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Tuning the electrical percolation threshold of polymer nanocomposites with rod-like nanofillers
Polymer 97 (2016) 295e299
Contents lists available at ScienceDirect
Polymer journal homepage: www.elsevier.com/locate/polymer
Short communication
Tuning the electrical percolation threshold of polymer nanocomposites with rod-like nanofillers Vijay Kumar 1, Amit Rawal* Department of Textile Technology, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 January 2016 Received in revised form 13 May 2016 Accepted 14 May 2016 Available online 15 May 2016
In this communication, a stochastic approach based upon Komori and Makishima’s work has been used for predicting the electrical percolation threshold of cylindrical nanofillers with three-dimensional (3D) spatial orientations in a typical nanocomposite system. Specifically, the proposed model was able to predict the volume fraction based percolation threshold of nanofillers with a wide range of aspect ratios (10e1000), which was substantiated with a variety of experimental data sets obtained from the literature. The anisotropic behavior of 3D aligned nanofillers was successfully introduced via in-plane and outof-plane orientation distributions. The percolation threshold values of nanofillers were also found to be comparable with those obtained from the stochastic model that has incorporated the excluded volume effect. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Percolation threshold Nanofiller Stochastic
1. Introduction Nanoscale conductive particles, such as carbon nanotubes, carbon nanofibers, metallic nanowires, graphene flakes, etc. have gained considerable attention in the field of nanocomposites due to their excellent electrical properties. Such characteristics are the gateway to replace monolithic metals in numerous emerging applications including flexible microelectronics, e-paper, organic light emitting diodes, sensors, antistatic coatings and touch screens [1e5]. These targeted applications can be practically achievable when maximum electrical conductivity of nanocomposites is attained with the lowest possible filler concentration, which inevitably reduces the cost. Enhancing the electrical conductivity of the nanocomposites in a cost-effective manner is triggered by means of percolation threshold that describes the minimum loading of conducting nanorods in a polymeric/colloidal medium, which are capable of forming a connected network through rod-rod contacts. The transition in the electrical conductivity at the percolation threshold level has been observed to be several orders of magnitude [6]. Alternatively, the concentration of nanorods at which the material exhibits an abrupt transition from insulator to a conductor
* Corresponding author. Fraunhofer Institute of Industrial Mathematics (ITWM), Fraunhofer-Platz 1, D-67663 Kaiserslautern, Germany. E-mail addresses: [email protected], [email protected] (A. Rawal). 1 Present address: University of Borås, SE-501 90 Borås, Sweden. http://dx.doi.org/10.1016/j.polymer.2016.05.041 0032-3861/© 2016 Elsevier Ltd. All rights reserved.
is characterized by the percolation threshold. In conducting nanocomposites containing nanofillers, the percolation threshold is dependent upon the filler attributes including aspect ratio, electrical properties, and dispersity in filler properties [7e9]. Percolation phenomena in polymer nanocomposites have been critically investigated through numerical [10e18], analytical [19e21] and semi-empirical modeling [22e25] techniques. Some of these modeling techniques were compared and subsequently, combined together to predict the percolation threshold of nanocomposites. For instance, the excluded volume model is a universally accepted continuum analytical model that has been applied in combination with Monte Carlo simulations for predicting the percolation threshold of a wide variety of fillers in nanocomposites [10,13,16,18,19]. However, the classical excluded volume model has been primarily developed for fillers with large aspect ratios, which can be computationally intensive for numerical modeling techniques [10]. Furthermore, the nanofillers with modest aspect ratios (<100) either grown inherently or as a result of sonication, are present in nanocomposites [7,18,26]. In addition, the excluded volume model considers the random distribution of fillers but in reality, these fillers can align in preferred in-plane or out-of-plane directions as a result of polymer processing techniques (extrusion, molding, drawing, fiber spinning) [27] or due to the application of electrical or magnetic fields [28]. Therefore, it is of paramount importance that the 3D orientation distribution of fillers should be considered for predicting the percolation threshold. In the present work, the analytical model to predict the
296
V. Kumar, A. Rawal / Polymer 97 (2016) 295e299
percolation threshold of 3D aligned nanofillers covering a wide range of aspect ratios based upon stochastic approach has been proposed. The stochastic approach is essentially based upon Komori and Makishima’s geometrical probability approach [29] which has been successfully applied in the past for predicting various properties of different types of fibrous systems [30e33]. The theoretical results have been compared with the other stochastic approach that accounted for the excluded volume effect [34]. The analytical model has been validated with a variety of experimental data sets, obtained from the literature.
