multiscale modeling and simulation of soft matter materials · ima : development and analysis of...

Multiscale Modeling and Simulation of

Soft Matter Materials

IMA : Development and Analysis of Multiscale Methods

November 2008

Paul J. AtzbergerDepartment of Mathematics

University of California Santa Barbara (UCSB)

In collaboration with :

Frank Brown (UCSB), Peter Kramer (RPI), Charles Peskin (NYU)

Support: NSF DMS – 0635535.

Outline

• Motivation for Studying Soft Matter Materials

• Our Modeling and Simulation Framework

• Incorporating the Role of Thermal Fluctuations

• Important Issues with Numerical Approximation

• Stochastic Time Step Integrators (Stiffness)

• Physical Validity of the Framework

• Applications in Rheology of Soft Matter Materials

Examples of Soft-Matter Materials

• Energy scale of interactions / deformations comparable to thermal energy (KBT)

• Thermal fluctuations play an important role in structure of material.

• Hydrodynamic coupling of components often important.

• Material properties depend on multiple temporal and spatial scales.

• Presents challenges for modeling and simulating soft-matter systems.

ColloidsGels (Actin)

Lipid Bilayer Membranes Polymeric Fluids

Molecular Structure of Typical Lipid Amphiphiles

Phosphogylceride

• Molecules consist of polar head group and non-polar tail group.

• Polar head groups have an affinity to be in contact with water molecules.

• Non-polar tails are favored to be out of contact with water molecules.

• Molecules form aggregates with orientational (not nec. positional order).

• Phases consist of rich structures with sometimes subtle dependence on

on hydrophilic vs hydrophobic group size and overall molecular shape.

micelle

water

air

bilayer

Phases of Lipid Systems (Surfactants / Amphiphiles)

Fig. Source: D. Cleaver Sheffield Hallam University

Long Hydrophilic Head Group

Short Hydrophilic Head Group

• Biological systems typically have relatively short hydrophilic head group.

• head group ~0.2nm

• hydrocarbon tails ~1-3nm.

• Lamellar and Micellar phases appear to be most common in biology.

• Many proteins anchor in lipid bilayer membranes and serve many

roles, for example enzymatic or mechanical roles.

• Fluidity appears important to biological function: mobility of surface

proteins, resistance to rupture, vesicle budding and fusion.

• Species of artic fish actually modify lipid mixture composition (increase

number double carbon bonds in lipid tails) to lower the critical

temperature of the fluid to gel phase transition.

Biological Lipid Systems (Fluid Mosaic)

Role of Membranes in Cellular Processes

.

• Cell organelles such as the Golgi apparatus make extensive use of lipid

bilayer membranes.

• Proteins are sorted and packaged into vesicles.

• Vesicles are transported and fuse with membranes of other organelles

or the plasma membrane.

• Understanding the mechanisms responsible for organelle function will

likely require insight into the fundamental physics of membrane bilayers.

Membrane Deformation Protein-Protein Interactions & Diffusion

• Deformations are expected to induce flows both of the lipids and of the bulk

solvent.

• “Diffusivity” of membrane embedded proteins may be effected by these flows

(via thermal fluctuations of the membrane shape / or active deformations).

• Exerting a force on a protein within a membrane induces motions at large

distances mediated by the hydrodynamics of both the lipids of the membrane

and bulk solvent fluid (if treated as only 2D fluid “Stokes paradox”) (Saffman-

Delbruck Theory).

• Hydrodynamics and membrane geometry may also play a role in collective

diffusion of proteins (aggregation phenomena / observed bleaching in exp.).

