Grafting polymer chains to the surface of nanoparticles overcomes the challenge of nanoparticle dispersion within nanocomposites and establishes high-volume fractions that are found to enable enhanced material mechanical properties. This study utilizes coarse-grained molecular dynamics simulations to quantify how the shear modulus of polymer-grafted nanoparticle (PGN) systems in their glassy state depends on parameters such as strain rate, nanoparticle size, grafting density, and chain length. The results are interpreted through further analysis of the dynamics of chain conformations and volume fraction arguments. The volume fraction of nanoparticles is found to be the most influential variable in deciding the shear modulus of PGN systems. A simple rule of mixture is utilized to express the monotonic dependence of shear modulus on the volume fraction of nanoparticles. Due to the reinforcing effect of nanoparticles, shortening the grafted chains results in a higher shear modulus in PGNs, which is not seen in linear systems. These results offer timely insight into calibrating molecular design parameters for achieving the desired mechanical properties in PGNs.

1.
M. J.
Hore
,
L. T.
Korley
, and
S. K.
Kumar
, “
Polymer-grafted nanoparticles
,”
J. Appl. Phys.
128
,
030401
(
2020
).
2.
S. K.
Kumar
,
B. C.
Benicewicz
,
R. A.
Vaia
, and
K. I.
Winey
, “
50th anniversary perspective: Are polymer nanocomposites practical for applications?
,”
Macromolecules
50
,
714
731
(
2017
).
3.
S. K.
Kumar
,
N.
Jouault
,
B.
Benicewicz
, and
T.
Neely
, “
Nanocomposites with polymer grafted nanoparticles
,”
Macromolecules
46
,
3199
3214
(
2013
).
4.
S. K.
Kumar
and
R.
Krishnamoorti
, “
Nanocomposites: Structure, phase behavior, and properties
,”
Annu. Rev. Chem. Biomol. Eng.
1
,
37
58
(
2010
).
5.
A. C.
Balazs
,
T.
Emrick
, and
T. P.
Russell
, “
Nanoparticle polymer composites: Where two small worlds meet
,”
Science
314
,
1107
1110
(
2006
).
6.
Z.
Wu
,
S.
Pal
, and
S.
Keten
, “
Implicit chain particle model for polymer-grafted nanoparticles
,”
Macromolecules
56
,
3259
3271
(
2023
).
7.
D.
Dukes
,
Y.
Li
,
S.
Lewis
,
B.
Benicewicz
,
L.
Schadler
, and
S. K.
Kumar
, “
Conformational transitions of spherical polymer brushes: Synthesis, characterization, and theory
,”
Macromolecules
43
,
1564
1570
(
2010
).
8.
N. J.
Fernandes
,
H.
Koerner
,
E. P.
Giannelis
, and
R. A.
Vaia
, “
Hairy nanoparticle assemblies as one-component functional polymer nanocomposites: Opportunities and challenges
,”
MRS Commun.
3
,
13
29
(
2013
).
9.
W. R.
Lenart
and
M. J.
Hore
, “
Structure–property relationships of polymer-grafted nanospheres for designing advanced nanocomposites
,”
Nano-Struct. Nano-Objects
16
,
428
440
(
2018
).
10.
K.
Ohno
,
T.
Morinaga
,
S.
Takeno
,
Y.
Tsujii
, and
T.
Fukuda
, “
Suspensions of silica particles grafted with concentrated polymer brush: Effects of graft chain length on brush layer thickness and colloidal crystallization
,”
Macromolecules
40
,
9143
9150
(
2007
).
11.
J.
Midya
,
M.
Rubinstein
,
S. K.
Kumar
, and
A.
Nikoubashman
, “
Structure of polymer-grafted nanoparticle melts
,”
ACS Nano
14
,
15505
15516
(
2020
).
12.
N. K.
Hansoge
,
A.
Gupta
,
H.
White
,
A.
Giuntoli
, and
S.
Keten
, “
Universal relation for effective interaction between polymer-grafted nanoparticles
,”
Macromolecules
54
,
3052
3064
(
2021
).
13.
N. K.
Hansoge
and
S.
Keten
, “
Effect of polymer chemistry on chain conformations in hairy nanoparticle assemblies
,”
ACS Macro Lett.
