RESEARCH ARTICLE | JUNE 16 2022
Electrical and thermal percolation in two-phase materials: A
perspective 
Special Collection: Non-Invasive and Non-Destructive Methods and Applications Part I — Festschrift
I. Y. Forero-Sandoval
J. A. Ramírez-Rincón
; A. P. Franco-Bacca ; F. Cervantes-Álvarez
; J. Ordonez-Miranda ; J. J. Alvarado-Gil 
; C. L. Gómez-Heredia
;
Journal of Applied Physics 131, 230901 (2022)
https://doi.org/10.1063/5.0091291
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PERSPECTIVE
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Electrical and thermal percolation in two-phase
materials: A perspective
Cite as: J. Appl. Phys. 131, 230901 (2022); doi: 10.1063/5.0091291
Submitted: 14 March 2022 · Accepted: 23 May 2022 ·
Published Online: 16 June 2022
I. Y. Forero-Sandoval,1
J. A. Ramírez-Rincón,3
View Online
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A. P. Franco-Bacca,2
F. Cervantes-Álvarez,2,a)
C. L. Gómez-Heredia,2
4
2,b)
J. Ordonez-Miranda,
and J. J. Alvarado-Gil
AFFILIATIONS
1
Institut Pprime, CNRS, Université de Poitiers, ISAE-ENSMA, Futuroscope Chasseneuil F-86962, France
2
Applied Physics Department, CINVESTAV-IPN, Mérida, Yucatán C.P. 97310, México
3
Grupo de Fisicoquímica y Análisis Matemático, Facultad de Ciencias y Humanidades, Fundación Universidad de América,
Avda. Circunvalar No. 20-53, Bogotá DC, Colombia
4
LIMMS, CNRS-IIS UMI 2820, The University of Tokyo, Tokyo 153-8505, Japan
Note: This paper is part of the Special Topic on Non-Invasive and Non-Destructive Methods and Applications Part I: Festschrift.
fernando.cervantes@cinvestav.mx
a)
b)
Author to whom correspondence should be addressed: juan.alvarado@cinvestav.mx
Electrical percolation in two-phase materials involves a very singular behavior, manifested as a huge change in the electrical conductivity,
for a given volume or mass fraction of the phase with higher conductivity. In contrast, in the case of heat transfer, in two-phase composite
systems, analogous percolative phenomena are far more elusive and have been rather difficult to observe in various physical systems. In this
Perspective, we present a critical analysis of experimental results and the application of theoretical models aimed to study the effects of percolation phenomena on the thermal and electrical properties of two-phase materials. Our attention will be focused on composites made of
high conductivity particles in a polymeric matrix. The effect of several factors, such as the geometrical and physical characteristics of fillers
and their connectivity with the matrix, the proportion between the conductivity of filler and the matrix, as well as the crucial role of interfacial thermal resistance, is considered. In particular, the differences between the thermal and electrical thresholds and the physical and geometrical conditions that should be fulfilled to observe thermal percolation are discussed. Future trends, to be followed in the development of
new materials, in order to enhance the thermal conductivity as well as in making the thermal percolative effects notable, based on including
additional phases and 2D fillers, are also discussed.
08 August 2023 20:26:24
ABSTRACT
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0091291
I. INTRODUCTION
Percolation is a concept commonly used to describe how a
fluid spreads or diffuses inside a medium.1 This term refers to a
fluid going through a solid material from one end to the other,
crossing the existing pores in its structure.2 Percolation is also used
to describe diverse phenomena due to similarities in their conceptualizations, such as diffusion of solutes in a solvent and spread of
diseases in communities, as well as flux of electrons, phonons, etc.,
inside a material. The diffusion of a species through a medium can
be random or driven by external forces, as the disease spreads
inside a community due to the mobility of their individuals or the
J. Appl. Phys. 131, 230901 (2022); doi: 10.1063/5.0091291
Published under an exclusive license by AIP Publishing
flux of water produced by gravity or capillary forces, respectively.3
One of the most interesting transport processes is related to heat
flux, exhibiting a time and space dependence that can be treated
using formalisms based on heat carrier dynamics.4 Therefore, it is
expected that heat transfer phenomena show percolation, while the
flux is driven through a temperature gradient, acting as the driving
force.
It is convenient to differentiate between diffusive and percolative approaches. Diffusive processes consider that the fluid disperses within the medium, while during percolation is the medium
easing the movement of the fluid. In the percolative case, the
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J. Appl. Phys. 131, 230901 (2022); doi: 10.1063/5.0091291
Published under an exclusive license by AIP Publishing
scitation.org/journal/jap
between fillers and the matrix, along with the interfacial thermal
resistance, are presented. Our analyses allow us to highlight the difference between the thermal and electrical percolation thresholds
and the conditions in which they become notable.
II. COMPOSITES MADE OF FILLERS IN A POLYMERIC
MATRIX
A composite consists of two or more phases, whose final
properties differ from those of the original materials. Specifically,
composites formed by a polymeric matrix are used due to their versatility and the benefits that they offer, as lightweight, flexibility,
corrosion stability, low cost, and ease of manufacturing; particularly, the ones with inorganic fillers are the most employed in the
industry.18 Moreover, the polymer properties, such as the mechanical, electrical, thermal, optical, and magnetic ones, among others,
are enhanced by the addition of fillers. There is a great variety of
applications of polymer-based composites such as compact electronic devices,19 conducting adhesives,20 anti-static materials,21
electromagnetic interference shielding,22,23 as well as many
others.24,25 Figure 1 presents a diagram of the characteristics and
applications of some representative composites combining different
fillers and polymeric matrices.
Micro- and nanometric metallic particles offer good results in
enhancing thermal and electrical conductivities in polymeric composites. They have been used for a long time due to their availability and variety.18 Particles from different oxide materials generally
present high electrical resistivity, as fillers they have been used in
several applications that go from biological and medical applications to the microelectronic industry.26 In the case of particles of
different nitrides, they have the advantage of combining an intrinsic high thermal conductivity and a low electrical conductivity,
which makes them very useful for electronic packaging.27
Moreover, carbon-based particles have motivated their use as
fillers in polymeric composites to modify their electrical and
thermal conductive responses and use them for diverse purposes
in different fields of science and industry. Specifically, nanostructured carbon materials with large aspect ratios like tubes, wires,
fibers, and rods have attracted much attention as fillers because of
their ease in the assembly of long conductive pathways for efficient
thermal and electrical energy transport in polymeric
composites.28–30 In recent years, graphene has gained attention
due to the advantage over carbon nanotubes, related to its greater
coupling capability in polymeric matrices.31 Since we want to
address the percolation phenomena associated with electrical and
thermal conductivities in polymeric composites, our attention will
be mainly focused on polymers with fillers of high thermal and
electrical conductivities such as metallic particles and carbon
nanostructures.
08 August 2023 20:26:24
spreading or dispersion of the fluid is determined by the intrinsic
and random characteristics of the medium, which are structural
and randomly interrupted or blinded interconnections.3
Additionally, the action of external forces on the system influencing
the flux must also be considered.1 The percolative approach, used
to explain how water seeps through a solid, has been used to study
the electrical resistivity of a binary mixture of metallic particles.5 At
present, this concept is relevant due to the design and development
of composite materials with suitable physical properties, particularly those related to the electron flow6–8 as well as heat dissipation
and transfer.9–11 The most studied and simplest systems that manifest electrical percolation are those constituted by two phases, the
dispersed one with high electrical conductivity and the continuous
one with an insulating quality. Percolation occurs when increasing
the amount of the dispersed phase and reaches a point where the
electrical conductivity increases abruptly, called the percolation
threshold.1 Percolation has generated great interest in the scientific
community for decades and has promoted the development of theoretical models and experimental research work of high impact in
understanding transport phenomena.12,13 These advancements, in
turn, have been stimulating a considerable development of composite materials of two or more phases, focused on structural manipulation at the microscopic level to reach the macroscopic required
physical properties.
