1 Simulations of Particle Suspensions at the Institute for ...

1**Simul ations**

2 Jens Harting, Martin Hecht, and Hans Herrmannknowledge **of** elementary processes and to enable us to find **the** a**for**ementionedrel**at**ions from simul**at**ions instead **of** experiments.In this paper we will present two approaches which have been appliedduring **the** last year: A combined Molecular Dynamics and Stochastic Rot**at**ionDynamics method has been applied to **the** simul**at**ion **of** claylike colloids and**the** l**at**tice-Boltzmann method was used as a fluid solver to study **the** behaviour**of** glass spheres in creeping shear flows.1.2 Simul**at**ion **of** Claylike Colloids: Stochastic Rot**at**ionDynamicsWe investig**at**e properties **of** dense suspensions and sediments **of** small sphericalAl 2 O 3 particles by means **of** a combined Molecular Dynamics (MD) andStochastic Rot**at**ion Dynamics (SRD) simul**at**ion. Stochastic Rot**at**ion Dynamicsis a simul**at**ion method developed by Malevanets and Kapral [17, 18]**for** a fluctu**at**ing fluid. The work this chapter is dealing with is presented inmore detail in references [7, 6].We simul**at**e claylike colloids, **for** which in many cases **the** **at**tractive Vander-Waals**for**ces are relevant. They are **of**ten called “peloids” (Greek: claylike).The colloidal particles have diameters in **the** range **of** some nm up tosome µm. The term “peloid” originally comes from soil mechanics, but particles**of** this size are also important in many engineering processes. Our modelsystems **of** Al 2 O 3 -particles **of** about half a µm in diameter suspended in w**at**erare **of**ten used ceramics and play an important role in technical processes. Insoil mechanics [22] and ceramics science [20], questions on **the** shear viscosityand compressibility as well as on porosity **of** **the** microscopic structure whichis **for**med by **the** particles, arise [26, 16]. In both areas, usually high volumefractions (Φ > 20%) are **of** interest. The mechanical properties **of** **the**se suspensionsare difficult to understand. Apart from **the** **at**tractive **for**ces, electrost**at**icrepulsion strongly determines **the** properties **of** **the** suspension. Depending on**the** surface potential, one can ei**the**r observe **for**m**at**ion **of** clusters or **the** particlesare stabilized in suspension and do sediment only very slowly. The surfacepotential can be adjusted by **the** pH-value **of** **the** solvent. Within Debye-Hückel**the**ory one can derive a so-called 2pK charge regul**at**ion model which rel**at**es**the** simul**at**ion parameters with **the** pH-value and ionic strength I adjustedin **the** experiment. In addition to **the** st**at**ic interactions hydrodynamic effectsare also important **for** a complete description **of** **the** suspension. Since typicalPeclet numbers are **of** order one in our system, Brownian motion cannot beneglected.The colloidal particles are simul**at**ed with molecular dynamics (MD),whereas **the** solvent is modeled with stochastic rot**at**ion dynamics (SRD).In **the** MD part **of** our simul**at**ion we include effective electrost**at**ic interactionsand van der Waals **at**traction, a lubric**at**ion **for**ce and Hertzian contact

