Article
Modeling Percolation in Polymer Nanocomposites by Stochastic Microstructuring Matias Soto 1 , Milton Esteva 1 , Oscar Martínez-Romero 1 , Jesús Baez 2 and Alex Elías-Zúñiga 1, * Received: 2 July 2015 ; Accepted: 23 September 2015 ; Published: 30 September 2015 Academic Editor: Biqiong Chen 1
2
*
Escuela de Ingeniería y Ciencias, Tecnologico de Monterrey, Campus Monterrey, Ave. Eugenio Garza Sada 2501, Monterrey, N.L. 64849, Mexico;
[email protected] (M.S.);
[email protected] (M.E.);
[email protected] (O.M.-R.) Centro de Estudios de Energía, Dpto. de Ing. Eléctrica, Tecnologico de Monterrey, Campus Monterrey, Ave. Eugenio Garza Sada 2501, Monterrey, N.L. 64849, Mexico;
[email protected] Correspondence:
[email protected]
Abstract: A methodology was developed for the prediction of the electrical properties of carbon nanotube-polymer nanocomposites via Monte Carlo computational simulations. A two-dimensional microstructure that takes into account waviness, fiber length and diameter distributions is used as a representative volume element. Fiber interactions in the microstructure are identified and then modeled as an equivalent electrical circuit, assuming one-third metallic and two-thirds semiconductor nanotubes. Tunneling paths in the microstructure are also modeled as electrical resistors, and crossing fibers are accounted for by assuming a contact resistance associated with them. The equivalent resistor network is then converted into a set of linear equations using nodal voltage analysis, which is then solved by means of the Gauss–Jordan elimination method. Nodal voltages are obtained for the microstructure, from which the percolation probability, equivalent resistance and conductivity are calculated. Percolation probability curves and electrical conductivity values are compared to those found in the literature. Keywords: carbon nanotubes; electrical properties; conductivity; percolation; Monte Carlo; nanocomposites; nodal voltage; simulations
1. Introduction The intrinsic electrical properties of carbon nanotubes (CNTs) have been found to be comparable to most metals in terms of conductivity and current density [1,2]. Depending on the nanotube chirality, its electrical conductivity can be classified as metallic or semiconducting [3–6]. Within a given group of single-walled carbon nanotubes (SWCNTs) grown inside the same furnace, researchers have found that approximately one-third are categorized as metallic, while two-thirds as semiconductors [3,5]. Metallic SWCNTs have a conductivity ranging from 104 to 107 S/m [6,7], while those classified as semiconductors have a conductivity of approximately 10 S/m [7]. Carbon nanotubes have been used as fillers to reinforce polymeric materials, and improvements in mechanical, thermal and electrical properties have been reported [2,3,8,9]. The properties of polymer nanocomposites (PNCs) can be engineered by manipulating the type and amount of fillers within the polymer matrix. Furthermore, these parameters can be used to tailor the electrical properties of the nanocomposite [8,9]. Having a model that incorporates the abovementioned parameters is essential to quantify their impact in the final electrical properties of the PNC. Previous theoretical and computational work has been reported in the open literature, which focuses on the prediction of the percolation threshold and electrical conductivity of PNCs [10–19].
Materials 2015, 8, 6697–6718; doi:10.3390/ma8105334
www.mdpi.com/journal/materials
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1.1. Percolation Behavior A polymer nanocomposite is composed of carbon nanotubes embedded in a polymer matrix. The fillers can be well dispersed, bundled and dispersed, or completely agglomerated. Either the bundles or the singular CNTs can be aligned in a particular direction or misaligned. When a specific electrical current source is supplied to a polymer nanocomposite with carbon nanotubes as fillers, there will be a defined transition from an insulator to a conductor in the electrical conductivity. This transition occurs because a conductive path has been established within the carbon nanotube network. The term percolation refers to this onset of transition to a network where connectivity suddenly appears [15,20]. For composites with conductive fillers dispersed in an insulating matrix, the percolation of a polymer nanocomposite may be directly related to the concentration of fillers inside the matrix. As this concentration is increased, the nanotubes are able to interact with each other to form the conductive network necessary for the transport of electrons across the material, and the concentration at which this occurs is known as the percolation threshold [3]. The contact percolation limit, which is associated with the establishment of a nanotube conductive path from one boundary to another in a polymer nanocomposite, has been estimated between 0.12 and 4.5 wt. % for different polymer matrices (polyimide, epoxy, etc.). In addition to the effect on electrical conduction due to an increase in the filler concentration within the polymer matrix, it is important to mention that percolation is also relevant when describing the mechanical properties of PNCs, so that a change in the behavior is observed around the percolation threshold, according to [21,22]. 1.2. Electrical Conductivity In general, the conductivity of a medium is defined as the inverse of resistivity, which is an intrinsic property of that same medium. Materials considered as conductors have a low electrical resistivity, while those considered as insulators have high values of resistivity. Semiconductors are between conductors and insulators [23]. In the case of polymer nanocomposites, the electrical conductivity is dependent on the percolated network because the nanotubes are embedded inside the polymer matrix. Unless percolation is established, the transport of electrons from one boundary to another will hardly occur. After percolation has been obtained within the PNC, the conductivity continues to be dependent on the concentration of the nanotubes [3]. It is well-known that not only the concentration of nanotubes has an effect on the conductivity of a PNC, but also their dispersion and agglomeration because this relates to the formation of contact conductive networks [12,14]. The electron hopping or quantum tunneling phenomenon also affects the electrical conductivity of a PNC [16], such that in cases where the filler concentration does not allow for the direct contact among nanotubes, if the separation distance between them is small enough, electrons have been detected to be transported within the polymer material [16,24]. The quantum tunneling barrier or resistance depends on the properties of the polymer, the distance between the two segments showing tunneling behavior, and the voltage applied at the junction. This characteristic behavior in polymer nanocomposites enhances its electrical conductivity [16]. In this work, we present a methodology capable of simulating the electrical properties of a polymer nanocomposite with carbon nanotube as fillers. The simulation can calculate values of percolation probability and electrical conductivity. In addition, the results of the nodal voltage analysis can be graphed to create a map of the voltage distribution within the electrical network. Our work shares several characteristics with previous works and also proposes some novel methods when modeling the microstructure and finding the conductive nanotube network within it. Although many recent works have a three-dimensional microstructure, this work along with that of Li et al. [13] employs a two-dimensional approach. Nanotube waviness has been previously used in the model of Dalmas et al. [11] and Li et al. [13], and also incorporated into our model. The concept of electron tunneling has already been added to the models reported in [12,16,18], while 6698
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contact resistance is part of the models previously reported in [11,16,17]; both of which were also incorporated into our model. The works of [12,14,17,18] use a special algorithm to find the conductive path in the microstructure; instead of using a similar approach, in our work we make use of linear algebra methodology in order to discriminate when a nanotube is connected to the network that conducts electric current. Aside from the aforementioned characteristics that our work shares with previous models, this methodology makes use of length and diameter distributions when generating the microstructure. Additionally, the waviness of a nanotube depends on its length based on a linear relation. Finally, the intrinsic conductivity of the nanotubes in our model is assigned according to a distribution found experimentally in [2]. 2. Procedure
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A methodology was established to simulate the electrical behavior of a polymer nanocomposite with carbon nanotubes as fillers. Firstly, the generation of the geometrical parameters of the PNC interactions to explained. an equivalent electrical the equivalent circuitare is then used in the model is Secondly, the circuit. networkFinally, interactions within theelectrical microstructure discussed, nodalwhich analysis. includes crossings and tunneling behavior. This is followed by the conversion of the microstructure anduse its of interactions an equivalent electrical the equivalent electrical of the As we will see, the the nodaltovoltage analysis methodcircuit. allowsFinally, us to determine percolation circuit is then used in the nodal analysis. fiber network as well as its electrical conductivity. This nodal voltage analysis is coupled with the use As we will see, the use of the nodal voltage analysis method allows us to determine percolation of linear algebra methodsasneeded to obtain theconductivity. voltage at each node in all the nanotubes in thewith network. of the fiber network well as its electrical This nodal voltage analysis is coupled the use of linear algebra methods needed to obtain the voltage at each node in all the nanotubes in 2.1. Generating the network.and Analyzing the Microstructure Generating and Analyzing the Microstructure A 2.1. network of fibers was generated within a representative volume element (RVE). The area of the network ofto fibers generated within a representative volume (RVE).1). TheThe areathickness, of the RVE was A determined be awas square (for simplicity) of side length (b–a)element (see Figure t, RVE was determined to be a square (for simplicity) of side length (b–a) (see Figure 1). The thickness, t, of the RVE was chosen as the maximum nanotube diameter. When using a specific distribution for the of the RVE was chosen as the maximum nanotube diameter. When using a specific distribution for nanotube diameter,diameter, the maximum diameter to a 95% level was used. the nanotube the maximum diameter to a confidence 95% confidence level was used.
Figure 1. Top view of Representative Volume Element (RVE).
