Percolation Modeling of Conductance of Self-Healing Composites
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Percolation Modeling of Conductance of Self-Healing Composites 
Alexander Dementsov and Vladimir Privman * 
Center for Advanced Materials Processing, and 
Department of Physics, Clarkson University, 
Potsdam, New York 13699, USA 
PACS: 62.20.Mk, 46.50.+a, 72.80.Tm 
Key Words: self-healing materials, composites, conductance, material fatigue 
Abstract 
 We explore the conductance of self-healing materials as a measure of the material 
integrity in the regime of the onset of the initial fatigue. Continuum effective-field 
modeling and lattice numerical simulations are reported. Our results illustrate the general 
features of the self-healing process: The onset of the material fatigue is delayed, by 
developing a plateau-like time-dependence of the material quality. We demonstrate that 
in this low-damage regime, the changes in the conductance and similar transport/response 
properties of the material can be used as measures of the material quality degradation. 
________________________   
* www.clarkson.edu/Privman 
Physica A 385, 543-550 (2007)    arXiv:0704.0935 
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 Recently a significant research effort has been devoted to the design of “smart 
materials.” In particular, self-healing composites [1-10] can restore their mechanical 
properties with time or at least reduce material fatigue caused by the formation of 
microcracks. It is expected that microcracks propagating through such materials can 
break embedded capsules/fibers which contain the healing agent — a “glue” that 
heals/delays further microcrack development — thus triggering the self-healing 
mechanism. In recent experiments [1,7-10], an epoxy (polymer) was studied, with 
embedded microcapsules containing a healing agent. Application of a periodic load on a 
specimen with a crack, induced rupture of microcapsules [1]. The healing glue was 
released from the damaged microcapsules, permeated the crack, and a catalyst triggered a 
chemical reaction which re-polymerized the crack.  
 Defects of nanosizes are randomly distributed throughout the material. 
Mechanical loads during the use of the material then cause formation of craze fibrils 
along which microcracks develop. This leads to material fatigue and, ultimately, 
degradation. Triggering self-healing mechanism at the nanoscale might offer several 
advantages [10] for a more effective prevention of growth of microcracks. Indeed, it is 
hoped [10] that nanoporous fibers with glue will heal smaller damage features, thus 
delaying the material fatigue at an earlier stage than larger capsules [1,9] which basically 
re-glue large cracks. Furthermore, on the nanoscale, the glue should be distributed/mixed 
with the catalyst more efficiently because transport by diffusion alone will be effective 
[10,11], thereby also eliminating the need for external UV irradiation [9], etc. 
 Theoretical and numerical modeling of self-healing materials are only in the 
initiation stages [10,12,13]. Many theoretical works and numerical simulations [14-17] 
consider formation and propagation of large cracks which, once developed, can hardly be 
healed by an embedded nano-featured capsules. Therefore, we have proposed [10] to 
focus the modeling program on the time dependence of a gradual formation of damage 
(fatigue) and its manifestation in material composition, as well as its healing by 
nanoporous fiber rupture and release of glue. 
 We will shortly formulate rate equations [10] for such a process. In addition to 
continuum rate equations for the material composition, numerical modeling can yield 
useful information on the structure, and, later in this article, we report results of Monte 
Carlo simulations. We also point out that the calculated material composition and 
structure must be related to macroscopic properties that are experimentally probed. The 
relation between composite materials composition and properties is an important and 
rather broad field of research [18].   
 Recently, it has been demonstrated experimentally [19] that a rather dilute 
network of carbon nanotubes, incorporated in the epoxy matrix, can provide a percolation 
cluster the conductance of which can not only reflect the degree of the fatigue of the 
material but also shows promise for probing the self-healing process. The main purpose 
of the present article is to initiate continuum effective-field, as well as numerical lattice 
modeling of percolation properties for materials with self-healing. 
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 Different transport properties can be used to probe material integrity (damage 
accumulation due to the formation of cracks). These include thermal conductivity 
[20,21], photoacoustic waves [22,23], electrical conductivity [19,24-27]. Generally, 
transport properties can be highly nonlinear as functions of the degree of damage. For 
example, the conductance can sharply drop to zero if the conducting network density 
drops below the percolation threshold. However, for probing the initial fatigue, in the 
regime of low levels of damage, one expects most transport properties to decrease 
proportionately to the damage.  