8
,
1209
1215
(
2019
).
14.
F.
Müller-Plathe
, “
Coarse-graining in polymer simulation: From the atomistic to the mesoscopic scale and back
,”
ChemPhysChem
3
,
754
769
(
2002
).
15.
N. K.
Hansoge
,
T.
Huang
,
R.
Sinko
,
W.
Xia
,
W.
Chen
, and
S.
Keten
, “
Materials by design for stiff and tough hairy nanoparticle assemblies
,”
ACS Nano
12
,
7946
7958
(
2018
).
16.
J. G.
Ethier
and
L. M.
Hall
, “
Structure and entanglement network of model polymer-grafted nanoparticle monolayers
,”
Macromolecules
51
,
9878
9889
(
2018
).
17.
J. G.
Ethier
,
L. F.
Drummy
,
R. A.
Vaia
, and
L. M.
Hall
, “
Uniaxial deformation and crazing in glassy polymer-grafted nanoparticle ultrathin films
,”
ACS Nano
13
,
12816
12829
(
2019
).
18.
P.
Akcora
,
H.
Liu
,
S. K.
Kumar
,
J.
Moll
,
Y.
Li
,
B. C.
Benicewicz
,
L. S.
Schadler
,
D.
Acehan
,
A. Z.
Panagiotopoulos
,
V.
Pryamitsyn
et al, “
Anisotropic self-assembly of spherical polymer-grafted nanoparticles
,”
Nat. Mater.
8
,
354
359
(
2009
).
19.
T.
Lafitte
,
S. K.
Kumar
, and
A. Z.
Panagiotopoulos
, “
Self-assembly of polymer-grafted nanoparticles in thin films
,”
Soft Matter
10
,
786
794
(
2014
).
20.
G. D.
Hattemer
and
G.
Arya
, “
Viscoelastic properties of polymer-grafted nanoparticle composites from molecular dynamics simulations
,”
Macromolecules
48
,
1240
1255
(
2015
).
21.
T.-Y.
Tang
and
G.
Arya
, “
Anisotropic three-particle interactions between spherical polymer-grafted nanoparticles in a polymer matrix
,”
Macromolecules
50
,
1167
1183
(
2017
).
22.
K.
Kremer
and
G. S.
Grest
, “
Dynamics of entangled linear polymer melts: A molecular-dynamics simulation
,”
J. Chem. Phys.
92
,
5057
5086
(
1990
).
23.
H.
Chao
and
R. A.
Riggleman
, “
Effect of particle size and grafting density on the mechanical properties of polymer nanocomposites
,”
Polymer
54
,
5222
5229
(
2013
).
24.
M.
Giovino
,
J.
Pribyl
,
B.
Benicewicz
,
S.
Kumar
, and
L.
Schadler
, “
Linear rheology of polymer nanocomposites with polymer-grafted nanoparticles
,”
Polymer
131
,
104
110
(
2017
).
25.
C.
Chevigny
,
F.
Dalmas
,
E.
Di Cola
,
D.
Gigmes
,
D.
Bertin
,
F.
Boué
, and
J.
Jestin
, “
Polymer-grafted-nanoparticles nanocomposites: Dispersion, grafted chain conformation, and rheological behavior
,”
Macromolecules
44
,
122
133
(
2011
).
26.
J. F.
Moll
,
P.
Akcora
,
A.
Rungta
,
S.
Gong
,
R. H.
Colby
,
B. C.
Benicewicz
, and
S. K.
Kumar
, “
Mechanical reinforcement in polymer melts filled with polymer grafted nanoparticles
,”
Macromolecules
44
,
7473
7477
(
2011
).
27.
H.
Li
,
R.
Ma
,
W.
Zhang
,
S.
Hu
,
X.
Zhao
,
L.
Zhang
, and
Y.
Gao
, “
Stress overshoot behavior in polymer nanocomposites filled with spherical nanoparticles under steady shear flow via molecular dynamics simulation
,”
Mater. Today Commun.
35
,
105573
(
2023
).
28.
D.
Maillard
,
S. K.
Kumar
,
B.
Fragneaud
,
J. W.
Kysar
,
A.
