Dynamics of Polymers and Membranes P. B. Sunil Kumar

Transcription

1 Dynamics of Polymers and Membranes P. B. Sunil Kumar Department of Physics, Indian Institute of Technology Madras, Chennai,

2 Dissipative Particle Dynamics (DPD) - Basic model P. Espagnol and P. Warren, Europhys. Lett. 30, 191 (1995) N DPD" particles interacting via conservative, dissipative and random pair wise forces S'10"%"(+)53#)%* &1#3("1@"7#+:"CCC DPD (mesoscopic) Soft-repulsive +@@+)(#7+"conservative 313,-13.+."8F[F,?"-+%.$ Molecular Dynamics (MD) Special (DPD-)thermostat, #0&*+0+3(+."-2"0+%3$"1@! dissipative stochastic DPD beadsr '+&'+$+3(")*4$(+'$"1@" +*+0+3($ MD with special thermostat 8:1'A$"43.+'"+64#*#-'#40"B 313,+64#*#-'#40")13.#(#13$/ +C=C/"$5+%'#3=? DC"E1..+0%33/"9C"FG3:+=/"%3."HC"H'+0+'/ Phys. Rev. E 68 8IJJK?"JLMNJI" OP2.'1.23%0#) #3(+'%)(#13$Q 8@*1:"+@@+)($?"%'+"@4**2 (%A+3"#3(1"%))143(>

3 Conservative non-bonded pair forces C F i j r C ij Fi j :!(# ij &( rij ) eij, with &( rij ) :! (1" ) rc ' ' conservative force strength "weight function" r i e i j r i j r j F C ji!" F C ij r :! r " r ri j :! ri j e :! r / r i j i j i j i j i j F C ij 0$ # i j % e ij cut-off radius r c r ij 3

4 Conservative pair forces for for bonded interactions B$C$'2/'6%-'%(#$%2$6)$$%-*%,-+)3$I6)+/'/'6%/EC5/$2%/'%(#$%BGB%E-2$5J FAH%%&'%+22/(/-'J%G-($'(/+53%(#+(%$EC#+3/M$%(#$%entropy-dominated character -'$%,+'%,#--3$J!!#+)E-'/, C-($'(/+5!! OPQP%FO/'/($%PR($'3/N5$%Q-'5/'$+)%P5+3(/,H C-($'(/+5!! $'()-C/,%3C)/'6 C-($'(/+5 S%%N$'2/'6%C-($'(/+539%(-)3/-'%C-($'(/+53 4

5 Thermostat forces: Dissipative pair forces vi D F i j r i e i j r ij r j 2 2 D F :#! $! %( r ) ( e " v ) e #! $! %( r ) ( e & e v j r :# r! r ri j :# ri j e :# r / r i j i j i j i j i j!v j!"#$%&'($)*+,$%-*%./*$0%1 &'2/+'%&'3(/(4($%-*%"$,#'-5-67%8+2)+39%:#$''+/9%&'2/+9%;<1=>%?+'@%A;;> =; ) v ij ij ij ij ij ij ij ij ij " dissipative force strengh t F D ji #! F D ij %! ( r ij )% is another weight function (see below) v :# v! v i j i j v e i j i j vi i j 5 ij v j ( e "v ) e i j

6 Thermostat forces: Random pair forces (continuum limit) R!" ( r Fi j : $ " "!( r ij )# i j e ij )% is the same weight ij function as in the case! random force strength of the dissipative force (see below) # % % % ij are random variables with the following properties : symmetric for each interacting particle pair at all times : # ( t) $ # ( t) ij zero mean for all particle pairs at all times : ji &# ( t) '$ 0 ij unit variance for all particle pairs at all times, and uncorrelated for different pairs of particles and different times, : &# ( t) # ( t( )' $ () ) * ) ) ) )+ t, t( ) ij kl ik jl il jk 2 2 This can be fixed by the substitution!! /, % I% Gaussian%+'2%+% uniform %% C)-J+J/5/(7%2$'3/( 7%*-)%(#$ %%%%)+'2-E%D+)/+J5$3%K-)L% %% $M4+557%K$ 55@ % "#$%2$*/'/(/-'%*-)%(#$% %%%%)+'2-E%*-),$%+3%3#-K'% %%%%#$)$%/3%*-)%(#$% continuum limit FNO/$' $) C)-, $ 33NH@ %%%% P-)% discrete (/E$%3($C3% %%%% 0-.t # 1 (#/3%#+3%(-%J$% %%%%E-2/*/$2%F3$$%'$Q(%C+ 6$HR ( # t i.e., by! R R!! (, and F "! ( F " # t # t! 6

