BROWNIAN DYNAMICS SIMULATIONS WITH HYDRODYNAMICS Juan J. Cerdà 1 1 Institut für Computerphysik, Pfaffenwaldring 27, Universität Stuttgart, 70569 Stuttgart, Germany. (Dated: July 21, 2009) Abstract - 1
I. MAIN RESULTS WE WILL OBTAIN Under the following assumptions τ < τ a = a2 D o Spherical particles of radius a. On colloidal time scales τ τ B the motion of the solvent can be described by the so-called Creeping flow equations (aka Stokes equations ), i.e. p(r) + η o 2 u(r) = 0 (1) u(r) = 0 (2) where p(r) is the pressure of the fluid at point r, and u(r) is the fluid velocity in such point. This assumption implies that the Reynolds number Re = ρ a v η o (3) is low. Stick boundary conditions between the fluid and the surface of the particles. we will show that: 2
It is possible to get the following finite difference algorithm (known as Ermak- McCammon scheme r i (t 0 + τ) = r i (t o ) + vi D (X o )τ + (2τ) 1/2 d(x o ) n + o(τ) (4) where v D i (X o ) = [βd ij (X o ) f j (X o ) + j D ij (X o )] (5) and X o = (r 1 (t o ),...r N (t o )) X = (r 1 (t o + τ),...r N (t o + τ)) β = 1 k B T n is a Gaussian Random vector of independently distributed components of mean zero and variance one. Usually computed using a uniform random number χ, such that n R = 12(χ 0.5) d(x) is square-root matrix of the positive definite matrix D which are 3N 3N. D is the translational hydrodynamic diffusivity tensor (or the product β D is known as the translational hydrodynamic mobility tensor (we will usually refer to them as diffusivity tensors. D(X 0 ) = d(x o ) d(x o ) (6) When there is no hydrodynamics: D ij (X) = D o δ ij 1 When there is hydrodynamics, D can be very complex, and obtaining analytical expressions is very difficult. If we take the approximation of just considering pairwise interactions, i.e. given a pair of spheres interacting i and j the other N 2 we assume do not exert any influence on the interaction between i and j, then we will show that it is possible to write where D ij (X) = D far ij (r ij ) + Dij near (X) (7) 3
D far ij (r ij ) is the far-field part, valid when the distances between particles i and j is large. A usual expression for the far-field is the so know as Rotne-Prager (RP) approximation Dij RP (r ij ) given by [ ( Dij RP 3 a (r ij ) = δ ij D o 1+(1 δ ij ) D o 4 we will see later where it comes from. D near ij r ij ) [1 + ˆr ij ˆr ij ] + 1 ( ) 3 a [1 3ˆr ij ˆr ij]] 2 r ij (X) is the near-field part (all terms not accounted in Dij RP (r ij )) plus the lubrication effects. It is important when the distances between particles i and j become smaller. We can neglect it if we have for instance a diluted system of particles that repeal each other. We cannot neglect it for instance when we have attraction between particles, or in the case of hard spheres. Another thing we will learn is that, within the approaches we do, the static properties of a system can be computed without any need of including Hydrodyamic Interactions. So, BD without HI can be still very useful!!! One should notice that in general: HI is not pairwise-additive They are long range, decaying as 1/r. They can even diverge for certain types of motion when particles approach very closely (lubrication effects) Implementation of Ermak scheme is in principle O(N 3 ) bottleneck in BD. Few particles can be simulated, typically 30 < N < 300. Let s proof the different results we have stated... (8) 4
