Dynamics of polyelectrolytes and living polymers in shear flow P. B. Sunil Kumar Department of Physics, IIT Madras. 1 Tuesday 10 August 2010
Hydrophilic Hydrophobic (a) Spherical Micelle, (b) Cylindrical Micelle, (c) Reverse Micelle, (d) Bilayer, (e) Hexagonal phase, (f) Vesicles Cylindrical micelles also called living polymers Can undergo scission/recombination under thermal fluctuations 2 Tuesday 10 August 2010
Micellar solutions are examples of living polymers Oscillating solid sphere Shear induced gelation Jayaraman et al. PRE, 67, 65301 (2003) Shear banding Liu et al. PRL, 77, 2121 (1996) Lerouge et al. PRL, 81, 5457 (1998) Rheocaos : R. Bandyopadhyay, G. Basappa, and A. K. Sood, Phys. Rev. Lett. 84 , 2022 (2000). Tuesday 10 August 2010
Padding and Boek (2004): Atomistic MD simulations erucate than for EHAC surfactants to compute the mechanical properties of atomistic micelles such as persistence length, elastic modulus and scission energy [8] Padding J T and Boek E S 2004 Europhys. Lett. 66 756 [9] Padding J T and Boek E S 2004 Phys. Rev. E 70 031502 [10] Kr¨ oger M and Makhloufi R 1996 Phys. Rev. E 53 2531 MD simulations of micelles consisting of coarse-grained (CG) mode surfactants to optimize the CG model with respect to the structure factor S(q) of the atomistic micelle at large values of the wavevector q. The CG model is used to extrapolate the structure factor for small q in order to obtain reliable values for the micelle bending rigidity and persistence length . [16] Boek E S, den Otter W K, Briels W J and Iakovlev D 2004 Phil. Trans. R. Soc. A 362 1625 [17] den Otter W K, Shkulipa S A and Briels W J 2003 J. Chem. Phys. 119 2363 Padding, Boek and Briels ((2005): a mesoscopic model of wormlike micelles, represented by chains which can break and recombine and can be subjected to shear flow. For this model, where the smallest length-scale is the persistence length, the elastic modulus , scission energy and persistence length are taken 4 as input parameter from the atomistic and CG MD. Tuesday 10 August 2010
Dissipative Particle Dynamics ( DPD) is a particle based “meso-fluid” model ! !"#"$%&&'()*)+''($,-.$#"/"0"1"$2&(34,-5$ Europhys. Lett. !" 67889:$7;; ! !"$<=>,?&3 ,-.$!"$@,))(-5$ Europhys. Lett. #$ 6788;:$787 S'10"%"(+)53#)%* &1#3("1@"7#+:"CCC DPD beads R Soft-repulsive +@@+)(#7+" con- '+&'+$+3(")*4$(+'$"1@" servative pair @1')+$"-+(:++3" %(10$;01*+)4*+$/"1'" 313,-13.+."8F[F,?"-+%.$ @*4#. +*+0+3($ DPD Molecular Dynamics (MD) MD with special thermostat (mesoscopic) 8:1'A$"43.+'"+64#*#-'#40"B 313,+64#*#-'#40")13.#(#13$/ Special (DPD-)thermostat, +C=C/"$5+%'#3=? #0&*+0+3(+."-2"0+%3$"1@ ! dissipative pair @1')+$ DC"E1..+0%33/"9C"FG3:+=/"%3."HC"H'+0+'/ ! stochastic pair @1')+$ Phys. Rev. E 68 8IJJK?"JLMNJI" OP2.'1.23%0#) #3(+'%)(#13$Q 8@*1:"+@@+)($?"%'+"@4**2 (%A+3"#3(1"%))143(> Tuesday 10 August 2010
DPD-‑ ¡ ¡Basic ¡model N "dpd" particles interacting via conservative, dissipative and random pair wise interactions Dissipative & random forces related through fluctuation-dissipation theorem Pair-wise random and dissipative forces conserve momentum Resulting in correct description of hydrodynamics Soft interactions allow for longer time steps, and therefore much longer times can be probed via DPD as opposed to MD • P. J. Hoogerbrugge and J. M. V. A. Koelman, Europhys. Lett. 19 , 155 (1992) Tuesday 10 August 2010
DPD model for living polymers ( LP) For, W – W, W – I W – A, I – I, I – A For, A-A r 1 r 5 r 3 9 Tuesday 10 August 2010
U 3 body = k 3 The three-body bending potential 2 (1 − r jk /r 3 ) 2 (1 − ˆ r ij . ˆ r jk ) I 1 – A’ 1 – A 2 & A’ 1 – A 2 – I 2 Three body potentials U ijk = k 3 (1 − ˆ r jk ) for A 1 I 1 A 1 ’ and A 2 I 2 A 2 ’ r ij · ˆ For LP For solvent • Simulation box size L = 40 , Periodic Boundary Condition • density ρ = 3 , number of particles 192000 • Resting length A-I = 1.0 , A-A = 2.0 • Spring constant k = 200 • Time step = 0.01 τ , with τ of the order of 10 -6 sec 10 S. Thakur et al Soft Matter 6 , 489, 2010 Tuesday 10 August 2010
