Pressure Transferable Coarse-Grained Potentials for Polymers … · 2017. 2. 7. · Forbes, Jerry W. Shock wave compression of condensed matter: a primer. Springer, 2013. 2015 LAMMPS

2015 LAMMPS Users’ Workshop

Pressure Transferable Coarse-Grained Potentials for Polymers under Isothermal and Shock

Compressions

Vipin Agrawal and Jay OswaldArizona State University

http://compmech.lab.asu.edu


Finite element analysis

Overall goals

Advance fundamental understanding of polymer stress response from materials-by-design perspective under extreme loading conditions

Improve thermodynamically-consistent, coarse-grained molecular dynamics for polymer materials

Representability: predict more than just structural properties

Transferability: predict properties across a range of thermodynamic states

Molecular dynamics

Coarse graining


Outline

• Transferability problem for CG potentials of soft materials: the CG potential is only accurate at the thermodynamic state where it was calibrated.

– Temperature transferability

– Pressure transferability

• IBI-EAM and many body CG potential

– Pressure optimization algorithm

• Model verification and validation

– Bulk modulus

– Shock hugoniot response


Coarse-Grained mapping

Groups of atoms represented by super atoms

All-atom molecule

Coarse-Grained molecule

Coarse-Grained mapping

What coarse-grained potential will reproduce the same thermodynamics of the MD system?


Transferability and representability• Representability: does the CG model accurately reproduce

thermodynamic properties?

• Transferability: is the CG potential accurate at a different thermodynamic state than where it was calibrated?

Polyethylene

Equation of state of polyethylene at T=300K computed from CG models derived by iterative Boltzmann inversion.

For CG models, naïve application of IBI-

derived potentials are overly soft in

volumetric compression.


Transferability of IBI polyethylene potential

Calibration pressure

• Probability of finding a particle at distance r from a reference point

( )( ) exp

B

u rg r A

k T

U(r) – potential mean force


Many body Coarse-Grained potential

1( ) ( )

2i i j ij ij ij

j i j i

E F r r

2

( ) 1ij

c

rr

r

Energy of atom i

Embedding energy function

local densityfunction

PairPotential

1D density profiles

2D density profile (1 atom)

Multi-atom density profile

Arbitrary density function

( Electron densityFunction )


Calibration of the IBI-EAM Potential

Pressure (MPa)

Density(g/cc)

-50.7 0.74857

-10.1 0.78123

0.1 0.78417

1.0 0.78585

10.0 0.79213

100 0.82962

1000 0.97115

2000 1.04353

5000 1.16120

10000 1.27479

Calibrated state points

Pair Potentials

Angle Potentials

( ) ln( ( ))BV r k T g r

( )( ) ln

sin( )B

PV k T

Bond Potentials

2

( )( ) lnB

P lV l k T

l


1

3i i i i i

i

p mV

v v r f

ke Fp pp p

1

3ke i i i

i

p mV

v v

'1ˆ ( )

3i ij ij ij

i j i

p rV

r r

' '1ˆ( ) ( )

3F i i ij ij ij

i j j i

p F r rV

r r

Calibration of Embedding Energy

Virial pressure equation

Decomposition of pressure

Kinetic pressure

Pairwise pressure

Embedded pressure


Pressure optimization algorithm

2

0

( ) ( ) ( )4cr

i ij i

j

Nr g r r r dr

V

1

nk

k

k

F a

( 1)'

1

nk T

k

k

F ka

C P

1×m[ ]kaC( 1)

n m

kAk

P

1

T T

C P P P Δp

Embedding energy coefficients are solved by least-squares minimization

The mean density at a particle can be computed from the radial density function g(r)

Taylor series expansion of the embedding energy

Derivative of embedding energy and corresponding vector form

Taylor polynomial derivatives Taylor polynomial coefficients


Calibrated IBI-EAM model for polyethylene

Pressure vs Density Optimized Embedded Function


Calibrated IBI-EAM model for polyethylene

Optimized Angle Potentials Optimized Pair Potentials


Transferability of IBI-EAM model for PE

Pair and angle distributions are well-reproduced over a wide pressure range!


MODEL VALIDATION AND VERIFICATION


Isothermal Bulk modulus comparison for PE

0

( )1 ln 1

V P PC

V B

( )T

T

Pk P V

V

2( )

B

T

k T Vk P

V

Tait EOS

Isothermal Bulk Modulus

Heydemann, P. L. M., and J. C. Houck. "Bulk modulus and density of polyethylene to 30 kbar." Journal of Polymer Science Part A‐2: Polymer Physics 10.9 (1972)

( ) 1 ln 1T

P B Pk P C

B C


Hugoniot us-up Calculations for PE

1/2

0 0

0

1

ln 1

su P

c k PC

k C

1/2

0 0 0

0

1ln 1

ln 1

pu P PC

c k k CPC

k C

Marsh, Stanley P. LASL shock Hugoniot data. Vol. 5. Univ of California Press, 1980

0

1/2

0 0( / )c

1/2

0

0 0

1

1 /s

P Pu

V V

01 /pu V V

Sound Speed

Jump Conditions

“pseudo” us-up plane


Shock Hugoniot Curve for PE

Marsh, Stanley P. LASL shock Hugoniot data. Vol. 5. Univ of California Press, 1980.

Experimental PE density – 0.916 g/cc MD PE density = 0.81 g/cc


Mie-Gruneisen EOS

/

/ dT

V

V V

dP dTdP

V dE dE

Gruneisen EOSγ is Gruneisen parameter

v

V V

E SdE TdS PdV C T

T T

V T

S STdS T dT T dV

T V

V T

P S

T V

v

PTdS C dT T dV

T


Temperature Rise Along the Hugoniot

0

( )( )H

v

f vT v T e e e dv

C

dvv

v v H

H H

dE dTC T C P

dv dv v

0

1( ) ( )

2H H

dE dPv v f v

dv dv

( )H

v

dT f vT P

dv v C

From Hugoniot conditions

Forbes, Jerry W. Shock wave compression of condensed matter: a primer. Springer, 2013


∂P/∂T and ∂E/∂T for PE

• ∂P/∂T and ∂E/∂T are significantly different between MD and CG• The ratio of these two are not very high• For most metals, ϒ/v assume to be constant while calibrated ratio from MD

simulations increases with volumetric compression


Summary of IBI-EAM shock study

IBI-EAM PE potential Matches mechanical properties reasonably well Temperature rise along the Hugoniot shows deviations

Documents

Pressure Transferable Coarse-Grained Potentials for Polymers … · 2017. 2. 7. · Forbes, Jerry W. Shock wave compression of condensed matter: a primer. Springer, 2013. 2015 LAMMPS