II. HOW TO GET THE FAR-FIELD : THE ROTNE-PRAGER (RP) APPROX- IMATION A. Steps we will do (1) Derive expressions for the velocity and pressure in a fluid that obeys the Creeping flow equations due to the action of external forces f ext : the Oseen tensor, and the pressure vector. (2) Suppose that the surface elements of the N colloidal spheres cause the external forces f ext that act on the fluid, and we have stick boundary conditions on the surface of the particles. (3) Simplify the Oseen tensor by assuming that the distance between two particles is much larger than the diameter of the particles 2a: the Oseen approximation for the microscopic diffusion matrices. (4) Assume the two-particle level approach. Get the general form of the diffusion matrices. Show that if we stay cut the series at first order we get the Rotne-Prager (RP) approximation. 5
B. Derive expressions for the velocity and pressure in a fluid that obeys the Creeping flow equations due to the action of external forces f ext : the Oseen tensor, and the pressure vector. We assume the fluid obeys the so-called Creeping flow equations (aka Stokes equations ), i.e. p(r) + η o 2 u(r) = 0 (9) u(r) = 0 (10) which are linear dif. equations. If we have an external force acting on point r f ext = f o δ(r r ), where the prefactor f o is the total force acting on the fluid f o = dr f ext (r ). (11) (12) then, due to the linearity of the Creeping flow equations it is possible to show that velocity and pressure in the fluid will be proportional to f o, ie v(r) f o p(r) f o and we can write this as the following equalities v(r) = T (r r )f o p(r) = g(r r )f o where T (r r ) is known as the Oseen Tensor (because it seems that Oseen was the first to derive an expression for this tensor). g(r r ) is known as the pressure vector. 6
In short, the Oseen Tensor and the pressure vector are the Green s functions of the creeping flow equations. Let s suppose now that our force is not just acting on r but over the entire fluid, then, once again thanks to the linearity of the Creeping equations the velocity and pressure at one given point r is the sum of the contributions acting on that point that arise from all points of the fluid, i.e., we can write the velocity and pressure at point r as: v(r) = dr T (r r ) f ext (r ) (13) p(r) = dr g(r r ) f ext (r ) (14) By substituting previous expressions of v(r) and p(r) into the creeping flow equations, we get (see B1.25) T (r) = g(r) = 1 4π 1 1 8 π η o r [ 1 + rr r 2 ] (15) r r 3 (16) 7
C. Suppose that the surface elements of the N colloidal spheres cause the external forces f ext that act on the fluid, and we have stick boundary conditions on the surface of the particles. Let s suppose we have N particles, and V j denotes the surface of the spheres. Therefore the velocity and pressure of the fluid at a point r is given by where v(r) = p(r) = ds T (r r ) f j (r ) (17) V j ds g(r r ) f j (r ) (18) V j f j is the force per unit area that a surface element ds of the sphere j which is located at position r exerts on the fluid. We assume stick boundary conditions, i.e. the velocity of the fluid at the sphere surfaces is the same than the velocity of the surfaces. This mathematically can be expressed as v particle = v fluid (r) when r V i (19) v i + Ω (r r i ) = ds T (r r ) f j (r ) when r V i V j (20) Due to symmetry (if there is no external torques) Ω (r r i ) = 0. And we get v i = V j ds T (r r ) f j (r ) when r V i (21) Now we can integrate the previous expression over the whole surface of particle (r V i ) and we get v i = 1 4πa 2 1 4πa 2 V i ds j i V i ds ds T (r r ) f i (r ) + (22) V i ds T (r r ) f j (r ) (23) V j Since we now T, we know that ds T (r r ) = 1 2a V i 3η 0 for r V i (24) 8
and therefore V i ds ds T (r r ) f i (r ) = 1 V i 6πη 0 a F i h (25) Fi h (t) = ds f i (r ) (26) V i i.e., F h i (t) is the total force that the fluid exerts on the particle i. Now still we need to compute, 1 4πa 2 j i V i ds V j ds T (r r ) f j (r ) (27) Let s write it in terms of the position vectors of the centers of the spheres br i and br j and the local vectors R = r r i and R = r r j, (plot B1.27), we can get for each term of the sum 1 ds ds T (R R + r 4πa 2 i r j ) f j (R + r j ) (28) V i V j 9
