Coarse-Grained Modeling and Simulations of Thermoresponsive …
63
In this study, the force constant is kept at k angle = 0 kT/rad
2 and θ o = π radians to
represent the case of fully flexible graft and matrix chains. Non-bonded graft (G) and
matrix (M) monomers interact via LJ [69] potential with σ i j = 1.0d and r cut = 2.0d ,
ε GG = ε MM = 0.5 kT (in reduced energy units) and ε GM is varied from 0.5 to 0.2 kT.
This combination of ε GG , ε GM and ε MM leads to a model parameter χ GM defined as
χ GM =
−ε GM +
ε MM +ε GG
2
taking on values from χ GM = 0 (ε GM = 0.5) to χ G M
= + 0.3 (ε GM = 0.2), i.e., purely entropic graft-matrix interaction to unfavorable
graft-matrix attraction.
Acceptor sites (A) and donor sites (D) are placed on all G and M monomer
beads, respectively, as seen in Fig. 9 to enable h-bonding between graft and matrix
monomers. The size of A and D beads (0.3d) is kept relatively small compared to
G and M beads (d) and they are bonded tightly to their parent G and M beads,
respectively, using harmonic bonds with k bond = 1000 kT/d
2 and r o = 0.37d. A
and D sites are also maintained at a specific angle with respect to their graft/matrix
polymer chains using harmonic angle potentials between A/D beads, their parent G/M
beads and successive G/M beads along that polymer chain with k angle = 50 kT/rad
2
and θ o = π/2 radians. Non-bonded interactions between A-D beads are modeled
using LJ potential with σ AD = 0.3d, r cut = 2σ AD and ε AD = 13 kT. We note that
the value of ε AD can be tuned to mimic the h-bonding interaction strength in any
acceptor–donor pair chemistry, currently, it mimics the maximum strength of OH:N
h-bond pair. The small size of A-D beads relative to their parent G and M beads
along with their placement on parent beads through the chosen harmonic bonded
potential together brings about directionality in A-D interaction. A-A and D-D interactions are modeled as purely repulsive using Weeks-Chandler-Andersen (WCA)
[71] potential with σ AA = σ DD = 2.3 σ AD and r cut = 1.1225 σ AA and ε AA = ε DD =
0.5 kT which increases the preference for A-D interaction over D-D or A-A interaction and prevents the possibility of A-D-A or A-D-D attractive interactions, thereby
making the A-D interaction specific as in the case of hydrogen bonds. In principle, we
could also include dihedral A-G-G-A potentials to model restricted movement and the
torsional penalty the monomers pay as they make the h-bonds. This would be implemented numerically similar to the stacking interactions in the ONA model with force
constants chosen to effectively mimic the torsional restraint in the selected polymer
chemistry. All other pairwise non-bonded interactions which include nanoparticlegraft/matrix beads, acceptor/donor beads-graft/matrix beads, acceptor/donor beadsnanoparticle and also acceptor–donor interaction in the absence of h-bonds (purely
entropic case) are modeled as purely repulsive using WCA potential.
4.3 Method: Simulation and Analyses
Using the above CG model for PNCs, we perform MD simulations in the NVT
ensemble within the LAMMPS [73] package. For the results described in the next
section, we simulate a single polymer grafted nanoparticle of diameter D P with the
desired graft chain length, N G , grafting density, , in a matrix of polymers of chain
63
In this study, the force constant is kept at k angle = 0 kT/rad
2 and θ o = π radians to
represent the case of fully flexible graft and matrix chains. Non-bonded graft (G) and
matrix (M) monomers interact via LJ [69] potential with σ i j = 1.0d and r cut = 2.0d ,
ε GG = ε MM = 0.5 kT (in reduced energy units) and ε GM is varied from 0.5 to 0.2 kT.
This combination of ε GG , ε GM and ε MM leads to a model parameter χ GM defined as
χ GM =
−ε GM +
ε MM +ε GG
2
taking on values from χ GM = 0 (ε GM = 0.5) to χ G M
= + 0.3 (ε GM = 0.2), i.e., purely entropic graft-matrix interaction to unfavorable
graft-matrix attraction.
Acceptor sites (A) and donor sites (D) are placed on all G and M monomer
beads, respectively, as seen in Fig. 9 to enable h-bonding between graft and matrix
monomers. The size of A and D beads (0.3d) is kept relatively small compared to
G and M beads (d) and they are bonded tightly to their parent G and M beads,
respectively, using harmonic bonds with k bond = 1000 kT/d
2 and r o = 0.37d. A
and D sites are also maintained at a specific angle with respect to their graft/matrix
polymer chains using harmonic angle potentials between A/D beads, their parent G/M
beads and successive G/M beads along that polymer chain with k angle = 50 kT/rad
2
and θ o = π/2 radians. Non-bonded interactions between A-D beads are modeled
using LJ potential with σ AD = 0.3d, r cut = 2σ AD and ε AD = 13 kT. We note that
the value of ε AD can be tuned to mimic the h-bonding interaction strength in any
acceptor–donor pair chemistry, currently, it mimics the maximum strength of OH:N
h-bond pair. The small size of A-D beads relative to their parent G and M beads
along with their placement on parent beads through the chosen harmonic bonded
potential together brings about directionality in A-D interaction. A-A and D-D interactions are modeled as purely repulsive using Weeks-Chandler-Andersen (WCA)
[71] potential with σ AA = σ DD = 2.3 σ AD and r cut = 1.1225 σ AA and ε AA = ε DD =
0.5 kT which increases the preference for A-D interaction over D-D or A-A interaction and prevents the possibility of A-D-A or A-D-D attractive interactions, thereby
making the A-D interaction specific as in the case of hydrogen bonds. In principle, we
could also include dihedral A-G-G-A potentials to model restricted movement and the
torsional penalty the monomers pay as they make the h-bonds. This would be implemented numerically similar to the stacking interactions in the ONA model with force
constants chosen to effectively mimic the torsional restraint in the selected polymer
chemistry. All other pairwise non-bonded interactions which include nanoparticlegraft/matrix beads, acceptor/donor beads-graft/matrix beads, acceptor/donor beadsnanoparticle and also acceptor–donor interaction in the absence of h-bonds (purely
entropic case) are modeled as purely repulsive using WCA potential.
4.3 Method: Simulation and Analyses
Using the above CG model for PNCs, we perform MD simulations in the NVT
ensemble within the LAMMPS [73] package. For the results described in the next
section, we simulate a single polymer grafted nanoparticle of diameter D P with the
desired graft chain length, N G , grafting density, , in a matrix of polymers of chain