64
A. Jayaraman et al.
length N M . The initial configuration of the PNC is created by first placing a single
polymer grafted particle (PGP) (devoid of matrix chains) at the center of a large
simulation box of size much larger than the final intended simulation box size. First,
to allow the graft chains to relax from the chosen initial configuration, this single
PGP undergoes an MD simulation run in the NVT ensemble at T
*
= 1 where all
interactions (graft-graft and graft-particle) are maintained as purely repulsive via
a WCA potential over 1 million-time steps where each time step is 0.001 σ (
m
ε
)
0.5 .
Next, we add matrix chains around this equilibrated single PGP. The use of a box size
larger than the intended cubic simulation box size ensures minimal overlaps during
the addition of matrix chains. The equilibrated PGP with matrix polymer chains in the
large simulation box is simulated using MD in the NVT ensemble at T
*
= 1 for over
another 1 million-time steps to facilitate equilibration and promote mixing of graft
and matrix chains. Thereafter, the simulation box is gradually compressed till we
achieve the desired box size. This reduction in box size takes over at least 2 milliontime steps for the chosen system described in the results section next. In general, the
desired box size is chosen based on the design parameters of the PNC (i.e., D P , N G ,
and N M ) and the desired total volume packing fraction, η. η is calculated as the
volume of all CG beads in the simulation box divided by the simulation box volume.
We maintain η = 0.367 to model the PNC in melt state. At the chosen η, one could
also simulate a PNC with multiple PGPs at a desired PGP fraction, φ G , calculated as
the volume of the beads that make up the PGP divided by the total occupied volume
within the simulation box, φ G =
V graft +V particle
V graft +V particle +V matrix
, where V graft , V particle and V matrix
represent the volume occupied by graft, particle and matrix beads, respectively.
For the results shown in the next section, we equilibrate the PNCs at the desired
box size at η = 0.367 in NVT ensemble at T
*
= 1 for 5 million-time steps and use
the subsequent configurations after equilibration collected every 10 [5] time steps
to obtain an ensemble of production run trajectories. These configurations are then
analyzed for calculating concentration profiles, brush height and chain conformations
as discussed in the next paragraph.
To determine wetting or the degree of interpenetration (mixing) of matrix chains
into the grafted layer one can calculate monomer concentration profiles of graft (C G )
and matrix (C M ) beads as a function of distance from the particle surface. [155]
Thickness of grafted layer can be quantified by the average brush height H b. Graft
and matrix chain conformations can also be described by the probability distribution
P(R ee ) of end-to-end distance, R ee , or probability distribution P(R
2
g ) of squared
radius of gyration,R
2
g of the graft and matrix chains.
We direct the reader to work by Martin and Jayaraman [123, 124, 156] which
describe detailed simulation protocols for PNCs with single and multiple PGPs; we
note that these past studies did not use the above CG model but focused on simulating
similar systems of PGPs in polymer matrix. Our recent work [154] uses the simulation
protocol of Martin and Jayaraman [123, 124, 156] along with the new CG model
described above to study the effect of directional interactions in PNCs; we present
some of the key results from this recent work here.
A. Jayaraman et al.
length N M . The initial configuration of the PNC is created by first placing a single
polymer grafted particle (PGP) (devoid of matrix chains) at the center of a large
simulation box of size much larger than the final intended simulation box size. First,
to allow the graft chains to relax from the chosen initial configuration, this single
PGP undergoes an MD simulation run in the NVT ensemble at T
*
= 1 where all
interactions (graft-graft and graft-particle) are maintained as purely repulsive via
a WCA potential over 1 million-time steps where each time step is 0.001 σ (
m
ε
)
0.5 .
Next, we add matrix chains around this equilibrated single PGP. The use of a box size
larger than the intended cubic simulation box size ensures minimal overlaps during
the addition of matrix chains. The equilibrated PGP with matrix polymer chains in the
large simulation box is simulated using MD in the NVT ensemble at T
*
= 1 for over
another 1 million-time steps to facilitate equilibration and promote mixing of graft
and matrix chains. Thereafter, the simulation box is gradually compressed till we
achieve the desired box size. This reduction in box size takes over at least 2 milliontime steps for the chosen system described in the results section next. In general, the
desired box size is chosen based on the design parameters of the PNC (i.e., D P , N G ,
and N M ) and the desired total volume packing fraction, η. η is calculated as the
volume of all CG beads in the simulation box divided by the simulation box volume.
We maintain η = 0.367 to model the PNC in melt state. At the chosen η, one could
also simulate a PNC with multiple PGPs at a desired PGP fraction, φ G , calculated as
the volume of the beads that make up the PGP divided by the total occupied volume
within the simulation box, φ G =
V graft +V particle
V graft +V particle +V matrix
, where V graft , V particle and V matrix
represent the volume occupied by graft, particle and matrix beads, respectively.
For the results shown in the next section, we equilibrate the PNCs at the desired
box size at η = 0.367 in NVT ensemble at T
*
= 1 for 5 million-time steps and use
the subsequent configurations after equilibration collected every 10 [5] time steps
to obtain an ensemble of production run trajectories. These configurations are then
analyzed for calculating concentration profiles, brush height and chain conformations
as discussed in the next paragraph.
To determine wetting or the degree of interpenetration (mixing) of matrix chains
into the grafted layer one can calculate monomer concentration profiles of graft (C G )
and matrix (C M ) beads as a function of distance from the particle surface. [155]
Thickness of grafted layer can be quantified by the average brush height H b. Graft
and matrix chain conformations can also be described by the probability distribution
P(R ee ) of end-to-end distance, R ee , or probability distribution P(R
2
g ) of squared
radius of gyration,R
2
g of the graft and matrix chains.
We direct the reader to work by Martin and Jayaraman [123, 124, 156] which
describe detailed simulation protocols for PNCs with single and multiple PGPs; we
note that these past studies did not use the above CG model but focused on simulating
similar systems of PGPs in polymer matrix. Our recent work [154] uses the simulation
protocol of Martin and Jayaraman [123, 124, 156] along with the new CG model
described above to study the effect of directional interactions in PNCs; we present
some of the key results from this recent work here.