Figure 1. Top view of Representative Volume Element (RVE). 2.1.1. Generating the Geometry
2.1.1. Generating the Geometry
Each fiber was generated as a series of straight segments with initial and final nodes. A pseudorandom number generator is used for this purpose. The generator uses a uniform distribution Each fiber was generated as a series of straight segments with initial and final nodes. A pseudorandom of numbers in the open interval (a, b). In this case, the generator was first used to obtain two number generator this purpose. usesgives a uniform distribution numbers numbers, the is X used and Yfor coordinates of a The singlegenerator point, which the location of the of first node of in the open the interval (a, b). In this thisbasic case,coordinate the generator wasnode first isused to asobtain two numbers, the X and Y segment. With pair, this saved an Object, using hierarchical object-oriented programming in Matlab. The Object, which can be simply called Node, may coordinates of a single point, which gives the location of the first node of the segment. With also this basic
coordinate pair, this node is saved as an Object, using hierarchical object-oriented programming in Matlab. The Object, which can be simply called Node, may also save other information about the node; 6699 this proved to be advantageous when administering large data structures in the computer algorithm. The second node is created at a distance l from the first node. The segment distance, l, depends on the fiber distance and the number of segments, such that,
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save other information about the node; this proved to be advantageous when administering large data structures in the computer algorithm. The second node is created at a distance l from the first node. The segment distance, l, depends on the fiber distance and the number of segments, such that, L “ kl
(1)
where k is the number of segments and L is the sum of all the individual segment lengths in a fiber. The proportion of the total length of the fiber to the RVE side length (b–a) will be referred to as the Materials 2015, 88 Materials 2015, 55 parameter g in some portions of this article. The same pseudorandom number generator is used to determine the orientation of the second angle of orientation, ϕ, is then used tothis findcase, thethe X–Y coordinates of the the second nodesuch according to the the node with respectϕ, to is thethen first used one. In open interval covers allsecond 360 degrees that the to angle of orientation, to find the X–Y coordinates of node according second node can be randomly-oriented in any direction with respect to the first node, as shown in following equations: equations: following Figure 2. The angle of orientation, φ, is then used to find the X–Y coordinates of the second node (2) cos ϕ ϕ according to the following equations: (2) cos
sinφϕ ϕ Xi`1 “ Xi ` l cos sin
(2)
(3) (3)
The second second node node is is also also saved saved as as aa Node NodeY Object. Object. As As part of the hierarchy, an Object called Segment The part l sin φ of the hierarchy, an Object called Segment (3) i`1 “ Yi ` is created, created, which which stores stores the the two two Node Node objects. objects. The The next next segment segment makes makes use use of of the the second second node node is The second node is also saved as a Node Object. As part of the hierarchy, an Object called (nodeSegment inisFigure Figure 2), fromstores which a third third node is generated. generated. The angle of orientation orientation of this this next next (node 11 in from which node is angle of of created,2), which theatwo Node objects. The next The segment makes use of the second segment is(node limited by theFigure boundaries that define the maximum maximum deviation ofangle the next next segment with with respect node i `by 1 the in 2), from which a third node is generated. Theof of orientation of this segment is limited boundaries that define the deviation the segment respect segmentangular is limited by the boundaries that the3.maximum deviation of theisnext segmentfor the to the thenext previous deviation, as shown shown indefine Figure The same same procedure followed to previous angular deviation, as in Figure 3. The procedure is followed for the with respect to the previous angular deviation, as shown in Figure 3. The same procedure is followed generation of the remaining segments for each fiber. As mentioned before, each Segment object is generation the remaining segments segments for each for fiber. mentioned before,before, each each Segment object is for theof generation of the remaining eachAs fiber. As mentioned Segment composed of two Node Nodeofobjects; objects; in objects; turn aa Fiber Fiber object can be created, created, whichwhich is composed composed of kkof Segment composed two in turn object can be which is of Segment object of is composed two Node in turn a Fiber object can be created, is composed k objects. The complete hierarchy of the Objects is shown in Figure 4. Segment objects. The complete hierarchy of the Objects is shown in Figure 4. objects. The complete hierarchy of the Objects is shown in Figure 4.
Figure 2. Angle of deviation of the second segment with respect to the previous one.
Figure 2. 2. Angle Angle of of deviation deviation of of the the second second segment segment with with respect respect to to the the previous previous one. one. Figure
Figure 3. Object hierarchy of the computer algorithm. Figure 3. Object Object hierarchy hierarchy of of the the computer computer algorithm. algorithm. Figure 3.
Multiple Fiber Fiber objects objects can can be be generated generated to to form form aa network. network. The The volume volume fraction fraction of of the the fillers fillers is is Multiple 6700 calculated by by summing summing the the volumes volumes of of all all the the individual individual fibers and and dividing dividing by by the the volume volume of of the the RVE. RVE. calculated fibers (4) (4)
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Multiple Fiber objects can be generated to form a network. The volume fraction of the fillers is calculated by summing the volumes of all the individual fibers and dividing by the volume of the RVE. Vall f ibers VRVE
c“
(4)
The volume of each fiber is given by the following equation:
Materials 2015, 8 Vf “
k ÿ
π i “1
6
d2 l 4 i
(5)
where d is the diameter of the fiber, and the volume of the RVE is given by the following equation:
(6)
Additionally, the aspect ratio (AR) of a fiber is defined 2as the ratio of its total length to its diameter, VRVE “ t pb ´ aq (6) which is assumed to be constant. Additionally,the thegeometry, aspect ratio of a diameter, fiber is defined as the ratio of itsoftotal length (waviness, to its When generating the(AR) length, and maximum angle deviation diameter, which is assumed to be constant. in essence), among other parameters, can be defined in our methodology. Following the work done by When generating the geometry, the length, diameter, and maximum angle of deviation Spanos and Esteva [2], the lengths of theparameters, fibers and the a distribution,Following and the maximum (waviness, in essence), among other candiameters be definedfollow in our methodology. the angle work of deviation can beand computed from linearofrelationship. fiber length, x, afollows a Weibull done by Spanos Esteva [2], the alengths the fibers andThe the diameters follow distribution, and theof maximum distribution the form:angle of deviation can be computed from a linear relationship. The fiber length, x, follows a Weibull distribution of the form:
1
(7)
1
x “ β p´ ln p1 ´ uqq α
(7)
where α and β are the shape and scale parameters found to be 2.4 and 161.74 by analyzing the data of and β are the shape and scale parameters found to be 2.4 and 161.74 by analyzing the data of Wangwhere et al.α[25], and where the variable is a uniformly distributed random number between 0 and 1. Wang et al. [25], and where the variable u is a uniformly distributed random number between 0 and 1. The mean valuevalue (length) for for thethe aforementioned nm.The Themicrostructures microstructures shown in The mean (length) aforementionedparameters parameters is is 143 143 nm. shown Figurein4Figure were 4obtained for 0.01 andand 0.02 volume parametersα α= 2.4 = 2.4 β = 161.74. were obtained for 0.01 0.02 volumefractions fractions with with parameters andand β = 161.74. Figure 4 how fiber length varies. NoticeNotice fromfrom Figure 4 how thethe fiber length varies. 1
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Figure 4. Network of fibers with lengths following a Weibull distribution (a) c = 0.01 and (b) c = 0.02. Figure 4. Network of fibers with lengths following a Weibull distribution (a) c = 0.01 and (b) c = 0.02.
The diameters of the fibers are defined according to a log-normal distribution from a fitting study by Spanos and Esteva [2]. The log-normal probability density function (PDF) determined in [3] The diameters of the fibers follows a distribution givenare as: defined according to a log-normal distribution from a fitting study by plnx´ µq2 Spanos and Esteva [2]. The log-normal probability function (PDF) determined in [3] follows a 1 density ´ 2σ2 ? e f pxq “ (8) xσ 2π distribution given as:
1 6701
√2
with a mean
0.02847 and standard deviation
(8) 0.3363 of the variable’s natural logarithm.
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with a mean µ “ 0.02847 and standard deviation σ “ 0.3363 of the variable’s natural logarithm. The mean of the diameter can be found by using the following relation, 2 Meandia “ eµ`σ {2
(9)
Materials Using 2015, the 8 above values of µ and σ, the mean diameter is found to be 1.089 nm (nanometers)
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with a standard deviation of 0.3767, which falls within the range of single-walled carbon nanotubes (SWCNTs). The histogram of such a distribution is shown in Figure 5 for 1000 values generated function lognrnd in Matlab and along a log-normal fit. The fit was generated usingfitthewas dfittool randomly using the function lognrnd in Matlab and alongcurve a log-normal fit. The curve command. Notice Figurecommand. 5, that there good agreement between randomly-generated generated usingfrom the dfittool Noticeisfrom Figure 5, that there is goodthe agreement between the randomly-generated log-normal data and the log-normal fit. log-normal data and the log-normal fit.
Figure 5. Log-normaldistribution distribution ofof thethe nanotube diameters. Figure 5. Log-normal nanotube diameters.
