Fabrication and Electrical Conductivity of Poly(methyl methacrylate

However, not much attention has been paid to the influence that processing .... Figure 4 SEM images of fractured cross sections of PMMA/CB composite ...
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J. Phys. Chem. B 2006, 110, 22365-22373

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Fabrication and Electrical Conductivity of Poly(methyl methacrylate) (PMMA)/Carbon Black (CB) Composites: Comparison between an Ordered Carbon Black Nanowire-Like Segregated Structure and a Randomly Dispersed Carbon Black Nanostructure Runqing Ou, Sidhartha Gupta, Charles Aaron Parker, and Rosario A. Gerhardt* School of Materials Science and Engineering, Georgia Institute of Technology, 771 Ferst DriVe, Atlanta, Georgia 30332-0245 ReceiVed: July 15, 2006; In Final Form: September 7, 2006

Poly(methyl methacrylate) (PMMA)/carbon black (CB) composites were fabricated using two different mixing methods: (1) mechanical mixing and (2) solution mixing of the precursors, followed by compression molding. The microstructures obtained were examined by optical and scanning electron microscopy. Electrical properties were measured using impedance spectroscopy over a wide frequency range (10-3 to +109 Hz). With the mechanical mixing method, a segregated structure is produced with PMMA particles forming faceted grains with carbon black particles aligning to form a network of 3D-interconnected nanowires. This microstructure allows percolation to occur at a low volume fraction of 0.26 vol % CB. In contrast, specimens made by the solution method have a microstructure where carbon black is distributed more randomly throughout the bulk, and thus, the percolation threshold is higher (2.7 vol % CB). The electrical properties of the PMMA/CB composites fabricated by the mechanical mixing method are comparable to those obtained with single-wall nanotubes as fillers.

I. Introduction Polymer composites are used in many different applications. They have been used as electromagnetic shielding materials,1,2 such as conductive plastic sheets useful for the packaging of integrated circuit devices.3,4 Other interesting applications include self-regulating heaters5 and electronic noses.6 More recently, there has been interest in using polymer composites as chemical sensors7 and supercapacitors for charge storage.8 A crucial aspect in the production of conducting polymer composite materials is the filler content, which must be as low as possible and still allow the composite to fulfill its electrical requirements; otherwise, the processing becomes difficult, the mechanical properties of the composite are poor, and the final cost is high. Carbon black (CB) has been used as a conductive filler in conductive polymer composites for many years. For an excellent review, refer to Balberg.9 Carbon black has been proven to be one of the most versatile functional fillers for rubber and plastics. In addition to its traditional use as a reinforcing agent in rubber tires, it can also be used as a toner for copy machines and printers,10 or as a pigment for plastics, producing various shades of black in the final products. A more recent and fast growing use of carbon black is in conductive polymer composites in which carbon black renders the composites the desired level of conductivity. The advantage of carbon black is that it can easily be obtained in sizes ranging from 10 nm to over 100 nm. Its properties and that of composites made with them will fall in the realm of nanomaterials, whose properties may or may not follow classical structure-property relationships. Polymer composite formulations are often chosen taking into account primarily the nature of the precursor materials and their * To whom correspondence should be addressed. Telephone: (404) 8946886. Fax: (404) 894-9140. E-mail: [email protected].

properties. However, not much attention has been paid to the influence that processing conditions may have on the resultant microstructure of a given material system. Experiments have shown that processing can influence the physical properties of composites substantially.11 This is because the microstructure of the composites can vary significantly when different processing techniques are used. Because the electrical resistivities of conducting fillers and most insulating polymeric materials vary by as much as 18 orders of magnitude, the electrical properties of insulator-conductor composites are especially sensitive to the microstructure, that is, whether a continuous network of contacting conducting filler particles is formed in the material and how the conducting filler is distributed in the materials and its local concentration. The electrical conductivity of mixtures of conductive and insulating materials is best explained by percolation theory. When a critical volume fraction of the conducting particles is reached, an interconnected network of conducting particles is formed, and thus, the electrical resistivity experiences several orders of magnitude change. A lot of percolation models have been developed to account for the percolation phenomenon, and a comprehensive review was recently given by Lux.12 Percolation models can be classified into statistical models,13,14 thermodynamic models,15,16 geometrical models,17-19 and structure-oriented models.20,21 The lowest percolation threshold predicted for monosized spheres is around a 16% volume fraction, while lower percolating volume fractions can be obtained for anisotropic fillers. Actual percolation thresholds depend on a lot of other factors, which include the nature of the precursor materials, the connectivity of the two phases,22 the size of both phases,23 the shape of the phases,24,25 the wetting behavior of one phase on the other,26 and so forth. The morphological features of the conductive filler phase in conductive polymer composite materials can be separated into