In contrast to the electrical transport case, for heat transfer in
a mixture consisting of a dispersed phase with high thermal conductivity within a continuous phase or matrix of low thermal conductivity, different research groups have shown interesting but
frequently contradictory results and, although thermal percolation
could be present, the phenomenon has not been fully understood.14
Similarities between thermal and electrical transport have been
reported by some research groups, even finding a thermal percolation threshold,14,15 while others report the simple monotonic
growth of the properties related to thermal transport.16
From a critical point of view, there are several significant and
crucial differences in heat and electron transfer. The contrast
between the thermal conductivities of the composing phases
cannot be as huge as the corresponding differences in electrical
conductivities. Several thermal transfer mechanisms are always
present in any composite material, in contrast to the electrical conduction case.17 Additionally, thermal and electrical transport phenomena are due to different transport mechanisms.4 Although
electrons in both cases have a preponderant role, in the case of
metallic materials, their influence is significantly much smaller in
heat transport in most non-metallic materials. The proper interpretation and description of the thermal percolation process and
threshold, using theoretical models, require considering all the different heat transfer mechanisms and the ubiquity of the thermal
flux.
In this Perspective, a critical analysis of relevant experimental
results on the thermal and electrical properties of two-phase materials is presented, along with theoretical models that can be used to
describe their percolative behavior. We will focus our attention on
composites of high conductivity particles in a polymeric matrix.
The effects of concentration, connectivity, geometry, as well as the
influence of the matrix and fillers on percolation are discussed. In
the case of heat transport, the crucial effects of the thermal contrast
PERSPECTIVE
III. PERCOLATION THEORY AND MODELING
A. Percolation theory
The percolation theory deals with the connectivity of objects
that belong to a structured network and how such connectivity is
related to the bulk properties of systems through cluster sizes
formed by these connected objects. Percolative models have been
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P1 / ( p pc )β ,
FIG. 1. Some representative examples of composites made of different fillers in
polymeric matrices and their applications.32–35
developed to study critical phenomena, which describe sharp
transitions on a particular property due to a spanning network
structure formed in a system.36 They are very successful in explaining the behavior of electrical conductivity in polymeric composites
loaded with conductive filler materials that present a sharply
insulator–conductor phase transition as a function of the volume
fraction of fillers added.37
J. Appl. Phys. 131, 230901 (2022); doi: 10.1063/5.0091291
Published under an exclusive license by AIP Publishing
(1)
08 August 2023 20:26:24
In its basic form, the percolation theory works with networks
like the ones formed in a checkerboard. Assuming that each box in
the checkerboard is in one of the two possible states: empty or
occupied, such state is randomly selected, and it is independent of
the state of their neighbors. Then we can define the probability p as
a parameter that helps to deal with the problem in any network,
and it establishes the probability for each box of being occupied.
This p is the same for all the boxes of the network. If p has a low
value, occupied sites will be individually isolated or forming tiny
groups named clusters.38 As p grows, clusters begin to grow too,
and their average sizes increment monotonically up to a critical
value pc. With p getting closer to pc from below, the larger clusters
begin to merge between them, generating some extremely long
clusters, in such a way that for a system with an infinite number of
sites, the average size of the clusters diverges at p ¼ pc . If the
system has a finite number of sites, the last condition implies the
existence of a complete pathway of occupied sites across the system.
This critical value pc is known as the percolation threshold. When
p is higher than pc and the system is big enough, only a large
cluster remains along with other tiny ones. As p increments its
value beyond pc, this large structure grows very fast, absorbing the
remaining isolated clusters. In summary, if p . pc , there is a percolative network, while for p , pc there is not, and pc is only defined
in an infinite system.39
The percolation probability P1 is the fraction of occupied
sites, belonging to the infinite percolative cluster, from all the available sites in the network, with P1 ¼ 0 if p , pc and P1 = 0 if
p . pc . This percolation probability is the perfect quantity to determine whether or not there is conduction in the system because it is
directly related to the volume fraction of the conductive phase.40
Finally, if p pc , the probability follows a power law, governed by the critical exponent β as follows:
with p slightly higher than pc. The topological critical exponent β,
known as the percolation cluster exponent, has been extensively
reported in the literature, and it has been shown to have universal
values of 5/36 ≈ 0.14 and 0.41 for two and three dimensions,
respectively. A compilation of various critical exponents and their
universal values can be found in the books by Stauffer and
Aharony41 and Sahimi.2
All the concepts of percolation in lattices have been extended
to continuum media, which are considered as systems of randomly
distributed objects interacting with each other but with different
forms and sizes. In this case, there are two main topics of interest,
the first one is the connectivity, which directly influences the percolation threshold, and the second one is the absence of lattice-like
critical behavior due to the lack of regular physical parameters, as
the regular and geometric distribution and the constant or invariant surroundings. In continuum media, the fillers’ volume fraction
(w) plays a significant role because it is considered as the equivalent
to probability p and is given by the relation w ¼ pf , where f is the
filling factor. Consequently, the percolation threshold takes place at
wc ¼ pc f , and the percolation probability P1 characterizes the size
of the clusters, which will have an infinite extension if w . wc , as
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occurs in discrete lattices. A thoroughly and current discussion
about these topics is given by Balberg.42
B. Electrical and thermal transport modeling
FIG. 2. A simplified scheme of some representative models for studying the
electrical and thermal transport in composites.44,47,48
J. Appl. Phys. 131, 230901 (2022); doi: 10.1063/5.0091291
Published under an exclusive license by AIP Publishing
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composites as a function of a discontinuous phase volume fraction,
afterward Cheng and Vachon extended this model to a discontinuous parabolic phase distribution.32
From a percolative standpoint, a composite can be studied
using different models. Most of them rely on the classical percolation theory40,51 (see Sec. III A for more details) also known as
scaling law or power law,
σ / (w wc )t ,
(2)
where σ is the effective electrical conductivity, w is the filler concentration in volume fraction, wc is the percolation threshold, and t
is the critical exponent, which depends on the connectivity and
aspect ratio of the fillers.52 Theoretically, it is expected that this
critical exponent has universal values of 1.33 and 2 for two and
three dimensions, respectively.37,39,53,54 Particularly, in carbon
nanotubes-based composites, the obtained t values range from 0.7
to 3.1. This discrepancy is related to the complexity of transport
processes in real systems.55 In particular, the electrical conductivity
is modulated by diverse transport phenomena in which quantum
tunneling of charge carriers can take place.8
It is worthwhile to mention that Eq. (2) is not accurate in predicting the electrical conductivity, and it is only valid for values
slightly above the percolation threshold. Although Eqs. (1) and (2)
have the same form, it is not valid to state that the conductivity is
proportional to the percolation probability. Experimentally, Last
and Tholuess showed that the behavior of electrical conductivity is
quite different from that of the percolation probability, and the curvatures are opposite.5 Based on the review of 147 different experimental results reported in the literature from various research
groups, Bauhofer and Kovacs concluded that is not possible to
extract reliable geometrical information of the carbon nanotube
network from most of the t values. However, they found that t is
predominantly in the range of 1.3–4, peaking around 2.56 Recently,
an analysis of these cases has been presented in the literature by
Balberg.52 In this work, it has been shown that the non-universality
of t is a consequence of the fact that this parameter is not only
related to the fillers’ aspect ratio, but also with interparticle electrical conduction mechanisms. For this reason, more sophisticated
schemes than the scaling law model, based on a similar approach,
have been developed. One of them is the one proposed by
Mamunya et al.,12 which includes the polymer–filler interaction.