Importance of Hydrodynamics and Thermal Fluctuations:

Some Illustrative Problems

Resolution of Modelingan

gstr

om

nan

om

ete

rm

icro

ns

mil

lem

ete

r

femtosecond picosecond nanoseconds milleseconds

Quantum

chemistry

Molecular

Dynamics

Coarse-Grained

Modeling I

Continuum

Mechanics

Coarse-Grained

Modeling II

time-scale

len

gth

-sc

ale

Resolution of Modelingan

gstr

om

nan

om

ete

rm

icro

ns

mil

lem

ete

r

femtosecond picosecond nanoseconds milleseconds

Quantum

chemistry

Molecular

Dynamics

Coarse-Grained

Modeling I

Continuum

Mechanics

Coarse-Grained

Modeling II

time-scale

len

gth

-sc

ale

hydrodynamics and thermal

fluctuations

Outline








Soft Matter Materials Modeling Approach

• Soft Matter Material Model:

(Newtonian Solvent Fluid + Microstructures)

• Membranes.

• Polymers.

• Lipid molecules.

• Particles: Colloids, Proteins, Ions, etc...

• Fluid flow accounted using hydrodynamic

description in an Eulerian reference frame.

• Microstructure accounted in a Lagrangian

reference frame moving with the structures.

• How should these descriptions be coupled?

(conserve energy and momentum)

• How can thermal fluctuations be

incorporated consistently?

Polymeric Fluid

Euler-Lagrange Framework (IB)

Stokes’ Fluid Equations (R << 1):

Euler-Lagrange Coupling (IB Approximation):

How can we use this approach in practice to model soft matter systems?

Polymer / Membrane

particle

polymer

particle

Peskin, Immersed Boundary Method, Acta Numerica, 11, 2002.

Polymer / Membrane

particleparticle

structure ! particles

Coarse-Grained Lipid Model (Mechanics)

Interface

Head

Tail

• U_core repulsion between all beads.

• U_tail attractions between all interface and tail beads.

• U_interface attraction between only interface beads (hydrophobic effect).

Interactions:

Lipid Mechanics:

• U_bend acts on all sequential triples.

• bond length constraint on all sequential beads.

Interactions: Potentials:

G. Brannigan, P. F. Philips and F. L. H. Brown, Phys. Rev. E, 72, 011915 (2005).

Coarse-Grained Lipid Model (Self-Assembly)Lipid Model Forms Bilayer Membranes

G. Brannigan, P. F. Philips and F. L. H. Brown,

Phys. Rev. E, 72, 011915 (2005).

Control Point Energy:

Control Point Dynamics:

Control Point Forcing of Fluid:

Representation within the SIB Framework

Hydrodynamic Coupling

Control

Points:

curvature stretching

stretching

Polymers

• Rouse and Worm-like Chains can

within the SIB framework.

• Rouse Chain:

(very flexible polymers)

• Worm-like Chain:

(DNA, unstructured RNA, proteins, filaments)

• Modeled by discretization into n nodes {X[j]}.

Outline








Euler-Lagrange Framework



How should thermal effects be taken into account?

Polymer / Membrane

particle

polymer

particle

Atzberger, Kramer, & Peskin JCP 2007.

Euler-Lagrange Framework



How should these terms be chosen to account for the physics?

Polymer / Membrane

particle

polymer

particle

• Equilibrium fluctuations should be Boltzmann distributed:

• Thermal forcing will be modeled by Gaussian stochastic processes -correlated in time.

• The thermal forcing should account appropriately for numerical approximation of the differential operators.

(ex: under-damping of high-freq modes by numerical Laplacian)

Determining the Thermal Fluctuations

Z is the normalization constant.

(fluid kinetic energy)

(“probability density”)

How should thermal forcing be introduced into these equations? (need to ensure physically meaningful / statistical mechanics)

Let the thermal forcing be of the form (ansantz):

then using Ito Calculus the covariance is:

The fluid equations can be written as:

The covariance of the fluid velocity is:

Determining the Thermal Fluctuations

.

Fluctuation-dissipation relations will be sufficient to determine

forcing.

(for reference):

By Ito Calculus the covariance of the fluid satisfies:

At statistical steady-state we have for the equilibrium fluctuations:

This determines the thermal forcing via covariance factor Q:

, where .

Important Issue: Given C and operator L can Q be derived and its action

numerically computed efficiently in practice?

Determining the Thermal Forcing

.