The maximum of deviation (see Figure 2) must specified.AsAs proposedby by[2], [2],aalinear The maximum angle angle of deviation (see Figure 2) must alsoalso be be specified. proposed linear relationship exists between the maximum angle of deviation of a segment, and the length of relationship exists between the maximum angle of deviation of a segment, and the length of the nanotube. the nanotube. The relation is given by, The relation is given by, y “ 0.375x ´ 7.5
0.375
7.5
(10)
(10)
where x is the nanotube length in nm and y is the maximum angle of deviation in sexagesimal degrees.
where x is the nanotube length in nm and y is the maximum angle of deviation in sexagesimal degrees. 2.1.2. Network Interactions
2.1.2. Network Interactions
A differentiation control is performed on the Segment objects to find the different network interactions by selecting a fiber and comparing its segments with all the other segments of the other A fibers. differentiation control is two performed the Segment objects to findthethe different network First, the radius of the segments on being compared is calculated as half distance between interactions by selecting a fiberSecond, and comparing its segments with all ofthe other segments of the other their initial and final nodes. the coordinates of the midpoints both segments are located. between two midpoints is calculated Figure as 6); half this the distance is then fibers.Third, First,the thedistance radius of the twothesegments being compared is (see calculated distance between compared with the sum of the radii of both segments, as in the following equation:
their initial and final nodes. Second, the coordinates of the midpoints of both segments are located. Third, the distance between the two midpoints is calculated (see D “ r1 ` r2 Figure 6); this distance is then compared (11) with the sum of the radii of both segments, as in the following equation: If the distance between midpoints (dm1 m2 ) is less than or equal to the parameter D, then the segments are considered to be “close enough”.
If the distance between midpoints (
(11)
) is less than or equal to the parameter D, then the segments
are considered to be “close enough”.
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the distance between the two midpoints is calculated (see Figure 6); this distance is then compared with the sum of the radii of both segments, as in the following equation: (11) If the2015, distance between Materials 8, 6697–6718
midpoints (
) is less than or equal to the parameter D, then the segments
are considered to be “close enough”.
Figure 6. Calculation of distance between midpoints for two segments. 6. Calculation of distance between midpoints for two segments. Materials 2015,Figure 8
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If two segments are considered to be “close enough”, then a second comparison is used to find
If two segments are considered to be “close enough”, then a second comparison is used to find if the if the two segments are crossed. To do that, first we calculate all the distances between the nodes of two crossed. To do that, first we calculate all the distances between the nodes of both bothsegments segmentsare using the following equation: segments using the following equation: |y| ´ mx ´ b| | (12) d“ ? (12) 2 √m ` 11 where mmand andb bareare slope y-intercept the straight line segment, and x and y are the where thethe slope and and y-intercept of theofstraight line segment, and x and y are the coordinates coordinates of the location of the node. By removing the absolute value of Equation (12), sign of the location of the node. By removing the absolute value of Equation (12), the sign of thethe number of the number provides additional information that can be used in the algorithm. If both nodes with provides additional information that can be used in the algorithm. If both nodes with x and y coordinates x and y coordinates are on the same side of any given line described by the parameters m and b of are on the same side of any given line described by the parameters m and b of Equation (12), then their Equation (12), then their distances will have equal signs; for nodes on opposite sides of such straight distances will have equal signs; for nodes on opposite sides of such straight line, their distances will have line, their distances will have opposite signs. For two segments to be crossed, the distances between opposite For twoone segments to be crossed, distances between the nodes segment one and the nodessigns. of segment and straight segmentthe two will have opposite signs; of and the distances straight twoofwill have opposite and the distances between the opposite nodes of signs, segment and betweensegment the nodes segment two and signs; straight segment one will also have as two shown in Figuresegment 7. straight one will also have opposite signs, as shown in Figure 7.
Figure7.7.Sign Sign differences differences between distances. Figure betweennode-segment node-segment distances.
For two segments crossed, computer algorithm separates bycrossing the crossing For thethe two segments thatthat areare crossed, thethe computer algorithm separates themthem by the point point into two new segments. These Segment objects will be composed of one of its original Node into two new segments. These Segment objects will be composed of one of its original Node objects and objects and the crossing as the will otherbeNode, will partFiber of theobject same as Fiber object before,all and the crossing point as the point other Node, part of thebesame before, andasinherit the inherit all the properties of the original “mother” segment. properties of the original “mother” segment. Contact with left and right boundaries were also accounted for in the model by simply Contact with left and right boundaries were also accounted for in the model by simply comparing the comparing the x-coordinate of the segment nodes to the x-coordinate values of the RVE’s left and right x-coordinate of the segment the x-coordinate valuesmarked of the RVE’s and right boundaries. boundaries. Figure 8 below nodes shows to boundary contact points with a left crossmark and crossing Figure 8 below segments shows boundary marked with a crossmark and crossing points between points between markedcontact with anpoints asterisk.
segments marked with an asterisk. 1 0.9 0.8 Contact point 0.7 0.6
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properties of the original “mother” segment. Contact with left and right boundaries were also accounted for in the model by simply comparing the x-coordinate of the segment nodes to the x-coordinate values of the RVE’s left and right boundaries. Figure 8 below shows boundary contact points marked with a crossmark and crossing points between Materials 2015, 8, 6697–6718 segments marked with an asterisk. 1 0.9 0.8 Contact point 0.7 0.6
Crossing point
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Materials 2015, 8 Figure 8. Network of fibers with crossing points between their segments. Figure Materials 2015, 8 8. Network of fibers with crossing points between their segments.
9 9
Tunneling segments were included model by comparing if distances between segments Tunneling segments were included intointo the the model by comparing if distances between segments were Tunneling segments were included into thetwo model by comparing if distances between segments were were less than a predefined threshold. For non-parallel straight segments in a two-dimensional less than a predefined threshold. For two non-parallel straight segments in a two-dimensional plane, the less thanthe a predefined threshold. For two non-parallel straight segments in a two-dimensional plane, the plane, shortest distance between them will always include an end point of one of these segments. shortest distance between them will always include an end point of one of these segments. This is the This is the basic between principlethem underlying the process tunneling paths. two segments have shortest distance will always includeof anfinding end point of one of theseIfsegments. This is the basic principle underlying the process of finding tunneling paths. If two segments have been found to be been found to be “close enough” by the previous differentiator, then the tunneling path testing begins. basic principle underlying the process of finding tunneling paths. If two segments have been found to be “close previous differentiator, the tunneling path testing begins. For this test, For thisenough” test, the by fourthe distances the fourthen end nodes and the corresponding opposite “close enough” by the previousbetween differentiator, then the tunneling path testing begins. Forsegments this test, the four distances between the four end nodes and the corresponding opposite segments are calculated, are calculated, as shown in Figure 9.
the four distances between the four end nodes and the corresponding opposite segments are calculated, as shown in Figure 9. as shown in Figure 9.
Figure 9. Shortest distance between an end node and another segment. Figure 9. Shortest distance between an end node and another segment. Figure 9. Shortest distance between an end node and another segment.
When of these distances arethan lessorthan to thedistance, defined d, distance, d, then When anyany of these foursfours distances are less equalortoequal the defined then a tunneling When any of these fours distances are less than or equal to the defined distance, d, then a tunneling a tunneling path created. Thisa path becomesobject a newwith Segment object with initial andother finalsegment nodes path is created. Thisispath becomes new Segment initial and final nodes as any path is created. This pathinbecomes a new Segment object and final as any other segment as any other segment the microstructure. There arewith twoinitial possible casesnodes for tunneling paths: one in the microstructure. There are two possible cases for tunneling paths: one with a path between a node with a path between a node and a segment (Figure 10) and another one with a path between two in the microstructure. There are two possible cases for tunneling paths: one with a path between a node and a segment (Figure 10) and another one with a path between two nodes (Figure 11). nodes (Figure (Figure 11). and a segment 10) and another one with a path between two nodes (Figure 11).
Figure 10. Tunneling path betweenaanode node anda segment. a segment. Figure 10.Tunneling Tunneling path path between Figure 10. between a nodeand and a segment.
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Figure 10. Tunneling path between a node and a segment.
Figure 11. Tunneling path between two nodes. Figure 11. Tunneling path between two nodes.
The distance threshold establishment a tunneling different polymer materials The distance threshold forfor thethe establishment of aoftunneling pathpath for for different polymer materials was was taken from Sun et al. [16], and shown in Table 1. According to Sun et al. [16], the tunneling distance taken from Sun et al. [16], and shown in Table 1. According to Sun et al. [16], the tunneling distance is is the point at which an increase in length beyond a specific threshold gives a conduction equal to zero. the point at which an increase in length beyond a specific threshold gives a conduction equal to zero. Below the distance threshold, tunneling conduction is a significant contributor to the overall electrical resistivity of the network and can be considered as part of the equivalent electrical circuit. Table 1. Minimum distances for tunneling paths. All values taken from Sun et al. [16]. Matrix Material
Distance (nm)
Polyethylene (PE) Polyimide (PI) Polyvinyl Alcohol (PVA)
2.00 2.50 2.27
2.2. Solution via Node-Voltage Analysis Node-voltage analysis consists of deriving equations for each node in the circuit (except for a reference node) by applying Kirchhoff’s current law. This is the principle underlying the methodology used to calculate the voltage at each node in the microstructure. In this methodology, each element in the microstructure is treated as a resistor, and then the resistor network is analyzed at each node. 2.2.1. Converting Fiber Network into Set of Equations According to several authors [11,12,17,26,27], a polymer nanocomposite may be modeled as a resistor network for simulation purposes. In this network, each nanotube is modeled as a resistor. The polymer matrix may be modeled as an insulating material by assigning it an electrical resistance much higher than the nanotubes. In our case, each segment is modeled as an electrical resistor, depending on its intrinsic resistivity (ρ), diameter, and length. If such resistivity is assumed to be constant along the material, as well as having a material with a constant cross-sectional area (assuming a circular solid rod as a model for the SWCNT), then the electrical resistance may be calculated by the following relation: ρL (13) R“ An where L is the length of the medium; and An is the cross-sectional area of the circular solid rod modeling the CNT; and ρ is the electrical resistivity. Tunneling resistance paths are included in the resistor network by using the following relation from Xu et al. [18]: ˆ ˙ h2 d 4πd ? ? Rtunnel “ exp 2mλ (14) h At e2 2mλ where e is the quantum of electricity; h is Planck’s constant; m is the electron mass; At is the cross sectional area of the tunnel formed between CNTs; d is the distance of the tunnel; and λ is the average
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barrier height, which depends on thebarrier materialheight of the of polymer matrix. A listmaterials. of average barrier heights Table 2. Average various polymer taken from Sun et al. [16] and Xu et al. [18] is summarized in the following Table 2.
