Multiscale modeling – Bridging the gap

Macroscale: •  typical times ~ 1 sec and lengths ~ 1microns •  phenomena: instabilities, pattern formation, phase separation •  methods: phase field models, FEM, etc.

Atomistic scale: •  times ~ 10–15 – 10-9 sec and lengths ~ 1-10 Å •  phenomena: microscopic mechanisms and interactions •  methods: molecular dynamics, Monte Carlo

Subatomistic scale: •  electronic structure, excitations, ab initio, Green functions

Mesoscale: •  typical times ~ 10–8 – 10-2 sec and lengths ~ 10-1000 Å •  phenomena: instabilities, patterns,complexation •  methods: phase field models, lattice Boltzmann, DPD, etc.

Recap: Molecular Dynamics •  The Lennard-Jones pair potential:

energy scale f depends only on this ratio

u(r) = 4ε σr

⎛⎝⎜

⎞⎠⎟

12

−σr

⎛⎝⎜

⎞⎠⎟

6⎡

⎣⎢⎢

⎤

⎦⎥⎥

•  i.e., it can be written in a general form

•  Let’s use the following transformations in order to convert the potential to a dimensionless form:

r* ≡ r /σu* ≡ u / ε

⎧⎨⎪

⎩⎪

Physically:

r

u(r)

σ: unit of length ε: unit of energym: unit of mass

⎧

⎨⎪

⎩⎪

Constant temperature MD: Langevin thermostat

mi

!""ri = −∇U !ri{ }( ) − Γ!"r +!ξi (t)

•  The typical method of generating a canonical ensemble is by introducing artificial degrees of freedom or coupling the system to heat bath.

•  Instead of Newton’s equation of motion solve the Langevin equation

•  Here, is a Gaussian correlated noise source with its first and second moments given as

and

Langevin thermostat: The bad news

mi

!""ri = −∇U !ri{ }( ) − Γ!"r +!ξi (t)

and

This basic version of the Langevin thermostat DESTROYS momentum conservation. To see where this property arises, look at the random forces part.

The physics of momentum conservation: • Momentum cannot just vanish but it must be transported away

• Momentum transport gives rise to long-range hydrodynamic interactions

• Langevin thermostat does not reproduce hydrodynamics correctly. Instead. hydrodynamics is screened (it decays)

Langevin thermostat: The good news

mi

!""ri = −∇U !ri{ }( ) − Γ!"r +!ξi (t)

and

This basic version of the Langevin thermostat DESTROYS momentum conservation.

THE GOOD NEWS: IT IS POSSIBLE TO FIX MOMENTUM CONSERVATION BY CONSIDERING THE RANDOM TERM IN A PAIRWISE FASHION. THIS LEADS TO DISSIPATIVE PARTICLE DYNAMICS.

What is DPD?

1.  Dissipative Particle Dynamics, DPD, can be seen as coarse-grained molecular dynamics with added noise similar to Brownian or Langevin dynamics.

2.  DPD can be seen as particle based flow solver.

•  DPD is a fairly new method. It was developed by Hoogerbrugge and Koelman in –92 - -93 and refined by Espanol and Warren –95.

•  DPD has become one of the most common coarse-grained methods.

•  DPD includes hydrodynamics.

bending

stretching

torsion

Success stories: Problems treated with DPD 1992: Hoogerbrugge & Koelman, flow in porous medium

1993: Koelman, hard sphere suspensions under shear flow

1994: Madden, Confined polymer in solution

1995: Schlijper, dilute polymer solutions

1996-1997: Domain growth and phase separation in a binary fluid

1996-1998: Boek, rheology of dense colloidal suspensions

1998-1999: Groot & Warren, Dynamics of microphase separation

1999: Venturoli & Smit: Simulation of amphiphilic particles

Coarse-graining: The concept No ”real” microscopic particles but ”fluid” particles representing:

1.  Clusters of a few real particles

2.  Segments of a polymer chain (< persistence length)

3.  A fluid element (continuum hydrodynamics)

Coarse-graining: •  Microscopic degrees of freedom are not taken explicitly into account

•  Energy can dissipated, or transferred, to the microscopic degrees of freedom; fluctuations

Forces that are needed (for a minimal model): •  Effective conservative forces between the particles

•  Dissipative forces

•  Random forces must reproduce canonical ensemble

Include interactions with fast modes

Dissipative Particle Dynamics: Forces

!Fi =

!Fij

C +!Fij

D +!Fij

R( )j≠ i∑

DPD, i.e., coarse-grained, particles: The total force: •  mass mi

•  mass vi

•  position ri

Conservative forces:

!Fij

C =aij (1− rij )r̂ij if r<rc

0 if r ≥ rc

⎧⎨⎪

⎩⎪

r

u(r)

cut-off

FINITE!

r

FCij

cut-off

aij

Dissipative Particle Dynamics: Forces Dissipative force: Random force:

!Fij

R =σwR (rij )θ yjr̂ij if r<rc

0 if r ≥ rc

⎧⎨⎪

⎩⎪

!Fij

D =−γ wD (rij )(r̂ij ⋅

!vij )r̂ij if r<rc

0 if r ≥ rc

⎧⎨⎪

⎩⎪

!Fij

R = −!Fji

R

Physically: Physically:

!Fij

D = −!Fji

D

Observations: •  Individual atoms show rapid fluctuations. NOT INTERESTING.

•  Average shape of a chain changes in longer time scale. INTERESTING.

• Averaging gives soft repulsive forces (Forrest & Suter) – Lennard-Jones was used as a starting point

• Potential of mean force:

Importance of soft repulsive forces

Soft repulsive!

How to choose the DPD parameters? •  DPD has three independent variables:

1.  timestep dt 2.  amplitude for random noise 3.  repulsion parameter for conservative forces

•  Is there a systematic way to choose them? Let’s see…

How to parameterise DPD? •  Length scale larger than atoms -> density becomes important

•  Reproduce local thermodynamics -> this must hold and we can use it to obtain parameters

•  Amplitude of density fluctuations is determined by compressibility

•  Concept of hydrophobicity is helpful

Choosing the parameters: Conservative force

1.  Simple fluid. Strategy: Match relative compressibilities. Isothermal compressibility

Dimensional analysis gives:

!κT ≡ κT ⋅kBT ⋅ρ

For water in 1 atm and 300 K: ( !κT )−1 = 15.9835

Once the equation of state p=p(r,T) is known from simulations, the dimensionless isothermal compressibility, , which contains the force strength becomes known.

-> match the strength of the conservative force in such a way that matches the value for the real fluid.

Dimensionless compressibility:

R.D. Groot in “Novel Methods in Soft Matter Simulations”

Choosing the parameters

From simulations we get (C=0.101)

Plug in the value for water:

Message: It is possible to relate the DPD parameters to real physical measurements. Note, that similar procedure can be made for polymers as well using the Flory-Huggins theory.

R.D. Groot in “Novel Methods in Soft Matter Simulations”

Fokker-Planck – Master - Langevin

Master equation

Fokker-Planck equation Langevin equation

Kramers-Moyal expansion

•  The state of the system: phase space

•  The probability to find the system at a particular point

•  The time-evolution of the density of states: the Liouville equation

Fluctuation-dissipation

Fokker-Planck Equation

A B

Stochastic differential equations -> Fokker-Planck equation

This can be written in a more suggestive form:

Require detailed balance and search for a stationary solution

and

That yields

Fokker-Planck Equation

and

Conservative part: Lcr = 0 for

Boltzmann distribution

Dissipative part: Ldr = 0 for

P. Español and P.B. Warren, Europhysics Letters 30, 191-196 (1995)

How to write a DPD code •  Take your favourite MD code

•  Use soft potentials

•  Include pairwise random and dissipative forces

•  Measure thermodynamic quantities in the usual way

Integration: DPD-velocity-Verlet

Update velocities and positions

Compute forces

Morphological evolution –lipid water system

- Box size: 10 x 10 x 10

- Particles: 200 lipids, 1800 waters

- Number density: 3

- No bending rigidity, no torsional potential

Morphological evolution - block co-polymer melt

R.D. Groot and T.J. Madden, J. Chem. Phys. 108, 8713 (1998).

Hydrodynamic modes are responsilbe for large-scale collective modes.

Structure of A-B dividing interface after 8000 time steps

A5B5

A5B5

A3B7

A3B3

A B

A4B6 A2B8

Should you get married to DPD?

Flekkøy and P.V. Coveney, Phys Rev. E (2000)

There is certainly more to DPD & coarse-graining than meets the eye!

The picture on the left shows the idea of Flekkoy and Coveney on coarse-graining. It is based on Voronoi tesellation and in the picture there are 4 different levels particles. Can you guess the 4 levels?

The big challenge is the have a systematic approach to coarse-graining. The Flekkoy-Coveney approach is a step into that direction.

Another alternative formulation is by C.P. Lowe; based on Andersen thermostat & gets rid of the weight function w.

Note:

There is a no single approach that is always applicable!

Summary

–  DPD is a very promising method for mesoscopic soft matter simulations

–  However, it has certain intrinsic problems due to the softness of the potentials and the dissipative coupling

–  Most of the problems can be solved!

–  More work needed on mapping between DPD and microscopic systems