(Fluctuation-Dissipation Principle)

Euler-Lagrange SIB Framework



Polymer / Membrane

particle

polymer

particle

Complex fluid

Atzberger, Kramer, & Peskin, Stochastic Immersed Boundary Method, JCP 2007.

Outline








Thermal Fluctuations for Discretized Equations

Continuum Laplacian

Eigenvalue Magnitude (Damping Strength)

Discretized Laplacian

wavenumber k

eig

en

va

lue®k

• Eigenvalues agree well for

small wavenumbers.

• Higher frequency modes are

“underdamped” by the

discretized Laplacian.

• Using thermal forcing from

continuum model gives

incorrect fluctuations

(too large for high freq. modes).

• Must be careful in how thermal fluctuations are introduced when discretizing

the equations.

• Fluctuations should be considered in relation to the approximate dissipative

operator of the discrete system.

Central Difference Laplacian on Uniform Periodic Mesh

Central Difference Approximation on

Uniform Periodic Mesh

.

Stokes’ Fluid Equations (3D periodic lattice):

(projection method is used)

• Will be helpful to work in Fourier space (diagonalize Laplacian).

• Discrete Fourier Transform (DFT) will be used:

Determining Thermal Forcing for

Central Difference Approximation on Periodic Mesh

.

Stokesian Fluid Equations (DFT on N3 lattice points):

where,

(incompressibility)

(real-valuedness)

This gives the numerical Laplacian (projection not included):

(diagonal matrix)

Represent

by

Fourier Modes

Determining Thermal Fluctuations for


.

• The energy can be expressed as:

• This gives a Boltzmann distribution of the form:

Gaussian with mean 0 and covariance:

• Fluctuation-dissipation relations (under the constraints):

(fluid kinetic energy)

(Parseval’s Lemma)

(incompressibility)

(real-valuedness);

(Kramer and Peskin 2003)

(Atzberger et al. 2007)

(self-conjugate modes)

Determining Thermal Fluctuations for


.

• Thermal forcing for the fluid (Fourier space):

where, : Brownian motion path (complex valued)

: Strength of forcing of the kth mode.

• Constraints are handled by operations on the generated thermal force.

Represent

by

Fourier Modes

Summary: Euler-Lagrange SIB Formalism (Uniform Periodic Mesh)

Fluid Equations (Fourier-Space):

Euler-Lagrange Coupling:

(particle force)

(thermal force)

(incompressibility)

(real-valuedness)

(viscous damping)

Uniform Mesh

Periodic Boundary Conditions

Polymer / Membrane

particle

polymer

particle

Euler-Lagrange SIB Formalism (Adaptive Mesh)

Fluid Equations (MAC Laplacian):


(particle force)

(thermal force)

(viscous damping)

Adaptive Mesh

Dirichlet Boundary Conditions

• For MAC Laplacian we have method to generate thermal forces directly 2D/3D:

(challenge to generate efficiently)

polymer

particle

Sampler for Adaptive Meshes

• Gauss-Siedel vs Multigrid for stochastic sampling.

• Multigrid only requires a few iterations to generate

nearly decorrelated variates.

• Can be used for uniform or adaptive non-periodic

meshes with imposed boundary conditions.

Time Scales of a Typical System

• Water at physiological temperature.

• L = 1 m (length-scale), N = 128 (number of modes)

• Time scales indicate a stiff set of stochastic equations.

• Resolving fastest time-scales of the fluid is expensive.

• One Solution: Treat fluid modes as relaxed on the time scales of the particles and track just the particles. (Brownian-Stokesian Dynamics).

• May be of interest to resolve fluctuations of hydrodynamic modes.

(Brownian-Stokesian Dynamics pre-computes Green’s functions,

presents challenges when imposing boundary conditions.)

Relaxation Time Scale (kth mode) Diffusion Time Scale

Outline








Numerical Methods for Dynamics Over

Long Time Steps (Basic Problem)

t

X

t

XConsequences of non-smoothness of

the system dynamics:

• Not efficient to derive schemes by

simply asymptotically expanding

system dynamics for small times t.