Matrix Material Barrier Height (eV) TablePolyethylene 2. Average barrier materials. (PE)height of various polymer 4.43 [16] Polyimide (PI) 4.56 [16] Matrix Material Barrier Height (eV) Polyvinyl Alcohol (PVA) 2.58 [16] Polyethylene (PE) 4.43 [16] Epoxy 0.5–2.5 Polyimide (PI) 4.56[18] [16]
Polyvinyl Alcohol (PVA) account for the Epoxy increase in electrical
2.58 [16] 0.5–2.5 [18]segments when two
In order to resistance are in contact or in close proximity, the electrical resistance for such segments was calculated according to Equation (13), In order to account for the increase in electrical resistance when two segments are in contact or in and increased 200% from its original value. This must be included in the model because the crossed close proximity, the electrical resistance for such segments was calculated according to Equation (13), segments provide a200% higher electrical resistance to the flow of electrons. increase percent was taken and increased from its original value. This must be included in theThe model because the crossed from segments a molecular dynamics study by Buldum et al.to[28]. provide a higher electrical resistance the flow of electrons. The increase percent was taken from a molecular dynamics study by Buldum al. [28]. The process of modeling the nanotube segmentset and tunneling paths as resistors is represented The process of modeling the nanotube segments and tunneling paths as resistors is represented schematically in Figure 12, in which R1 and R2 are calculated using Equation (13), R3 is a tunneling schematically in Figure 12, in which R1 and R2 are calculated using Equation (13), R3 is a tunneling path path and is using Equation R4 through throughR7R7have have a resistance according to andcalculated is calculated using Equation(14), (14), and and R4 a resistance according to Equation but increased 200% becauseof ofcontact contact resistance. Equation (13) (13) but increased 200% because resistance.
12. Schematic of modeling a fiber network as a resistor network. FigureFigure 12. Schematic of modeling a fiber network as a resistor network.
Afterelement each element in the network beenassigned assigned aa value the following matrixmatrix After each in the network hashasbeen valueofofresistance, resistance, the following equation taken from [27], is formed: X \ X \ equation taken from [27], is formed: rG s V “ I (15) N
N
N
X \ ̅ where the term V N is a column vector containing the nodal voltages; I N is also a column vector (15) of all the current sources into the circuit; and GN is the conductance matrix. For the case of a circuit where the term is a column vector containing the nodal voltages; ̅ is also a column vector of all with only electrical resistors and sources, this matrix is also symmetric [29]. the currentTosources into circuit; and GN the is the conductance matrix. For the case of a in circuit with illustrate thethe process of deriving set of equations, the resistor network shown Figure 13 only will be analyzed. current sources vector is formed using[29]. the current sources connected at each electrical resistors andThe sources, this matrix is also symmetric In the example below, of the deriving current entering nodeof2 isequations, obtained bythe transforming circuit from Tonode. illustrate the process the set resistor the network shown in having a voltage as a source to a resistance in parallel with a current source, as shown in Figure 14. Figure 13 will be analyzed. The current sources vector is formed using the current sources connected at After this transformation, the current source is equivalent to V s /R1 . X
\
each node. In the example below, the current entering node 2 is obtained by transforming the circuit from having a voltage as a source to a resistance in parallel with a current source, as shown in Figure 14. After this transformation, the current source is equivalent to Vs/R1.
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12 12
Figure 13. Example case of a resistor network. Figure 13. Example case of a resistor network.
Figure 13. 13. Example Example case case of of aa resistor resistor network. network. Figure
Figure 14. Transformed example resistor network. In our case, the left-hand side boundary is connected a current source, while the right-hand side Figure Transformed exampleto network. Figure 14.14. Transformed exampleresistor resistor network. Figure 14. Transformed example resistor network. boundary is connected to the ground (voltage equal to zero). The current source column vector is formed In ourthe case, the left-hand side boundary is connected to a current source, while the right-hand by In theour algebraic sum of the source currents entering each to node, while source, using the same numbering case, left-hand side boundary boundary is connected connected a current current while thenodal right-hand side In our case, the left-hand side is to a source, while the right-hand side boundary is connected to the ground (voltage equal to zero). The current source column vector side to arrange the vector in the correct order, as seen in Figure 15. If a node is connected to the left-hand boundary connected tothe theground ground (voltage equal tozero). zero). The current source column vector formed boundary isisconnected to to The current column isisformed is formed by the algebraic sum of (voltage the sourceequal currents entering each node,source while using thevector same nodal side boundary via one or more resistors, Ohm’s Law is used to add the value of each source current by the thenumbering algebraic sum sum of the thethe source currents entering each node, while using the same nodal numbering numbering to arrange vectorcurrents in the correct order, as seen inwhile Figureusing 15. Ifthe a node is nodal connected to by algebraic of source entering each node, same entering node. the the left-hand sidein via one or more resistors, Ohm’s used is add the value of each to arrange arrange the vector the correct correct order, as seen seen in Figure Figure 15.Law node connected to the the left-hand to the vector inboundary the order, as in 15. IfIf aaisnode istoconnected to left-hand source current entering the node. the current sources vector is formed, the conductance matrix can be populated. Nodal sideAfter boundary via one one or more more resistors, Ohm’s Law isis used used to to add add the the value value of each each source sourceanalysis current side boundary or resistors, current Aftervia the current sources vector isOhm’s formed,Law the conductance matrix can be of populated. Nodal states that the diagonal elements of the conductance matrix are formed by adding the conductance of the entering the node. node. entering the analysis states that the diagonal elements of the conductance matrix are formed by adding the resistors connected to each node [29] (conductance is[29] defined as thecan inverse ofasresistance). For After the current sources vector formed, the conductance conductance matrix be populated. Nodal analysis analysis conductance of sources the resistors connected to each node (conductance defined the Nodal inverse of the After the current vector isis formed, the matrix canis be populated. network 13elements and 14,ofof the ofare theelements conductance matrix arematrix shown For the network Figures 13 andelements 14 thematrix diagonal of the conductance arebelow statesresistance). thatof theFigures diagonal thediagonal conductance formed by adding theconductance conductance ofthe the states that the diagonal elements of the conductance matrix are formed by adding the of shown below in Figure 15. in Figure connected 15. resistors to each each node node [29] [29] (conductance (conductance isis defined defined as as the the inverse inverse of of resistance). resistance). For For the the resistors connected to network of of Figures Figures 13 13 and and 14, 14, the the diagonal diagonal elements elements of of the the conductance conductance matrix matrix are are shown shown below below network in Figure Figure 15. 15. in
Figure 15. Diagonal elements of conductance matrix for example case.
Figure 15. Diagonal elements of conductance matrix for example case. Figure 15. 15. Diagonal Diagonal elements elements of of conductance conductance matrix matrix for for example example case. case. 6707 Figure
Materials 2015, 8 Materials 2015, 8 8, 6697–6718 Materials 2015,
13 13
Once the diagonal elements of the conductance matrix have have been filled,filled, the non-diagonal elements Once the diagonal elements the conductance the non-diagonal Once the diagonal elements of the of conductance matrixmatrix have been been filled, the non-diagonal elements must also be identified to theand matrix. to nodal analysis,tothenodal non-diagonal elements must alsoand beadded identified addedAccording to the matrix. According analysis, elements the must also be identified and added to the matrix. According to nodal analysis, the non-diagonal elements elements of the j-thmatrix row ofare the zero, conductance zero, except those of the columns of thenon-diagonal j-th row of the conductance exceptmatrix those are of the columns corresponding to nodes of thecorresponding j-th row of the are to zero, of thewhich columns to nodes to conductance nodes directlymatrix connected the jexcept node bythose a resistor, valuecorresponding is the conductance directly connected to the j node by a resistor, which value is the conductance of the connected resistor, of the connected with [29].value Updating conductanceof matrix, the diagonal directly connected to resistor, the j node bya anegative resistor,sign which is thetheconductance the connected resistor, with aand negative sign [29]. Updating the conductance matrix, the diagonal and top non-diagonal elements top non-diagonal elements the of the matrix are matrix, shown in 16. and Thetop bottom non-diagonal with a negative sign [29]. Updating conductance theFigure diagonal non-diagonal elements of theelements matrix are Figure 16.byThe bottom can simply added by canshown simplyinbe added noting thatnon-diagonal the matrix is elements symmetric about the be diagonal (seenoting of the matrix are shown in Figure 16. The bottom non-diagonal elements can simply be added by noting Figure 17).is symmetric about the diagonal (see Figure 17). that the matrix that the matrix is symmetric about the diagonal (see Figure 17).