Moreover, sigmoidal-type models have been created based on
the characteristic shape curve of the electrical conductivity as a
function of filler concentration.13,57 Interestingly, some of them
agree with the threshold predicted by the classical percolation
theory. A particular characteristic of these models is that they can
be usable for the whole concentration range of fillers, although a
deeper analysis of the obtained parameters is necessary.
Additionally, a relationship between them and the structural characteristic of the composite is also pending.
In the literature, extensive reviews on electrical conductivity
for diffusive and percolative models can be found; see, for example,
the articles by Clingerman et al.,58 Lin,44 Zhai et al.,47 and Yang
et al.48 Taherian et al. have shown another categorization considering micro- and macro-mechanical models, image processing
methods, and analytical models.59
08 August 2023 20:26:24
The prediction of electrical and thermal properties in multiphase materials is a recurrent problem in materials science.
Modeling these properties is a challenging task because the resultant composite properties depend on many factors, like their intrinsic properties and filler concentration, distribution, orientation,
size, shape, and matrix-filler interface properties.43,44 Several empirical, semi-empirical, and fundamental theoretical models have been
developed to understand the transport phenomena in these materials. Figure 2 presents some of these models, considering diffusive
and percolative approaches.45 The models, where the diffusive phenomena are dominant, are those based on effective medium theories. The most well-known is the Maxwell–Garnett model, which
was originally developed for studying the effective polarizability of
spherical inclusions at very low concentrations46 and has been subsequently extended for the study of thermal and electrical conductivities. One of the most successful models is the one proposed by
Bruggeman and its variations,47,48 which can be used even at a
high concentration of fillers. Bruggeman’s model was formerly
developed to study the electrical properties of systems made of
spherical inclusions embedded in a medium with a high contrast in
permittivity between them. Afterward, this model has been successfully adapted to study several physical properties in systems with
different geometries.49,50 In order to consider the volumetric
packing fraction, shape, and orientation of the particles, Lewis and
Nielsen developed a well-known approach based on the Halpin–
Tsai model, which was originally designed for filaments of uniform
cross section in a parallel configuration at low concentrations.48 In
contrast, using a probabilistic approach, Tsao developed a model
that considers a discrete phase in a continuous polymeric matrix,
allowing to study the thermal conductivity in two-phase
PERSPECTIVE
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Regarding thermal conductivity, the interpretation of the
experimental data, in some cases, has been performed using the
electrical percolation concepts as well as from methodologies developed from the classical percolation theory.58 Multiple studies and
models based on Bruggeman’s approach consider the built-up of
the composite through a process of incremental homogenization,
as the general effective media model (GEM) from McLachlan. This
model considers the formation of the fillers’ network, including an
exponential factor and the critical concentration of percolative behavior.49 Otherwise, based on a generalization of parallel and series conduction models, Agari and Uno60 developed a model considering the
ease of constructing conductive chains by the fillers, the crystallinity,
and the polymer crystal size in a uniform distribution. These authors
use two adjusting parameters, nevertheless being a semi-empirical
model, it is impossible to predict the values of the parameters. More
recently, Zhang et al.61 proposed a model considering the shape and
distribution of the filler in the matrix. It has been shown that compared to all the other models, this approach predicts extremely well
the effective thermal conductivity, with the great advantage that it
does not involve any empirical or adjustable parameters, and all the
quantities are extracted from the experimental data.62 For an extensive
account of thermal conductivity models, the reader can consult the
review of Progelhof et al.63 and Zhai et al.47
IV. ELECTRICAL AND THERMAL TRANSPORT
MECHANISMS
J. Appl. Phys. 131, 230901 (2022); doi: 10.1063/5.0091291
Published under an exclusive license by AIP Publishing
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FIG. 3. Schematic diagram of a composite depicting percolation phenomena in
electrical conductivity (blue line) as a function of particle volume fraction (w): (a)
The formation of particle clusters at low w (green circles). (b) Subsequent clusters structuring induces the formation of conductive pathways of connected
fillers as w increases. (c) Totally interconnected structure with several conductive
pathways formed at higher. Note that the y-axis (related to electrical conductivity) is represented in a logarithmic scale and the volumetric fraction axis is
linear.
is called the percolation threshold [see Fig. 3(b)]. For concentrations above this threshold, the contribution by hopping gets
reduced, and a cooperative behavior between hopping and conduction by contact is established [see Fig. 3(c)].
Heat conduction in most polymers is hindered due to the lack
of long-range order of the polymeric chains and their surroundings, producing scattering of heat carriers and reducing the net
energy flux.45 The interactions between the polymeric chains have
great importance for heat transport.73 In general, semi-crystalline
polymers present higher thermal conductivities than totally amorphous ones.74 Several strategies aimed to enhance heat transport in
polymers, such as chain crosslinking by hydrogen bonding, polymeric chains orientation, improvement of crystallinity, among
many others, have been devised.75–77 However, the simplest
method to increase the thermal conductivity of polymers is to mix
them with high thermal conductivity particles. These fillers generally include metals and carbon-based micro- and nanostructures in
their different crystalline and semi-crystalline phases.68,78 In these
cases, it is also necessary to consider the phonon coupling mechanisms at both types of interfaces: polymer–filler and filler–filler.
The coupling of the transport mechanisms at the interfaces is
related and depicted by the parameter known as interfacial thermal
resistance. This parameter is one of the most crucial factors in
determining the thermal conductivity of composites. A good interparticle thermal coupling by decreasing the filler–filler interfacial
thermal resistance is necessary to facilitate the heat transfer along
the percolative network, while maintaining or increasing the
polymer–filler interfacial thermal resistance, due to the fact that
heat transfer from fillers to the polymeric matrix acts in detriment
of the functionality of the effective thermal network due to the generation of heat loses. The size and quality of the heat-conducting
network are affected by particle concentration too.79
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Polymers are formed by chains of monomers, and they have
different degrees of crystallinity. The charge transfer in these materials has been extensively studied, and it is shown that in most
cases, the transport is mainly carried out by the hopping mechanism between neighboring polymeric chains.64,65 Generally,
hopping includes jumping over potential barriers and quantum
mechanical tunnelling.64 Strong differences in electrical conductivity of polymers are attributed to variations in the overlap between
molecular orbitals.6,7 It is possible to substantially improve the electrical conductivity of a polymer by embedding fillers with high
electrical conductivity, such as metals,11,66–68 carbon nanotubes,15,56,69 graphene,70,71 among others.72 In these composites,
three different regions can be identified:64 the first zone corresponds to the clusters, each one consists of a set of particles whose
conductivity is the highest in the composite; the second refers to
regions between adjacent clusters, whose conductivity value is near
to that of the composite; and finally, the last one that only contains
polymer, as shown in Fig. 3. Within the clusters, electronic conduction takes place, while hopping-mediated conductivity occurs in
the other two regions [see Fig. 3(a)]. In these composites, the electrical conductivity shows a strong dependence on the size and concentration of the filler particles, in such a way that at low
concentrations, the predominant process is hopping. By increasing
the number of metallic particles, the clusters grow, and in the same
way, the space between adjacent ones becomes smaller, thus
increasing the surface area for electronic transport and facilitating
transport by hopping, which produces the highest rate of increase
in electrical conductivity in the composite.8 The volume or mass
fraction concentration at which the macroscopic chains of the conductive material are formed, facilitating the transport of electrons,
PERSPECTIVE
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PERSPECTIVE
V. FACTORS DETERMINING THE COMPOSITE THERMAL
CONDUCTIVITY
Heat and electrical transport and percolation in composite
materials depend on several factors such as the type and geometry
of fillers, their concentration, as well as on the interaction between
the polymeric matrix and fillers at their interphases. Below, a brief
review and discussion of some relevant aspects is presented.
