Supplementary Material
Reducing Conduction Losses In High Energy Density Polymer Using Nanocomposites
Yash Thakur1, Meng H. Lean2, Q.M. Zhang1*
1
School of Electrical Engineering and Computer Science, Materials Research Institute,
The Pennsylvania State University, University Park, PA 16802, USA
2
QEDone LLC, 4174 Marston Lane, Santa Clara, CA 95054, USA
*Contact authors: [email protected]
Figure S1. Dielectric data of neat THV as a function of temperature.
Multiscale Modeling
Multiscale simulations involve both continuum 1D and particle 3D models and use empirical data
together with consistent sets of parameters adjusted for the appropriate dimensional treatments to
predict results suited for comparison with measurement and for cross-validation with each other.
Continuum Model
(a)
Anode
(b)
Computational Cell for Particle Simulation
Anode
∆𝑝 =
THV+Al2O3
𝐽𝐴
∆𝑡
𝑞
Anode
Film
𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑠𝑖𝑑𝑒 𝑤𝑎𝑙𝑙𝑠:
/𝑛 = 0
Dielectric
Fluid
Cathode
Amorphous polymer
Cathode
∆𝑛 =
𝐽𝐴
∆𝑡
𝑞
Cathode
Figure S2: Simulation geometry for: (a) 1D continuum model; (b) 3D particle model.
A. 1D Continuum Model
The drift-diffusion equations are used to simulate bipolar charge transport resulting in the
creation of a robust and rapid hybrid time-dependent algorithm. More details on the continuum 1D
simulation model, including axisymmetric versions for divergent field configurations with gaseous
voids and charge packet formation are available in the literature.1–3 This comprehensive, selfconsistent, simulation method is used for the continuum solution through the nanocomposite THV
film shown in Figure S2(a) assuming an effective permittivity from the Lichtenecker logarithmic
rule:4
𝑙𝑛  = 𝑓 𝑙𝑛 𝜀𝑓 + (1 − 𝑓 ) 𝑙𝑛 𝜀𝑏
(S1)
where εf and εb are, respectively, the permittivities of the filler and polymer binder, and f is the
volume fraction of inorganic filler. Measured I-V data, setup geometry, and measurement
conditions such as Voltage-time diagrams are used to infer J-E curves as input with mobility
derived from Hopping Conduction parameters extracted from the empirical fit. Direct Tunneling
injected charges migrate with hopping mobility and undergo bulk charge trapping/de-trapping, and
trap-assisted Shockley-Read-Hall species recombination. A 4th order total variation diminishing
Runge-Kutta (TVD-RK4) method with upwind differencing is used to integrate the current
continuity and rate equations for trapping, de-trapping, and recombination.
The drift-diffusion, charge conservation, and current continuity equations are given by:
𝐉n = ρ− μn 𝐄 − Dn ∇
(S2)
𝐉p = ρ+ μp 𝐄 + Dp ∇+
−
= ∇𝐉n + Un
t
(S3)
+
= −∇𝐉p + Up
t
∇(∇) = −(+ + − + i )
(S4)
where + and  are the positive and mobile negative charge densities, Up and Un are the
recombination rates, p and n are the positive and negative hopping charge mobilities derived
from measurement, Dp and Dn are the positive and negative charge diffusion coefficients, i is the
intrinsic impurity charge density,  (=or) is the permittivity, and the other notations have the
usual meanings. The mobility and diffusion coefficients are related by the Einstein equation at
equilibrium:
Dn
n
=
kB T
q
=
Dp
μp
(S5)
with kB (=1.38065x1023 m2kg/s2K) as the Boltzmann constant, T the absolute temperature, and q
the Coulomb charge. This set of equations is similar to those used in conventional semiconductor
device modeling where a suitable choice of variables include the natural set: , p, and n with the
latter two being, respectively, the positive and negative charge number densities. Using a Gummellike method, the drift-diffusion equations are decoupled and solved sequentially.5
𝑷=3
1 2 3
𝜀−1
𝑬
𝜀+2
P
E
E
P: Induced polarization
Figure S3: Illustration of mobile charge interaction with dielectric fillers and meandering through
interspaces.
B. 3D Particle Model
In the 3D particle model shown in Figure S2(b), spherical Al2O3 nanofillers are randomly dispersed
in the THV binder. More details on the particle model are available in the literature.6–8 The classical
electrical double layer (EDL) representation is modified for the filler by substitution of a dipolar
core where the induced polarization is aligned with the bias electric field as shown in Figure S3.