Rungta
,
B. C.
Benicewicz
,
H.
Deng
,
L. C.
Brinson
, and
J. F.
Douglas
, “
Mechanical properties of thin glassy polymer films filled with spherical polymer-grafted nanoparticles
,”
Nano Lett.
12
,
3909
3914
(
2012
).
29.
A. P.
Thompson
,
H. M.
Aktulga
,
R.
Berger
,
D. S.
Bolintineanu
,
W. M.
Brown
,
P. S.
Crozier
,
P. J.
in’t Veld
,
A.
Kohlmeyer
,
S. G.
Moore
,
T. D.
Nguyen
,
R.
Shan
,
M. J.
Stevens
,
J.
Tranchida
,
C.
Trott
, and
S. J.
Plimpton
, “
LAMMPS - A flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales
,”
Comput. Phys. Commun.
271
,
108171
(
2022
).
30.
G. S.
Grest
and
K.
Kremer
, “
Molecular dynamics simulation for polymers in the presence of a heat bath
,”
Phys. Rev. A
33
,
3628
(
1986
).
31.
G. S.
Grest
,
M.
Pütz
,
R.
Everaers
, and
K.
Kremer
, “
Stress–strain relation of entangled polymer networks
,”
J. Non-Cryst. Solids
274
,
139
146
(
2000
).
32.
E. R.
Duering
,
K.
Kremer
, and
G. S.
Grest
, “
Dynamics of model networks: The role of the melt entanglement length
,”
Macromolecules
26
,
3241
3244
(
1993
).
33.
V.
Goel
,
J.
Pietrasik
,
H.
Dong
,
J.
Sharma
,
K.
Matyjaszewski
, and
R.
Krishnamoorti
, “
Structure of polymer tethered highly grafted nanoparticles
,”
Macromolecules
44
,
8129
8135
(
2011
).
34.
V. V.
Ginzburg
, “
Modeling the morphology and phase behavior of one-component polymer-grafted nanoparticle systems
,”
Macromolecules
50
,
9445
9455
(
2017
).
35.
H.
Yun
,
J. W.
Yu
,
Y. J.
Lee
,
J.-S.
Kim
,
C. H.
Park
,
C.
Nam
,
J.
Han
,
T.-Y.
Heo
,
S.-H.
Choi
,
D. C.
Lee
et al, “
Symmetry transitions of polymer-grafted nanoparticles: Grafting density effect
,”
Chem. Mater.
31
,
5264
5273
(
2019
).
36.
H.
Zhang
,
W.
Wang
,
M.
Akinc
,
S.
Mallapragada
,
A.
Travesset
, and
D.
Vaknin
, “
Assembling and ordering polymer-grafted nanoparticles in three dimensions
,”
Nanoscale
9
,
8710
8715
(
2017
).
37.
Y. N.
Pandey
,
G. J.
Papakonstantopoulos
, and
M.
Doxastakis
, “
Polymer/nanoparticle interactions: Bridging the gap
,”
Macromolecules
46
,
5097
5106
(
2013
).
38.
Y.
Zhu
,
A.
Giuntoli
,
W.
Zhang
,
Z.
Lin
,
S.
Keten
,
F. W.
Starr
, and
J. F.
Douglas
, “
The effect of nanoparticle softness on the interfacial dynamics of a model polymer nanocomposite
,”
J. Chem. Phys.
157
,
094901
(
2022
).
39.
M. J.
Hamer
,
B. V.
Iyer
,
V. V.
Yashin
,
T.
Kowalewski
,
K.
Matyjaszewski
, and
A. C.
Balazs
, “
Modeling polymer grafted nanoparticle networks reinforced by high-strength chains
,”
Soft Matter
10
,
1374
1383
(
2014
).
40.
H.
Kamberaj
,
R.
Low
, and
M.
Neal
, “
Time reversible and symplectic integrators for molecular dynamics simulations of rigid molecules
,”
J. Chem. Phys.
122
,
224114
(
2005
).
41.
S. K.
Sukumaran
,
G. S.
Grest
,
K.
Kremer
, and
R.