A. Filler concentration effect
polymeric composites is determined by
η¼
kc km
,
km
(3)
where kc is the thermal conductivity of the polymeric composite
and km is the thermal conductivity of the matrix. Note that Eq. (3)
is also applicable for any physical quantity, including electrical conductivity. Generally, adding micrometric particles to a polymer
induces a monotonic increase in thermal conductivity. For
example, for silver microparticles, an enhancement of nearly 170%
at a concentration of 50 vol. % can be obtained, while for carbon
nanotubes, the enhancement is commonly around 200% at a fraction of 7 vol. %.31 Percolative phenomena depend on the formation
of a connected fillers network.43,88 From the geometrical point of
view, it is necessary to reach a certain concentration of fillers to
guarantee the formation of the heat conduction network.
Therefore, to observe the percolation threshold and its effects, it is
necessary to scan a broad range of concentrations along with the
implementation of high-performance techniques. Adequate methodologies include Raman optothermal techniques,31 the bridge
method,87 3ω and steady-state methods,9 double-guarded plate
technique,15 transient plane source technique,89 and laser flash,
among several others.11,31
B. The role of the aspect ratio
The effect on the thermal and electrical properties of composites, made of different filler materials embedded in insulating
matrices, has been extensively reported in the literature (see
Table I). Although the effective conductivities and percolation processes in composite materials strongly depend on the physical
properties of the filler, it has been reported that the aspect ratio has
a preponderant role.14 This quantity defines the ratio between the
axial and transversal dimensions of a particle (length/diameter).
08 August 2023 20:26:24
Reports on the electrical properties of composites made of
electrically conductive fillers in a polymeric matrix are numerous,
along with the observation of the percolation effect.56 In these composites, the electrical conductivity grows abruptly in several orders
of magnitude as the fillers’ volume or mass concentration is
increased. The concentration value where this behavior occurs is
related to the percolation threshold concept. The dependence of
the electrical percolation threshold on the kind, size,80 and aspect
ratio of the fillers has been discussed in the literature. Larger values
of the aspect ratio have been related to percolation thresholds
occurring at lower concentrations.56,81,82 This characteristic is especially remarkable for composites based on carbon nanotubes,15,83
fibers,84 metallic nanowires,85,86 and other filamentous structures.43
The changes in the electrical conductivity were experimentally
measured using standard multiple probe methods.9,15,85,87 The
remarkable measured changes in electrical conductivity can be
attributed to significant contrast (∼1015) between the high electrical
conductivity of the fillers and one of the insulating matrices, which
promote the electronic transport along the percolative network.17
Similarly, the effect of filler concentration on the thermal conductivity of polymers with high thermal conductivity particles has
been extensively studied.35 Composites with different fillers, in the
range of high concentrations, are the most studied due to the
numerous advantages offered and their applications (see Fig. 1).
The enhancement η of the thermal conductivity in this kind of
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TABLE I. Comparison between the electrical and thermal thresholds for different composites.
Filler
Matrix
Size
MWCNTs
MWCNTs
SWNTs
MWCNTs
Graphene flakes
RET
PDMS
PMMA
RET
Epoxy
Carbonyl-iron particles
CIP
Silver nanowires
Cu
Graphene nanoplatelets
Polyester
resin
PVDF
YIG
Epoxy
L ∼ 1.6 μm, D ∼ 45 nm
L ∼ 40–110 μm, D ∼ 15 nm
NR
L ∼ 1.4 μm, D ∼ 40 nm
L ∼ 2–8 μm, T ∼ 0.35–12
nm
D ∼ 1.67 μm
NR
NR
L ∼ 20 μm, T ∼ 15 nm
wtc
(vol)
a
jwtc wec j 100/wec
(%)
wec (vol)
a
0.022
0.000 35
0.3c
0.0183
0.30
0.023, 0.033
0.000 45
0.3
0.020
0.10
0.38
0.022c
0.21
0.17
b
a
Reference
4.54
22.2
…
8.5
200
9
103
15
102
14
0.46
17.4
11
0.022
0.16
0.05
…
31.2
240
87
10
106
a
Parallel.
Perpendicular, NR: not reported; RET: reactive ethylene terpolymer; PDMS: poly (dimethylsiloxane); PMMA: polymethylmetacrylate; PVDF: poly
(vinylidene fluoride); YIG: yttrium iron garnet; L: length; D: diameter; T: thickness.
c
In these cases, the authors assumed that the threshold for thermal conductivity is the same as that observed for electrical conductivity.
b
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The aspect ratio has become a tunable parameter to improve the
thermal and electrical conductivities in composites made of microand nanostructured fillers in polymeric matrices.90 Recent studies,
considering the aspect ratio of copper, silver, and gold nanowires,
have shown remarkable effects even at low concentrations, predominantly in the axial component of the heat conduction plane.85 It
has been shown that for concentrations of around 1–5 wt. %, these
fillers can improve the thermal conductivity of the polymeric
matrix, reaching values of around 1.5 W/mK, which can be considered adequate for several applications in electronics.89 In general,
1D nanomaterials, such as wires,85 tubes,91,92 and fibers,93,94 have
especially attracted attention in developing composites because of
the benefits that their high aspect ratio can bring, in achieving
lower percolation thresholds due to the construction of large heat
conductive pathways, which allow acquiring high thermal conductivity values at small fillers content.