Charge motion through the dielectric nanofillers follow a logic where incoming charge is repelled
by the end of the dipole with similar polarity and attracted towards the opposite polarity end and
vice-versa. Initial charge is assumed to attach on impact forming the bound Stern-Helmholtz layer.
Sequential incoming charges are repelled to form the diffuse outer Gouy-Chapman transport layer.
Cumulative charge buildup on opposing ends of the dipole contributes to the Maxwell-WagnerSillars (MWS) polarization. The interaction zone is created by gradual charge deposition and
formation of the diffuse layers as charge migrates while undergoing bulk charge trapping/detrapping, and recombination with Monte Carlo selection through the polymer film. Conduction
paths are formed due to overlap of the diffuse double layers by increase in nanoparticle loading
within the dielectric composite. Charges that make it through the film to the counter-electrode have
trajectories that are curvilinear paths which meander through the interspaces. Arriving charges at
the counter-electrode are neutralized and therefore only contribute to the conduction of the film,
but not to the field.
The electrodynamic simulation of 3D charge particle migration through the nanocomposite
film involves the time-dependent integration of the two following equations of motion:
𝑑𝒗
𝑚 𝑑𝑡 = 𝑞𝑬
(S6)
𝑑𝒙
𝑑𝑡
=𝒗
(S7)
where m and q are the mass and the charge of the particle, v is its velocity and E is the electric
field at the particle location. The equations above are followed in time as they evolve in velocity
and position (phase space) to determine the trajectory of each particle in an ensemble. The driving
term in (S6) is the Coulomb force on the particle with contributions to the E field from point and
dipolar charge, and applied bias. Due to the short acceleration time and small mean free path,
implementation of (S6) and (S7) is typically simplified by assuming that charges migrate with a
drift velocity, v=E, where  is the hopping mobility in the polymer inferred from fits to
measurement. At each time interval, a two-step field solve-particle push procedure solves the
Poisson equation for charge conservation and integrates the equations of motion for particle
simulation in the computational cell. A random distribution of nanofillers is dispersed within the
cell to the prescribed wt% loading using “hard sphere” logic; i.e. allowing contact but no over-lap.
Using this 3D particle-in-cell (PIC) model, charge particle-particle interactions and particlenanofiller attachments are tracked from injection through the amorphous polymer and
nanocomposite layers until arrival at the counter-electrodes. A self-consistent particle-particle,
particle-mesh (P3M) scheme may be implemented for very large numbers of particles.9–11 The
computational cell used for the nano-scale simulations is comprised of sandwiched layers of 500
nm thick nanocomposite between two 250 nm thick amorphous polymer layers. The cuboid of
linear dimension 1 m is bounded by 4 vertical zero flux condition (/n = 0) side walls, and top
(anode) and bottom (cathode) electrodes at assigned potentials to maintain E field bias. Following
the empirical J-E curves, discrete numbers of charge particles are continuously injected from the
electrodes during each time-step from injection locations that are randomly dispersed over the
electrode surface. Periodic boundary conditions are imposed on the 4 vertical walls to force exiting
particles to re-enter at the opposite side walls.