Fluid-gel transition C -- % ratio of number of monomers to the total number of particles Phase behavior was characterized by diffusivity of trimers • Fluid phase at low C • Gel phase at higher C • Transition point C = Cp = 3.5 • Branched point ---- coordination number of particle is > 2 11 Tuesday 10 August 2010
Organization of polymers C = 2 Fluid C = 3 Increasing C C = 4 Gel C = 5 12 Tuesday 10 August 2010
Segment length distribution Cluster size distribution branching of polymers lead to random percolation • Exponential distribution at low C < Cp l Average polymer length Φ Monomer concentration • Power law distribution at C = Cp: random percolation Branching reduces the average segment • Gel phase contains spanning clusters length and small segments 13 • M. E. Cates and S. J. Candau, J. Phys 2,6869 (1990) Tuesday 10 August 2010
Self-intermediate scattering function N: total number of monomers Relaxation time Zimm dynamics at large q: S. Thakur et al Soft Matter 6 , 489, 2010 14 Diffusive at low q: Tuesday 10 August 2010
First recombination time τ R = scission time (t 2 )–recombination time (t 1 ) Two classes of recombination kinetic: Mean field (MF) and Diffusion controlled (DC) • Ben O’ Shaughnessy and Jane Yu, PRL, 74, 4329 (1995) t d t d t s t d 15 Tuesday 10 August 2010
stress relaxation • We find a residual stress is due to the spanning clusters with stretched bonds • Early time oscillation is the result of bond potential within a trimer • G. Faivre and J. L. Gardissat, • G(t) decays as: Macromolecule, 19, 1988 (1986) 10 10 1 1 G(t) G(t) 0.1 0.1 0.01 0.01 0.1 1 10 100 0.1 1 10 100 t t C=3.0 C=4.0 16 Tuesday 10 August 2010
Shear V* z y x -V* • Density of wall = 20* Density of fluids • Periodic boundary in x & y directions • Results were verified using Lees-Edward method to rule out wall effect 17 Tuesday 10 August 2010
Shear induced Lamellar phase • First layer forms near the shear boundaries • All the polymers within a range of r 5 is pulled towards the boundary • Next layer forms exactly at a distance r 5 from the first • S. Thakur et al Soft Matter 6 , 489, 2010 19 Tuesday 10 August 2010
Shear induced structures x z x y • Lamellar phase in the system with C = 5 18 y Tuesday 10 August 2010
Shear induced structures x z y x • Columnar phase in the system with C = 5 • Layer spacing is decided by r 5 20 y Tuesday 10 August 2010
Stress and viscosity • Alignment of polymers leads to decrease in viscosity 21 Tuesday 10 August 2010
1000 Cluster size distribution for a tetramer 2% 2.5% building block. 3% 100 4% 5% Percolation at 3% concentration n(s) 10 1 0.1 0.01 1 10 100 1000 10000 s 1000 2% 2.5% 2.6% Cluster size distribution for a pentamer 100 2.75% building block. 3% n(s) 4% 10 Percolation at 2.75% concentration 1 0.1 For larger segments the percolation is at a lower concentration 0.01 1 10 100 1000 10000 s 20 Tuesday 10 August 2010
Diffusion coefficient as a function of concentration for different building blocks 0.001 0.0001 D 1e-05 Trimer Tetramer Pentamer 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1/C wt % 21 Tuesday 10 August 2010
Trimer a 1 = a 2 = 0 . 1 a 1 = a 2 = 0 . 5 a 1 = a 2 = 1 . 0 a 1 = a 2 = 0 . 1 Tetramer Pentamer a 1 = a 2 = 1 . 0 a 1 = a 2 = 1 . 0 γ = 1 ˙ Tuesday 10 August 2010
Sc of DPD fluid is around 1 , while that for a real fluid is about 1000 !! Higher Sc fluid ? !"#$%& '(()"'*+,-!"#$./01$)&$0,2+$)3"&2'24,5 !"#$%&'()*$ +,-.%'/)01-%$('")2$-2$(34)")$5-63%-7'%")4($-7')%5-6%3&-'-8'9/$((-.)5"%)+:")30; '((3/5-)0-7%)04)7($-"3-'.<:5"-"#$-=)0$&'")4-2)5435)",-53-"#'"-"#$- *"))$*2,6*+3722,0839$) > Sc :,;< = 63%-'-()?:).@-)5-3+"')0$.A 5, BC-DC-E3/$F- Europhys. Lett. >? !GHHHA-GIJ @A0'37*'B,(')'3$2$) ! 'BB"#&,2",280$,2+$,C7&*"&72A Tuesday 10 August 2010
E a r l y s t a g e ˙ Sc=7, =1 γ s t e a d y s t a t e Tuesday 10 August 2010
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