Everaers
, “
Identifying the primitive path mesh in entangled polymer liquids
,”
J. Polym. Sci., Part B: Polym. Phys.
43
,
917
933
(
2005
).
42.
T.
Vettorel
and
K.
Kremer
, “
Development of entanglements in a fully disentangled polymer melt
,”
Macromol. Theory Simul.
19
,
44
56
(
2010
).
43.
W. B.
Lee
and
K.
Kremer
, “
Entangled polymer melts: Relation between plateau modulus and stress autocorrelation function
,”
Macromolecules
42
,
6270
6276
(
2009
).
44.
R. S.
Hoy
and
G. S.
Grest
, “
Entanglements of an end-grafted polymer brush in a polymeric matrix
,”
Macromolecules
40
,
8389
8395
(
2007
).
45.
N. C.
Karayiannis
,
V. G.
Mavrantzas
, and
D. N.
Theodorou
, “
A novel Monte Carlo scheme for the rapid equilibration of atomistic model polymer systems of precisely defined molecular architecture
,”
Phys. Rev. Lett.
88
,
105503
(
2002
).
46.
S. W.
Sides
,
G. S.
Grest
,
M. J.
Stevens
, and
S. J.
Plimpton
, “
Effect of end-tethered polymers on surface adhesion of glassy polymers
,”
J. Polym. Sci., Part B: Polym. Phys.
42
,
199
208
(
2004
).
47.
R.
Auhl
,
R.
Everaers
,
G. S.
Grest
,
K.
Kremer
, and
S. J.
Plimpton
, “
Equilibration of long chain polymer melts in computer simulations
,”
J. Chem. Phys.
119
,
12718
12728
(
2003
).
48.
W.
Tao
,
J.
Shen
,
Y.
Chen
,
J.
Liu
,
Y.
Gao
,
Y.
Wu
,
L.
Zhang
, and
M.
Tsige
, “
Strain rate and temperature dependence of the mechanical properties of polymers: A universal time-temperature superposition principle
,”
J. Chem. Phys.
149
,
044105
(
2018
).
49.
J.
Wang
and
T.
Ge
, “
Crazing reveals an entanglement network in glassy ring polymers
,”
Macromolecules
54
,
7500
7511
(
2021
).
50.
J.
Rottler
and
M. O.
Robbins
, “
Growth, microstructure, and failure of crazes in glassy polymers
,”
Phys. Rev. E
68
,
011801
(
2003
).
51.
D.
Meng
,
S. K.
Kumar
,
T.
Ge
,
M. O.
Robbins
, and
G. S.
Grest
, “
Crazing of nanocomposites with polymer-tethered nanoparticles
,”
J. Chem. Phys.
145
,
094902
(
2016
).
52.
M. J.
Stevens
, “
Manipulating connectivity to control fracture in network polymer adhesives
,”
Macromolecules
34
,
1411
1415
(
2001
).
53.
T.
Ge
,
M. O.
Robbins
,
D.
Perahia
, and
G. S.
Grest
, “
Healing of polymer interfaces: Interfacial dynamics, entanglements, and strength
,”
Phys. Rev. E
90
,
012602
(
2014
).
54.
T.
Ge
,
G. S.
Grest
, and
M. O.
Robbins
, “
Tensile fracture of welded polymer interfaces: Miscibility, entanglements, and crazing
,”
Macromolecules
47
,
6982
6989
(
2014
).
55.
T.
Ge
,
G. S.
Grest
, and
M. O.
Robbins
, “
Structure and strength at immiscible polymer interfaces
,”
ACS Macro Lett.
2
,
882
886
(
2013
).
56.
T.
Ge
,
F.
Pierce
,
D.
Perahia
,
G. S.
Grest
, and
M. O.
Robbins
, “
Molecular dynamics simulations of polymer welding: Strength from interfacial entanglements
,”
Phys. Rev. Lett.
110
,
098301
(
2013
).
57.
M. J.
Stevens
, “
Interfacial fracture between highly cross-linked polymer networks and a solid surface: Effect of interfacial bond density
,”
Macromolecules
34
,
2710
2718
(
2001
).
58.
J.
Choi
,
C. M.
Hui
,
M.
Schmitt
,
J.
Pietrasik
,
S.
Margel
,
K.