C. Field Solution with the BIEM
A source distribution technique (SDT) based on physically intuitive charge species is used to solve
the inhomogeneous Poisson equation with the boundary integral equation method (BIEM) and the
free space Green function. Instead of deriving a custom geometry and material dependent Green
function, an equivalent problem is solved where conducting and insulating boundaries and material
interfaces are represented by free and bound charge distributions of unknown magnitude that
collectively satisfy all local and far-field boundary conditions. Fields and potentials are defined by
superposing integral contributions from all these source types and their distributions. Once the
charge distributions are ascertained, field parameters are evaluated by superposition of the integral
contributions from all sources. A Green function method is used where G, the free space Green
function, is the fundamental solution to a point charge, or Dirac delta, :
2G = – (r  r’)
(S8)
The solution to the Poisson equation given by
2 = – /
(S9)
in 1D is the free space Green function:
(b−x)(x−a)
(b−a)
; (𝑎 ≤ x < x ′ )
G[xx ′ ] = {(x′ −a)(b−x)
(b−a)
(S10)
; (x′ < x ≤ b)
where x ∈ [a, b] denotes the observer, a and b are the edges of the layer, and x’ is the source
location or distribution. The E field is derived from the potential by:
E = – 
(S11)
with potential, , given by:
(b−x)
𝑥
(b−x)
(x−a)
𝑏
(x−a)
(x) = (b−a) ∫𝑎 (x ′ − a)f(x ′ )dx′ + (b−a) (a) + (b−a) ∫𝑥 (b − x ′ )f(x ′ )dx′ + (b−a) (b)
(S12)
where f(x’) are the trapped and mobile bipolar charge distributions. The normal derivative is given
by:
d(x)
dx
1
𝑥
(a)
1
𝑏
(b)
= − (b−a) ∫𝑎 (x ′ − a)f(x′)dx′ − (b−a) + (b−a) ∫𝑥 (b − x ′ )f(x ′ )dx′ + (b−a)
(S13)
which allow computation of the E field within the layer by integration of the analytically
differentiable kernels. The integrals in (S12) and (S13) are evaluated using numerical integration
by mapping Gauss-Jacobi quadrature into the two partial integrals.12
For the 3D particle model, the composite field due to the bias voltage, space charge, and
polarization charge is given by:
𝑬 = 𝑬𝑏𝑖𝑎𝑠 + 𝑬𝑞 + 𝑬𝑓𝑖𝑙𝑙𝑒𝑟
(S14)
where the second and third terms are due, respectively, to point charges and filler polarization. For
small to moderate particle counts, n, fields from the space charge are computed from superposing
or summing the point source solutions for all particles considered:
1
1
0
0
𝑬𝑞 = − ∫𝑉 ′ ∇𝐺 𝜀 𝜌(𝑣 ′ ) 𝑑𝑉′ − ∑𝑛𝑖=1 ∇𝐺 𝜀 𝑞𝑖
(S15)
with G=1/4r-r’ as the 3D Green function fundamental solution to a point charge as shown in
(S8). The polarization, P, from sphere-shaped dielectric materials in a dielectric medium is:
−1
𝑷 = 3 +2 𝑬
(S16)
where  is the ratio of the permittivity of the sphere to the polymer matrix, and (1)/(+2) is the
Clausius-Mossotti factor which vanishes when the material and medium permittivities are
identical. The material dipole moment is given by:
𝒑=
−1
4
𝜋𝑎3 𝑷 = 4𝜋 +2 𝑎3 𝑬
3
(S17)
and the corresponding dipole field is:
𝑬𝑓𝑖𝑙𝑙𝑒𝑟 = 
1
4𝜀0
(
3(𝒑𝒓)𝒓−𝒑
𝑟3
)
(S18)
where r is the distance of evaluation, a is the nanofiller radius, and (S18) is evaluated at the
centroids and summed over all nanofillers. Ebias arises from the free charge, , distributed on the
electrodes and bound charge on the zero flux side walls. The expressions for potential and normal
derivatives are given by:
(𝑟) = ∫𝑆′ 𝐺
𝜕(𝑠)
𝜕𝑛
σ(𝑟,𝑠′)
𝜀0
𝜕𝐺 σ(𝑠′)
= {∫𝑆′ 𝜕𝑛
𝜀0
1
1
𝒓−𝒓𝒊
0
0
𝒓−𝒓𝑖 
𝑑𝑆 ′ + ∑𝑛𝑖=1 𝐺 𝜀 𝑞𝑖 + ∑𝑚
𝑖=1 𝐺 𝜀 𝒑𝑖 ∙
𝑑𝑆 ′ −
σ(𝑠)
𝜕𝐺 1
𝜕𝐺 1
2𝜀0
0
0
(S19)
3
} + ∑𝑛𝑖=1 𝜕𝑛 𝜀 𝑞𝑖 + ∑𝑚
𝑖=1 𝜕𝑛 𝜀 𝒑𝑖 ∙
𝒓−𝒓𝒊
𝒓−𝒓𝑖 
3
(S20)
Enforcement of boundary conditions for potential and flux result in integral equations for
Dirichlet and Neumann conditions, respectively, which are then solved simultaneously. These
equations incorporate superposition of contributions from sources that include: free charge on
electrodes; bound charge on vertical walls; trapped charge on nanofillers; and mobile and trapped
volume space charge, . The collocation method is used to discretize these equations resulting in
a matrix that is inverted to determine the magnitude of the free and bound source distributions.