2. Theoretical analysis For a disordered system such as nanocomposites to be percolative, the fundamental criterion is the continuity to transport electrons should be maintained, which can be modulated through rod-rod contacts, as shown in Fig. 1. Although, the thin layer of the polymer matrix in the range of few nanometers between the nanofillers can still allow the passage of electrons via quantum mechanical tunneling mechanism [35]. Neglecting the tunneling effect in this analysis due to the fact that the electrical conductivity of the network primarily results from rod-rod contacts [36]. Therefore, the rod-rod contacts of nanofillers which are responsible for (dis)continuity in the electron path should be critically analyzed. Considering the orientation of each cylindrical rod defined by a pair of orientation angles ðq; 4Þ in a spherical coordinate system, where q and 4 are polar and azimuthal angles, respectively. The probability density function or the orientation distribution function, Uðq; 4Þ is the probability of a rod lying in an infinitesimal range of angles q and q þ dq, and 4 and 4 þ d4 is given by Uðq; 4Þsin qdqd4, which satisfies the following normalization condition.
Zp 0
d4
Zp
sin qUðq; 4Þdq ¼ 1
(1)
0
Komori and Makishima [29] described a stochastic approach by assuming the rods to be straight cylindrical entities with constant length and diameter and the rod-rod contact condition would only occur if the center of mass of rod A of defined orientation ðq; 4Þ enters into the neighbourhood region of rod B with orientation ðg; zÞ, as shown in Fig. 1. Based on defined number of contacts in a volume (V) along with the probability of formation of a contact, the mean distance between the rod-rod contacts (bKM ) is given by [29],
bKM ¼
V 2DLI
where Jðq; 4Þ ¼
Rp 0
(2)
dz
Rp 0
I¼
Rp 0
dq
Rp 0
Jðq; 4Þsin qUðq; 4Þd4;
sin cðq; 4; g; zÞUðg; zÞsin gdg
and
sin c ¼ ½1 fcos q cos g þ cosð4 zÞsin q sin gg where D is the nanofiller diameter, L is the total length of the nanofiller defined in volume V, I is an orientation parameter defining the orientation characteristics of nanofillers in the assembly, c is the angle between the two axes of nanofillers having defined types of orientation distributions, Uðq; 4Þ and Uðg; zÞ. Also, the volume fraction of rods (m) can be calculated through the following equation. 2 1=2
m¼
pD2 L
(3)
4V
Combining Equations (2) and (3),
bKM ¼
pD 8I m
(4)
It is worth mentioning that Komori and Makishima [29] have not accounted for the changes in the probability of a contact with the successive contacts, i.e. excluded volume effect has been neglected [34]. In other words, an existing contact reduces the effective contact length of a rod, which diminishes the probability of formation of new contacts. On the other hand, the existing rodrod contact also tends to reduce the free volume of the nanofiller assembly that results in enhancing the chances of making new contacts [37]. Therefore, the excluded volume effect is a trade-off between the reductions in effective contact length and that of free volume [38]. In reality, this effect may not significantly alter the number of rod-rod contacts as the critical volume fractions for attaining desired percolation thresholds are often realized at significantly lower levels. Nevertheless, we also calculated the mean distance between the contacts based upon Pan’s approach [34] that considered the excluded volume effect by modifying and extending Komori and Makishima’s model [29], as shown below.
bPan ¼
ðp þ 4mjÞD 8mI
where Kðq; 4Þ ¼
Rp 0
(5)
j¼
Rp Uðg;zÞsin g dg 0 dz sin cðq;4;g;zÞ
Rp 0
Rp dq 0 d4Jðq; 4ÞKðq; 4ÞUðq; 4Þsin q;
Next, it is pertinent to note that each rod should make atleast
Fig. 1. A cartoon depicting percolating and non-percolating network of electrically conductive nanofillers. Here, each nanofiller is displayed as a cylindrical rod. The magnified image illustrates rod A defined by a pair of orientation angles ðq; 4Þ in contact with rod B which is aligned with a pair of orientation angles ðg; zÞ.
