A 2 ] 2 (t) 6 2 k4 α2 A (t). 6 (17) Thus, both effects, cage effect and structural effect, are ncluded n the self-ntermedate scatterng functon FS A (k, t). 5. Summary In the present paper, we have performed the Brownan-dynamcs smulaton on the bnary collodal suspensons wth the Kob-Anderson type Lennard- Jones potentals. We have then nvestgated several physcal quanttes at dfferent temperatures, such as

5 Brownan-Dynamcs Smulaton of Collodal Suspensons wth Kob-Andersen Type Lennard-Jones Potentals 89 Sports, Scence and Technology of Japan. Numercal computatons for ths work were performed on the SGI Altx3700 machne at the Insttute of Flud Scence, Tohoku Unversty. References [1] P. G. Debenedett and F. H. Stllnger: Supercooled Lquds and the Glass Transton, Nature, Vol.410 (2001), pp [2] Proc. 3rd Int. Symposum on Slow Dynamcs n Complex Systems (eds., M. Tokuyama and I. Oppenhem), AIP, Melvlle, Vol.708, [3] U. Bengtzelus, W. Götze and A. Sjölander: Dynamcs of Supercooled Lqud and the Glass Transton, J. Phys. C, Vol.17 (1984), pp Fg.6 The self-ntermedate scatterng functon of the partcles of type A versus tme for dfferent temperatures at k = The detals are the same as n Fg. 3. the pressure P, the radal dstrbuton functon g AA (r), the mean-square dsplacement M2 A (t), and the selfntermedate scatterng functon FS AA (k, t). Thus, we have shown that as the temperature decreases, the system smoothly becomes a supercooled state from a lqud state wthout crystallzaton. In fact, the pressure P decreases smoothly wthout any jumps as T decreases. The splt of the second peak of g AA (r) are observed at temperatures lower than T = 0.6. Ths s one of sgns for a supercooled state. Ths stuaton s the same as that n the Lennerd-Jones bnary mxtures [15]. We have not dscussed the long-tme self-dffuson coeffcent and several characterstc tmes. In order to study whether the system s n a supercooled state or not and also how the present system s dfferent from Lennard-Jones bnary mxtures, we must explore them from a unfed pont of vew. Ths wll be dscussed elsewhere together wth self-consstent analyses of smulaton results by the mean-feld theory. Acknowledgements The authors are grateful to T. Narum and Y. Terada for frutful dscussons. Ths work was partally supported by Grants-n-ad for Scence Research wth No.(C) from Mnstry of Educaton, Culture, [4] E. Leutheusser: Dynamcal Model of the Lqud- Glass Transton, Phys. Rev. A, Vol.29 (1984), pp [5] M. Tokuyama: Mean-Feld Theory of Glass Transtons, Physca A, Vol.364 (2006), pp [6] M. Tokuyama: Smlartes n Dversely Dfferent Glass-Formng Systems, Physca A, Vol.378 (2007), pp [7] M. Tokuyama and I. Oppenhem: Dynamcs of Hard-sphere Suspensons, Phys. Rev. E, Vol.50 (1994), pp.r16 R19. [8] M. Tokuyama: Slow Dynamcs of Equlbrum Densty Fluctuatons n a Supercooled Collodal Lqud, Physca A, Vol. 289 (2001), pp [9] W. Kob and H. C. Andersen: Scalng Behavor n the β-relaxaton Regme of a Supercooled Lennard-Jones Mxture, Phys. Rev. Lett., Vol.73 (1994), pp [10] W. Kob and H. C. Andersen: Testng Mode-couplng Theory for a Supercooled Bnary Lennard-Jones Mxture: The van Hove Correlaton Functon, Phys. Rev. E, Vol.51 (1995), pp [11] W. Kob and H. C. Andersen: Testng Mode-couplng Theory for a Supercooled Bnary Lennard-Jones Mxture. II. Intermedate Scatterng Functon and Dynamc Susceptblty, Phys. Rev. E, Vol.52 (1995), pp

6 90 KIMURA and TOKUYAMA / Rep. Inst. Flud Scence, Vol.19 (2007) [12] E. Flenner and G. Szamel: Relaxaton n a Glassy Bnary Mxture: Comparson of the Modecouplng Theory to a Brownan Dynamcs Smulaton, Phys. Rev. E, Vol.72 (2005), pp [13] E. Flenner and G. Szamel: Relaxaton n a Glassy Bnary Mxture: Mode-couplng-lke Power Laws, Dynamc Heterogenety, and a New Non-Gaussan Parameter, Phys. Rev. E, Vol.72 (2005), pp [14] W. S. Jodrey and E. M. Tory: Computer Smulaton of Close Random Packng of Equal Spheres, Phys. Rev. A, Vol.32 (1985), pp [15] T. Narum and M. Tokuyama: How Dfferent s the Dynamcs of a Lennard-Jones Bnary Flud from One-Component Lennard-Jones Flud?, Rep. Inst. Flud Sc., Vol.19 (2007), pp