• Schemes must carefully account for

cancellation of the fluctuating terms

over the time step.

Integrators for the Stochastic Dynamics

Fluid Equations (Stokes Flow):


(incompressibility)

(particle force)

(thermal force)

(viscous damping)

Uniform Mesh

Adaptive Mesh

Stochastic Integrator for the Fluid Dynamics

(Semigroup Approach)

Integrating the fluid equations gives:

where,

Assuming forces change on slow time scale relative to fluid dynamics:

Statistics of the thermal fluctuations can be computed by Ito Calculus:



[for reference]:

Substituting into the above we obtain the integrator:

where,

is a Gaussian random variable with mean 0 and covariance

Important Issue: To obtain viable numerical method we must be able to

efficiently compute (or approximate) the operators and .

Two cases we shall consider:

1) Operator L is easily diagonalizable (FFT).

or

2) Large separation of times scales between dynamics of fluid and particles.



Uniform Periodic Mesh: (Laplacian diagonalizable via FFT)

This gives the integrator:

;

Integrator for the Lagrangian Dynamics

Fluid Equations (Stokes Flow):


(incompressibility)

(particle force)

(thermal force)

(viscous damping)

Uniform Mesh

Adaptive Mesh

Stochastic Integrator for the Lagrangian Dynamics


Integrating the particle equations gives:

Assuming particle positions change only a small amount over time step:

Substituting above we obtain the integrator:

Important Issue: To obtain a viable numerical method we must be able to

efficiently compute (or approximate) the statistics of .


Semigroup Method

Statistics of the fluctuations of the time integrated fluid velocity

can be computed by Ito Calculus.

Mean:

Covariance:

Fluid-Microstructure Covariance:


Semigroup Method

Uniform Periodic Mesh: (Laplacian diagonalizable via FFT)

• This gives the numerical integrator:

;

(samples time integrated

fluctuations of the fluid)

Outline








Physical Validity of the Formalism

Some basic checks:

• Do particles as represented by diffuse with the appropriate dependence on the physical parameters of the system? (i.e. particle size, fluid viscosity, temperature, …)

• Do such interacting microstructures exhibit the correct hydrodynamic correlations (i.e. Oseen “hydrodynamic” coupling tensor)?

• Does the joint fluid-microstructure system have the correct equilibrium fluctuations? (i.e. Gibbs-Boltzmann statistics)

.

Does the SIB formalism reproduce well-known physics of microscopic systems?

±a

Diffusion of Immersed Boundary Particles

Theory (red curve):

Simulation (data points):

• Theory and simulation agree giving the correct scaling in the physical

parameters. (particle size, fluid viscosity, temperature)

• Overdamped treatment of the fluid fluctuations still yields the correct

diffusion of particles.

Diffusion Coefficient:Diffusion Coefficient

Resolution of the fluid potentially allows for subtle features of the

diffusive dynamics of particles to be considered. [hydrodyamic memory]

Diffusion of SIB Particles(hydrodynamic memory effects)

SIB Analytic Theory (green curve):

Numerical Simulations (blue curve).

• Power law for decay of the velocity autocorrelation function predicted by

MD simulations (Alder & Wainright 1950’s).

• SIB captures hydrodynamic memory effects contribution to the particle

dynamics.

Velocity Autocorrelation Function

Equilibrium Fluctuations of Particles

• Fluctuation-dissipation consider only the fluid equations invoked to

derive the thermal forcing.

• Must check this is sufficient to ensure Gibbs-Boltzmann is equilibrium.

Results: Numerical simulations

indicate correct equilibrium

fluctuations of particles.

soft-wall potential

Statistical mechanics requires Gibbs-Boltzmann distributed

fluctuations of fluid and particles.

particleparticle

Equilibrium Fluctuations of Polymer (WC)

Longitudinal & Transverse Fluctuations

, : variances of kth mode (Boltzmann Statistics)

Periodic

Do polymers exhibit Gibbs-

Boltzmann fluctuations?