V. Kumar, A. Rawal / Polymer 97 (2016) 295e299
297
two contacts for percolating the electrons within a nanocomposite [33]. Alternatively, the mean distance between the contacts should be at least half of the rod length to attain percolation threshold. Accordingly, we define a ‘coverage parameter’, i.e. G ¼ bl (where b is the distance between the contacts and l is the rod length), which represents the number of rod-rod contacts formed on a given rod length. In order to achieve percolation threshold in nanofillers, G 0:5 and for non-percolating fillers, G > 0:5. Therefore, we define a ‘mean coverage parameter’, i.e. G ¼ bl which refers to the probability of percolation of the network. In the past, the distance between the contacts was found to exhibit an exponential distribution [39,40] and is given by,
. b PðbÞ ¼ 1 b e b
(6)
In the similar fashion, the mean coverage parameter ðGÞ can take any positive real value and the contact parameter (G) of a nanofiller accordingly follows an exponential distribution defined by its mean G . Thus, the probability distribution of G can be calculated as,
. G PðGÞ ¼ 1 G e G
(7)
Zheng et al. [41] have benchmarked the statistical percolation threshold when 50% of the sample percolates. Following this percolation threshold concept, the critical volume fraction (mc ) is achieved when percolation probability is 50%, R 0:5 eGG R 0:5 0:5 dG ¼ 0:5 or G ¼ lnð10:5Þ ¼ 0:72. Therei.e. 0 PðGÞdG ¼ 0 G
fore, in order to attain percolation threshold, bl should be 0.72 or the mean contact per rod (nc ¼ l=b) should be 1.38. Finally, the critical volume fraction of nanofillers can be computed using Komori and Makishima [29] and Pan [34] models based upon above considerations,
mKM ¼ c
p 5:77sI
Fig. 2. Comparison of critical volume fraction of nanofillers obtained through Komori and Makishima [29], Pan [34] and Shim et al. [33] models for a wide range of aspect ratios.
; mPan ¼ c
p 5:77sI 4j
(8)
where s is the aspect ratio (ratio of length to diameter) of the rod, mKM and mPan are the critical volume fractions (percolation threshc c olds) obtained through Komori and Makishima [29] and Pan [34] models, respectively. MATLAB® programme was used for computing the value of I by considering the 3D orientation distributions of nanofillers.
as depicted in Fig. 2. This analysis clearly confirmed that Komori and Makishima’s model [29] can be applied to predict percolation threshold of nanocomposites having nanofillers with cylindrical geometries. The next question arises whether this model is capable of handling nanofillers with a wide variety of aspect ratios. A comparison was made between the theoretical results obtained from Komori and Makishima’s model [29], and the experimental data sets collected for percolation threshold pertaining to a wide variety of nanocomposites, as shown in Fig. 3. In general, a good agreement has been obtained between the theoretical and experimental data results. Ostensibly, the theoretical results have also been benchmarked with the simulated results of White et al. [18] based upon the fact that the Monte Carlo simulations for percolation threshold values of cylindrical nanofillers with small aspect ratios (<60) have been successfully predicted. Although, the previously developed analytical models including soft core and
3. Discussion As aforementioned, the percolation threshold in a typical nanocomposite can be attained when the mean number of contacts per rod is 1.38. According to Balberg et al. [19], the mean number of contacts per rod based upon excluded volume effect was computed to be 1.41 and 1.49 through analytical and Monte Carlo simulation techniques, respectively. On the other hand, Shim et al. [33] have considered the mean number of contacts per rod to be 2 and subsequently, the percolation threshold was found to be equal to that of inverse of aspect ratio based upon Komori and Makishima’s approach [29]. This consideration can yield large overestimations of percolation threshold in nanocomposites specifically with lower aspect ratio nanofillers (s < 100), as illustrated in Fig. 2. Since, some of the important nanofillers have modest aspect ratios (<100) due to their inherent nature or as a result of composite fabrication techniques [7]. Nonetheless, the excluded volume effect considered in Pan’s model [34] has resulted in percolation threshold values similar to those obtained from Komori and Makishima’s model [29],
Fig. 3. Critical volume fraction of cylindrical nanofillers as a function of aspect ratio. Comparison of results from Monte Carlo simulations (circles) [18], predicted (solid line) using Komori and Makishima’s model [29], and experimental data sets obtained for a wide variety of nanofillers (plus sign [8], cross [9], squares [18], stars [23], triangles [24], diamonds [25]).