 Let us summarize our recently proposed model [10] of the material composition 
in the continuum rate equation approach. We denote by ( )u t  the fraction of material that 
is undamaged, by ( )g t  the fraction of material consisting of glue-carrying capsules, by 
( )d t  the fraction of material that is damaged, and by ( )b t  the fraction of material with 
broken capsules, so that we have 
( ) ( ) ( ) ( ) 1u t g t d t b t+ + + = . (1)
We consider the regime of small degree of degradation of the material, i.e., we assume 
that at least for small times, t , we have ( ) 1u t ≈ , whereas ( )d t , ( )b t  and ( )g t  are 
relatively small. In fact, (0) 0b = . 
 For the purposes of simple modeling, we assume that on average the capsules 
degrade with the rate P , which is somewhat faster than the rate of degradation of the 
material itself due to its continuing use (fatigue), p , i.e., P p> . The latter assumption 
was made to mimic the expected property that a significant amount of microcapsules 
embedded in the material may actually weaken its mechanical properties and, were it not 
for their healing effect, reduce its usable lifetime (though it was noted [1,11] that a small 
amount of microcapsules actually increased the epoxy toughness); the density of the 
“healing” microcapsules is one of the important system parameters to optimize in any 
modelling approach. Thus, we approximately take 
( ) ( )g t Pg t= − ,           yielding         ( ) (0) Ptg t g e−= . (2)
One can write a more complicated rate equation for ( )g t , but the added, nonlinear terms 
are small in the considered regime.  
 However, for the fraction of the undamaged material, we cannot ignore the 
second, nonlinear term in the relation 
( ) ( ) ( )u t pu t H t= − + . (3)
Here we introduced the healing efficiency, ( )H t , which can be approximated by the 
expression 
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( ) ( ) ( ) (volume healed by one capsule)H t d t g t∝ × . (4)
The healing efficiency is proportional to the fraction of glue capsules, as well as to the 
fraction of the damaged material, because that is where the healing process is effective. 
The latter will be approximated by ( ) 1 ( )d t u t≈ − , which allows us to obtain a closed 
equation for ( )u t . Indeed, in Eq. (3) we can now use 
( ) [1 ( )]PtH t Ae u t−= − . (5)
The healing efficiency is controlled by the parameter 
(0) (volume healed by one capsule)A g∝ × . (6)
 While the model just formulated is quite simple, “minimal,” and many 
improvements can be suggested, it has the advantage of offering an exact solution, 
1(1 ) ( )( ) (0)
Pt Pt P
pt AP e pt AP e P p AP eu t u e Ae d e
− − − − −−− − − − + − − −= + ∫ . (7)
This result is illustrated by the solid curves in Fig. 1, where we set (0) 1u =  for simplicity. 
The main feature observed is that even when the healing efficiency parameter A  is rather 
small (here 0.02) but nonzero, the decay of the fraction of the undamaged material is 
delayed for some interval of time. This represents the self-healing effect persisting until 
the glue capsules are used up. 
 Equation (6) suggests that an important challenge in the design of self-healing 
materials will be to have the healing effect of most capsules cover volumes much larger 
than a capsule, in order to compensate for a relatively small value of (0)g , which is the 
fraction of the material volume initially occupied by the glue-filled capsules. Since the 
glue cannot “decompress,” its healing action, after it spreads out and solidifies, should 
have a relatively long-range stress-relieving effect in order to prevent further crack 
growth over a large volume. 
 The present simple continuum modeling cannot address the details of the 
morphological material properties and glue transport; numerical simulations will be 
needed to explore this issue. It is interesting to note that most material properties will 
also depend on the specific morphological assumptions; their derivation within various 
approximation schemes, will require more information than that provided by the average, 
“effective field” approximate “materials quality” measures such as ( )u t . Here we are 
interested, specifically, in the material conductivity. 
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Figure 1: The solid curves illustrate the fraction of the undamaged 
material, ( )u t , calculated according to Eq. (7) with 0.02A = , 0.003p = , 
0.008P =  — the top curve, and without self-healing: 0A = , 0.003p =  
— the bottom curve. The dashed curves illustrate the behavior of the 
mean-field conductance for these two cases, respectively, with the 
conductance decreasing slower with self-healing present, eventually 
reaching zero at the percolation transition at 0.5u = , see Eq. (8). 