C. Interfacial thermal resistance
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coupling between the vibrational spectra of graphene and polymer,
resulting in a low interfacial thermal resistance through covalent
and hydrogen bonding (around 10−9 m2 K/W).98,99
VI. PERCOLATION OVERVIEW
Several strategies to enhance the thermal conductivity of composites and to observe percolation effects have been implemented.14,31 Interesting approaches include the use of carbon
nanostructures as fillers in polymers, inducing significant increments in thermal conductivity at low concentrations using CNTs,
or even better by using graphene that induces three-dimensional
interconnected networks with the matrix, improving heat transfer
through the entire composite.31,100 Nevertheless, the improvement
in thermal conductivity (up to 1–30 W/mK) is much lower than
the values that could be expected.43,101 This is associated with the
relatively low contrast (∼105) between the thermal conductivity of
the fillers and the polymeric matrices. Additionally, the large interfacial thermal resistance results in a low net heat flux through the
composite, and particularly, a high filler–filler interface resistance
tends to act as a detrimental factor for heat transfer along the effective percolative network.17
A. Connectivity and percolation
Electrical and thermal percolation processes show notable differences in the shape and growth rate of conductivities, as a function of fillers’ volume fraction. Numerous studies report the
existence of percolative phenomena, especially in the electrical
cases, due to the large contrast between the properties of the matrix
and fillers. In the thermal case, some authors associate the sudden
increase in conductivity, at a specific filler concentration, with the
thermal percolation threshold,9,10,102,103 in a similar way to its electrical counterpart. Other research groups have reported that as the
filler concentration increases, a first stage appears where the
thermal conductivity grows linearly, then at a given concentration
value, the curve presents a relatively large increase in the slope.14
This last concentration value has been also identified as the
thermal percolation threshold using the criterion of the second
derivative.62
According to our previous discussion (see Secs. II and III), the
geometric percolative network is always formed in the polymeric
matrix if the filler content reaches the appropriate concentration
above a critical value. Then, it is expected that the thermal and
electrical transport can flow through the fillers network, giving rise
to percolation phenomena. Even if the formation of this network
takes place, the physical transport mechanisms must be accessible
to let the heat and charge carriers to move along the created conductive paths. Otherwise, different studies report how to induce the
formation of fillers’ networks in polymeric matrices through the
application of electric and magnetic external fields.28,29,95,104
However, in some cases, forming an oriented pathway of fillers
inside the composite does not guarantee an outstanding increase in
the thermal conductivity compared with a composite at the same
fillers’ concentration but randomly distributed.29 The main effect
of orienting the filler-formed structures is achieving their connectivity at lower concentrations compared with a composite system
randomly dispersed. Boudenne et al.16 have reported the study of
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Interfacial thermal resistance is one of the most relevant
aspects, deserving special attention due to its strong influence on
the thermal conductivity of composites. Developing composites
using a polymeric matrix and embedding a reasonable quantity of
fillers with a very high thermal conductivity or using fillers with a
large aspect ratio, like carbon nanotubes, or even aligning them are
not enough to get a high thermal conductivity composite.28,29,31,95
The reason for this behavior is the presence of the interfacial
thermal resistance between the matrix and the filler. This physical
property quantifies the hindrance to the thermal flow simply due
to the presence of the interface. Interfacial thermal resistance is
present even at atomically perfect interfaces and appears because of
the physical differences between neighboring materials, generating
heat carriers scattering. In the particular case of composites made
of fillers embedded in polymers, it has been reported that the interfacial thermal resistances are typically of the order of 10−3–
10−4 m2 K/W for glass microspheres, 10−5–10−6 m2 K/W for nickel
nanowires, or 10−7 m2 K/W for carbon nanotubes and can be as
small as 10−9 m2 K/W for graphene.31,36 Analyses of the complex
process associated with interfacial thermal resistance have been a
subject of a variety of theoretical works. The most basic models,
such as the acoustic mismatch and the diffuse mismatch, including
their interpolations, assume that the scattering processes at the
boundary are elastic.96 This assumption establishes serious limitations for their use in real systems, especially in the case of composites; therefore, it has been necessary to develop more sophisticated
approaches. Particularly, for carbon nanotubes embedded in a polymeric matrix, it has been shown, using the finite element method,
that the correspondingly high interfacial thermal resistance
(10−7 m2 K/W) is a consequence of the mismatch between the
phonon density of states of the carbon nanostructure and the corresponding matrix polymeric chains.17,31 Additionally, the theoretical
analysis using steady state non-equilibrium molecular dynamics
was used to calculate the interfacial thermal resistance of graphene–graphite–polymer systems. It was found that longwavelength phonons in graphene play an important role in thermal
transport across the graphene–polymer interfaces.97,98 Additionally,
it has been reported that the out-of-plane motion induces strong
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concentration and the values of the conductivities of the filler and
the matrix reported by Boudenne et al., the “ideal” values for each
fillers’ concentration reported for that composite system were
calculated.
Figure 4(b) presents the plot of the percentage from the ratios
between the experimental and the “ideal” values for both conductivities. The electrical conductivity percentage decreases from
100%, representing the electrical conductivity of the “ideal” system,
to a minimum value around the percolation threshold at
w ¼ 0:025. Then, it grows and stabilizes, showing a constant
growth trend from w ¼ 0:05 up to w ¼ 0:20 and finally exhibiting
an incremental behavior, reaching a value of 10−5% at w ¼ 0:35.
This final value is too small to be considered an improvement compared to the optimum value achieved by an “ideal” system,
although the increment is almost ten orders of magnitude, referring
to the minimum value at the percolation threshold. In contrast, the
percentage of thermal conductivity shows a decrease from 100% to
a minimum of 5.5% at the maximum volume fraction at 0.35,
without presenting a detectable thermal percolation threshold. The
behavior of both curves indicates that the two conductivities are
closer to that of an “ideal” system only at low volume fractions. As
volume fraction grows, conductivities behave oppositely. The
thermal conductivity gets worst while the electrical one improves.
Nevertheless, from this result, we infer that although the electrical
conductivity increases several orders of magnitude, this increment
is small compared to its thermal counterpart when both are compared to an “ideal” composite system. In other words, the net
change of the electrical conductivity can be too small when compared with the change in thermal conductivity if both of them are
weighted with an “ideal” composite system. This performance evaluation approach of composites can be useful in the electronic
industry, where the main objective is to have at hand materials with
high thermal conductivity and electrical insulation behavior at the
same time for cooling devices with efficient thermal dissipation.
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samples composed of micrometric nickel particles in polymers
aligned by using magnetic fields previously to the curing process.
Thermal and electrical conductivities were measured parallel and
perpendicular to the chainlike structures formed. These authors
found that electrical percolation occurs at a lower filler concentration value for the parallel measurement than for the perpendicular
one; nevertheless, they do not observe percolative phenomena in
thermal conductivity in any case.
The enhancement in electrical and thermal properties on
composites is commonly determined from the comparison with the
values of the pristine polymeric matrices, see Eq. (3). Although the
improvements reached in those properties are huge, taking the
matrices as a baseline or reference, these increments can be tiny for
some practical applications. For example, larger increments in electrical properties of around ten orders of magnitude in a highly
insulating matrix could be irrelevant in some practical cases
because the composite remains behaving as an insulator. Therefore,
a better way to classify or quantify the improvements in thermal
and electrical conductivities is needed.
An alternative way for estimating the enhancement produced
by fillers in the composite is to compare it with an “ideal” system,
having the possibility of reaching the maximum possible enhancement at each concentration. To illustrate our approach, let us consider the case of a composite, in which the filler particles are
aligned [Fig. 4(a)], defining the “ideal” configuration in such a way
that the completely aligned fillers can ideally form chain-like or
fibrillar structures with a parallel arrangement, as sketched in the
inset of Fig. 4(b). For practicality to evaluate this approach, let us
consider again the results reported by Boudenne et al.16 for aligned
micrometric particles of nickel embedded in a silicone matrix. The
optimum thermal (electrical) conductivity of the “ideal” system is
calculated using the effective parallel model (Voigt model):59
k ¼ kf w þ km (1 w), where kf (km ) is the thermal conductivity of
the filler (matrix) and w is the fillers’ volume fraction. Taking the
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FIG. 4. (a) Schematic diagram of the induction of alignment by an external magnetic field. Note that the particles tend to align themselves along the field lines, forming
pathways. (b) Normalized thermal and electrical conductivities with optimal configuration (parallel model), for the composite silicone/nickel, Figs. 5 and 7 in Ref. 16. Inset
in (b), “ideal” equivalent system to the formation of chains. Note that the right y-axis (related to electrical conductivity) is represented in a logarithmic scale.
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Note that in the range of volume fractions presented here, the
efficiency is higher for thermal transport when compared with an
ideal system. However, as the volume fractions of fillers grow, the
efficiency in the electrical transport increases, while the thermal
one decreases. Unfortunately, the lack of experimental data for
higher volume fractions,16 probably due to the difficulty in manufacturing, does not allow the efficiency evaluation when the conductivities reach their maximum value at the saturation level of
fillers’ volume fraction.