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V. Kumar, A. Rawal / Polymer 97 (2016) 295e299
core shell cases based upon excluded volume effect are only valid for nanofillers with very large aspect ratios (s / ∞) [10,42,43]. Fig. 3 clearly shows an excellent agreement between the simulated results of White et al. [18] and the theoretical results obtained from Komori and Makishima’s model [29]. This clearly demonstrates that the analytical approach of Komori and Makishima [29] can be used for computing the percolation thresholds of cylindrical nanofillers with a wide range of aspect ratios. One of the major merits of Komori and Makishima’s model [29] is that it enables us to introduce the alignment of rods in the inplane and out-of-plane directions. This not only allows us to compute the effect of alignment of rods on percolation threshold in 3D but individual orientation distributions of the rods as a result of processing conditions or due to the application of electrical or magnetic fields can be introduced. Zheng et al. [41] pointed out that the conventional excluded volume effect allows the rods to be added into the system with specified orientation distributions until the percolation occurs However, the orientation distribution functions are interrelated with the volume fraction and the processing conditions. Accordingly, the orientation distribution is altered when the new rods are being introduced into the system. Thus, it is pertinent to either simulate or determine the orientation distribution of the rods in the in-plane and out-of-plane directions experimentally, and analyze their effects on the percolation threshold of nanocomposites. In this research work, a virtual experiment was conducted by systematically varying the anisotropic behavior of nanofillers in the form of Gaussian distributions in the out-of-plane direction by keeping the same in-plane orientation distribution (Gaussian distribution) to analyze their effects on the percolation threshold of nanocomposites, as shown in Fig. 4. These percolation threshold values were also compared with the 3D randomly aligned rods. These simulated results clearly revealed that the anisotropic behavior of rods inevitably led to higher percolation threshold values corresponding to 3D randomly aligned rods specifically with lower aspect ratios (<100). These results matched well with the previously reported research works that demonstrated lower percolation thresholds occur for isotropic or even with slightly anisotropic behavior of nanofillers [21,44e47]. For two-dimensional (2D) systems, the rod orientation in the defined space can be easily varied to obtain its effect on the percolation threshold of nanofillers [45]. However, in 3D systems, the same value of orientation parameter (I) can be obtained for different orientation distributions in the in-plane and out-of-plane directions. For example, the value of I for a typical 2D random system is 0.64 [29] whereas the same value of I can be obtained by introducing anisotropic (Gaussian) distributions in the in-plane and out-of-plane directions (see Fig. S1). Therefore, it would be incorrect to make any inferences about the percolation threshold of nanofillers from either in-plane or out-of-plane orientation distribution. Since, the different combinations of orientation distributions in the in-plane and out-of-plane directions can lead to the same magnitude of I. Accordingly, we have provided the algorithm in the form of pseudocode for computing the value of I based upon orientation distributions in the in-plane and out-of-plane directions (see supplementary information for details). 4. Conclusions In summary, a stochastic approach based upon Komori and Makishima’s model [29] has been applied to predict the percolation threshold of cylindrical nanofillers using 3D orientation distributions of the rods. In general, a good agreement has been obtained between the theoretical and experimental results obtained from the literature. Furthermore, the values of critical volume fraction of nanofillers have been benchmarked with the simulated results
Fig. 4. Effect of (a) out-of-plane orientation distribution of rods on the (b) the critical volume fractions of nanofillers keeping the same in-plane orientation distributions. Here ‘G1’, ‘G2’ and ‘G3’ represent Gaussian distributions with mean of 90 and standard deviations of 15 , 30 and 60 , respectively and ‘R’ refers to a random alignment in the out-of-plane direction. In-plane orientation distribution of rods is represented by a Gaussian distribution with mean and standard deviations of 90 , and 15 , respectively. The results of critical volume fractions have also been compared with a 3D randomly aligned network of nanofillers, which are referred as ‘3D-R’.
obtained from White et al. [18] that effectively proved the usefulness of the analytical model for nanofillers covering a wide range of aspect ratios. The percolation threshold values of nanofillers were also comparable with those obtained from Pan’s stochastic model [34] that considered the excluded volume effect. The anisotropic behavior of nanofillers in 3D was successfully introduced in order to predict the percolation threshold values. It is anticipated that the simple analytical approach presented in this research work can be considered as a toolkit to tune and guide the percolation threshold of nanofillers in nanocomposites.
Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.polymer.2016.05.041.
V. Kumar, A. Rawal / Polymer 97 (2016) 295e299
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Contents lists available at ScienceDirect
Polymer journal homepage: www.elsevier.com/locate/polymer
Short communication
Tuning the electrical percolation threshold of polymer nanocomposites with rod-like nanofillers Vijay Kumar 1, Amit Rawal* Department of Textile Technology, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 January 2016 Received in revised form 13 May 2016 Accepted 14 May 2016 Available online 15 May 2016
In this communication, a stochastic approach based upon Komori and Makishima’s work has been used for predicting the electrical percolation threshold of cylindrical nanofillers with three-dimensional (3D) spatial orientations in a typical nanocomposite system. Specifically, the proposed model was able to predict the volume fraction based percolation threshold of nanofillers with a wide range of aspect ratios (10e1000), which was substantiated with a variety of experimental data sets obtained from the literature. The anisotropic behavior of 3D aligned nanofillers was successfully introduced via in-plane and outof-plane orientation distributions. The percolation threshold values of nanofillers were also found to be comparable with those obtained from the stochastic model that has incorporated the excluded volume effect. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Percolation threshold Nanofiller Stochastic
1. Introduction Nanoscale conductive particles, such as carbon nanotubes, carbon nanofibers, metallic nanowires, graphene flakes, etc. have gained considerable attention in the field of nanocomposites due to their excellent electrical properties. Such characteristics are the gateway to replace monolithic metals in numerous emerging applications including flexible microelectronics, e-paper, organic light emitting diodes, sensors, antistatic coatings and touch screens [1e5]. These targeted applications can be practically achievable when maximum electrical conductivity of nanocomposites is attained with the lowest possible filler concentration, which inevitably reduces the cost. Enhancing the electrical conductivity of the nanocomposites in a cost-effective manner is triggered by means of percolation threshold that describes the minimum loading of conducting nanorods in a polymeric/colloidal medium, which are capable of forming a connected network through rod-rod contacts. The transition in the electrical conductivity at the percolation threshold level has been observed to be several orders of magnitude [6]. Alternatively, the concentration of nanorods at which the material exhibits an abrupt transition from insulator to a conductor
* Corresponding author. Fraunhofer Institute of Industrial Mathematics (ITWM), Fraunhofer-Platz 1, D-67663 Kaiserslautern, Germany. E-mail addresses: [email protected], [email protected] (A. Rawal). 1 Present address: University of Borås, SE-501 90 Borås, Sweden. http://dx.doi.org/10.1016/j.polymer.2016.05.041 0032-3861/© 2016 Elsevier Ltd. All rights reserved.
is characterized by the percolation threshold. In conducting nanocomposites containing nanofillers, the percolation threshold is dependent upon the filler attributes including aspect ratio, electrical properties, and dispersity in filler properties [7e9]. Percolation phenomena in polymer nanocomposites have been critically investigated through numerical [10e18], analytical [19e21] and semi-empirical modeling [22e25] techniques. Some of these modeling techniques were compared and subsequently, combined together to predict the percolation threshold of nanocomposites. For instance, the excluded volume model is a universally accepted continuum analytical model that has been applied in combination with Monte Carlo simulations for predicting the percolation threshold of a wide variety of fillers in nanocomposites [10,13,16,18,19]. However, the classical excluded volume model has been primarily developed for fillers with large aspect ratios, which can be computationally intensive for numerical modeling techniques [10]. Furthermore, the nanofillers with modest aspect ratios (<100) either grown inherently or as a result of sonication, are present in nanocomposites [7,18,26]. In addition, the excluded volume model considers the random distribution of fillers but in reality, these fillers can align in preferred in-plane or out-of-plane directions as a result of polymer processing techniques (extrusion, molding, drawing, fiber spinning) [27] or due to the application of electrical or magnetic fields [28]. Therefore, it is of paramount importance that the 3D orientation distribution of fillers should be considered for predicting the percolation threshold. In the present work, the analytical model to predict the
296
V. Kumar, A. Rawal / Polymer 97 (2016) 295e299
percolation threshold of 3D aligned nanofillers covering a wide range of aspect ratios based upon stochastic approach has been proposed. The stochastic approach is essentially based upon Komori and Makishima’s geometrical probability approach [29] which has been successfully applied in the past for predicting various properties of different types of fibrous systems [30e33]. The theoretical results have been compared with the other stochastic approach that accounted for the excluded volume effect [34]. The analytical model has been validated with a variety of experimental data sets, obtained from the literature.