D. Simplify the Oseen tensor by assuming that the distance between two particles is much larger than the diameter of the particles 2a: the Oseen approximation for the microscopic diffusion matrices. If R R < 2a, then T (R R + r i r j ) T (r i r j ) we thus get for each term of the sum that 1 ds ds T (r r ) f 4πa 2 j (r ) T (r i r j ) Fj h (29) V i V j Thus finally we get v i = 1 6πη 0 a F i h j i T (r i r j ) F h j (30) the diffusion matrix elements can be obtained by recalling that D ij must satisfy v i = β D ij (X) F h j (31) which leads to the Oseen approximation for the microscopic diffusion matrices if we just take the leading order expansion respect a/r ij, namely D ii = D o 1 (32) D ij k B T T (r i r j ) = 3 4 D a o [1 + ˆr ij ˆr ij ] r ij when i j (33) where r ij = r i r j, and D o = k B T/(6πη o a) (the Stokes-Einstein diffusion coefficient). Remember this expression is only valid for large values of r ij (i.e. small values of a/r ij ) otherwise we need to add more terms to the expansion. 10
E. Assume the two-particle level approach. Get the general form of the diffusion matrices. Show that if we stay cut the series at first order we get the Rotne-Prager (RP) approximation If we assume that in the interaction between particle i and j no other of the N 2 particles is able to modify the interaction (i.e. low densities are assumed), then, in this two-particle approach it is possible to show that the general expressions for the diffusion matrices are N D ii = D o 1 + D o [A s (r ij ) ˆr ij ˆr ij + B s (r ij ) [1 ˆr ij ˆr ij ]] (34), j i D ij = D o [A c (r ij ) ˆr ij ˆr ij + B c (r ij ) [1 ˆr ij ˆr ij ]] (35) where the so called mobility functions (s=self, c=cross) are A s (r ij ) = 15 ( ) 4 a + 11 ( ) ( 6 ( ) ) 8 a a + O 4 r ij 2 r ij r ij A c (r ij ) = + 3 ( ) ( ) 3 a a + 75 ( ) ( 7 ( ) ) 9 a a + O 2 r ij r ij 4 r ij r ij B s (r ij ) = 17 ( ) ( 6 ( ) ) 8 a a + O 16 r ij r ij B c (r ij ) = 3 ( ) a + 1 ( ) ( 3 ( ) ) 9 a a + O 4 2 r ij r ij r ij (36) (37) (38) (39) (40) 11
III. HOW TO GET THE ERMAK-MCCAMMON SCHEME A. Steps we will do (1) Derivation of the Smoluchowski equation with HI (2) Recast the Smoluchowski equation in terms of the particle drift velocities vi D (X). (3) Assume that the time step τ τ a and therefore the configuration has changed so little that D(X) D(X o ) and vi D (X) vi D (X o ). Get a solution for the simplified Smoluchowski equation. (4) Derive the finite difference algorithm. 12
B. Derivation of the Smoluchowski equation with HI What we want to proof is that given a pdf P (X, t) is governed by the equation P (X, t) dt = Ô (...) Ô P (X, t) (41) [ ri D i,j (X) (β (...) rj Φ(X) + rj (...) )] (42) i, There are different ways of getting the Smoluchowski equation with HI (a) Integration of the Liouville equation P (X, t) dt = ˆL P (X, t) (43) where ˆL (...) [ p ] j m r j (...) + [ rj Φ] pj (...) (44) is the Liouville operator. But integrating the Liouville equation respect the phase space coordinates and the momenta of the colloids is very involved (see JCP 54, 3547, (1971); JCP 57, 2098, (1972).) (b) As we will do... Let be now X = (r 0,..., r N ) (only the positions). Then N(t) = dx P (X, t) (45) W where W is an arbitrary volume of the positional phase space. Therefore, given that only the component of dx/dt can lead to a in-/out- flow of probability across the surface W [ ] dn(t) P (X, t) dx = dx = ds (X, t) (46) dt t dt W W Now using Gauss integral theorem [ [ ]] P (X, t) dx dx + X P (X, t) = 0 (47) t dt W where X is respect the 3N position coordinates. Given W is arbitrary, we get [ ] P (X, t) dx = X P (X, t) t dt 13 (48)