Kernel functions, including the Green function and its analytic derivatives (G, G/n, and G),
are integrated numerically using Gauss-Legendre quadrature.12 Singular kernels are accurately
computed using minimum order sampling by tying the quadrature weight function to the
singularity. Details on the computation of these integrals are discussed in detail for axisymmetric
and 2D geometries.8,13 Large portions of the discretized matrices may be saved and reused in
matrix algebra for rapid computation of the sources, or independent variable, .13 With A as
electrode area, and empirical J-E, finite numbers of positive and negative charge particles, n, are
injected at each time-step, t, from randomized locations on the surface of the cathode and the
anode into the respective adjacent polymer according to:
∆𝑛 =
𝐽𝐴
𝑞
∆𝑡
(S21)
For large numbers of particles, n, and spatially varying distributions, a “scatter-gather”
method may be implemented to scatter the space charge, , to qi on the vertices of the inscribed
volume mesh using, for example, trilinear interpolation:
𝜌=
∑8𝑖=1 𝛼𝑖 𝑞𝑖
𝑉
(S22)
where i are the trilinear interpolation functions and V is the mesh volume. This particle-mesh
scheme facilitates integration by gathering the contributions from the vertices, i, of all volume
meshes, j:
1
1
0
0
𝑬 = ∫𝑉 ′ ∇𝐺𝜌(𝑣 ′ ) 𝜀 𝑑𝑉 ′ ≈ ∑𝑘𝑗=1 ∑8𝑖=1 ∇𝐺 𝜀 𝛼𝑖 𝑞𝑖
(S23)
to compute the E field at the  position. This method requires only nk (<<n2) calculations for a
mesh of size k<<n. Following (S22), the current density, J, can be similarly expressed as:
𝑱=
∑𝑖 𝛼𝑖 𝑞𝑖 𝒗𝑖
𝑉
(S24)
with vi (=E) as the drift velocity. Contributions of these volume sources to the E-field are
evaluated as the algebraic summations of the field from each volume mesh as shown in (S23).
D. Time Integration Strategy
A total variation diminishing version of the 4th order Runge-Kutta scheme that guarantees
convergence is used to integrate the current continuity and recombination equations for the
continuum model.14 The upwind scheme which has guaranteed stability is implemented for spatial
differencing.15 The TVD-RK4 scheme is also used for bulk trapping/de-trapping. To illustrate, the
1st order upwind scheme for u/t + au/x=0 is given by:
(un+1
− uni )
(uni − uni−1 )
i
+a
= 0 for a > 0
∆t
∆x
(S25)
(un+1
− uni )
(uni+1 − un )
i
+a
= 0 for a < 0
∆t
∆x
where a is the velocity, u is the independent variable, subscript i refers to spatial grid index, and
superscript n refers to iteration time level. The TVD-RK4 advances temporal integration as:
u(1) = un + ∆t L(un )
(S26)
with u(1) as the first of the 4th order terms, and du/dt = L(u), where L is an operator.16 The total
variation (TV) is given by:
u
TV = ∫ |x | dx ≈ ∑|ui+1 − ui |
(S27)
which integrates the incremental change u/x over the entire range of x, and is a property that
ensures that TV(un+1)  TV(un). The TVD scheme prevents spurious oscillations in the solution by
preserving monotonicity, enabling sharper shock predictions on coarse grids thereby saving
computation time.
The continuum algorithm proceeds through: (1) bipolar current injection; (2) solve Poisson
equation with the BIEM; (3) temporal integration of continuity and recombination equations using
TVD-RK4 and upwind scheme; (4) compute changes in bulk trapping/de-trapping; and (5) update
charge arrays and return to (1). To minimize local error, mesh size, h < LD, the Debye length,
where LD = √k B T⁄q2 Ni , and Ni is the largest charge number density. The time step, t < , the
dielectric relaxation time, where =/qNi, characteristic of charge fluctuations to decay. The
stability criterion of the explicit algorithm is given by the Courant-Friedrichs-Levy (CFL) limit, c:
v∆t
c = | ∆h | ≤ 1
(S28)
representing the ratio of mobile charge velocity, , to mesh velocity, h/t, where for stability, the
mesh velocity cannot be faster than the charge speed.17
For the particle model, the two differential equations given by (S6) and (S7) are solved
numerically using a predictor-corrector method of order t2, or 2nd order accurate in time. The
dependent variable, x, is iteratively improved at each time level. To start at time level n, the set
(xn-1, xn) is assumed to be known. Then, the predictor-corrector equations are:
Predictor:
𝒗𝑛 =  𝑬𝑛
𝒙𝑛+1 = 𝒙𝑛−1 + 2∆𝑡 𝒗𝑛
1
𝒙𝑛−1′ = 𝒙𝑛 + (𝒙𝑛+1 − 𝒙𝑛−1 )
4
Corrector:
(S29)
𝒗𝑛+1 = 𝑬𝑛+1
1
𝒙𝑛+1 = 𝒙𝑛−1′ + 𝑡 𝒗𝑛+1
2
Update:
𝒙𝑛−1 = 𝒙𝑛
𝒙𝑛 = 𝒙𝑛+1
The initial location for the (n-1) iteration level is improved in the predictor stage, and used
in the corrector stage to obtain the updated location for the next time level. The tacit assumptions
are: the field is “frozen-in” relative to the time step of charge migration, and the transient time to
attain terminal velocity is much shorter than the mean free path or the time between collision
events of oppositely charged particles
Charge Transport and Interactions
Bipolar charge transport incurs trapping/de-trapping and recombination.