Matyjazsewski
, and
M. R.
Bockstaller
, “
Effect of polymer-graft modification on the order formation in particle assembly structures
,”
Langmuir
29
,
6452
6459
(
2013
).
59.
P. J.
Daivis
and
B.
Todd
, “
A simple, direct derivation and proof of the validity of the SLLOD equations of motion for generalized homogeneous flows
,”
J. Chem. Phys.
124
,
194103
(
2006
).
60.
A.
Giuntoli
,
N. K.
Hansoge
,
A.
van Beek
,
Z.
Meng
,
W.
Chen
, and
S.
Keten
, “
Systematic coarse-graining of epoxy resins with machine learning-informed energy renormalization
,”
npj Comput. Mater.
7
,
168
(
2021
).
61.
U.
Gurel
and
A.
Giuntoli
, “
Shear thinning from bond orientation in model unentangled bottlebrush polymer melts
,”
Macromolecules
56
,
5708
5717
(
2023
).
62.
A.
Giuntoli
and
S.
Keten
, “
Tuning star architecture to control mechanical properties and impact resistance of polymer thin films
,”
Cell Rep. Phys. Sci.
2
,
100596
(
2021
).
63.
T.-H.
Li
,
V.
Yadav
,
J. C.
Conrad
, and
M. L.
Robertson
, “
Effect of dispersity on the conformation of spherical polymer brushes
,”
ACS Macro Lett.
10
,
518
524
(
2021
).
64.
W.
Hu
,
Polymer Physics: A Molecular Approach
(
Springer Science & Business Media
,
2012
).
65.
A.
Stukowski
, “
Visualization and analysis of atomistic simulation data with OVITO–the open visualization tool
,”
Modell. Simul. Mater. Sci. Eng.
18
,
015012
(
2009
).
66.
X.
Huang
and
C. B.
Roth
, “
Optimizing the grafting density of tethered chains to alter the local glass transition temperature of polystyrene near silica substrates: The advantage of mushrooms over brushes
,”
ACS Macro Lett.
7
,
269
274
(
2018
).
67.
W.
Xia
,
J.
Song
,
N. K.
Hansoge
,
F. R.
Phelan
, Jr.
,
S.
Keten
, and
J. F.
Douglas
, “
Energy renormalization for coarse-graining the dynamics of a model glass-forming liquid
,”
J. Phys. Chem. B
122
,
2040
2045
(
2018
).
68.
A.
Wang
,
F.
Vargas-Lara
,
J. M.
Younker
,
K. A.
Iyer
,
K. R.
Shull
, and
S.
Keten
, “
Quantifying chemical composition and cross-link effects on EPDM elastomer viscoelasticity with molecular dynamics
,”
Macromolecules
54
,
6780
6789
(
2021
).
69.
T.
Prisk
,
M.
Tyagi
, and
P.
Sokol
, “
Dynamics of small-molecule glass formers confined in nanopores
,”
J. Chem. Phys.
134
,
114506
(
2011
).
70.
W.
Nie
,
J. F.
Douglas
, and
W.
Xia
, “
Competing effects of molecular additives and cross-link density on the segmental dynamics and mechanical properties of cross-linked polymers
,”
ACS Eng. Au
3
,
512
526
(
2023
).
71.
J.
Midya
,
Y.
Cang
,
S. A.
Egorov
,
K.
Matyjaszewski
,
M. R.
Bockstaller
,
A.
Nikoubashman
, and
G.
Fytas
, “
Disentangling the role of chain conformation on the mechanics of polymer tethered particle materials
,”
Nano Lett.
19
,
2715
2722
(
2019
).
72.
W.
Voigt
, “
Ueber die beziehung zwischen den beiden elasticitätsconstanten isotroper körper
,”
Ann. Phys.
274
,
573
587
(
1889
).
73.
M.
Jhalaria
,
Y.
Cang
,
Y.
Huang
,
B.
Benicewicz
,
S. K.
Kumar
, and
G.
Fytas
, “
Unusual high-frequency mechanical properties of polymer-grafted nanoparticle melts
,”
Phys. Rev. Lett.
128
,
187801
(
2022
).

Supplementary Material

You do not currently have access to this content.