B. Phenomenology
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fact that graphene tends to form better composites with higher
thermal conductivities, even at higher concentrations than other
carbon nanostructures,31 because graphene provides a better coupling for the phononic vibrational modes with the polymeric
matrix, resulting in a reduction in interfacial thermal resistance.31
Therefore, the observed differences between both thresholds of the
composites as shown in Table I are a direct consequence of the fact
that heat and electric charge carriers as well as the mechanisms
that drive their movement in the composite are not the same.
From a morphological-topological point of view (see Sec. II),
it is only necessary to reach a certain concentration of fillers to
form a network that allows the percolation to occur. In the case of
electronic transport, electrons can hop between conductive filler
particles through the polymeric matrix by tunnel effect, following a
conductive pathway through the whole fillers network from one
side to the other in the sample, giving rise to electric percolative
phenomena. However, in the case of heat transport, even though
the percolative morphological-topological network is formed, the
physical processes are very different to the ones involved in electronic transport. Here, the polymeric matrix, even without fillers,
transports heat by means of phonons. Additionally, at filler–filler
interfaces, heat carriers experience strong scattering processes,31
which drastically reduce heat transfer, being able to hide or shield
the percolative behavior.
C. Manufacturing processes
The manufacturing process of polymeric composites is one of
the most difficult aspects to manage, in the sense that even though
each research group can control the reproducibility of their own
samples, unambiguous comparison of measurements of the conductivity performed by different research groups of apparently the
same kind of samples is not always possible. Given that the electrical percolation involves well-established phenomena, sample manufacturing conditions affecting them can be helpful in
understanding their impact on thermal percolation. Crucial factors
in the preparation of samples include fillers manufacturing, pretreatment and functionalization,56 the dispersion method used,31,56
post-curing treatments as thermal annealing,107,108 among many
others affect significantly the resultant effective conductivity. In
particular, it has been shown that the manufacturing process can
induce the shifting of the electrical percolation threshold to higher
volume fractions when the fillers are better dispersed, indicating
that self-aggregation induces notable changes.56,109,110 Specifically,
Li et al.110 have reported variations ranging from 0.06 to 0.26 vf%
in the electrical percolation threshold for composites with CNTs as
a function of dispersion quality in epoxy resin matrices. In contrast,
in matrices obtained from aqueous-based polymers, the hydrophobicity of carbon nanostructures can play a crucial role77,111 due to
clustering effects that could shift the percolation threshold.
Moreover, annealing and other thermal treatments are also factors
that must be considered because they offer the possibility of
improving the contact at the filler–matrix interface. Zhang et al.
report that the electrical percolation threshold can be decreased to
0.13 wt. % by hot-pressing, while for extruded samples the threshold appears around 4 wt. % in composites made of carbon nanotubes in a polyurethane matrix.107 Gao et al.108 studied composites
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We consider that the heat flux in the composites comes from
two contributions: electrons and phonons. It is crucial to understand how the heat transfer phenomena are taking place in these
materials to determine which one is mainly responsible for the heat
flux. Applying the Wiedemann–Franz law, it is possible to estimate
the contribution of electrons to the total thermal conductivity.105
For example, for a composite made of a polyester resin matrix and
fillers of carbonyl iron particles (CIP), with a concentration of 54%,
Forero-Sandoval et al.11 found that the electronic contribution to
the total thermal conductivity is almost negligible (lower than 1%).
Therefore, the heat, in this composite system, is carried primarily
by phonons. This finding can explain why the thermal conductivity
does not grow as expected when a metallic filler is embedded in a
polymeric matrix and why the percolation threshold is not
observed as a sudden increment. This result, based on
Wiedemann–Franz law, suggests that nevertheless, the heat carriers
in the metallic fillers are electrons, the low efficiency in the transference of energy between electrons and phonons in the polymeric
matrix produces an interfacial thermal resistance. Then, the percolative phenomena get shielded and only appear as a light slope
steepening in the thermal conductivity curve as a function of the
concentration of the fillers.11 In contrast, considering again the
case of the aligned micrometric particles of nickel embedded in a
silicone matrix reported by Boudenne et al.,16 the alignment
induced by a magnetic field puts the particles into contact forming
a chain-like structure, improving the thermal conductivity by the
formation of a heat transfer network. However, from the experimental data, it is not possible to observe the characteristic sudden
increase, corresponding to the thermal percolation threshold, as
could be expected. The contact between the fillers in the chain-like
structures is not metal–metal because each filler particle has a polymeric layer covering it. This layer regulates the electron–phonon–
electron interactions determining the thermal transport between
two neighboring fillers. Additionally, the contribution of electrons
to thermal conductivity is negligible, as discussed above, and once
again, the percolative phenomena get shielded.
In order to get a deeper overview of the differences between
the thermal and electrical properties, in the analyzed composites,
results from some significant works in which both thresholds were
reported are shown in Table I. As can be observed, the differences
between both percolation thresholds are around 5% for MWCNTs
in RET9 to 240% for graphene nanoplatelets in epoxy resin.106 In
the case of carbon nanotubes, it goes from 5% in RET to 22% in
PDMS.9,103 Note that both thresholds differ notably in the case of
graphene (200% and 240%).14,106 This behavior results from the
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D. Characterization methodologies
A large variety of methods have been used to analyze the percolation and filler effects on the thermal properties of multiphase
composites (see Refs. 31 and 107). The kind of composite, the
dependence of its thermal properties on the composition, the
geometry, and size of the sample, defines which techniques are
more suitable for providing reliable data. It must also be considered
whether the rate of change in thermal conductivity, as a function of
the concentration near the percolation threshold, is big enough to
be distinguishable by the technique used in its characterization.
Then, selecting the characterization method with appropriate resolution becomes necessary.9,15,31,112 Another crucial requirement is
the use of experimental approaches as less invasive as possible, in
such a way that they do not interfere with synergy effects mediated
by the fillers, which could appear in the form of percolation
effects.9 Additionally, the case in which the sample shows anisotropy must be considered, such as induced by the alignment of the
fillers in a preferred direction. In this case, percolation effects can
be different depending on the measurement direction.113 Therefore,
using a methodology or a combination of them to determine the
in-plane and cross-plane thermal conductivities would be the most
convenient to obtain complete thermal characterization.97 Kim
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et al.9 have reported the characterization of anisotropic plateshaped composites made of carbon nanotubes in a polymeric
matrix. These authors found that along the cross-plane direction,
the measurements only reveal a monotonous growth of the thermal
conductivity as a function of the concentration. In contrast, for the
in-plane direction, the observed behavior is completely different
because around 2 vf% of carbon nanotubes, clear evidence of percolation in thermal and electrical conductivities was found.
All along this section, we have discussed that even though the
percolative network must be formed in two-phase composites
made of thermally conductive fillers embedded in a polymer
matrix, contradictory and non-consistent results have been reported
by different research groups about the thermal percolation phenomenon. However, several aspects have been made clear during
the years of dealing with this kind of experiments, first of all that
the strategies for inducing the increase in the thermal conductivity
are well defined and that the limitations for making these changes
larger are strongly related with the dimensionality of the fillers and
their interaction with the polymers. Surmounting these limitations
can also be the key to find unquestionable thermal percolation
effects as we see in Sec. VII.
