Simulations of transport in porous media

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1 Simulations of transport in porous media Igor Stanković, Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia url:

2 Simulations of transport in porous media 1. Diffusion as continuous time random walk. Types of diffusion 3. Creation of the model porous structures 4. Diffusion in model porous structures Igor Stanković, Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia url:

3 Diffusion as continuous time random walk collision collision collision r=0,t =0 collision collision collision collision with: (1) another particle, () porous structure walls, (3) crystal surface, (4) bubble surface, etc.

4 Diffusion as continuous time random walk probability that particle is found at position r at time t p r, t probability of jump with length r & time step t r,t r=0,t =0 r t = d r r p r, t RD mean square displacement

5 Diffusion as continuous time random walk probability of previous position p r ; t = d r ' dt ' K r r ',t t ' p r ',t ' t R 0 kernel D What is form of kernel?

6 Diffusion as continuous time random walk p r ; t = d r ' dt ' K r r ',t t ' p r ',t ' t R 0 D up k ;u 1= K k ; u p k ; u

7 Diffusion as continuous time random walk p r ; t = particle moves first time particle already moved

8 Diffusion as continuous time random walk p r ; t = dr ' dt ' r r ' r ', t ' RD probability of jump with length r and time step t t t dr ' dt ' p t t ' ; r r ' r ',t ' RD 0 1 u p k ; u = p k ; u k,u u

9 Diffusion as continuous time random walk up k ;u 1= K k ; u p k ; u 1 u p k ; u = p k ; u k,u u k ;u u K k ; u =u 1 u

10 Diffusion as continuous time random walk k ;u u K k ; u =u 1 u assume decoupled time and length u u K k ; u = p k 1 1 u

11 Diffusion as continuous time random walk probability of jump with length r & time step t r,t = dt t d r r,t 0 RD = dt d r r r,t 0 waiting time RD space step

12 Diffusion as continuous time random walk = dt t d r r,t 0 RD = dt d r r r,t 0 t, k 0, u 0 RD, =, u =1 u, p k =1 k u =1 c1 u 0 1, p k =1 k

13 Diffusion as continuous time random walk u =1 u, p k =1 k, up k ;u 1= k p k ;u normal diffusion p r ; t = p r,t ' t r u =1 c1 u 0 1, =, anomalous diffusion p k =1 k p r ; t = p r,t ' t t r

14 Types of diffusion normal diffusion, r t = d r r p r, t =6Dt RD sub-diffusion =, r t = d r r p r, t ~t, 0 1 RD super-diffusion time and length have to be coupled r t = d r r p r, t ~t, 1 RD

15 How frequent are different types of the diffusion? sub-diffusion (limiting case) =, normal diffusion, super-diffusion (turbulence) r t =t 3

16 3 super-diffusion r t =t (turbulence) normal diffusion r t =t sub-diffusion (limiting case) r t =t 1/ 3

17 Some applications: A B AB selective membranes

18 Some applications:

19 Creation of the model porous structures Igor Stanković, Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia url:

20 Total energy in the system is expressed as sum of two contributions: 4 6 E tot = r ij F i i 3 j r = 0 r 0 3 r r cut 4 r cut r min r cut r, r r cut r =0, r r cut binary contribution short range attractive potential (SHRAT) r min = 1/6,r cut =1.6, w 0 = F i = 0 105, des = r cut 3k k k r 0 F k i des w 0 des many body contribution embedding functional k =,4, i = w r ij j Igor Stanković, Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia igor.stankovic@scl.rs url: r r w r = w r cut r cut Lucy's weight function 3

21 Molecular dynamics method: Basics molecules, atoms 6 4 metals, semiconductors 3 r = 0 r 0 3 r r cut 4 r cut r min r cut r, r r cut r =0, r r cut Paulie principle local density van der Waals

22 Molecular dynamics method: What can we get out of it? potential energy kinetic energy total energy temperature from kinetic energy pressure

23 Molecular dynamics method: How to keep temperature constant?

24 Molecular dynamics method: How to simulate bulk material? periodic boundary conditions

25 url: t/tref=000 t/tref=00 t/tref=500 t/tref=100 Igor Stanković, Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia

(a) (b) (c) (a) U.Mähr, H. Purnama, E. Kempin, R.

26 (a) (b) (c) (a) U.Mähr, H. Purnama, E. Kempin, R. Schomäcker, and K. H. Reichert, J. Membr. Sci., 171 (000), 85. (b) M. Meyer, A. Fischer, and H. Hoffman, J. Phys. Chem. B, 106 (00) 158. (c) Igor Stanković, Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia url:

27 Monte Carlo integration: volume points from a random sample are counted, if they are within certain radius around the particles (r): V/V0=Nwithin/Ntot. surface is calculated as numerical differential of volume over radius: V(r)/ r. Structre factor and fractal dimension: structure factor S(k) is related to the pair distribution function, g(r), by the Fourier sin kr transformation: S k =1 4 n g r 1 r dr M sites R R Dm S k k 0 D m Igor Stanković, Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia igor.stankovic@scl.rs url: kr

28 M sites R R.5 M sites R R 3

$Mass and surface fractals: M sites R R Dm M sites on the R R surface g r r Ds for small r!$

29 Mass and surface fractals: M sites R R Dm M sites on the R R surface g r r Ds for small r! Dm 3 g r 1 r 3 D s S k k D m S k k 6 D s in reality structures are fractal only on a certain range of scales: S(k) S k k D m S k k Igor Stanković, Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia igor.stankovic@scl.rs url: 6 D s k

30 GEAM (F=1, Fk=0 k>), n=0.5, T=0.01 k π/k k DS=.4 time k k k Ds.44().49(5).3(1).1() k3 R. Vacher et al., Phys. Rev. B, 37 (1988), Igor Stanković, Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia igor.stankovic@scl.rs url:

31 T n F Ds=.7(4) Ds=.5() Ds=.() D Dm=.5()

32 Volume and surface: GEAM (F=1, Fk=0 k>), n=0.5 Monte Carlo integration: volume V/V0=N/Ntot, surface V(r)/ r. distribution of free flight paths e x / e x /

33 gas particles do not interact gas wall interaction SHREP» 4 Φ(r)=φr-4 (r r ), r rcut 0 cut thermostat acts only on wall Φ(r)=0, r>rcut particles! (repulsive)

34 Knudsen self diffusion (small densities): ~1 r t =6Dt kbt r t = t m

35 r t Dt for t/tfree>>1 T Vpore /V0 D α (1) 1.008(1) () 1.019() (4) 1.055() () 0.976() (6) 1.058() (3) 1.0(1)

36 Knudsen self diffusion (small densities): fractal dimension.4

37 r t=1000. t=1500. t=000. r - distance P(r,t) - probability that random walker has displaced to distance r after time t 3 / P r, t =C 1t e r Dt

38 gas gas interaction SHRAT (attractive) gas wall interaction SHREP r t ~t, 0 1 sub-diffusion

39 particles group mass increases/ speed decreases =, as result we have sub-diffusion

6 Hydrophobic interactions

The Physics and Chemistry of Water 6 Hydrophobic interactions A non-polar molecule in water disrupts the H- bond structure by forcing some water molecules to give up their hydrogen bonds. As a result,