________________________________________________________________________ 
 Since our numerical calculations reported below, assume square lattice 
(coordination number 4z = ) bond percolation, the conductance, ( )G t , shown as the 
dashed curves in Fig. 1, was calculated by using the bond-percolation mean-field formula 
[28],  
[ ]1 ( )( ) max 1 , 0 max 2 ( ) 1, 0
G t z u t
− = − = − − 
. (8)
Here the conductance is normalized to have ( 0) 1G t = = , and for simplicity we assumed 
that the bond percolation probability is given by ( )u t , i.e., we consider the situation when 
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the conductance of the healthy/healed material is maximal, whereas the other areas do not 
conduct at all. In the regime of a relatively low damage, which is likely the only one of 
practical interest, and also the one where the mean-field expressions are accurate, we note 
that the conductance provides a convenient, proportional measure of the material 
degradation, 
[ ](0) ( ) (0) ( )G G t K u u t− − , (9)
where the constant /( 2)K z z= −  depends on the microscopic details of the material 
conductivity. Here 2K = , but in practical situations this parameter can be fitted from 
experimental data.  
 In order to further explore the self-healing process, we carried out Monte Carlo 
simulations on square lattices of varying sizes, with periodic boundary conditions. All the 
bonds in the lattice were initially present, and the healing cells were a small fraction of 
the lattice (square) unit cells, distributed uniformly over the lattice, with the built in 
constraint that they do not touch each other (including no corner contact). Our 
simulations reported here, were carried out with this fraction (0) 0.15g = , i.e., the 
probability that a cell was designated glue-carrying was 15%. 
 At times 0t > , bonds were randomly broken with the probability (rate per unit 
time) p  for ordinary bonds, and ( )P p>  for bonds of healing cells (those with “glue”). 
If at least two bonds are broken in a healing cell, the glue leaks out and restores broken 
bonds. Here we assumed local healing, with the glue only spreading to the 8 neighboring 
cells before solidifying, thus restoring all the 24 bonds of the 3 3×  square group of cells 
that includes the healing cell as its center. Furthermore, once the glue leaks out, the 
healing cell becomes inactive, but its bonds still have the larger probability, P , to be re-
broken. 
 We further assumed that all the original or healed bonds have the same, maximal 
conductance, whereas all the broken bonds do not conduct at all. Since the periodic 
boundary conditions induce a torus geometry, the conductance of a system of N N×  
square cells, was calculated between two parallel lines, each N  lattice bonds long (which 
were really circles due to periodicity), at the distance / 2N  from each other, by using a 
standard algorithm [29]. Note that these two lines are connected by two equal-size system 
halves (we took N  even for simplicity), and the conductivities via these two pathways 
were included in the overall calculation. Our typical results are illustrated in Fig. 2, where 
we plot the number (fraction) of unbroken bonds, ( )n t , with initially (0) 1n = , as well as 
the normalized conductance, ( )G t . 
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Figure 2: The solid curves illustrate the fraction of the unbroken bonds, n  
(top curve), and the normalized conductance, G  (bottom curve), for the 
following choice of the parameters: 32N = , 0.003p = , 0.008P = , and 
the initial fraction of the healing cells 15%. The dashed curves illustrate 
similar results with the same parameters but with no healing cells. The 
data were averaged over 40 Monte Carlo runs for the case with self-
healing, and over 20 runs for the case without self-healing. 
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 The two lower curves in Fig. 2, showing the conductance with and without self-
healing, do not vanish at finite times due to finite-size effects [30]. In fact, without self-
healing the simulation of the conductance is just the ordinary numerical evaluation for 
square-lattice uncorrelated bond percolation. As the lattice size increases, the finite-lattice 
conductance, as well as other percolation properties, develop critical-point behavior at the 
percolation transition that occurs, for this particular morphology, when the fraction of the 
broken bonds reaches 0.5. Thus, the conductance, ( )G t , shows significant lattice-size 
dependence even without self-healing, as illustrated in Fig. 3, whereas the fraction of the 
unbroken bonds, ( ) ptn t e−= , not shown in the figure, has no size dependence.  
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Figure 3: Size dependence of the conductance without self-healing, with 
otherwise the same parameters as in Fig. 2. From top to bottom, the results 
shown correspond to lattice sizes 8N = , 16, 32. The data were averaged 
over 500, 100 and 20 Monte Carlo runs, respectively. Note that the 
percolation transition occurs at (ln 2) 231/t p= , from which time value 
on, the N →∞  limiting value of the conductance is zero. 