VII. PERSPECTIVE AND NEW TRENDS
As discussed above, thermal percolation in typical two-phase
composites has been studied theoretically and experimentally by
several research groups. This phenomenon is present as a sudden
increase in thermal conductivity or as a slope steepening in the
thermal conductivity curve as a function of the concentration of
the fillers. It has been proposed that to guarantee the observation
of the thermal percolation phenomenon, the composites must
fulfill two correlated requirements: the first refers to the high
enough contrast in thermal conductivity between the filler and the
matrix, and the second to reduce as much as possible the interparticle thermal resistance.17 Particularly, in the case of thermal conductivity, this contrast is at most of the order of 105, while in the
electrical conductivity, this contrast easily achieves values ranging
from 1012 to 1016. Under these conditions and considering the
charge carrier modes involved in electrical conductivity already discussed before, electrical percolation is a phenomenon that can be
easily observed. In contrast, the relatively small difference between
thermal conductivities of the polymer and filler indicates that it is
crucial to reduce as much as possible the interfacial thermal interparticle resistance in order to avoid the shielding of the effects
induced by the formation of the percolative network.
Several strategies can be followed to attain such objective. First
of all, by improving the contact between the particles. This possibility has been explored in the literature by assembling an ordered
nano- or microstructure in which the filler particles touch each
other, and after that, the polymerization of the composites was
induced. The arrangement can be generated by external electric or
magnetic fields depending on the kind of used fillers. Magnetic
responsive fillers are some of the best options; however, the results
reported have not provided enough evidence of conclusive thermal
percolation phenomena.16,29,114 In order to improve the interparticle contact, choosing the adequate fillers, a combination of electric
and magnetic field, as well as applying successive DC/AC fields and
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made of graphite nanoplatelets of different aspect ratios and dispersed in the polylactic acid (PLA) matrix after an annealing treatment. They found that the fillers with the smallest lateral size
produced a significant improvement in the electrical conductivity
and in the percolation threshold. In contrast, composites with
larger fillers or at very high concentrations did not show significant
changes, even considering different aspect ratios or giving an
annealing treatment.
In the case of the thermal percolation thresholds, as can be
seen in Table I, the thresholds can vary in a very wide range of
concentrations, as low as wtc ¼ 0:00035 for MWCNTs in PDMS103
to higher values as wtc ¼ 0:38 for CIP micrometric spheres in a
polyester resin matrix.11 These works suggest a trend consisting of
the observation of thermal percolation thresholds for lower concentrations when the aspect ratio of the fillers is increased. The results
shown in Table I indicate that generally for 1D fillers, the thermal
threshold seems to appear at lower concentrations.87 Otherwise, the
case of 2D fillers is much more complicated, in particular for graphene in epoxy, the thermal threshold is in the range from 0.17106
to 0.3.14 These results show that it is difficult to unambiguously
compare the values of the thermal percolation thresholds, even for
materials of similar composition, if the preparation conditions as
well as the properties of the forming phases are not well defined.
Also, there has been various reports of composites in which the
corresponding thermal threshold is not observed.31 Therefore, it
can be inferred that the observation of the thermal percolation
depends on the careful consideration and fine tuning of the parameters involved in the preparation of the samples.14 Additionally, the
introduction of additional phases, arrangement of the fillers, or
morphology manipulation on these binary systems during the manufacturing processes, possibly exhibit or shield the percolation
effects on the thermal conductivity, as will be shown in the last
section of this Perspective.
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FIG. 5. Schematic diagram of some of the main new trends to develop a new
generation of composites with improved thermal properties.
obtaining high-performance materials. In Fig. 5, we present a
scheme of some of the main new trends that the research community is following, in order to develop a new generation of composites with better features and capabilities in thermal transport and
its management. We will discuss some examples of recent and relevant works following these novel strategies.
The study performed by Kargar et al.14 exemplifies the synergistic effects of fillers and interfacial thermal resistance. These
authors have shown that using 2D fillers embedded in an epoxy
matrix, such as graphene or hexagonal boron nitride, a high contrast in thermal conductivity between the matrix and the filler can
be reached. They found that both composites present a clear
thermal percolation threshold. The graphene composite showed the
thermal percolation threshold at a loading fraction of 20 vol. %,
well above the electrical one, located around 10 vol. %. The key idea
here is that the thermal transport properties of the high-loading
composites are strongly influenced by the cross-plane thermal conductivity of the quasi-two-dimensional fillers. This behavior has
been investigated and is known as hybridization, which is a synergistic effect generated by a set of fillers having different morphologies and interactions with the matrix, which cooperate in such a
way that at certain proportion of the constituents, the thermal conductivity of the composite is enhanced, and the percolation effects
can be observed. In the case of graphene when having a second
filler, this last one can prevent the agglomeration of graphene,
helping to interconnect the graphene flakes and favoring the heat
transport along them.122
Recently, a novel development in the fabrication of composite
materials has emerged, and it is helping to shed some light on the
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with increasing intensities, before and during the first stages of
polymerization, can be helpful in removing the remnant polymer
between touching particles.114 A further step in improving the
contact between fillers involves the generation of high conductivity
links between particles. This has been explored by diverse authors
in special for carbon nanostructures.115,116 In the cases discussed
up to this point, considering binary systems, the links should be of
the same material as the one of the fillers. Bonding via
boron-induced nanojunctions has demonstrated to be an adequate
method to join nanotubes.115 A similar method to join graphene
sheets using nanotubes has shown to be performed successfully.116
Using materials of these characteristics for assembling the fillers
would be an excellent option in developing a high thermal conductivity network. If the restriction of the binary system is not
required, the linking materials117 and the possibilities are much
broader as will be discussed later in this article.
It is also important to mention that in the case of carbon
nanostructures, the availability of fillers of a broad range of geometries and the number of layers opens an ample spectrum for the
development of new composites. In particular, the use of singlelayer graphene and single-walled carbon nanotubes against the possibility of using multilayered/multiwalled structures of the same
kind has been explored. It has been shown in the literature that the
interparticle thermal resistance between carbon nanostructures
decreases substantially as a function of the number of layers.118,119
This can be the cause of remarkable differences in percolation
thresholds in MWCNTs and SWCNTs, which have been reported
in the literature (see Table I), even though the polymeric matrices
are different, and therefore up to now, they can only be interpreted
as possible trends. The tendency of a more efficient transfer of heat
has been observed in smart materials, when carbon nanotubes
become thicker.120 However, it has been shown that contact resistance also depends on the tube curvature and that a balance
between the number of walls and curvature can be necessary to
minimize thermal interparticle resistance.119 Recently, it has been
shown that interparticle thermal resistance between carbon nanothreads can be diminished substantially when comparing with
CNTs.121 Additionally, it has been found that introducing interfacial crosslinkers can enhance the performance of nanothreads as
fillers with a great potential in the development of highperformance conductive networks in composites.121
Systematic studies, using low thermal conductivity matrix with
a variety of carbon nanostructures, increasing their number of
layers/walls, inducing functionalization, as well as the arrangement
of the fillers in a polymer matrix, can have a great impact in revealing the nature of thermal percolation phenomena as well as the
tendency of these kind of systems to produce materials with lower
thermal conductivities as the values expected. Therefore, it can be
inferred that by choosing the appropriate matrix, achieving a high
contrast between the thermal conductivities of the matrix and
fillers, and reducing the interparticle thermal resistance to improve
the connectivity, it would be possible to observe the thermal percolation threshold. Nevertheless, achieving these conditions in typical
two-phase composites is complicated, and even in some cases, it is
impossible. This situation has motivated the development of a new
generation of materials where the design of multiphase composites
with 2D fillers and sophisticated matrices is fundamental in
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Recently, new highly promising materials based on carbides
and nitrides of transition metals, known as MXenes, have emerged.