Figure 16. Top non-diagonal elements of the conductance matrix
Figure 16. Top non-diagonal elements of the conductance matrix Figure 16. Top non-diagonal elements of the conductance matrix
Figure 17. 17. Complete matrix of example Figure Completeconductance conductance matrix of example case. case. Figure 17. Complete conductance matrix of example case. Boundary Conditions 2.2.2.2.2.2. Boundary Conditions 2.2.2. Boundary Conditions The RVE shown in Figure 1 is a three-dimensional space with an electrical potential between
The RVE in Figure 1 isboundaries. a three-dimensional space with an electrical potential between the leftleft- shown and right-hand In the work by Dalmas al. [11], this potential difference ThetheRVE shown in Figureside 1 is a three-dimensional space with anetelectrical potential between the leftwas set to 200V, while all the other considered electrically insulated (zero electrical and right-hand side boundaries. In theboundaries work by are Dalmas et al. as [11], this potential difference was set to and right-hand side boundaries. In the work by Dalmas et al. [11], this potential difference was set to current flow through them). The same potential difference was used in this work while solving 200V, while all the other boundaries are considered as electrically insulated (zero electricalthe current 200V,system whileofallEquation the other areside considered electrically insulated electrical current (15).boundaries The left-hand boundaryasacts as the voltage source (zero and the right-hand flow through them). The same potential difference was used in this work while solving the system of side boundary be considered as connected to the ground. flow through them).can The same potential difference was used in this work while solving the system of Equation (15). The left-hand side boundary acts as the voltage source and the right-hand side boundary Equation (15). TheSystem left-hand side boundary acts as the voltage source and the right-hand side boundary Solving of Equations can be2.2.3. considered as connected to the ground. can be considered as connected to the ground. Once the set of equations has been formed and the boundary conditions defined, as shown in previous sections, then the system can be solved for the nodal voltages. The method used in this 2.2.3.the Solving System of Equations 2.2.3.work Solving Systemthe of system Equations for solving of linear equations is Gauss-Jordan elimination. The main advantage of this method is that it can discriminate between thethe independent non-independent Once the set of equations has been formed and boundaryand conditions defined,equations as showninin the Once the set of equations has been formed and the boundary conditions defined, as shown a linear system. When generating a conductance matrix for a random network of fibers, it is highlyin the previous sections, then the system can be solved for the nodal voltages. The method used in this work probable that then some the of the fiberscan willbe not be connected a “main” network. The “main” previous sections, system solved for the to nodal voltages. The method usednetwork in this work for solving the system of linear equations is Gauss-Jordan elimination. The main advantage of this refers to the group of fibers that are connected to each other and where at least one is connected for solving the system of linear equations is Gauss-Jordan elimination. The main advantage toof this theisvoltage method that itsource. can discriminate between the independent and non-independent equations in a linear method is The thatnodes it canofdiscriminate betweenconnected the independent and non-independent equations in a linear the afibers that are not the “main” network or to the system. When generating conductance matrix for a to random network of fibers, it issource highlyboundary probable that system. When generating a conductance matrix forset. a random network of fibers, is highly probable will not have an independent equation in the In previous works, other itauthors resorted to that some of the fibers will not be connected to a “main” network. The “main” network refers to the group algorithm to determine a conduction path network. that would eliminate the setrefers of equations some using of theanfibers will not be connected to a “main” The “main”from network to the group of fibers connected to each other where atpath leastbefore one isgenerating connectedthetoconductance the voltage matrix. source. the that fibersare that are not connected to thisand conduction of fibers that are connected to each other and where at least one is connected to the voltage source. In nodes this way, the fibers conductance is guaranteed to be nonsingular. of resorting to such will The of the that arematrix not connected to the “main” networkInstead or to the source boundary
The nodes of the fibers that are not connected to the “main” network or to the source boundary will not have an independent equation in the set. In previous works, other authors resorted to using an not have an independent equation in the set. In6708 previous works, other authors resorted to using an
conductance matrix is guaranteed to be nonsingular. Instead of resorting to such algorithm for path finding, a pure mathematical solution, Gauss-Jordan method, can be used to solve the system of equations. The Gauss-Jordan method generates a matrix of zeroes and ones only. The non-independent 2015, 8, 6697–6718 because they will composed of only zeroes after being reduced by the rows Materials can be identified Gauss–Jordan elimination method. These rows of zeroes are sent to the bottom of the system of equations, afterfor which, newly-modified conductance matrix is multipliedmethod, by the current source algorithm path the finding, a pure mathematical solution, Gauss-Jordan can be used tovector solve the system of equations. The Gauss-Jordan method generates a matrix of zeroes and ones only. to find the nodal voltages of Equation (15). The non-independent rows can be identified because they will composed of only zeroes after being The information obtained from the nodal voltage analysis can be used to visualize the electrical reduced by the Gauss–Jordan elimination method. These rows of zeroes are sent to the bottom of the properties RVE. The Figures 18 and 19 show the voltage distribution: colorbar is systemofofthe equations, afterdiagrams which, thein newly-modified conductance matrix is multiplied by the a current vector findvoltage the nodal voltages of Equation (15). used source to show howtothe varies in the microstructure; those fibers that are not connected to the The information obtained from the nodal voltage analysis used to visualize electrical voltage source or the “main” network are colored in black; and,can thebeinsulated polymerthe matrix is simply properties of the RVE. The diagrams in Figures 18 and 19 show the voltage distribution: a colorbar shown in white. Additionally, electrical currents incoming on left-hand side boundary and electrical is used to show how the voltage varies in the microstructure; those fibers that are not connected to currents outgoing on right-hand sidenetwork boundary shown (Figure 18b), where thepolymer units are in Amperes. the voltage source or the “main” are are colored in black; and, the insulated matrix is simply shown in white. Additionally, electrical left-hand side are boundary and to the The sum of incoming currents must be equal to thecurrents sum of incoming outgoing on currents, which also equal electrical currents outgoing on right-hand side boundary are shown (Figure 18b), where the units are along total current in the circuit. The equivalent resistance of the RVE is also shown on the diagram, in Amperes. The sum of incoming currents must be equal to the sum of outgoing currents, which are with also the corresponding electrical conductivity. This information summarizes the electrical properties of equal to the total current in the circuit. The equivalent resistance of the RVE is also shown on the RVE. the diagram, along with the corresponding electrical conductivity. This information summarizes the properties the RVE. A electrical microstructure of of straight nanotubes is shown first in Figure 18. The nanotubes not connected to A microstructure of straight nanotubes is shown first Figurein18. The nanotubes not connected the voltage source directly or through another nanotube are in shown black, which means that the voltage to the voltage source directly or through another nanotube are shown in black, which means that the distribution across them is zero. The nanotube that is connected to the voltage source but not connected voltage distribution across them is zero. The nanotube that is connected to the voltage source but not to theconnected ground istoastheinground an open and the voltage across it isacross 200 V. As200 expected, the is circuit as in anconfiguration, open circuit configuration, and the voltage it is V. As theRVE total current and RVE of both Figurezero. 18 are both zero. total expected, current and conductivity of conductivity Figure 18 are 1
1 ITot = 0 A Req = Inf
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Figure 18. (a) Microstructure of straight nanotubes, c = 0.01, AR = 130; and (b) corresponding nodal
Figure 18. (a) Microstructure of straight nanotubes, c = 0.01, AR = 130; and voltage diagram. (b) corresponding nodal voltage diagram. Figure 19 shows a microstructure with length and diameter distributions, generated using the
Figure 19 shows a microstructure with the length and and diameter distributions, generated using the parameters described in Section 2.1.1. Here, incoming outgoing electrical currents are shown parameters in Sectionand 2.1.1. Here, electrical the incoming andofoutgoing electricalthat currents are shown on on the described diagram. Incoming outgoing currents those nanotubes are connected to boundaries but in open circuit configuration are zero. In addition, the sum of incoming currents equals the sum of those outgoing, and both sums are equal to the total current in the circuit, which is shown on the top left corner of Figure 19b along the equivalent resistance and conductivity of the RVE. Nodal voltages of each crossing are not shown numerically because of the amount of crossings in the diagram, but the color code indicates how the voltage is being distributed in the RVE.
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boundaries but in open circuit configuration are zero. In addition, the sum of incoming currents equals the sum of those outgoing, and both sums are equal to the total current in the circuit, which is shown on the top left corner of Figure 19b along the equivalent resistance and conductivity of the RVE. Nodal voltages of each crossing are not shown numerically because of the amount of crossings in the diagram, Materials 2015, 8, 6697–6718 but the color code indicates how the voltage is being distributed in the RVE. 1
1 ITot = 4.1529e-07 A
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Figure 19. (a) Microstructure with length/diameter distributions, c = 0.015; and (b) corresponding Figure 19. (a) Microstructure with length/diameter distributions, c = 0.015; and nodal voltage diagram. (b) corresponding nodal voltage diagram.