Results: Variance of Fourier

modes indicate fluctuations are

accurately captured over scales

larger than Eulerian mesh

spacing.

At resolutions finer than the

fluid mesh, polymer

fluctuations are suppressed.

(polymer energy)

Outline








Applications in Rheology of

Soft Matter Materials

Rheological Experiments to Measure

Shear Viscosity of Materials

Rheomety Device

SIB Simulations of Sheared Materials

Membrane (directoinal shear viscosities / zero shear limit)

Polymeric Fluid (effective shear viscosity as function shear rate)

membrane_shear_view_diag.avi

Shearing Boundary Conditions

• Lees-Edwards Boundary Conditions (1972).

• MD particle interactions periodic boundary conditions are modified:

• Material frame is shifted when crossing periodic boundary.

• Jump in velocity occurs over the boundary.

• We shall now discuss how to handle these conditions for the Stokes fluid flows.


.

Stokes Equations:

Boundary Conditions:

• Using standard coordinate frame, u is no longer periodic (because of shift).

• We shall consider fluid flow described in the sheared coordinate frame:


.

Stokes Equations (sheared coordinate frame):

Boundary Conditions (become jump condition):

Change of coordinates (shear in x-direction along z-axis)

• Jump boundary condition can be accounted for by introducing forcing term

J on RHS.


.

Stokes Equations (with jump related forcing term J):

Boundary Conditions (become periodic for modified PDE):

• We shall discretize this PDE to obtain the fluid flow.

• Some important issues:

• Can the discrete operators be diagonalized by FFT?

• Can incompressibility be enforced efficiently?

• How should J be discretized?


.

Stokes Equations (with jump related forcing terms J):

• Yields cyclic matrix (grid-translation invariant) approximation for Laplacian and

Divergence (=> diagonalizable by FFT)

• This allows for previously discussed integration schemes to be used.

• Central Finite Differences used for and to obtain L(t) and .

Stokes Equations (discretized in space, projection method):

• J is obtained by plugging jump condition into the central difference scheme

when crossing top and bottom periodic boundaries.

Results for Lipid Bilayer Membranes

.

Coarse-Grained Lipid Model (Mechanics)

Interface

Head

Tail

• U_core repulsion between all beads.

• U_tail attractions between all interface and tail beads.

• U_interface attraction between only interface beads (hydrophobic effect).

Interactions:

Lipid Mechanics:

• U_bend acts on all sequential triples.

• bond length constraint on all sequential beads.

Interactions: Potentials:

G. Brannigan, P. F. Philips and F. L. H. Brown, Phys. Rev. E, 72, 011915 (2005).

Control Point Energy:

Control Point Dynamics:

Control Point Forcing of Fluid:

Representation within the SIB Framework

Hydrodynamic Coupling

Control

Points:

.

Bending Modulus

• Monte-Carlo Sampling of Equilibrium

Membrane Fluctuations (Metropolis).

• Fourier interpolation of the membrane surface.

• Elastic sheet model in the Monge gauge fit to

Fourier modes to deduce effective kc.

• For constant tension simulations must be careful

to average over the fluctuations of the projected

area L2.

• Within range of physical membranes: kc = 1 – 8 £ 10-20J.

• Flexible kc comparable to DGDG.

• Stiffer kc comparable to DLPC & DMPC.

Brown, Phys. Rev. E, 72, 011915 (2005).

.

Stresses within the Bilayer

Brown, Phys. Rev. E, 72, 011915 (2005).

• Monte-Carlo Sampling of Equilibrium

Membrane Fluctuations (Metropolis-Hastings).

• Stresses estimated from microscopic model

using the virial stress formula of Irving-

Kirkwood (1950).

• Surface tension for slab (fixed z) given by

• Effective tension of membrane bilayer

• (z) agrees qualitatively with fully atomic MD

simulations.

• is within range of theoretical estimates

32 – 65 mJ/m2.

• Hydrodynamic simulations of the membrane

model using the SIB method.

• Stresses estimated from volume averaged

microscopic stressed.