2. Theoretical analysis For a disordered system such as nanocomposites to be percolative, the fundamental criterion is the continuity to transport electrons should be maintained, which can be modulated through rod-rod contacts, as shown in Fig. 1. Although, the thin layer of the polymer matrix in the range of few nanometers between the nanofillers can still allow the passage of electrons via quantum mechanical tunneling mechanism [35]. Neglecting the tunneling effect in this analysis due to the fact that the electrical conductivity of the network primarily results from rod-rod contacts [36]. Therefore, the rod-rod contacts of nanofillers which are responsible for (dis)continuity in the electron path should be critically analyzed. Considering the orientation of each cylindrical rod defined by a pair of orientation angles ðq; 4Þ in a spherical coordinate system, where q and 4 are polar and azimuthal angles, respectively. The probability density function or the orientation distribution function, Uðq; 4Þ is the probability of a rod lying in an infinitesimal range of angles q and q þ dq, and 4 and 4 þ d4 is given by Uðq; 4Þsin qdqd4, which satisfies the following normalization condition.
Zp 0
d4
Zp
sin qUðq; 4Þdq ¼ 1
(1)
0
Komori and Makishima [29] described a stochastic approach by assuming the rods to be straight cylindrical entities with constant length and diameter and the rod-rod contact condition would only occur if the center of mass of rod A of defined orientation ðq; 4Þ enters into the neighbourhood region of rod B with orientation ðg; zÞ, as shown in Fig. 1. Based on defined number of contacts in a volume (V) along with the probability of formation of a contact, the mean distance between the rod-rod contacts (bKM ) is given by [29],
bKM ¼
V 2DLI
where Jðq; 4Þ ¼
Rp 0
(2)
dz
Rp 0
I¼
Rp 0
dq
Rp 0
Jðq; 4Þsin qUðq; 4Þd4;
sin cðq; 4; g; zÞUðg; zÞsin gdg
and
sin c ¼ ½1 fcos q cos g þ cosð4 zÞsin q sin gg where D is the nanofiller diameter, L is the total length of the nanofiller defined in volume V, I is an orientation parameter defining the orientation characteristics of nanofillers in the assembly, c is the angle between the two axes of nanofillers having defined types of orientation distributions, Uðq; 4Þ and Uðg; zÞ. Also, the volume fraction of rods (m) can be calculated through the following equation. 2 1=2
m¼
pD2 L
(3)
4V
Combining Equations (2) and (3),
bKM ¼
pD 8I m
(4)
It is worth mentioning that Komori and Makishima [29] have not accounted for the changes in the probability of a contact with the successive contacts, i.e. excluded volume effect has been neglected [34]. In other words, an existing contact reduces the effective contact length of a rod, which diminishes the probability of formation of new contacts. On the other hand, the existing rodrod contact also tends to reduce the free volume of the nanofiller assembly that results in enhancing the chances of making new contacts [37]. Therefore, the excluded volume effect is a trade-off between the reductions in effective contact length and that of free volume [38]. In reality, this effect may not significantly alter the number of rod-rod contacts as the critical volume fractions for attaining desired percolation thresholds are often realized at significantly lower levels. Nevertheless, we also calculated the mean distance between the contacts based upon Pan’s approach [34] that considered the excluded volume effect by modifying and extending Komori and Makishima’s model [29], as shown below.
bPan ¼
ðp þ 4mjÞD 8mI
where Kðq; 4Þ ¼
Rp 0
(5)
j¼
Rp Uðg;zÞsin g dg 0 dz sin cðq;4;g;zÞ
Rp 0
Rp dq 0 d4Jðq; 4ÞKðq; 4ÞUðq; 4Þsin q;
Next, it is pertinent to note that each rod should make atleast
Fig. 1. A cartoon depicting percolating and non-percolating network of electrically conductive nanofillers. Here, each nanofiller is displayed as a cylindrical rod. The magnified image illustrates rod A defined by a pair of orientation angles ðq; 4Þ in contact with rod B which is aligned with a pair of orientation angles ðg; zÞ.