3. Results and Discussion
3. Results and Discussion 3.1. Percolation Behavior
3.1. Percolation Behavior
First, the percolation behavior of the simulated microstructures was analyzed and compared to similar work in the literature. In order to do that, Monte Carlo simulations were performed for First, the percolation behavior of the simulated microstructures was analyzed and compared to similar different set of parameters. The percolation probability was calculated as the ratio of successfully work in the literature. In orderover to do Monte simulations were performed percolated microstructures thethat, number of Carlo simulations performed. Thus each for datadifferent point set of in the graphs (Figures probability 20–23) represents Monte simulations for a single set parameters. The below percolation was several calculated as Carlo the ratio of successfully percolated of parameters. microstructures over the number of simulations performed. Thus each data point in the graphs below The fillers in the polymer nanocomposite, carbon nanotubes in this case, are said to percolate (Figures 20–23) represents several Monte Carlo simulations for a single set of parameters. when they form a network which is able to conduct electricity across the polymer matrix. A sudden The fillers the polymer nanocomposite, nanotubes in this acase, are volume said to percolate increase in in percolation probability is expectedcarbon from theory, but because limited is being when a smooth increase in percolation seen instead (Figurematrix. 20), which is alsoincrease theysimulated, form a network which is able to conductprobability electricityisacross the polymer A sudden observed in the work of Flandin et al. [20]. in percolation probability is expected from theory, but because a limited volume is being simulated, The two curves in Figure 20 show the percolation behavior for the simulation of microstructures a smooth increase in percolation probability is seen instead (Figure 20), which is also observed in the with constant length and diameter distributions for all nanotubes with an aspect ratio of 140 and a work of Flandin et of al.deviation [20]. maximum angle of 45˝ . The only difference between the curves is in the length of the fibers, (a) curves 500 nmin and (b) 300 Figure 20 shows thebehavior typical percolation behavior described in The two Figure 20nm. show the percolation for the simulation of microstructures literature [3,15]. with constant length and diameter distributions for all nanotubes with an aspect ratio of 140 and The crossing point between the curves gives the percolation threshold for the given parameters, a maximum angle of deviation The onlyprobability differenceofbetween the curves is in the length of the which occurs at 3.25 vol. % andofat45°. a percolation approximately 0.5. This is consistent fibers, 500 nm and (b) makes 300 nm. Figure 20 shows the typical percolation described in with(a) Zeng et al. [19], which use of a percolation probability of 0.5 as an indicatorbehavior of the electrical percolation threshold in their Monte Carlo simulations. It can also be noted that the curve with literature [3,15]. g = 0.03 has a steeper slope at the critical area where the percolation probability rapidly increases. A sudden increase in percolation probability is expected from theory, thus it can be observed that decreasing the parameter g has the effect of generating a percolation curve, which tends to approach the theoretical sudden increase.
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Percolation Probability Percolation Probability
parameter g has theateffect of generating a percolation curve, probability which tendsrapidly to approach the theoretical has a steeper slope the critical area where the percolation increases. A sudden sudden increase. increase in percolation probability is expected from theory, thus it can be observed that decreasing the parameter g has the effect of generating a percolation curve, which tends to approach the theoretical Materials 2015, 8, 6697–6718 1 sudden increase. 0.8 g=0.5 g=0.3
1 0.6 0.8
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Volume Fraction 0 0 0.01 curves 0.02 intersect 0.03 0.04 0.06 0.07 Figure 20. Percolation probability at the0.05 percolation threshold, for AR = 140.
Volume Fraction
Figureparameters 20. Percolation probability curves the percolation threshold, for AR = 140. Keeping all the other constant, theintersect aspectatratio was increased from 140 to 240, and a range Figure 20. Percolation probability curves intersect at the percolation threshold, for AR = 140. of volume fractions were for gconstant, = 0.5theand g ratio = 0.3. From Figure it and cana be seen that Keeping all thesimulated other parameters aspect was increased from 14021, to 240, volume fractions were simulated for = 0.5 and g =% 0.3.and Figure 21, itapproximately can be seen that 1.9 vol. %. theKeeping curves intersect at two different points,the at gaspect 1.75 vol. again atfrom allrange theofother parameters constant, ratio wasFrom increased 140 to 240, and a range the curves intersect at two different points, at 1.75 vol. % and again at approximately 1.9 vol. %. The Thevolume corresponding percolation probability 0.386 and 0.55, respectively, closebetoseen the that 0.5 of fractions were simulated for gis 0.386 =is0.5 g0.55, = about 0.3. From Figure 21,percolation it can corresponding percolation probability andand about respectively, close to the 0.5 probability value, expected from [19]. percolation probability as expected from the curves intersect at value, twoasdifferent points, at [19]. 1.75 vol. % and again at approximately 1.9 vol. %.
Percolation Probability Percolation Probability
The corresponding percolation probability is 0.386 and about 0.55, respectively, close to the 0.5 percolation probability value, as1 expected from [19]. 0.8 1
g=0.5 g=0.3
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0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Volume Fraction
Figure 21. Percolation probability curves intersect at percolation threshold, AR = 240.
0 0.005 0.01 curves 0.015 0.02 0.025 0.03 0.035 0.04 threshold, AR = 240. Figure 21. Percolation probability intersect at percolation
Volume Fraction Figure 22 compares two percolation curves with the same fiber length parameter, g, but with different aspect ratios. Increasing the aspect ratio (AR) shifts the curve to the left, from which it can Figure 21. Percolation probability curves intersect at percolation threshold, AR = be observed that the increase in the aspect ratio decreases the volume fraction at which the percolation threshold occurs, which is consistent with previous literature [13]. Moreover, from the figure below, it can be noted that for the same volume fraction, a higher percolation probability can be obtained for the case of aspect ratio of 240, thus a higher electrical conductivity would in turn be achieved.
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240.
that the increase in the aspect ratio decreases the volume fraction at which the percolation threshold occurs, which is consistent with previous literature [13]. Moreover, from the figure below, it can be noted that for the same volume fraction, a higher percolation probability can be obtained for the case of 8, 6697–6718 aspect ratio ofMaterials 240,2015, thus a higher electrical conductivity would in turn be achieved.
Percolation Probability
1 0.8 AR=140 AR=240 0.6 0.4 0.2 0 0
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Volume Fraction Figure 22. Comparison of percolation probability curves with different aspect ratios, g = 0.3.
Figure 22. Comparison of percolation probability curves with different aspect ratios, g = 0.3.
The values for percolation threshold obtained are comparable to experimental values from the Table 3 showsthreshold the comparison with dataare takencomparable from Potschke et [30]. Additionally,values we The valuesliterature. for percolation obtained toal.experimental from the may compare these results to the pioneering work of Flandin et al. [20]. In their work, they simulated literature. Table 3 shows the comparison data takenmicrostructures from Potschke etup al.entirely [30]. Additionally, we may the percolation behavior of randomwith three-dimensional made of straight fibers with antoaspect of about 14, and found a percolation threshold 4.45% ˘ 0.05%. Although compare these results the ratio pioneering work of Flandin et al. [20].ofIn their work, they simulated the the models are different in several aspects and the results in percolation threshold comparable, the percolation behavior of could random three-dimensional microstructures up entirely of straight fibers disagreement be mainly attributed to the difference in aspect ratiomade of the fillers.
with an aspect ratio of about 14, and found a percolation threshold of 4.45% ± 0.05%. Although Table 3. Comparison of percolation threshold results with values from other authors. the models are different in several aspects and the results in percolation threshold comparable, Author Aspect Ratio CNT Type Matrix Type Percolation Threshold the disagreement could be mainly attributed to the difference in aspect ratio of the fillers. This work 140–240 SWCNTs PE (thermoplastic) 1.75–3.25 vol. % Potschke et al. [30]
100–667
MWCNTs
PC (thermoplastic)
1.47–2.94 vol. %
Table 3. Comparison of percolation threshold results with values from other authors. Additional to the results in percolation behavior shown above, an improvement upon previous
Author CNT Type Matrix Typewith length Percolation Threshold simulations was Aspect made toRatio portray the microstructure more accurately and diameter statistical distributions and length-dependence of the maximum angle of deviation. Monte Carlo This work 140–240 SWCNTs PE (thermoplastic) 1.75–3.25 vol. % simulations were performed to calculate the percolation probability for the microstructure with the Potschkeaforementioned et al. [30] distributions, 100–667 and theMWCNTs PC (thermoplastic) 1.47–2.94 vol. % results are shown in Figure 23. For these simulations, the
log-normal diameter distribution used had a mean diameter of 2.18 nm; and the Weibull fiber length had a mean of 0.29 µm. Fibers were generated with five per fiber in Additionaldistribution to the results in value percolation behavior shown above, an segments improvement upon previous an RVE of 1 µm side length and a thickness equal to the maximum diameter to a confidence level simulations was made to portray the microstructure more accurately withcanlength and diameter statistical of 95%, which was calculated to be 2.85 nm. The percolation threshold be estimated to be between 2.5 and 3 vol. %, which comparable to the angle experimental results by Potschke al. [30].simulations were distributions and length-dependence ofis the maximum of deviation. MonteetCarlo
performed to calculate the percolation probability for the microstructure with the aforementioned distributions, and the results are shown in Figure 23. For these simulations, the log-normal diameter distribution used had a mean diameter of 2.18 nm; and the Weibull fiber length distribution had a mean value of 0.29 µm. Fibers were generated with five segments per fiber in an RVE of 1 µm side length and a thickness equal to the maximum diameter to a confidence level of 95%, which was calculated to be 6712
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2.85 nm.2015, The8,percolation Materials 6697–6718 threshold can be estimated to be between 2.5 and 3 vol. %, which is comparable to the experimental results by Potschke et al. [30].
