Modeling of colloidal gels

Transcription

1 Modeling of colloidal gels rheology and contact forces 1 Ryohei Seto, TU München Heiko Briesen, TU München Robert Botet, LPS, Paris-Sud Martine Meireles, LGC, Univ. Paul Sabatier Bernard Cabane, ESPCI

2 Compaction of gels 2

3 How do particles interact in gels? 3

4 Can DLVO model be applied? DLVO potential Strong flocculation U/kT Weak flocculation 4

5 JKR model We have to consider the finite contact area to estimate the cohesive force. [Johnson 1971] Pull-off force ~ Size of particle 5

6 Various types of interactions In colloidal system - Van der Waals force - Depletion force - Multicharged cations - Polymer bridge - Nano particles - Surface roughness? 6

7 Tangential elements Normal element Tangential elements sliding Only potential force (DLVO type) bending torsion 7

8 Contact model for cohesive forces Normal (F Nc, k N ) Bending (M Bc, k B ) Sliding (F Sc, k S ) Torsion (M Tc, k T ) 8

9 Rheology and contact forces 9

10 Rheology Response to external forces Shear stress τ Compressive stress P 10

11 Shear rheology Shear stress Elastic Yielding Shear strain 11

12 Compressive rheology Compressive stress Close Packing Plastic? Consolidation? Yielding? Elastic? Compressive strain 12

13 Modeling for colloidal gels Particles are spherical and rigid Particles interact each other by contact forces, that resist the tangential forces (bending moment) Thermal agitation can be negligible Slow deformation limit 13

14 Simulation 14

15 2D simulation 2D simulation = Spheres in narrow layer 15

16 3D simulation 16

17 Simulation 1 Compare 2D and 3D simulations with the same type of bond. n ( ˆF Sc, ˆF Bc, ˆF Tc ), (ˆk N, ˆk S, ˆk B, ˆk T ) o = n (0.5, 0.1, 0.1), (10, 5, 1, 1) o Normal (F Nc, k N ) Bending (M Bc, k B ) Sliding (F Sc, k S ) Torsion (M Tc, k T ) 17

18 New contacts Contacting particles are always connected by the same type of sticky bond. 18

19 Simulation D: N samples 0.1 2D: N samples P / P D 2D e-05 Regular close packing 19 1e φ

20 Simulation 1 R. Buscall, P. D. A. Mills, J. W. Goodwin, and D. W. Lawson. Scaling behaviour of the rheology of aggregate networks formed from colloidal particles. J. Chem. Soc. Faraday Trans, 84(12): , Buscall D: N samples 2D: N samples P / P e-05 3D 2D Regular close packing 20 1e φ

21 What we see in Simulation 1 Compression curves can be reproduced by the quasi-static simulation with the contact model. (The range of volume fraction is wider than typical experiments) The ranges of pressure are very wide for both 2D and 3D simulations; ~10 5 It looks that the compression curve roughly follows a power law. 21

22 Compression curve is power-function? P( )? 22

23 Simulation 2 Compare 3 types of bonds The other conditions are the same as simulation 1. Bond 1 (used in simulation n ( ˆF Sc, ˆF Bc, ˆF Tc ), (ˆk N, ˆk S, ˆk B, ˆk o 1) n o T ) = (0.5, 0.1, 0.1), (10, 5, 1, 1) Bond 2 n ( ˆF Sc, ˆF Bc, ˆF Tc ), (ˆk N, ˆk S, ˆk B, ˆk o n o T ) = (0.5, 0.05, 0.05), (10, 5, 1, 1) Bond 3 n ( ˆF Sc, ˆF Bc, ˆF Tc ), (ˆk N, ˆk S, ˆk B, ˆk T ) o = n (0.5, 0.02, 0.02), (10, 5, 1, 1) o 23

24 Compression curves of 3 types of bonds 2D 3D 1 1 Bond 1 Bond 1 Bond 2 Bond Bond Bond

25 Local slopes D Bond 1 Bond 2 Bond 3 Least square fitting for the data in windows D 0.05 Bond 1 Bond 2 Bond

26 Bending regime and sliding regime Bond 1 Bond 2 Bond 3 2D EC CP EC D CP Bending regime (Power-law regime) Sliding regime (Deviation from CP EC Close packing Elastic compaction after gel point 26 power-law regime)

27 What we can know by simulation 2 Compare 3 types of bonds. A compression curve consists of the bending regime (power-law) and the sliding regime. 27 The range of power-law regime is specified.

