Transcript The Stillinger-Weber Potential

Biological fluid mechanics at the
micro‐ and nanoscale
Lecture 7:
Atomistic Modelling
Classical Molecular Dynamics Simulations
of Driven Systems
Anne Tanguy
University of Lyon (France)
Atomistic Modelling:
Classical Molecular Dynamics Simulations
of Driven Systems.
I. Description
II. The example of Wetting
III. The example of Shear Deformation
Classical Molecular Dynamics Simulations consists in solving the Newton’s equations
for an assembly of particles interacting through an empirical potentiaL;
In the Microcanonical Ensemble (Isolated system): Total energy E=cst
In the Canonical Ensemble: Temperature T=cst
with
if no external force
Different possible thermostats: Rescaling of velocities, Langevin-Andersen, Nosé-Hoover…
more or less compatible with ensemble averages of statistical mechanics.
Equations of motion: the example of Verlet’s algorithm.
Adapt the equations of motion, to the chosen Thermostat for cst T.
Thermostats:
after substracted the Center of Mass velocity, or the Average Velocity along Layers
dv
• Langevin Thermostat: mi i  G.vi  Fi  k (t )
dt
Random force k(t)
Friction force –G.v(t)
with <k(t).k(t’)>=cste.2GkBT.d(t-t’)
• Andersen Thermostat: prob. of collision nDt, Maxwell-Boltzman velocity distr.
• Nosé-Hoover Thermostat:
dH
0
dt '
• Rescaling of velocities:
• Berendsen Thermostat:
with
(
)1/2
Heat transfer. Coupling to a heat bath.
Examples of Empirical Interactions:
The Lennard-Jones Potential:
2-body interactions
cf. van der Waals
Length scales sij ≈ 10 Å
Masses mi≈10-25 kg
Energy eij≈ 1 eV ≈ 2.10-19J ≈ kBTm
Time scale t 
Time step Dt = 0.01t ≈ 10-14 s
106 MD steps ≈ 10-8 s = 10 ns
or
m.s 2
e
0.1s 2
1020
 8  1012 s
 10 s or t 
D(T  1) 10
12
106x10-4=100% shear strain in quasi-static simulations
N=106 particles, Box size L=100s ≈ 0.1 mm for a mass density r=1.
3.N.Nneig≈108 operations at each « time » step.
The Stillinger-Weber Potential:
For « Silicon » Si, with 3-body interactions
Stillinger-Weber Potential F. Stillinger and T. A. Weber, Phys. Rev. B 31 (1985)
4
ESW (1,2,...,N )  i , j ( A.r  B ). e
Melting T
Vibration modes
Structure Factor
( r  a ) 1
2-body interactions
(Cauchy Model)
 .(rij  a ) 1  .(rik a ) 1
 i , j ,k f (ijk ). e
3-body interactions
The BKS Potential:
For Silica SiO2, with long range effective Coulombian Interactions
B.W.H. Van Beest, G.J. Kramer and R.A. Van Santen, Phys. Rev. Lett. 64 (1990)
EBKS (r ) 
qi q j
4e 0 r
 Aij e
 Bij r

Cij
r
6
où(i, j )  Si,O
Ewald Summation of the long-range interactions,
or Additional Screening (Kerrache 2005, Carré 2008)
Example: Melting of a Stillinger-Weber glass, from T=0 to T=2.
Microscopic determination of different physical quantities:
-Density profile, pair distribution function
-Velocity profile
-Diffusion constant
-Stress tensor (Irwin-Kirkwood, Goldenberg-Goldhirsch)
-Shear viscosity
(Kubo)
II. The example of Wetting
Surface Tension: coexistence beween the liquid and the gas at a given V.
(L. Joly, 2009)
Surface Tension:
h
The Molecular Theory of Capillarity:
Intermolecular potential energy u(r).
Total force of attraction per unit area:

Fz h   r1.r 2  dz d 3r. f z r 
h

 2r1.r 2  r (r  h)u (r )dr
h
Work done to separate the surfaces:


h0
h0
W  2 S   Fz h dh   r1.r 2  dr .r 3 .u (r )
(Hautman and Klein, 1991)
(I. Israelachvili, J.S.Rowlinson and B.Widom)
3
 LV . cos   SV   SL  for  SV   SL   LV .
III. The example of Shear Deformation
Boundary conditions:
Example: quasi-static deformation of a solid material at T=0°K
Quasi-static shear
at T=0.
Fixed walls
Or biperiodic boundary conditions
(Lees-Edwards)
At each step, apply a small strain de ≈ 10-4 on the boundary,
And Relax the system to a local minimum of the Total Potential Energy V({ri}).
Dissipation is assumed to be total during de.
dt  a / c  1012 s
Quasi-Static Limit
de
de .c
e lim.c


 108 s 1 ( 10 4 u LJ ).
dt
a
a
ux
F
 s xy shear stress
S
Ly
st rain e xy 
Rheological behaviour:
Stress-Strain curve in the quasi-static regime
ux
2 Ly
F
 s xy shear stress
S
ux
Ly
st rain e xy 
ux
2 Ly
y
Local Dynamics:
Global and Fluctuating Motion of Particles
X
F
 s xy shear stress
S
ux
Ly
Local Dynamics:
Global and Fluctuating Motion of Particles
Transition from Driven to Diffusive motion
due to Plasticity, at zero temperature.
Dy _ max
st rain e xy 
cage effect
(driven
motion)
ux
2 Ly
ep
Diffusive
Dn ~ Dexy
Tanguy et al. (2006)
Driving at Finite Temperature:
The relative importance of Driving and of Temperature
must be chosen carefully.
Low Temperature Simulations: Athermal Limit
.
Typical Relative displacement due to the external strain
Typical vibration of the atom due to thermal activation
>>
larger than
a.  .t
k BT
kh
Convergence to the quasi-static behaviour, in the athermal limit:
At T=10-8 (rescaling of the transverse velocity vy et each step)
M. Tsamados
(2010)
.
s   .     
 
.
.
0 .4
s   .   cste
at finite T
Effect of aging
T= 0.2-0.5 Tg =0.435
Rescaling of transverse velocities in parallel layers
Non-uniform Temperature Profile at Large Shear Rate
Time needed to dissipate heat created by applied shear across the whole system
c 1 m
td  
L L r
.
dQ
 s xy . 
Heat creation rate due to plastic deformation
dt
tQ 
Time needed to generate kBT,
.
tQ  t d   
k BT
.
s xy . 
k BT .L
.
c.s xy   
 
Visco-Plastic Behaviour:
Non uniform T
Flow due to an external force (cf. Poiseuille flow)
F. Varnik (2008)
End