Percolation Probability
1 0.8 0.6 0.4 0.2 0 0
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Figure 23. Percolation probability with length and diameter statistical distributions. Figure 23. Percolation probability with length and diameter statistical distributions.
3.2. Tunneling Tunneling Behavior 3.2. Behavior The tunneling cut-off distances and the average barrier height values taken from [16] and the The tunneling cut-off distancesresistance, and the average height taken from [16] (14), and the equation equation to calculate tunneling takenbarrier from [18] andvalues shown as Equation were used to calculate tunneling resistance, taken from [18] and shown as Equation (14), were used to calculate the to calculate the tunneling resistance as a function of the distance between segments for two polymer matrices, PE and PI. as From the figure below (Figure 24), it can be seenfor that when the distance between tunneling resistance a function of the distance between segments two polymer matrices, PE and two nanotubes decreases, the tunneling electrical resistance decreases too. Such resistance seems to PI. From the figure below (Figure 24), it can be seen that when the distance between two nanotubes increase rapidly at a distance nearresistance 2 nm, which agreestoo. withSuch previous workseems [16]. to increase rapidly at decreases, the tunneling electrical decreases resistance The distance between two nanotube segments can affect the voltage distribution in the a distance near 2 nm, which agrees with previous work [16]. microstructure. A simple comparison it is shown to illustrate this point: a tunneling path is formed The distance between two nanotube segments can affect the voltage distribution in the microstructure. between two segments (formed inside the red circle in Figure 25a); for this particular case, the A simple comparison it is shown illustrate point: aoftunneling between two segments 7 Ω.isIfformed tunneling path is 0.45 nm longto and has a this resistance 1.0 ˆ 10path the separation between the 11 (formed inside the red circle in Figure 25a); for this particular case, the tunneling path is 0.45 nm long segments is increased to 0.89 nm, the resistance increases to 3.2 ˆ 10 Ω. As expected, the voltage Ω. Ifchanges the separation between the0.20 segments is increased to 190 0.89Vnm, the and a resistance of 1.0 path × 107also drophas across the tunneling dramatically: from V in the first case to in the 11 second case. Additionally, increasing the separation between the segments decreases the electrical resistance increases to 3.2 × 10 Ω. As expected, the voltage drop across the tunneling path also changes current from 1.77 1.23V A,inwhich in turn the RVE. dramatically: fromto0.20 the first casewould to 190affect V inthe theconductivity second case.ofAdditionally, increasing the 21 Ω and the voltage drop At a tunneling path length of 1.92 nm, the resistance is already 3.0 ˆ 10 separation between the segments decreases the electrical current from 1.77 to 1.23 A, which in turn is approximately 200 V. Keeping in mind that the total voltage drop across the RVE is 200 V, the flow would affect the conductivity of the RVE. of electrons would be nearly impossible for tunneling paths longer than 2 21 which agrees with the Materials 8 path length of 1.92 nm, the resistance is already 3.0 × 10 nm, At 2015, a tunneling Ω and the voltage drop is cut-off distances of Table 1.
Log tunneling resistance (log Ohms)
approximately 200 V. Keeping in mind that the total voltage drop across the RVE is 200 V, the flow 100 of electrons would be nearly impossible for tunneling paths longer than 2 nm, which agrees with the PE cut-off distances of Table 1.90 PI 80 70 60 50 40 30 20 10 0 -2.5
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Figure 24. Tunneling resistance as a function of distance between nanotubes.
Figure 24. Tunneling resistance as a function of distance between nanotubes. 6713
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Figure 24. Tunneling resistance as a function of distance between nanotubes. 1 0.9 1.2246e-09 0.8 1.2246e-09
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Figure 25. 25. Tunneling Tunneling path of (a) 0.45 0.45 nm nm and and (b) 0.89 0.89 nm. nm.
Electrical Conductivity 3.2.1. 3.3. Electrical Conductivity
TheThe conductivity thematerial material calculated fromCarlo Monte Carlo simulations, which conductivity of the waswas also also calculated from Monte simulations, which included included tunneling behavior, contact resistance, and microstructure with length and diameter tunneling behavior, contact resistance, and microstructure with length and diameter distributions and distributions and length-dependent angle deviation. The mean diameterwas of the length-dependent maximum angle of maximum deviation. The meanof diameter of the log-normal distribution log-normal distribution was setlength at 2.18 nm, the mean fiber length the µm, Weibull distribution set at 2.18 nm, the mean fiber of the Weibull distribution was set of at 0.29 with five segments was set atper 0.29 µm, with five segments per fiber. A side length of 1 µm and thickness of 2.85 nm was fiber. A side length of 1 µm and thickness of 2.85 nm was set for the RVE. The conductivity 5 set fordistribution the RVE. The distribution was set withofmetallic conductivityintrinsic of 105 S/m was conductivity set with metallic intrinsic conductivity 10 S/mintrinsic and semi-conductor and semi-conductor conductivity S/m, potentialcut-off difference ofof 200 V, tunneling cut-off conductivity of 10intrinsic S/m, potential differenceofof10 200 V, tunneling distance 2 nm, and a barrier distance of 2 nm, and a barrier height of 4.43 eV. height of 4.43 eV. Each data Figures the mean conductivity value of the several Each datapoint point plotted plotted ininFigures 26,26, 27, 27 andand 29 is29 theismean conductivity value of the several Monte Monte Carlo simulations performed for each set of parameters. These conductivities were compared Carlo simulations performed for each set of parameters. These conductivities were compared to the Materials 2015, 8 from simulations to thevalues values obtained from simulations by[12] Hu(see et al. [12]26). (see 26). Hu al. performed obtained by Hu et al. Figure HuFigure et al. performed 3-DetMonte Carlo 20 3-D Monte Carloofsimulations of with straight CNTs with an AR 100,diameter length of µm,and diameter 50 nm, simulations straight CNTs an AR = 100, length of 5= µm, 505nm, a constant 4 S/m. The values of conductivity 4 and a constant distribution of intrinsic CNT conductivity of 10 distribution of intrinsic CNT conductivity of 10 S/m. The values of conductivity obtained by Hu et al. obtained Hu than et al. the areones clearly largerinthan ones obtained incurves this work, although curves are clearlyby larger obtained this the work, although both follow a similarboth trend. This follow a similar trend. This is as expected, namely because [12] uses a constant4 CNT intrinsic is as expected, namely because [12] uses a constant CNT intrinsic conductivity of 10 S/m, while this conductivity of 104 S/m, while this work assumes that one-third of the CNTs are metallic5 and with work assumes that one-third of the CNTs are metallic and with a conductivity of 10 S/m and a conductivity of 105 S/m and two-thirds behave as electrical semi-conductors with an intrinsic two-thirds behave electrical semi-conductors withdoes an intrinsic 10contact S/m. Inresistance addition, conductivity of 10as S/m. In addition, Hu et al. [12] not takeconductivity into accountofthe Hu et al. [12] doesCNTs not take into account thewhich contact resistance CNTsofcome generated when come into contact, increases thegenerated electricalwhen resistance CNTsinto andcontact, of the RVE overall. which increases the electrical resistance of CNTs and of the RVE overall. 2
10
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Figure 26. Comparison of PNC conductivity to the work by Hu et al. [12].
Figure 26. Comparison of PNC conductivity to the work by Hu et al. [12]. To obtain a better comparison to [12], a range 6714of volume fractions between 0.01 and 0.05 was simulated with a constant fiber length of 500 nm and aspect ratio of 100, and using a constant intrinsic conductivity of 104 S/m for all the nanotubes generated in the microstructure and ignoring contact
Materials 2015, 8, 6697–6718
To obtain a better comparison to [12], a range of volume fractions between 0.01 and 0.05 was simulated with a constant fiber length of 500 nm and aspect ratio of 100, and using a constant intrinsic conductivity of 104 S/m for all the nanotubes generated in the microstructure and ignoring contact resistance. The RVE side length was set at 1 µm and the thickness, in this case, was set equal to the fiber diameter, which is 5 nm. Figure 27 shows the original values of conductivity along with the ones from the modified simulation, being compared with those obtained by [12]. It can be seen that, by defining a constant intrinsic conductivity for all the nanotubes and ignoring contact resistance, a higher PNC conductivity can be obtained, which tends towards the values obtained by [12]. As seen from the figure above, a discrepancy of at least one order of magnitude in the conductivity is seen between both studies. The RVE in our work is essentially two-dimensional, however, a thickness must be defined in order to calculate its conductivity. Such thickness was defined as the diameter of the fibers, namely 5 nm, while Hu et al., who generates a three-dimensional microstructure in their simulations, uses the dimensions of their three-dimensional RVE to calculate its conductivity. If the thickness of the RVE is increased in our work, a higher value of conductivity is obtained and similar values to those of Hu et al. [12] may be obtained. However, this solution to the discrepancy is arbitrary and this problem Materials 2015, 8 must be further studied.