A. Charge Attachment/Detachment
Trapping and de-trapping of space charge in polymeric materials are related to the microstructure
and morphology of the materials. Charge trapping takes place at a hopping site that requires energy
substantially greater than the average energy to release charge carriers. Trapping mechanisms
include: physical defects such as dangling bonds which lead to shallow traps; “self” traps due to
field modification which alters the length of the polymer chain and their potential well; and
chemical defects or impurities which result in deep traps. De-trapping mechanisms may be:
photon-assisted by illumination; phonon-assisted through lattice vibration; impact ionization; and
tunneling, with the latter two occurring at high fields.
For the continuum model, bulk trapping and de-trapping of bipolar mobile charge are given
by:
t+
ρt+
+
+
= k+
ρ
(1
−
) − k+
t
d ρt
t
∞
(S30)
t−
ρt−
− −
−
= k t ρ (1 − ) − k −
d ρt
t
∞
where the density of trapping states is given by , and kt is the trapping rate:
kt =
J(t)
q
(S31)
with  as the trapping cross-section. The de-trapping rate is given by:
k d = Nc th  e−Ea /kB T
(S32)
where Nc is the effective density of states in the LUMO, vth is the thermal velocity, and Ea is the
trap depth. The trapping and de-trapping time constants are the inverse of the rate coefficients.
For the particle model, attached charge on nanofiller surfaces are considered to be trapped
charge. In this modified EDL model, initial charge is assumed to attach on impact, or probability
Pt(t)=1. Subsequent drop-off has Pt(t)0 as the limiting behavior due to Coulomb force and dipole
field repulsion from build-up of the attached charge following the MWS effect. Charge detachment
is physically controlled by comparing the local field to the trap depth used in the de-trapping rate,
kd, in (S32). Bulk trapping is implicitly assumed in the use of the Hopping Conduction mobility.
B. Recombination
In the continuum model charge recombination is trap-assisted, analogous to the Shockley-ReedHall (SRH) model, involving four possible combinations of positive and negative mobile and
trapped charge.
The recombination equations are given by:
Up = S2 + t− + S3 + −
(S33)
Un = S1 − +
+ S 3 − +
t
where S1, S2, and S3 are recombination coefficients for mobile negative charge and trapped positive
charge, mobile positive charge and trapped negative charge, and mobile positive and negative
charge, respectively.
For the particle model, Monte Carlo collision is used to describe particle-particle events
between oppositely charged entities resulting in recombination. The probability of collision of the
ith charge particle with an opposite polarity particle in a time step t is given by:
𝑃𝑖 = 1 − 𝑒 −𝑛0 (𝐸𝑖 )𝑣𝑖 ∆𝑡
(S34)
where n0 is the number density of the opposite polarity mobile charge, (Ei) is the total collision
cross-section which in general depends on the kinetic energy of the ith particle (and could be larger
than the geometrical cross-section given by  = 𝜋(𝑟𝑖 + 𝑟𝑗 )2 ), and vi is the relative velocity of this
particle to the opposite polarity particle. Physically, no represents the number of collisions per
unit length, 1/no is the mean free path between collisions, and novi is the collision frequency.
The probability of a collision within a finite t, is determined by comparing Pi with Ri[0,1], a
uniformly generated random number. Monte Carlo selection returns a non-event unless Pi > Ri,
when particle i is considered to have sustained a collision within the time step t resulting in
recombination and neutralization of the charge pair.
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