• Kramer-Kirkwood Estimator of Stress:

• Shear viscosity obtained from simulations of

membrane sheared at rate .

• Shear viscosity given by:

• Shear Viscosity (SIB):

• Sckulipa, Otter, Briels (Biophys. J. 2005):

• Physiological Range:

.

Shear Viscosity [Preliminary]

_°

´m= ¾m= _°



membrane_shear_view_top.avi

• Motion of individual head groups and pairs

considered.

Head Group Diffusivity [Preliminary]

Lipid Head Group Position

Diffusion Tensor Components

Single Head Diffusion:

membrane_fluct_1_3_run13_upper_view.avi


Head Group Diffusivity [Preliminary]

Diffusivity of a Pair of Head Groups:

Lipid Head Group Positions

Diffusion Tensor for Separation Distance D(r)

membrane_fluct_1_3_run13_upper_view.avi


Polymeric Fluids

.

• Rouse and Worm-like Chains can

within the SIB framework.

• Rouse Chain:

(very flexible polymers)

• Worm-like Chain:

(DNA, unstructured RNA, proteins, filaments)

• Modeled by discretization into n nodes {X[j]}.

Polymers

curvature stretching

stretching

Polymeric Fluid (FENE Dimers)

.

Potential Energy U(r) Force |F(r)|

Finitely Extensible Nonlinear Elastic Dimers:

(non-zero rest length bonds)

Polymeric Fluid (FENE Dimers)

.

• SIB Methodology can be used to

study shear thinning of a FENE dimer

fluid.

• Preliminary studies being carried out

to compute shear stress with SIB with

hydrodynamic interactions.

• Kramers-Kirkwood Stress Tensor

FENE_Shear_BrownGroup1_2_run04.avi

Polymer Knots and SIB

.

Polymer Knots and Links

Consequences:

• Solution map of space surrounding polymer curve is a

homeomorphism.

• Topological invariants of curves are preserved under the stochastic

flow.

• Knottedness or linking of polymers retained under flow (up to

numerical accuracy).

• No need for excluded volume forces when simulating polymer knots

and links to impose topological constraints.

All structures derive motion by averaging a common velocity field.

Osmotic Wall-Pressure of Polymer Knots

Concentration of Monomers in Boundary Layer

Other “Soft Matter” Systems

.

• Force generation in cell motility and division

generated by polymerization reactions.

• Examples include:

• microtubule chromosome separation

• actin polymerization in leading edge of cell

• Motor proteins are involved in active transport

and force generation within cells.

• Examples include:

• Kinesins / Dyneins : active transport along

microtubules.

• Myosins : force generation (sliding) of actin

filaments, also active transport (myosin V).

Motor Proteins and Polymerization Processes

Brownian Ratchet (Hydrodynamic Load)

.

diffusion

• The ratchet is prevented from back-slipping to the left.

• Brownian fluctuations drive the ratchet to the next interval.

• Basic model for single headed Kinesin motors and microtubule / actin

(de)-polymerization.

• Many interesting questions about the role of the cargo (purple) and it’s

coupling to the motor on the rate of stochastic transport.

.

Vesicle Transport by Motor Protein

Brownian Ratchet (Hydrodynamic Load)

.

Related Papers and Codes: http://www.math.ucsb.edu/~atzberg/

Collaborators:

•Stochastic Immersed Boundary Method:

• Peter R. Kramer (Dept. Math, Rensselaer Polytechnic Institute)

• Charles S. Peskin (Dept. Math, New York University)

• Coarse-Grained Lipid Bilayer Membrane Modeling:

• Frank Brown (Chemistry and Biochemistry, University of

California Santa Barbara)

•Evgeni Penev (Chemistry and Biochemistry, University of

California Santa Barbara)

Acknowledgements

Support: NSF DMS – 0635535.

Mathematics Department

Graduate and Postdoctoral Research in Applied Mathematics

More information about positions: http://www.math.ucsb.edu/~atzberg/

Los Angeles: 90 Miles

UC Santa Barbara

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