These materials have a graphene-like structure and can mark the
beginning of a revolution in the development of high-conductivity
polymeric composites. In a 2D flake of MXene, n + 1 layers of early
transition metals (M) are interleaved with n layers of carbon or
nitrogen (X), with a general formula of Mn+1XnTx, where Tx represents the surface terminations, such as O, OH, F, and/or Cl, which
are bonded to the outer M layers. Some tens of species of MXenes
have been developed based on different M, and X elements and different proportions between them. These materials present a high
specific surface area, a large interlayer spacing, hydrophilicity, and
abundant highly active surface sites. These characteristics make
MXenes ideal fillers for developing high-performance polymeric
composites that in contrast to graphene can demand minimal functionalization because of the compatibility between the functional
groups of the Mxene (O, OH, and F) with the polymeric
matrix.128,129
Among the MXenes, the best known are the Ti3C2Tx materials
that have a thermal conductivity that is, at room temperature, only
ten times the thermal conductivity of a polymer (3 W/mK). A strategy to generate a polymeric matrix with high conductivity is to take
advantage of the highly convenient properties of Ti3C2Tx, which
allows us to introduce a high thermal conductivity phase. Wang
et al., using this approach, developed a novel composite formed by
micro- and nanometric particles of silver in epoxy resin with a
small quantity of Ti3C2Tx.130 They showed that the thermal conductivity of the composite epoxy/Ag/MXene increased around 25%
higher than that of Ag/epoxy composites. The thermal conductivity
enhancement in epoxy/Ag/MXene composites is due to a strong
interaction of the MXene flakes with Ag particles. When studying
the effect of the variation of the MXene content on the composites,
these authors found thermal and electrical percolative effects
around 0.12 vol. % (see the supplementary information of the
article of Wang et al.).130 Above this concentration value, the
thermal conductivity decreases due to a viscosity increment in the
pre-cured polymer, which produces a composite with undesirable
characteristics.
It is important to mention that one of the limitations of
MXenes is their low thermal conductivity; however, the development of high thermal conductivity MXenes has been predicted theoretically. In particular, it has been found using first principles
calculations that Sc2CF2 can reach a high thermal conductivity of
around 470 W/mK.131 Given the large variety of MXenes, it is
expected that in the coming years, new developments would
emerge, which can be useful in the development of highperformance polymeric composites.
08 August 2023 20:26:24
complex phenomena of thermal percolation. The method is based
on adding another phase and inducing order in the composite.
One of the most promising methodologies to produce this kind of
composites with a clear percolation threshold is the one proposed
by Huang et al.123 They devised a simple way to attach graphene
platelets at the interface of an immiscible PCL/PLA blend and
found that at 0.53 vol. % of fillers charge, the thermal conductivity
of the composite is about four times larger than that of the pristine
PCL/PLA polymer blend. In this case, the thermal (electrical) percolation threshold is observable at 0.11 vol. % of graphene platelets.
These results indicate that the co-continuous structure of the PCL/
PLA polymers can accommodate the graphene platelets, remaining
aligned along with the interface between the PCL and PLA phases,
which reduces the interfacial thermal resistance and results in a
composite with high thermal conductivity.
In the same way, more sophisticated systems are currently
being developed aimed to assemble 2D and 3D structured composites with high thermal conductivity. These methodologies consist in
manufacturing polymeric scaffolds to use them as matrices to
insert different nanostructures inside them. This approach is one of
the most attractive possibilities for developing high-performance
composites, aiming to improve heat transfer at low concentrations
of fillers, reducing the interfacial thermal resistance, and controlling
the thermal conductivity.124–127
Zhu et al. reported the development of a 2D nanocomposite
made of layered boron nitride nanosheets interconnected with
nano-fibrillated cellulose, where the nanosheets play the role of a
thermally conductive network.124 These nanocomposites have high
thermal conductivity, 145.7 W/mK at 50 wt. % boron nitride nanosheets in the nano-fibrillated cellulose matrix, with a marked
thermal percolation threshold under 10 wt. % of nanosheets. These
results are a consequence of the aligned and layered structure of the
nanosheets. In this case, heat transfer occurs preferentially along
the direction parallel to the interface formed by the nanosheets and
the cellulose, exhibiting a large surface area.
Otherwise, Chen et al.127 developed a cellulose nanofiber supported 3D interconnected boron nitride nanosheet aerogels in epoxy.
These composites present a thermal conductivity enhancement of
about 1400% at a low boron nitride nanosheet loading of 9.6 vol. %
with low electrical conductivity. At 1.2 vol. %, a clear transition is
present in the form of a slope change in the thermal conductivity
curve as a function of the concentration. Both effects are related to
the boron nitride nanosheets’ self-assembling on the cellulose nanofiber surface. This configuration results in a large contact area
between the nanosheets inducing a decrease in the interfacial
thermal resistance. Additionally, the epoxy nanocomposites are still
highly insulating, having an electrical resistivity of 1015 Ω cm.
In addition, using 3D scaffolds, Vu et al. developed a vertically
aligned aerogel formed by SiC scaffold in an epoxy resin.126 These
composites showed very high cross-plane thermal conductivity of
14.32 Wm−1 K−1 at a SiC sheet content of 3.71 vol. %. This provided a record-high thermal conductivity efficiency of 1651% per
1 vol. % filler. The composites clearly exhibit an impressive thermal
percolation threshold at low concentration of 1.78 vol. % of the SiC
sheet. Due to its high thermal conductivity with comparatively low
electrical conductivity, these novel composites have immense
potential in thermal management in microelectronic applications.
PERSPECTIVE
VIII. CONCLUDING REMARKS
Thermal and electrical percolations are effects that depend on
the formation of a network of high conductivity fillers in a matrix,
the geometry of these fillers, and on the physical mechanisms that
govern the movement of the corresponding carriers. Thermal percolation is limited by the small contrast between the filler and the
matrix and by the interparticle thermal resistance. Two strategies
have been shown to provide good results to make thermal
131, 230901-12
Journal of
Applied Physics
percolation visible. The first one consists in generating synergistic
effects mediated by the fillers with the matrix, which can cooperate
in such a way that at a certain proportion of the constituents, the
thermal conductivity of the composite is enhanced, and the percolation effects can be observed. This could be achieved by using 2D
structures as fillers favoring the heat transport along the plane. The
second strategy would be to emulate the electrical case and develop
manufacturing processes in which the thermal contrast between the
filler and the matrix is enhanced.
In the case of the binary systems, in which there is controversy
about the possibility of observing thermal percolation effects,
unambiguous comparison of the results can only be performed if
the manufacturing processes are standardized. In the meanwhile,
based on the more recent advances in the area, it can be inferred
that the observation of thermal percolation depends on the careful
fine tuning of the parameters involved during the preparation of
the samples. The presence of additional phases, arrangement and
functionalization of fillers, or induced morphology in binary
systems during the manufacturing process could have been decisive
in shielding or enhancing thermal percolation effects.
ACKNOWLEDGMENTS
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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08 August 2023 20:26:24
This work was partially funded by Project No.
SRE-AMEXCID-2016-1-278320 and Cinvestav Scientific Research
and Technological Development Fund No. 98. A. P. Franco-Bacca
and F. Cervantes-Alvarez are grateful to SRE-AMEXCID2016-1-278320 for financial support.
PERSPECTIVE
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