7 Thermostat forces: Random pair forces (discrete case) t 2 # $ 2 R (2) x( t) ) 5 F ( t% ) dt% * + 0, & ' ( & '!!!! Discretization: Divide [0, t]! into! n! steps of size! 0t " t/ n. 2 n n n 2 2 # R - t. $ R R - t. ( ) ) 6 Fi 1 2 Fi Fj. i" 1 n * ' " 66 & ' i" 1 j" 1 3 n4 / & x t ' ( & +, R R 2 Assume the discrete analogon t o & F ( t) F ( t% )' "! 7 ( t8 t% ) is & F F ' "! 7 R R 2 i j ij n t. 2 t 2 - t. 2 ( ). 6 i" 1. / & x t ' (! 1 2 "! "! t 1 2 "! t 0t 3 n 4 n 3 n 4 But x t of t; for an ordinary random walk & x t ' ( t Assu ' " 7 2 & ( ) ' must be independent 0 2 R R 2 ( ) must hold / mption & Fi Fj! ij was wrong. 7

8 Why is the DPD thermostat able to establish a chosen (set) temperature kbt* in the system? Dissipative forces always take energy out of the system. This mimics energy transfer to atomistic degrees of freedom that are not explicitly taken into account in the mesoscopic DPD model. Random kick forces on average transfer energy(from hidden atomistic degrees of freedom) into the system. A given temperature kbt* can only be established if the to and from energy transfers on average balance each other over time. 8

9 Lett. 30 G=QQRI%=Q=%% # S7%F$+'3%-*%(#$%CHC%(#$)F-3(+(%+%373($F%-*%D+)(/,5$3%G!CHCTK$+230I /3%2$3,)/K$2%/'%(#$%canonical ensemble9%/@$9%at constant temperature kt B * /**! () r G=I "#$%same!j$/6#(%*4',(/-'0 +DD$+)3% linear /'%(#$%)+'2-F%*-),$%F R 9%+'2%quadratic /'%(#$%2/33/D+(/E$%*-),$%F () r &'%D)/',/D5$%(#$%*-)F%-*%%%%%%%%%/3%+)K/()+)7%G+3%5-'6%+3%/(3%*/)3(%(J-%F-F$'(3 $L/3(I9%+'2%/'%(#$%standard formulation -'$%,#--3$3%(#$%same weight function as in the conservative forcem!! ( r) "! ( r). GAI "#$%2/33/D+(/E$%*-),$%3()$'6(#%$ +'2%(#$%)+'2-F%*-),$%3()$'6(#%% 3+(/3*7%(#$ fluctuation-dissipation relation kt* & B 2 % 2$ 9

10 Discretized equations of motion in DPD: (Modified) velocity-verlet integration scheme!"#$%&'()*"#+$#(,!"#$%&'()*+!,+-,./ &0+12+,&)-"!'&$0!-.-/"#$%&'() "#+$#(,!"#$%&'() 345%.!/+6+5-''.%78 9:;+<.,$"/*+=;+>!''-"!&0.0*+ Phys. Rev. E 01 3BCCC8+DEFGGH 10

11 How is a Lipid Simulated? Model Hydrophilic Head Group Choline Phosphate Glycerol Hydrophobic Tail Group: Fatty Acids e.g. Phosphatidylcholine

12 Lipids - Interaction parameters for conservative forces In addition the particles in a lipid are connected by springs We simulate 16,000 lipids in a box of 86x86x40, with a total of approximately one million particles.