1 **Simul ations**

4 Jens Harting, Martin Hecht, and Hans HerrmannFor **the** rot**at**ion, a simple Euler algorithm is applied:ω i (t + δt) = ω i (t) + δtT i , (1.5)ϑ i (t + δt) = ϑ i (t) + F(ϑ i , ω i , δt) , (1.6)where ω i (t) is **the** angular velocity **of** particle i **at** time t, T i is **the** torqueexerted by non central **for**ces on **the** particle i, ϑ i (t) is **the** orient**at**ion **of**particle i **at** time t, expressed by a qu**at**ernion, and F(ϑ i , ω i , δt) gives **the**evolution **of** ϑ i **of** particle i rot**at**ing with **the** angular velocity ω i (t) **at** time t.The concept **of** qu**at**ernions [1] is **of**ten used to calcul**at**e rot**at**ional motionsin simul**at**ions, because **the** Euler angles and rot**at**ion m**at**rices can easily bederived from qu**at**ernions. Using Euler angles to describe **the** orient**at**ion wouldgive rise to singularities **for** **the** two orient**at**ions with ϑ = ±90 ◦ . The numericalproblems rel**at**ed to this fact and **the** rel**at**ively high comput**at**ional ef**for**t **of** am**at**rix inversion can be avoided using qu**at**ernions.The Stochastic Rot**at**ion Dynamics method (SRD) introduced by Malevanetsand Kapral [17, 18] is a promising tool **for** a coarse-grained description**of** a fluctu**at**ing solvent, in particular **for** colloidal and polymer suspensions.The method is also known as “Real-coded L**at**tice Gas” [10] or as “multiparticle-collisiondynamics” (MPCD) [23]. It can be seen as a “hydrodynamiche**at** b**at**h”, whose details are not fully resolved but which provides **the** correcthydrodynamic interaction among embedded particles [15]. SRD is especiallywell suited **for** flow problems with Peclet numbers **of** order one and Reynoldsnumbers on **the** particle scale between 0.05 and 20 **for** ensembles **of** manyparticles. The method is based on so-called fluid particles with continuouspositions and velocities. Each time step is composed **of** two simple steps: Onestreaming step and one interaction step. In **the** streaming step **the** positions**of** **the** fluid particles are upd**at**ed as in **the** Euler integr**at**ion scheme knownfrom Molecular Dynamics simul**at**ions:r i (t + τ) = r i (t) + τ v i (t) , (1.7)where r i (t) denotes **the** position **of** **the** particle i **at** time t, v i (t) its velocity**at** time t and τ is **the** time step used **for** **the** SRD simul**at**ion. After upd**at**ing**the** positions **of** all fluid particles **the**y interact collectively in an interactionstep which is constructed to preserve momentum, energy and particle number.The fluid particles are sorted into cubic cells **of** a regular l**at**tice and only **the**particles within **the** same cell are involved in **the** interaction step. First, **the**irmean velocity u j (t ′ ) = 1N j(t ′ )∑ Nj(t ′ )i=1 v i (t) is calcul**at**ed, where u j (t ′ ) denotes**the** mean velocity **of** cell j containing N j (t ′ ) fluid particles **at** time t ′ = t + τ.Then, **the** velocities **of** each fluid particle in cell j are upd**at**ed as:v i (t + τ) = u j (t ′ ) + Ω j (t ′ ) · [v i (t) − u j (t ′ )] . (1.8)Ω j (t ′ ) is a rot**at**ion m**at**rix, which is independently chosen randomly **for** eachtime step and each cell. The mean velocity u j (t) in **the** cell j can be seen

1 **Simul ations**

6 Jens Harting, Martin Hecht, and Hans Herrmann**the** following we focus on three st**at**es: St**at**e A (pH = 6, I = 3 mmol/l) is in**the** suspended phase, st**at**e B (pH = 6, I = 7 mmol/l) is a point already in**the** clustered phase but still close to **the** phase border, and st**at**e C (pH = 6,I = 25 mmol/l) is loc**at**ed well in **the** clustered phase.a) suspended case b) layer **for**m**at**ionc) central cluster d) plug flowFig. 1.2. Images **of** four different cases. For better visibility we have chosen smallersystems than we usually use **for** **the** calcul**at**ion **of** **the** viscosity. The colors denotevelocities: Dark particles are slow, bright ones move fast. The potentials do notcorrespond exactly to **the** cases A–C in figure 1.1, but **the**y show qualit**at**ively **the**differences between **the** different st**at**es: a) Suspension like in st**at**e A, **at** low shearr**at**es. b) Layer **for**m**at**ion, which occurs in **the** repulsive regime, but also in **the**suspension (st**at**e A) **at** high shear r**at**es. c) Strong clustering, like in st**at**e C, soth**at** **the** single cluster in **the** simul**at**ion is de**for**med. d) Weak clustering close to**the** phase border like in st**at**e B, where **the** cluster can be broken into pieces, whichfollow **the** flow **of** **the** fluid (plug flow)Some typical examples **for** **the** different phases are shown in figure 1.2a)–d). These examples are meant to be only illustr**at**ive and do not correspondexactly to **the** cases A–C in figure 1.1 denoted by uppercase letters. In **the**suspended case (a), **the** particles are mainly coupled by hydrodynamic interactions.One can find a linear velocity pr**of**ile and a slight shear thinning.If one increases **the** shear r**at**e ˙γ > 500/s, **the** particles arrange in layers.The same can be observed if **the** Debye-screening length **of** **the** electrost**at**icpotential is increased (b), which means th**at** **the** solvent contains less ions(I < 0.3 mmol/l) to screen **the** particle charges. On **the** o**the**r hand, if oneincreases **the** salt concentr**at**ion, electrost**at**ic repulsion is screened even more

