Computational modelling for performance
improvement of polymer nanocomposites
Łukasz Figiel
International Institute for Nanocomposites
Manufacturing (IINM), WMG, University of Warwick
http://www2.warwick.ac.uk/fac/sci/wmg/research/multifunctional_systems/lfigiel
Warwick Centre for Predictive Modelling Seminar
28 May 2015
Ma et al. (2010), 41: 1345, Comp. Part A.
IINM Research
Understanding Materials at the Nanoscale &
Linking with Macroscale
Dispersion &
distribution;
Polymernanoparticle
interactions
Synthesis,
functionalisation,
defects
Controlled manufacturability at Scale
Tracking
morphology in
primary &
secondary
processing
In-situ
characterisation
& modelling
Delivering Material to the End User
IINM Team
Prof Tony McNally
Director
Polymer Science,
Processing,
Functional Materials
Dr Chaoying Wan
Polymer
Chemistry,
Graphene,
Biomimetics
Dr Claire Dancer
Electromagnetic Materials,
Ceramics
Dr Lukasz Figiel
Materials
Modelling,
Mechanics
Dr Vannessa Goodship
Polymer Processing
Dr Tara Schiller
Polymer
Characterisation,
Biomaterials
Modelling Methodology: overview
Polymer constitutive law
Interface/interphase
constitutive law
TEM, XRD, AFM
RVE
Morphology
reconstruction:
1 m
Homogenisation
MACRO
Nanofiller properties
Localisation
Reconstruction of initial morphology
1μm
30
30
25
25
20
20
Count
Count
ψ
ψ
ψ
Digital reconstruction
Experiment
TEM/XRD
15
  662.731
15
10
10
5
5
0
  0.989
  3.915
0
2
4
6
8
10
Number of clay platelets
12
14
2
4
6
8
10
12
Number of clay platelets
Acceptance-rejection algorithm
• Draw samples (e.g. orientation) from a given distribution
(e.g. skew-Gaussian)
• Intersection/overlap checks
• Ensures global/local periodicity
• Implementation: Fortran (2D) and Python (3D)
14
Examples of 2D and 3D models
2D models
3D models
Constitutive model: polymer matrix
Stress-strain behaviour around Tg
- entangled polymer
Glass-rubber model
‘conformational’ (C)
‘bond-stretching’ (B)
Inter/intraatomic
potentials
  ˆ B  ˆ C   m 1
Total Cauchy stress tensor:
Bond-stretching (B) stress tensor:
Conformational (C) stress tensor:
Stress-induced crystallisation:
Entangled
network
ˆ B 
ˆ C(i ) 
ˆ B
 2GB Dˆ
B
where
where
 m  K B ln J
 B   B aT , aS , a 
1  ˆ (i ) AC 1 3 ˆ ( k ) AC 
AC = AC ˆ (Ni ) ,T, Ns , , 
 N ˆ (i )   N

where
(
k
)

J
N 3 k 1
ˆN 

ˆ C  ˆ C,crit
where

ˆ C,crit  ˆ C,crit aT , as(S) , Dˆ


Constitutive model: interface/interphase/gallery
1
 i   ref aT aS a
2
 H
aT  exp 
 R
1
 1



INT
  T Tref

 
 
 CV

C
aS  exp 
 INT V

 T f  T Tref  T 
Qiao et al. (2011), 49:740, J. Pol. Phys. B
 (cohesive stress)
Clay platelet
tan  M (t , T
INT
ref
)

EM" (t , TrefINT )
EM' (t , TrefINT )
   max f ( )
max
(I)
(II) Slippage of platelets
(III)
 (slippage)
1
(I): Network stiffness
2
0
(III): Network failure
 
(I) f ( )  2   
1  1 
2
if   1 ;
(II) f ( )  1
if 1    2
3
2
   2 
   2 
(III) f ( )  2 
  3
 1






 0 2
 0 2
if 1     2 ,
Scale transitions: macro-to-RVE & RVE-to-macro
Mechanical field
uRVE  x    macro X  u f  x 
 macro  Fmacro  1
-
 macro 
1
 


 RVE x dVRVE
VRVE
VRVE

Non-mechanical field
ref
 RVE  x    RVE
 macromacro  x   f  x 
qRVE 
1
VRVE V 
RVE
qRVE  x  dVRVE
Tangent operator