13 13

14 DPD simulations of membranes in constant tension ensemble The method is similar to the Nos - Hoover method for NVT simulations. é We will introduce a variable ɛ = ln(v/v 0 ) p ɛ and associated mass W. The DPD integration scheme is now modified as, v ɛ v ɛ v ɛ v ɛ ɛ ɛ v i v i p ɛ p ɛ F ɛ t Update ɛ ɛ + p ɛ W t V,F C i, F D i, F R i and P = 1 dv [ i r i exp(ɛ ɛ ) {r i + v i t} p 2 i m i + i F C i r i ] Iterate for v i and p ɛ such that P is equal to P 0 ṽ i v i p ɛ p ɛ Initial guess : v i exp(ɛ ɛ )v i p ɛ p ɛ F ɛ t A. F. Jakobsen, JCP 122, ( 2005) ; 125, ( 2006) {( F C i m i + FD i m i 2 p ɛ W v i {( F C i m i 1+p ɛ t/w ) + FD i m i ) t + FR i m i t } Compute F D i and construct a piston with momentum } t + FR i t m i p ɛ p ɛ F ɛ t F e = dv [P P 0 ]+ d dn d 14 i p 2 i m i

15 Tension in a membrane is given by γ s =<L z [P z 1 2 (P x + P y )] > Lz is the dimension of the simulation box normal to the membrane. To obtain the desired values of the surface tension ϒs and the normal pressure PN, we will replace the parameter pε by a tensor. d The first term in the equation F e = dv [P P 0 ]+ dn d should then be, for α =(x, y), and for the normal direction F α = da [ γ 0 s L z (P 0 N P α ) ] + F α = dv [ P z P 0 N ] + d dn d d dn d i p 2 i m i i i p 2 i m i p 2 i m i It is also possible to control the tension by a Langevin piston method in which the force on the piston is given by d p 2 i F α = dv [P α P 0 ]+ γ p p αα + σ p ξ α dn d m i i with σ 2 p =2γ p Wk b T 15 A. F. Jakobsen, JCP 122, ( 2005) ; 125, ( 2006)

16 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

17 DPD model for living polymers ( LP) For, W W, W I W A, I I, I A For, A-A r 1 r 3 r 5 9

18 A trimer based model for living polymers The three-body bending potential I 1 A 1 A 2 & A 1 A 2 I 2 Linear spring connecting A 1 -I 1, A 1 -I 1 and A 1 -A 1 For LP For solvent Simulation box size L = 40, Periodic Boundary Condition density ρ = 3, number of particles 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 We use LAMMPS ( modified) 10

19 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

20 Organization of polymers 12

21 Organization of polymers C = 2 12

22 Organization of polymers C = 2 C = 3 12

23 Organization of polymers C = 2 Fluid C = 3 12

24 Organization of polymers C = 2 Fluid C = 3 Increasing C 12

25 Organization of polymers C = 2 Fluid C = 3 Increasing C Gel 12

26 Organization of polymers C = 2 Fluid C = 3 Increasing C C = 4 Gel 12

27 Organization of polymers C = 2 Fluid C = 3 Increasing C C = 4 Gel C = 5 12

28 Segment length distribution Cluster size distribution branching of polymers lead to random percolation l Φ Average polymer length Monomer concentration Branching reduces the average segment length M. E. Cates and S. J. Candau, J. Phys 2,6869 (1990) Exponential distribution at low C < Cp Power law distribution at C = Cp: random percolation Gel phase contains spanning clusters and small segments 13

29 Self-intermediate scattering function N: total number of monomers Relaxation time Zimm dynamics at large q: Diffusive at low q: 14

30 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

31 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

32 stress relaxation Residual stress is due to the spanning clusters with stretched bonds Early time oscillation is the result of bond potential within a trimer G(t) decays as: G. Faivre and J. L. Gardissat, Macromolecule, 19, 1988 (1986) 16

33 Shear V* z y -V* x 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

34 Shear induced structures x z x y Lamellar phase in the system with C = 5 y 18

35 Shear induced Lamallar phase After 2000 DPD time step Initiated by migration of polymers towards the higher velocity regions First layer forms near the shear boundaries After DPD time step 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 19