28 Why do they show large slopes? How does consolidation happens? 28

29 Simulation 3 Initial bonds are sticky (bond1). But, only volume excluding force acts between newly contacting particles. 29

30 Fragmentation by compaction 30

31 Compare the compression curves 2D 3D P / P P / P e e φ 1e φ

32 What we can deduce by simulation 3 Initial bonds are sticky (bond1). But, only volume excluding force acts between newly contacting particles. New bond generations lead to the large slopes, i.e. the rapid consolidation. 32

33 Again look at the consolidation Soft domains are crushed, and become to robust domains 33

34 34

35 Summary for compaction simulation The quasi-static simulation with the contact model reproduces the known compaction behavior. Compression curve involves bending regime and sliding regime. The bending regime shows a power-law behavior, which ends up in sliding regime by compaction. The rapid consolidation is caused by new bond generation. 35

36 Viscosity of breakable aggregate suspensions 36

37 How is a colloidal aggregate stressed in a shear flow? 37

38 Stokes equations + u ru u = 0 = rp + 0 r 2 u Reynolds number = (Typical length) (Typical velocity) Viscosity Stokes equations 0 = rp + 0 r 2 u u = 0 38

39 39 Drag forces, torques, and stresslets Single sphere system Stokes law F = 6 0 a(u U 1 (r)), T = 8 0 a 3 ( 1 ), S = a 3 E 1. Many-spheres system Stokesian dynamics F (1) F (2). T (1) T (2).. S (1) S (2). = U (1) U (r (1) ) U (2) U (r (2) ) (1) (2)... E E.

40 Drag forces, torques, and stresslets A rigid aggregate = SD + rigid-structure constraint [Harshe et.al 2010] F ag T ag S ag = (R ag ) FU (R ag ) FO (R ag ) FE (R ag ) TU (R ag ) TO (R ag ) TE (R ag ) SU (R ag ) SO (R ag ) SE U ag U (r 0 ) ag E Resistance matrix for a rigid aggregate (11 11) 40

41 Drag forces for a freely suspended z rigid-aggregate x 41

42 Viscosity of dilute suspension = 0 = 0 + n 0 hs ag i = 0 + ns r = (Einstein s equation) 42

43 Fractals aggregates Reaction limited cluster-cluster aggregation (CCA) N = 16 N = 64 N =

44 3 types of small fractal clusters Reaction limited cluster-cluster aggregation (CCA) Diffusion limited particle-cluster aggregation (DLA) Eden growth model (Eden) CCA DLA Eden Rg N Radius of gyration R 2 g 1 X (r (i) r 0 ) 2 N i

45 N dependences Stresslet acting on an aggregate Maximum bending moment S ag/af CCA DLA Eden max Mbend/aF CCA DLA Eden N Disturbance of flow N per an aggregate 45

46 Breakup by hydrodynamic stress low shear-rate high shear-rate 46

47 Shear rate dependence of the viscosity F Nc > 100 pn M Bc = 30 pn µm The bending moment causes the break up. r Volume fraction = CCA DLA Eden r = 5 2 (Einstein s equation) 47

48 Summary for viscosity of aggregates suspension The size and structure of aggregates appears in the effective viscosity of the suspension. The size of aggregates is determined by the hydrodynamic forces and the contact forces. We may estimate the contact forces from a measurement of the viscosity of dilute suspension. 48