1 **Simul ations**

8 Jens Harting, Martin Hecht, and Hans Herrmann**the** Navier-Stokes equ**at**ions [3]. We consider **the** time evolution **of** **the** oneparticlevelocity distribution function n(r,v, t), which defines **the** density **of**particles with velocity v around **the** space-time point (r, t). By introducing **the**assumption **of** molecular chaos, i.e. th**at** successive binary collisions in a dilutegas are uncorrel**at**ed, Boltzmann was able to derive **the** integro-differentialequ**at**ion **for** n(r,v, t) named after him [3]. The LB technique arose from **the**realiz**at**ion th**at** only a small set **of** discrete velocities is necessary to simul**at**e**the** Navier-Stokes equ**at**ions [5]. Much **of** **the** kinetic **the**ory **of** dilute gases canbe rewritten in a discretized version. The time evolution **of** **the** distributionfunctions n is described by a discrete analogue **of** **the** Boltzmann equ**at**ion[14]:n i (r + c i ∆t, t + ∆t) = n i (r, t) + ∆ i (r, t) , (1.9)where ∆ i is a multi-particle collision term. Here, n i (r, t) gives **the** density**of** particles with velocity c i **at** (r, t). In our simul**at**ions, we use 19 differentdiscrete velocities c i .To simul**at**e **the** hydrodynamic interactions between solid particles in suspensions,**the** l**at**tice-Boltzmann model has to be modified to incorpor**at**e **the**boundary conditions imposed on **the** fluid by **the** solid particles. St**at**ionarysolid objects are introduced into **the** model by replacing **the** usual collisionrules **at** a specified set **of** boundary nodes by **the** “link-bounce-back” collisionrule [19].When placed on **the** l**at**tice, **the** boundary surface cuts some **of** **the**links between l**at**tice nodes. The fluid particles moving along **the**se links interactwith **the** solid surface **at** boundary nodes placed halfway along **the** links.Thus, a discrete represent**at**ion **of** **the** surface is obtained, which becomes moreand more precise as **the** surface curv**at**ure gets smaller and which is exact **for**surfaces parallel to l**at**tice planes.The particle position and velocity are calcul**at**ed using Newton’s equ**at**ionsin a similar manner as in **the** section on SRD simul**at**ions. However, particlesdo not feel electrost**at**ic interactions, but behave like hard spheres in **the** casepresented in this section.The purpose **of** our simul**at**ions is **the** reproduction **of** rheological experimentson computers. We simul**at**e a represent**at**ive volume element **of** **the**experimental setup and compare our calcul**at**ions with experimentally accessibled**at**a, i.e. density pr**of**iles, time dependence **of** shear stress and shear r**at**e.We also get experimentally inaccessible d**at**a from our simul**at**ions like transl**at**ionaland rot**at**ional velocity distributions, particle-particle and particle-wallinteraction frequencies. The experimental setup consists **of** a rheoscope withtwo spherical pl**at**es. The upper pl**at**e can be rot**at**ed ei**the**r by exertion **of** aconstant **for**ce or with a constant velocity, while **the** complementary value ismeasured simultaneously. The m**at**erial between **the** rheoscope pl**at**es consist**of** glass spheres suspended in a sugar-w**at**er solution. The radius **of** **the** spheresvaries between 75 and 150 µm.We are currently investig**at**ing **the** occurrence **of** non-Gaussian velocitydistributions **of** particles **for** higher particle densities and higher shear r**at**es.