36 Shear induced Lamallar phase After 2000 DPD time step Initiated by migration of polymers towards the higher velocity regions First layer forms near the shear boundaries After DPD time step 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 19

37 Shear induced Lamallar phase After 2000 DPD time step Initiated by migration of polymers towards the higher velocity regions First layer forms near the shear boundaries After DPD time step 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 19

38 Shear induced Lamallar phase After 2000 DPD time step Initiated by migration of polymers towards the higher velocity regions First layer forms near the shear boundaries After DPD time step 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 19

39 Shear induced structures x z x y Columnar phase in the system with C = 5 Layer spacing is decided by r 5 y 20

40 Stress and viscosity Alignment of polymers leads to decrease in viscosity 21

41 Polyelctrolytes Macromolecules which dissociates into charged polymer and counterions (CI) in polar solvents are known as polyelectrolytes (PE). Anchored polyelectrolytes are of importance for many biologically and industrially relevant processes We will look at some of the equilibrium and dynamical properties of a single grafted polyelectrolyte using Brownian dynamics and Monte Carlo simulations.

42 Polyelctrolytes Macromolecules which dissociates into charged polymer and counterions (CI) in polar solvents are known as polyelectrolytes (PE). Anchored polyelectrolytes are of importance for many biologically and industrially relevant processes We will look at some of the equilibrium and dynamical properties of a single grafted polyelectrolyte using Brownian dynamics and Monte Carlo simulations.

43 The Model Total energy of the system consists of three parts, (i) Lennard-Jones potential Diameter of the bead a Length of the box L Number of monomers N Impenetrable, non-polarizable walls at x=0 and x=l Periodic minimum image boundary conditions along the y and z directions. (ii) harmonic spring potential between connected beads (iii) Electrostatic potential between any pair of beads,

44 The Model Total energy of the system consists of three parts, (i) Lennard-Jones potential Diameter of the bead a Length of the box L Number of monomers N Impenetrable, non-polarizable walls at x=0 and x=l Periodic minimum image boundary conditions along the y and z directions. (ii) harmonic spring potential between connected beads (iii) Electrostatic potential between any pair of beads,

45 Polyelectrolyte Phase Diagram K. Jayasree, P. Ranjith, Madan Rao and PBSK J. Chem. Phys. 130, (2009) In our simulation, we model the nonelectrostatic interactions by a Lennard-Jones potential monomer-monomer interaction ε= ε 1 = ε 2 CI-CI and monomer-ci interactions ε 1 =1, ε 2 =0 ε=3,a=3.0 ε=3,a=0.1

46 The free energy from umbrella sampling Monte Carlo

47 The free energy from umbrella sampling Monte Carlo

48 Fluctuations The asphericity Where, are the eigenvalues of the gyration tensor

49 Fluctuations The asphericity Where, are the eigenvalues of the gyration tensor

50 Brownian Dynamics Effect of counter-ion condensation on polyelectrolytes

51 Brownian Dynamics Effect of counter-ion condensation on polyelectrolytes

52 Brownian Dynamics Effect of counter-ion condensation on polyelectrolytes

53 Brownian dynamics with external shear The equation of motion for the monomers is dr i dt = j M ij rj U(t)+v(r i )+ 2k B TB ij f j (t) v(r) = γyˆx M ij = 1 6πηa I [ 1 3 6πηr ij is the solvent velocity profile M = B B T 4 (I + r ijr ij r 2 ij ] )+ a2 (I 3 r ijr ij ) 2rij 2 rij 2 is the Rotne-Prager tensor i=j i j, r ij 2a 1 6πηa [ (1 9r ij 32a )I +3r ijr ij 32ar ij ] i j, r ij 2a The components of B is obtained recursively by the Cholesky decomposition of the 36 mobility tensor M

54 Lees Edward periodic boundary condition Our simple periodic boundary condition should now be modified to the Lees-Edward periodic boundary condition. If the flow is in the X-direction and the velocity gradient in the Y-direction then, a particle leaving the top boundary at y=l with velocity (vx,vy,vz) will appear at the bottom at y=0 with new x-coordinate and new x-component of velocity given by, x new = x n t U t int[n t U t/l]l 37