C macro  macro
 macro
1
t  x d  RVE
VRVE
 RVE
N
=
1
VRVE
f (I )  x( I )
I =1
Case study 1: Secondary processing near Tg
After melt extrusion
After solid-state forming
Morphology evolution
1 m
2 m
Experiment
Viscous flow
Stress-induced crystallisation
Simulation
Morphology evolution with applied deformation (T=100C)
Unstretched
TEM image analysis
Unstretched
=0°
Strain ~1.5
Average tactoid orientation [degrees]
(Applied) Strain ~1
60
1% 2PLAT
1% 8PLAT
5% 2PLAT
50
40
30
20
10
0
0.5
1
Applied strain
1.5
Strain ~2
Nanocomposite sheet during thermoforming (1000C, 1s-1)
Macro model
CPE4
L
4
1
1
3
2
2
3
L
5L
4
4
1
5
3
2
Case study 2: Rate-dependent response
• CFRP laminates under impact loading
Opening of matrix cracks (brittle nature of epoxy)
→ delamination at the interfaces, structural degradation
• Hypothesis: CNT/epoxy surface coating will enhance the impact
resistance due to an increase in energy dissipation/absorption.
(2)
pull-out in
form of CNT
crack bridging
(1) CNT
an enhanced
nonlinear
deformation
of the matrix
σ22
Force
Matrix behaviour
MWCNT properties
Strain energy function
Elastic constants
E33zig-zag 
 31zig-zag 
4 3Cr
 rCNT / 2  9  3  C r
2
r 0

3  C r

/ C /  22  

 /  2 

Edwards & Vilgis (1986), 27: 483, Polymer
1  Cr r0 / C /  22 
r 0
2
2
/ C
G31zig-zag 
2
8 3n sin  / 2n  Cr
2
2 

AC = AC ˆ (Ni ) , T, NC , 
2
 rCNT / 2   2  6  Cr r02 / C 
Rejuvenation

 v
Tf  Tf 0  (Tf  Tf 0 ) 1  exp   v
 0

Buckley et al. (2004), 52: 2355, J. Mech. Phys. Sol.
14  12cos  / n   2cos 2  / n 
10  4cos  / n   6cos 2  / n 
Shen and Li (2004) Phys. Rev. B.


 
Adiabatic heating

cTf
1
 : D   b : De  
T
c
c

Buckley et al. (2004), 52: 2355, J. Mech. Phys. Sol.
MWCNTs parameters
Values of model parameters
Weidt & Figiel (2015), 115: 52, Comp. Sci. Techn.
e.g. Cr bond-stretching constant; Cθ bond-angle variation
constant; n - chirality
Weidt & Figiel (2015), 115: 52, Comp. Sci. Techn.
Interface behaviour
CNT pull-out via MD
Tangential traction-separation law
II
max
F

2.7 MPa
E II
x II
II
Fmax
S 
 dl
Li et al. (2011), 50: 1854, Comp. Mat. Sci
.
Weidt & Figiel (2015), 115: 52, Comp. Sci. Techn.
and
Fshear (a)  4
a


 12
6
12

6
2
1/3 2 
2
1/3 2 6
2
1/3 2 3 
a  2   (a  2  )
(a  2  ) 
Weidt & Figiel (2015), 115: 52, Comp. Sci. Techn.
max
Di  Fshear
(a)   0.225nm
RVE size:
Ensemble size:
Weidt & Figiel (2014), 82: 298, Comp. Mat. Sci.
Stress-strain rate response, and related parameters
Weidt & Figiel (2015), 115: 52, Comp. Sci. Techn.
Energy absorption
Quasi-static rate (110^-3 s^-1)
Impact rate (110^3 s^-1)
Weidt & Figiel (2013), In. Njuguna (Ed.), Structural Nanocomposites, 207-224, Springer, 2013.
Concluding remarks
• Computationally-efficient & accurate, multiscale approach
can assist in the optimisation of processing and property
enhancements for polymer nanocomposites
• Further work ongoing on:
– Linking with molecular simulations to account more accurately for
nanoparticle functionalization & nanoparticle-polymer interactions
– Description of nanoparticle functionalization-related uncertainty
– Computational efficiency enhancement for localisationhomogenisation scheme through model reduction and parallel
processing
– Incorporation of non-mechanical fields (e.g. thermal, electric) into the
scheme
Acknowledgements
•
•
•
•
•
Dr. C. Pisano, Dr. D. Weidt (University of Limerick, Ireland)
Prof. P. Buckley (University of Oxford, UK)
Prof. F. Dunne (Imperial College London, UK)
Dr. P. Spencer (University of Bradford, UK)
Dr. G. Menary (Queen’s University Belfast, UK), Dr. K. Soon, Dr. R. Rajeev
(formerly at Queen’s University Belfast, UK)
• Prof. A. Galeski (CMMS Lodz, Poland)
•
•
•
•
•
Engineering and Physical Sciences Research Council (EPSRC), UK
Materials & Surface Science Institute, University of Limerick, Ireland
Irish Research Council (IRC)
Irish Centre for High-End Computing (ICHEC)
National Science Centre, Poland