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297
two contacts for percolating the electrons within a nanocomposite [33]. Alternatively, the mean distance between the contacts should be at least half of the rod length to attain percolation threshold. Accordingly, we define a ‘coverage parameter’, i.e. G ¼ bl (where b is the distance between the contacts and l is the rod length), which represents the number of rod-rod contacts formed on a given rod length. In order to achieve percolation threshold in nanofillers, G 0:5 and for non-percolating fillers, G > 0:5. Therefore, we define a ‘mean coverage parameter’, i.e. G ¼ bl which refers to the probability of percolation of the network. In the past, the distance between the contacts was found to exhibit an exponential distribution [39,40] and is given by,
. b PðbÞ ¼ 1 b e b
(6)
In the similar fashion, the mean coverage parameter ðGÞ can take any positive real value and the contact parameter (G) of a nanofiller accordingly follows an exponential distribution defined by its mean G . Thus, the probability distribution of G can be calculated as,
. G PðGÞ ¼ 1 G e G
(7)
Zheng et al. [41] have benchmarked the statistical percolation threshold when 50% of the sample percolates. Following this percolation threshold concept, the critical volume fraction (mc ) is achieved when percolation probability is 50%, R 0:5 eGG R 0:5 0:5 dG ¼ 0:5 or G ¼ lnð10:5Þ ¼ 0:72. Therei.e. 0 PðGÞdG ¼ 0 G
fore, in order to attain percolation threshold, bl should be 0.72 or the mean contact per rod (nc ¼ l=b) should be 1.38. Finally, the critical volume fraction of nanofillers can be computed using Komori and Makishima [29] and Pan [34] models based upon above considerations,
mKM ¼ c
p 5:77sI
Fig. 2. Comparison of critical volume fraction of nanofillers obtained through Komori and Makishima [29], Pan [34] and Shim et al. [33] models for a wide range of aspect ratios.
; mPan ¼ c
p 5:77sI 4j
(8)
where s is the aspect ratio (ratio of length to diameter) of the rod, mKM and mPan are the critical volume fractions (percolation threshc c olds) obtained through Komori and Makishima [29] and Pan [34] models, respectively. MATLAB® programme was used for computing the value of I by considering the 3D orientation distributions of nanofillers.
as depicted in Fig. 2. This analysis clearly confirmed that Komori and Makishima’s model [29] can be applied to predict percolation threshold of nanocomposites having nanofillers with cylindrical geometries. The next question arises whether this model is capable of handling nanofillers with a wide variety of aspect ratios. A comparison was made between the theoretical results obtained from Komori and Makishima’s model [29], and the experimental data sets collected for percolation threshold pertaining to a wide variety of nanocomposites, as shown in Fig. 3. In general, a good agreement has been obtained between the theoretical and experimental data results. Ostensibly, the theoretical results have also been benchmarked with the simulated results of White et al. [18] based upon the fact that the Monte Carlo simulations for percolation threshold values of cylindrical nanofillers with small aspect ratios (<60) have been successfully predicted. Although, the previously developed analytical models including soft core and
3. Discussion As aforementioned, the percolation threshold in a typical nanocomposite can be attained when the mean number of contacts per rod is 1.38. According to Balberg et al. [19], the mean number of contacts per rod based upon excluded volume effect was computed to be 1.41 and 1.49 through analytical and Monte Carlo simulation techniques, respectively. On the other hand, Shim et al. [33] have considered the mean number of contacts per rod to be 2 and subsequently, the percolation threshold was found to be equal to that of inverse of aspect ratio based upon Komori and Makishima’s approach [29]. This consideration can yield large overestimations of percolation threshold in nanocomposites specifically with lower aspect ratio nanofillers (s < 100), as illustrated in Fig. 2. Since, some of the important nanofillers have modest aspect ratios (<100) due to their inherent nature or as a result of composite fabrication techniques [7]. Nonetheless, the excluded volume effect considered in Pan’s model [34] has resulted in percolation threshold values similar to those obtained from Komori and Makishima’s model [29],
Fig. 3. Critical volume fraction of cylindrical nanofillers as a function of aspect ratio. Comparison of results from Monte Carlo simulations (circles) [18], predicted (solid line) using Komori and Makishima’s model [29], and experimental data sets obtained for a wide variety of nanofillers (plus sign [8], cross [9], squares [18], stars [23], triangles [24], diamonds [25]).