1 **Simul ations**

10 Jens Harting, Martin Hecht, and Hans Herrmann4. E. Falck, J. M. Lahtinen, I. V**at**tulainen, and T. Ala-Nissila. Influence **of** hydrodynamicson many-particle diffusion in 2d colloidal suspensions. Eur. Phys. J.E, 13:267–275, 2004.5. U. Frisch, B. Hasslacher, and Y. Pomeau. L**at**tice-gas autom**at**a **for** **the** Navier-Stokes equ**at**ion. Phys. Rev. Lett., 56(14):1505–1508, 1986.6. M. Hecht, J. Harting, M. Bier, J. Reinshagen, and H. J. Herrmann. Shear viscosity**of** clay-like colloids: Computer simul**at**ions and experimental verific**at**ion.submitted to Phys. Rev. E, 2006. cond-m**at**/0601413.7. M. Hecht, J. Harting, T. Ihle, and H. J. Herrmann. Simul**at**ion **of** claylikecolloids. Physical Review E, 72:011408, 2005.8. M. Hütter. Brownian Dynamics Simul**at**ion **of** Stable and **of** Coagul**at**ing Colloidsin Aqueous Suspension. PhD **the**sis, Swiss Federal **Institute** **of** TechnologyZurich, 1999.9. M. Hütter. Local structure evolution in particle network **for**m**at**ion studiedby brownian dynamics simul**at**ion. Journal **of** Colloid and Interface Science,231:337–350, 2000.10. Y. Inoue, Y. Chen, and H. Ohashi. Development **of** a simul**at**ion model **for** solidobjects suspended in a fluctu**at**ing fluid. J. St**at**. Phys., 107(1):85–100, 2002.11. A. Komnik, J. Harting, and H. J. Herrmann. Transport phenomena and structuringin shear flow **of** suspensions near solid walls. J. St**at**. Mech: Theor. Exp.,P12003, 2004.12. A. J. C. Ladd. Numerical simul**at**ions **of** particul**at**e suspensions via a discretizedboltzmann equ**at**ion. part 1. **the**oretical found**at**ion. J. Fluid Mech., 271:285–309, 1994.13. A. J. C. Ladd. Numerical simul**at**ions **of** particul**at**e suspensions via a discretizedboltzmann equ**at**ion. part 2. numerical results. J. Fluid Mech., 271:311–339,1994.14. A. J. C. Ladd and R. Verberg. L**at**tice-boltzmann simul**at**ions **of** particle-fluidsuspensions. J. St**at**. Phys., 104(5):1191, 2001.15. A. Lamura, G. Gompper, T. Ihle, and D. M. Kroll. Multi-particle-collisiondynamics: Flow around a circular and a square cylinder. Eur. Phys. Lett, 56:319,2001.16. J. A. Lewis. Colloidal processing **of** ceramics. J. Am. Ceram. Soc., 83:2341–59,2000.17. A. Malevanets and R. Kapral. Mesoscopic model **for** solvent dynamics. J. Chem.Phys., 110:8605, 1999.18. A. Malevanets and R. Kapral. Solute dynamics in mesoscale solvent. J. Chem.Phys., 112:7260, 2000.19. N. Q. Nguyen and A. J. C. Ladd. Lubric**at**ion corrections **for** l**at**tice-boltzmannsimul**at**ions **of** particle suspensions. Phys. Rev. E, 66(4):046708, 2002.20. R. Oberacker, J. Reinshagen, H. von Both, and M. J. H**of**fmann. Ceramic slurrieswith bimodal particle size distributions: Rheology, suspension structure andbehaviour during pressure filtr**at**ion. Ceramic Transactions, 112:179–184, 2001.21. P. Mij**at**ović. Bewegung asymmetrischer Teilchen unter stochastischen Kräften.Master-**the**sis, Universität Stuttgart, 2002.22. S. Richter and G. Huber. Resonant column experiments with fine-grained modelm**at**erial - evidence **of** particle surface **for**ces. Granular M**at**ter, 5:121–128, 2003.23. M. Ripoll, K. Mussawisade, R. G. Winkler, and G. Gompper. Low-reynoldsnumberhydrodynamics **of** complex fluids by multi-particle-collision dynamics.Europhys. Lett., 68:106–12, 2004.

1 **Simul ations**