SUPPLEMENTAL MATERIAL
Molecular dynamics simulations of shock waves in hydroxyl-terminated
polybutadiene melts: Mechanical and structural response
Markus G. Fröhlich, Thomas D. Sewell, and Donald L. Thompson
Department of Chemistry, University of Missouri-Columbia, Columbia, MO 65211-7600, USA
I.
CALCULATIONS OF SHOCK AND TOTAL PRESSURES
Calculation of the shock- and total pressures in the reverse ballistic configuration for explicit
shock MD simulations is nontrivial but straightforward. As discussed in the main article, the
shock pressure immediately behind the shock front is dominated by the longitudinal stress. Thus
the pressure Px in Eq. (S1) is the negative longitudinal component of the stress tensor S.
Px   S xx
(S1)
Due to momentum and energy conservation the shock pressure Ps can also be calculated from the
Hugoniot jump conditionS1,S2 using Eq. (S2).
Ps  0  us  u p
(S2)
In general, values of Ps and Px are nearly equal.
In addition to the shock pressure, the hydrostatic (total) pressure Pt is given in Eq. (S3).
Pt  
1
 S xx  S yy  S zz 
3
(S3)
In the next two paragraphs we detail the calculation of the stress tensor required in Eqs. (S1) and
(S3).
1
We used the LAMMPS option compute stress/atom to calculate for each atom in a given bin
the instantaneous symmetric six-component per-atom stress tensor S:S3
S
N

1 p
1 Nb
1 Na
   mv v    r1 F1  r2 F2      r1 F1  r2 F2      r1 F1  r2 F2   r3 F3 
2 n 1
2 n 1
3 n 1


1 Nd
   r1 F1  r2 F2   r3 F3   r4 F4  +Kspace  ri , Fi  
(S4)
4 n 1

where  = xx, yy, zz, xy, xz, yz. We averaged this quantity over 1000 integration time steps
(equal to the data sampling rate of 10.0 ps–1 used for all other data in the study). As defined in
Eq. (S4), the per-atom stress S is actually a stress  volume quantity,S3 thus each component of
the total stress tensor S' for a given bin is the sum over the associated atomic stress component
for the atoms in that bin divided by three times the volume of the bin. From this definition plus
Eq. (S4) the stress tensor S' consists of both a kinetic contribution (mv2) and a virial contribution.
The virial contribution is given by the sum over the respective force field terms: 1/n  rnFn,
with n = 2 for pair and bond interactions, n = 3 for angle terms, n = 4 for dihedral terms; and kspace terms. Whereas the virial contribution calculated by LAMMPS can be used without
modification, the kinetic contribution has to be modified to remove the whole-body motion of
material in a given bin along the shock direction (the Cartesian vector (–up, 0, 0) for unshocked
and (0, 0, 0) for completely shocked material.) We accounted for this as follows.
For each chain length we performed two 10.0 ps NVE trajectories for 3-D bulk fully flexible
systems, that is, prior to introducing the vacant space and defining the rigid piston. In one
trajectory the center-of-mass velocity of the system was set to zero, corresponding to impact
velocity up = (0, 0, 0) (stationary system). In the second trajectory the impact speed up = 1.0, 1.5,
2.0, or 2.5 km·s1 was subtracted from the x-component of all atoms in the system such that the
entire system exhibited longitudinal flow along xˆ . Using the average total stress tensors from
2
these simulations, it is possible to calculate a correction tensor Scorr = Sstationary – Sflow, and hence
to correct the total stress tensor S of a given bin using S = S + Scorr.
II.
CALCULATIONS OF STRAIN RATE
We considered two ways to calculate the strain rate in the shock front. The first method is
described by Bringa et al.S4 and is based on the width of the shock rise. According to the
Hugoniot condition the strain  is defined by

up
us
,
(S5)
and the strain rate by
up
u
d
  s 
dt
x x
(S6)
where x is the width of the shock front. As discussed in Ref. S4, this approach provides
satisfactory results for ‘hard’ materials, for example, metals. The definition is difficult to apply
for ‘soft’ materials, for example, polymers, due to the requirement to define the shock width x.
The difficulty is that for soft matter the compressibility is large and the shock width narrow (only
2-3 bins in the present work), such that the spatial resolution is limited.
An alternative approach, which is more suitable for soft material, is to use the density
profiles as functions of the distance from the shock front. Specifically, in the Eulerian
representation in which material flows through fixed-width bins along the shock direction the
density increase is proportional to the decrease l of the bin width in a corresponding Lagrangian
representation. Thus, the strain  is defined by
3

l l (t )  l0 1  (t )  1 0


,
l0
l0
1 0
(S7)
where (t) and 0 are the mass density at time t relative to the shock front and in the unshocked
material, respectively. The time t ahead (–) and after (+) the shock rise can be calculated given
the shock speed us. Figure S1(a) shows the strain as function of time and Fig. S1(b) the strain
rate d/dt as a function of time, evaluated numerically from the gradient of the strain vs. time
plots for chain length nC = 128 and impact speeds up = 1.0, 1.5, 2.0, and 2.5 km·s–1. The second
and third columns of Table SI contain, respectively, the strain rates obtained using Eq. (S6) and
the peak strain rates obtained based on Eq. (S7); that is, the extreme values in Fig. S1(b).
REFERENCES
S1
Y. B. Zel’dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature
Hydrodynamic Phenomena (Dover, Mineola, NY, 2002).
S2
M. L. Wilkins, Computer Simulation of Dynamic Phenomena (Springer, Berlin, 1999).
S3
see http://lammps.sandia.gov/doc/compute_stress_atom.html
S4
E. M. Bringa, A. Caro, Y. Wang, M. Victoria, J. M. McNaney, B. R. Remington, R. F. Smith,
B. R. Torralva, and H. Van Swygenhoven, Science 309, 1838 (2005).
4
0.0

unshocked
shocked
0.2
0.4 (a)
2
0
t (ps)
2
0
t (ps)
2
d  / dt (ps 1 )
0
1
2
2
(b)
FIG. S1. (a) Compressive strain  and (b) strain rates d/dt as function of time t for nC = 128 and
up = 1.0, 1.5, 2.0, and 2.5 km·s–1 (light to dark blue). The peak strain rates shown in the third
column of Table SI are the minima in panel (b).
5
TABLE SI. Calculated strain rates d/dt for the nC = 128 system and all investigated impact
velocities.
up
d/dt
d/dt
(km·s–1)
(s–1)a,b
(s–1)a,c
1.0
–0.551012
–0.411012
1.5
–0.831012
–0.791012
2.0
–1.111012
–1.191012
2.5
–1.381012
–1.611012
a
The negative signs denote compressive strain.
b
Using Eq. (S6).
c
Using Eq. (S7).
6