which can be rephrased as P (X, t) t = rj [v cg j P (X, t)] (49) where cg means coarse-grained to the diffusive time scale. Ok. Now, what we need is an expression for the v cg j. Let s assume that in the scale we measure things the inertial forces are very small (relaxation has already passed) when compared to Forces arising from the total potential Φ The Brownian Force F brow j forces due to the friction with the solvent F h j induces a fluid flow which affects other particles. which make than when a fluid moves, it Given, we assume Creeping flow equations to be valid, its linearity allows to say that the force acting on a sphere j must be linearly proportional to the velocities of all the other spheres, namely F h j = i=1 B ij v cg j (50) where B is the hydrodynamic friction tensor. In the case there is no hydrodynamics, then we just have B ij = γ 1 δ Kronecker ij (51) i.e, the usual friction law. Therefore, in the diffusive time scale, the sum of all previous forces must be zero rj Φ B ij v cg j + F brown j = 0 (52) which can be rewritten in the 3N momenta space notation X as X Φ B ij v cg + F brown = 0 (53) where v cg (v cg i,..., v cg N ), idem for F brown. If we define the diffusion tensor as D k B T B 1 (54) 14
we can get by inverting the previous expression 53, and returning to the notation for each particle, that v cg i = D ij [ ] β rj Φ + βfj brown (55) Thus, we get the following expression for the Smoluchowski equation P (X, t) dt = Ô (...) Ô P (X, t) (56) [ ri D i,j (X) (β (...) rj Φ(X) + rj (...) )] (57) i, aka GSE= Generalized Smoluchowski equation with HI. Keep in mind that the previous equation you can just apply it on the Brownian time and length scales, when both particles and solvent are quasi-inertia-free. 15
C. Recast the Smoluchowski equation in terms of the particle drift velocities v D i (X). The Smoluchowski equation P (X, t) dt = Ô (...) Ô P (X, t) (58) [ ri D i,j (X) (β (...) rj Φ(X) + rj (...) )] (59) i, can be recast as where P (X, t) t v D i (X) = i (v i D (X) P (X, t) ) + i=1 i [ j (D i,j P (X, t)] (60) i, [β D ij (X) F j (X) + j D ij (X)] (61) v D i (X) are the drift velocities. A new term arises due to hydrodynamics: j j D ij, which physically acts as a repulsive force (tends to send particles to regions where they have more mobility). Notice that P eq (X) exp( βφ(x)) Z (62) where Z is the partition function, is a stationary solution of the Smoluchowski equation. So what? The last fact implies that Hydrodynamics do no have effect on static equilibrium properties!!! (within the approaches we have taken, keep that also in mind) Your simulations without HI should give the correct equilibrium properties, no need of doing for them the costly HI simulations. 16
D. Assume that the time step τ τ a and therefore the configuration has changed so little that D(X) D(X o ) and v D i (X) vd i (X o). Get a solution for the simplified Smoluchowski equation. In that case, the GSE reduces to a diffusion equation with constant coefficients, namely (in the 3N position space coordinates) P (X, τ X o ) t = v D (X o ) P (X, τ X o ) + [ (D(X o ) P (X, τ X o ) (63) is It can be proof that the solution to the previous equation, if P (X, 0 X o ) = δ(x X o ) P sol delta (X, τ X o ) = ( (4πτ) 3N/2 1 det(d(xo )) exp 4τ (64) [X X o τv D (X o )] T D 1 (X 0 ) [X X o τv D (X o )] ) (65) which is a Gaussian. 17
E. Derive the finite difference algorithm. We can see that the true solution for our system will be P (X, τ X o ) = P sol delta (X, τ X o ) + o(τ) (66) therefore we get the momentum relations (X X o ) o = vi D (X o )τ + o(τ) (67) (X Xo ) 2 o = 2DD ij (X o )τ + o(τ) (68) which can be proof that what we have then is stochastically equivalent, in the limit τ 0 to the Ermak-McCammon scheme r i (t 0 + τ) = r i (t o ) + v D i (X o )τ + (2τ) 1/2 d(x o ) n + o(τ) (69) where v D i (X o ) = [βd ij (X o ) f j (X o ) + j D ij (X o )] (70) 18