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Volume Fraction Figure 27. Comparison of [12] with modified simulation.
Figure 27. Comparison of [12] with modified simulation. The number of simulations needed for convergence was analyzed; for this purpose, the mean
1.09 1.07 1.05 1.03 1.01 0.99
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alized Mean nductivity
alized Mean nductivity
Theconductivity number ofafter simulations needed for convergence was analyzed; for this purpose, the mean each simulation was normalized with respect to the final value. For a volume conductivity eachafter simulation was normalized with respectthe to mean the final value. Fordoes a volume fraction fraction after of 0.05, about 700 Monte Carlo simulations, conductivity not vary more than 1% with the final value (see A volumedoes fraction 0.04more requires of 0.05, after about 700 respect Monte to Carlo simulations, theFigure mean28a). conductivity not of vary than 1% more simulations to find a converging mean conductivity. Figure 28b shows that after about 2500 with respect to the final value (see Figure 28a). A volume fraction of 0.04 requires more simulations to simulations, the mean value does not vary more than 1% with respect to the final mean conductivity. find aSimilarly, converging mean conductivity. Figure 28b shows that after about simulations, theat mean volume fraction of 0.03 converges at about 3300 simulations, and2500 0.02 volume fraction value about does not more than 1% with respect to the final mean conductivity. Similarly, volume fraction 7400vary simulations. When performing Montesimulations, Carlo simulations for avolume volume fraction fraction of percolation probability of 0.03 converges at about 3300 and 0.02 atlow about 7400 simulations. (less than 0.01), only the few percolated cases exhibit conductivity, thus the mean value becomes When performing Monte Carlo simulations for a volume fraction of low percolation probability somewhat volatile and close to zero. The number of simulations needed for convergence was too (less than theoffew cases presented exhibit conductivity, thus the mean value becomes great 0.01), for the only purpose thispercolated work. The value as the mean conductivity is only a rough somewhat volatile and close to zero. The number approximation of the estimated converged value.of simulations needed for convergence was too great for the purpose of this work. The value presented as the mean conductivity is only a rough approximation of the estimated converged value. 1.1 1 0.9 0.8
1.09 1.07 1.05 1.03 1.01 0.99 0.97 0.95
Normalized Mean Normalized Mean Normalized Mean Conductivity Conductivity
Materials 2015, 8
Normalized Mean Conductivity
of 0.03 converges at about 3300 simulations, and 0.02 volume fraction at about 7400 simulations. When performing Monte Carlo simulations for a volume fraction of low percolation probability (less than 0.01), only the few percolated cases exhibit conductivity, thus the mean value becomes somewhat volatile and close to zero. The number of simulations needed for convergence was too great Materials 8, 6697–6718 for2015, the purpose of this work. The value presented as the mean conductivity is only a rough approximation of the estimated converged value.
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1.1 0 200 400 600 800 1000 1 Number of Simulations (b) Materials 0.9(a)2015, 8 0.8 1.1 0.7 Figure 28. Cont. 1 0.6 0.9 0.5 0.8 0.4 0.7 0 1000 2000 3000 0.6
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c = 0.02. Figure 28. Normalized mean conductivity (a) c = 0.05; (b) c = 0.04; (c) c = 0.03; and (d) Figure conductivity(a) (a)c c==0.05; 0.05;(b) (b)cc==0.04; 0.04;(c) (c)cc==0.03; 0.03; and and(d) (d)cc==0.02. 0.02. Figure28. 28.Normalized Normalized mean mean conductivity
The conductivities were also compared to experimental values [30]. In their work, Potschke et al. conductivities were were also compared to to experimental valuesvalues [30]. In[30]. their In work, Potschke et al. conductivities also compared experimental their work, Potschke measured The theThe resistivity of PNCs with MWCNTs with diameters between 10 and 15 nm and length measured the resistivity of PNCs with MWCNTs with diameters between 10 and 15 nm and length et al. measured the resistivity of PNCs with MWCNTs with diameters between 10 and 15 nm and between 1 between and 10 1µm in polycarbonate (PC) canbebe seen in Figure 29 that the experimental and 10 aµm a polycarbonate (PC)matrix. matrix. ItItcan seen Figure 29 thatin the experimental length between 1 and 10inµm in a polycarbonate (PC) matrix. Itincan be seen Figure 29 that the data obtained Potschke et al. in agreement withthethe values obtained in this work datafrom obtained from Potschke etisal. iscloser in closer values obtained this values work than thosethan experimental data obtained from Potschke etagreement al. is in with closer agreement within the obtained in those from than the simulation byThe [12]. The discrepancy inthe the values ranging from 0.01values and 0.03 volume fractions from the by [12]. discrepancy values ranging 0.01 and 0.03from volume fractions thissimulation work those from the simulation by in [12]. The discrepancy infrom the ranging 0.01 and between our work and those obtained experimentally by [30] can be attributed to a difference in the exact 0.03our volume between ourexperimentally work and those obtained experimentally byto[30] can be attributed between workfractions and those obtained by [30] can be attributed a difference in the exact location of the percolation threshold.of Forthe thepercolation work in [30],threshold. a sudden drop in conductivity to a difference in the exact location For the work inoccurs [30], below a sudden location of0.04 the volume percolation threshold. For shows the work in [30], a sudden in conductivity occurs below fraction, whilebelow our work a steady decrease untilour 0.02drop volume fraction. Ultimately, drop in conductivity occurs 0.04 volume fraction, while work shows a steady decrease 0.04 volume work shows adiscrepancy steady decrease until volume fraction. Ultimately, discrepancy could our also be attributed to the difference incould polymer matrix. until this 0.02fraction, volume while fraction. Ultimately, this also be0.02 attributed to the difference in this discrepancy could also be attributed to the difference in polymer matrix. polymer matrix. 2
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4. Concluding Remarks
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This work contributes by developing a two-dimensional simulation of a microstructure with Weibull Figure 29. Comparison of conductivities to simulation and experimental data. distribution for the nanotube length, a log-normal distribution of the nanotube diameter, as data. well as a Figure 29. Comparison of conductivities to simulation and experimental relation used for the calculation of the maximum angle of deviation of a segment with respect to the 4. Concluding Remarks previous segment, taken by the work done by [2]. The main purpose of incorporating these improvements Concluding Remarks This contributes developing two-dimensional simulation ofofaa microstructure to thework simulation was to beby able to generate aa microstructure more representative carbon nanotubes with Weibull distribution the nanotube length, a log-normal of the nanotube network inside afor polymer matrix. Additionally, this work distribution includes tunneling behavior todiameter, the
This work contributes by developing a two-dimensional simulation of a microstructure with Weibull distribution for the nanotube length, a log-normal distribution of the nanotube diameter, as well as a 6716 relation used for the calculation of the maximum angle of deviation of a segment with respect to the previous segment, taken by the work done by [2]. The main purpose of incorporating these improvements
Materials 2015, 8, 6697–6718
as well as a relation used for the calculation of the maximum angle of deviation of a segment with respect to the previous segment, taken by the work done by [2]. The main purpose of incorporating these improvements to the simulation was to be able to generate a microstructure more representative of a carbon nanotubes network inside a polymer matrix. Additionally, this work includes tunneling behavior to the two-dimensional simulation, which has been included in previous works. The simulation is completed by incorporating an intrinsic nanotube conductivity distribution when calculating electrical resistance to each segment in the microstructure, and by adding contact resistance to the crossed segments. Finally, the simulation does not require the use of a path search algorithm to find the network of nanotubes that expand from one boundary to the other. Instead, a purely mathematical solution is employed to solving the set of equations generated from the nodal analysis, in which the non-independent equations are identified and removed from the set of equations by using the Gauss-Jordan method. The computational methodology identified and developed can be used as an aid in the study and development of polymer nanocomposites. It is capable of generating a microstructure to a specified CNT volume fraction or specific number of CNTs and with the capability of adjusting several parameters such as fiber length, diameter, waviness, number of segments per fiber, and even including distributions for fiber length and diameter. The methodology can calculate relevant parameters in the study of the electrical properties of PNCs, such as percolation probability, equivalent electrical resistance, total electrical current flow in the RVE, nodal voltages for the entire microstructure, and conductivity of the RVE. The relevant parameters were put together in a diagram that summarizes the electrical properties of a particular microstructure; some examples of these diagrams are shown in Figures 18 and 19. Acknowledgments: This work was funded by Tecnologico de Monterrey-Campus Monterrey, through the Research group of Nanotechnology and Devices Design. Additional support was provided by Consejo Nacional de Ciencia y Tecnología (CONACYT). Project Number 242269, México. Author Contributions: M.S., M.E. and A.E.-Z. formulated and developed the theoretical model. M.S. was responsible of performing the computational simulations. M.S., M.E. and A.E.-Z. analyzed the data generated by the computational simulations. O.M.-R. and J.B. provided significant guidance in the development of the model. M.S. wrote the paper and A.E.-Z. reviewed and edited the paper. Conflicts of Interest: The authors declare no conflict of interest. References 1. 2.
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