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core shell cases based upon excluded volume effect are only valid for nanofillers with very large aspect ratios (s / ∞) [10,42,43]. Fig. 3 clearly shows an excellent agreement between the simulated results of White et al. [18] and the theoretical results obtained from Komori and Makishima’s model [29]. This clearly demonstrates that the analytical approach of Komori and Makishima [29] can be used for computing the percolation thresholds of cylindrical nanofillers with a wide range of aspect ratios. One of the major merits of Komori and Makishima’s model [29] is that it enables us to introduce the alignment of rods in the inplane and out-of-plane directions. This not only allows us to compute the effect of alignment of rods on percolation threshold in 3D but individual orientation distributions of the rods as a result of processing conditions or due to the application of electrical or magnetic fields can be introduced. Zheng et al. [41] pointed out that the conventional excluded volume effect allows the rods to be added into the system with specified orientation distributions until the percolation occurs However, the orientation distribution functions are interrelated with the volume fraction and the processing conditions. Accordingly, the orientation distribution is altered when the new rods are being introduced into the system. Thus, it is pertinent to either simulate or determine the orientation distribution of the rods in the in-plane and out-of-plane directions experimentally, and analyze their effects on the percolation threshold of nanocomposites. In this research work, a virtual experiment was conducted by systematically varying the anisotropic behavior of nanofillers in the form of Gaussian distributions in the out-of-plane direction by keeping the same in-plane orientation distribution (Gaussian distribution) to analyze their effects on the percolation threshold of nanocomposites, as shown in Fig. 4. These percolation threshold values were also compared with the 3D randomly aligned rods. These simulated results clearly revealed that the anisotropic behavior of rods inevitably led to higher percolation threshold values corresponding to 3D randomly aligned rods specifically with lower aspect ratios (<100). These results matched well with the previously reported research works that demonstrated lower percolation thresholds occur for isotropic or even with slightly anisotropic behavior of nanofillers [21,44e47]. For two-dimensional (2D) systems, the rod orientation in the defined space can be easily varied to obtain its effect on the percolation threshold of nanofillers [45]. However, in 3D systems, the same value of orientation parameter (I) can be obtained for different orientation distributions in the in-plane and out-of-plane directions. For example, the value of I for a typical 2D random system is 0.64 [29] whereas the same value of I can be obtained by introducing anisotropic (Gaussian) distributions in the in-plane and out-of-plane directions (see Fig. S1). Therefore, it would be incorrect to make any inferences about the percolation threshold of nanofillers from either in-plane or out-of-plane orientation distribution. Since, the different combinations of orientation distributions in the in-plane and out-of-plane directions can lead to the same magnitude of I. Accordingly, we have provided the algorithm in the form of pseudocode for computing the value of I based upon orientation distributions in the in-plane and out-of-plane directions (see supplementary information for details). 4. Conclusions In summary, a stochastic approach based upon Komori and Makishima’s model [29] has been applied to predict the percolation threshold of cylindrical nanofillers using 3D orientation distributions of the rods. In general, a good agreement has been obtained between the theoretical and experimental results obtained from the literature. Furthermore, the values of critical volume fraction of nanofillers have been benchmarked with the simulated results
Fig. 4. Effect of (a) out-of-plane orientation distribution of rods on the (b) the critical volume fractions of nanofillers keeping the same in-plane orientation distributions. Here ‘G1’, ‘G2’ and ‘G3’ represent Gaussian distributions with mean of 90 and standard deviations of 15 , 30 and 60 , respectively and ‘R’ refers to a random alignment in the out-of-plane direction. In-plane orientation distribution of rods is represented by a Gaussian distribution with mean and standard deviations of 90 , and 15 , respectively. The results of critical volume fractions have also been compared with a 3D randomly aligned network of nanofillers, which are referred as ‘3D-R’.
obtained from White et al. [18] that effectively proved the usefulness of the analytical model for nanofillers covering a wide range of aspect ratios. The percolation threshold values of nanofillers were also comparable with those obtained from Pan’s stochastic model [34] that considered the excluded volume effect. The anisotropic behavior of nanofillers in 3D was successfully introduced in order to predict the percolation threshold values. It is anticipated that the simple analytical approach presented in this research work can be considered as a toolkit to tune and guide the percolation threshold of nanofillers in nanocomposites.
Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.polymer.2016.05.041.
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