________________________________________________________________________ 
 Results with self-healing, for the size dependence of the conductance, are shown 
in Fig. 4. We point out that the fraction of the unbroken bonds also has some variation 
with N  in this case. However, the differences in n -values are too small to be displayed 
in the figure. (The size dependence of ( ; )n t N  might become quite pronounced and 
interesting when the self-healing process is non-local, as discussed in [10].) For most 
practical purposes the self-healing process will be of interest as long as the material 
fatigue is small, i.e., in the regime of the initial plateau that develops in properties such as 
( )n t , or ( )u t  in the continuum model. Therefore, we did not attempt to study in detail the 
percolation transition for the conductance, which in this case should be a variant of some 
sort of a correlated bond percolation, though the universality class is likely not changed.  
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Figure 4: Size dependence of the conductance (the solid curves) with self-
healing, with the same parameters as in Figs. 2 and 3. From top to bottom, 
the solid curves correspond to lattice sizes 8N = , 16, 32. The data were 
averaged over 2000, 400 and 40 Monte Carlo runs, respectively. The 
dashed curve shows the fraction of the healthy bonds, the size-dependence 
of which leads to variations too small to be shown on the scale of the 
vertical axis in this plot (the curve shown is for 32N = ). 
________________________________________________________________________ 
 Let us now discuss the extent to which the continuum model can fit the results for 
the lattice model. Now, without self-healing, the mean-field approximation can provide 
rather accurate results for the conductance, except perhaps right near the percolation 
transition [31], as illustrated in Fig. 5. With self-healing, the situation is less consistent. 
The numerical lattice-model result (for our largest 32N = ), is compared to the 
continuum model expression with varying A , in Fig. 6.  
 While, especially for larger values of A , the continuum model curves show all 
the features of the self-healing conductance, including the initial drop followed by 
“shoulder,” the overall agreement is at best only qualitative. Thus, using A  as the 
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adjustable parameter, one cannot achieve a quantitatively accurate fit of the lattice-model 
data. We note that the continuum model considered, should be viewed as “minimal” in 
that it represents the simplest possible set of assumptions that yield the self-healing 
behavior and also offer exact solvability. Specifically, the continuum model assumes that 
the initial fraction of the glue-carrying capsules is very small, and the finite healing 
efficiency is achieved by each cell healing a large volume, see the discussion in 
connection with Eq. (6). On the other hand, to have a full-featured self-healing behavior, 
in the lattice case with short-range healing, we had to take the initial fraction of the 
healing cells at least of order 10% (we took 15% in our simulations). Thus, for better-
quality fit the continuum model will have to be modified, and will be more complicated, 
involving more than one quantity (now we only consider ( )u t , for which we obtain a 
closed equation) and likely nonlinear equations. We plan to consider this in our future 
work. 
________________________________________________________________________ 
Figure 5: The dashed curve shows the mean-field approximation for the 
conductance without self-healing, calculated according to Eq. (8), with 
0.003p = . The solid curve is the 32N =  lattice-model result, as in Fig. 3. 
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Figure 6: The dashed curves show the conductance calculated according 
to the continuum model, for 0.003p =  and 0.008P = , with, from bottom 
to top, 0.01A = , 0.02, 0.03, 0.04, 0.05. The solid curve is the 32N =  
lattice-model result with self-healing, as in Fig. 4. 
________________________________________________________________________ 
 In summary, we explored the conductance of self-healing materials, with several 
assumptions that include short-range healing, conductivity being directly proportional to 
the local material “health,” and the use of simple effective-field continuum model, as 
well as two-dimensional square lattice numerical simulations. While our assumptions 
may have to be modified for different, more realistic situations, our results illustrate the 
general features of the self-healing process. Specifically, the onset of the material fatigue 
is delayed, by developing a plateau-like time-dependence of the material quality at initial 
times. In this regime, the changes in the conductance, and likely in most other 
transport/response properties of the material that can be experimentally probed, measure 
the material quality degradation proportionately, whereas for larger damage at later times, 
transport properties may undergo dramatic changes, such as the vanishing of the 
conductance in our case, and they might not be good measures of the material integrity. 
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 We wish to thank Dr. D. Robb for bringing reference [19] to our attention and for 
helpful discussions, and we acknowledge support of this research by the US-ARO under 
grant W911NF-05-1-0339 and by the US-NSF under grant DMR-0509104. 
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