10.1021/jp064498o CCC: $33.50 © 2006 American Chemical Society Published on Web 10/17/2006

22366 J. Phys. Chem. B, Vol. 110, No. 45, 2006 “random” and “segregated” distribution models. It has been noted that using a segregated distribution of the conductive filler phase in the polymer matrix leads to the formation of a conductive network at a rather low critical content of conductive filler.27 One approach to attain the segregated distribution of the conductive filler phase is based on the compression molding of a dry-blended mixture consisting of a conductive filler (e.g., metal powder or carbon black) and a conventional polymer powder. In the absence of any shear, the conductive filler particles remain essentially located at the interfaces between the polymer particles, building up a continuous conductive network. Kusy and Turner27 found that percolation was reached at as low as 6 vol % metallic particles by compacting PVC and nickel particles at relatively low temperatures. Bouchet et al.28 manufactured conductive composites of ultra-high molecular weight polyethylene and ceramics on the basis of the segregated network concept and investigated the combined effects of temperature, pressure, and sintering time. Another approach to achieve a segregated structure and hence a low percolation threshold is to selectively localize a conductive filler such as carbon black in one polymer phase or at the interface of an immiscible polymer blend.29-36 Such systems often exhibit a double percolation effect which refers to the filler network formation in the filler rich phase and the continuity of this phase. The electrical properties of polymer/CB composites have been studied extensively, but investigations dealing with ac conductivities are limited. Percolation theory using scaling arguments has been applied to ac electrical properties of polymer/CB composites.37-39 According to classical percolation theory,40,41 near the percolation threshold pc, the dc conductivity, σdc, and the low-frequency dielectric constant, (0), observe the following power laws in relationship to the conducting filler concentration p: σdc ) σm|p - pc|t (for p > pc) and (0) ) d|p - pc|-s (for p > pc and p < pc), where σm and d are the conductivity and the dielectric constant of the conductor and the insulator, respectively. The frequency-dependent ac conductivity σ(ω) and dielectric constant (ω) follow σ(ω) ∝ ωx and (ω) ∝ ω-y, respectively. The exponents t, s, x, and y are related by x ) t/(t + s), y ) s/(t + s), and x + y ) 1. In real systems, the measured exponents s, t, x, and y may be quite different from the theoretical values. Thus, the theory has been modified to account for the complex geometry and structure of carbon black aggregates and the effects of tunneling mechanisms between CB aggregates.39 For a review of percolation and tunneling mechanisms in composite materials, please refer to Balberg et al.42 In this paper, we compare the microstructures obtained by fabricating poly(methyl methacrylate) (PMMA)/CB composites via two different processing methods. The first method is based on mechanical mixing of the polymer and CB filler followed by compression molding. The second involves dissolving PMMA in a solvent and mixing carbon black in the PMMA solution. The electrical conductivities of the composites made by the two different methods were acquired using dc resistivity measurements and ac impedance spectroscopy. An early account of this work appears elsewhere.43 II. Experimental Section The insulating polymer matrix used was PMMA obtained from Buehler Ltd. (Transoptic powder). The powder particle size ranged from 5 to 100 µm. The conductive filler used was carbon black (CDX975) obtained from Columbian Chemicals, which was developed for use in highly conductive applications.

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Figure 1. TEM micrograph depicting the microstructure of the highly aggregated carbon black used in this study.

TABLE 1: Carbon Black Concentrations Used to Fabricate PMMA/CB Composite Specimens phr

vol %

wt %

phr

vol %

wt %

0.1 0.2 0.4 0.6 0.8 1.0 2.0

0.066 0.13 0.26 0.39 0.52 0.65 1.29

0.10 0.20 0.40 0.60 0.79 0.99 1.96

3.0 3.5 5.0 6.5 7.5 10.0

1.93 2.24 3.17 4.09 4.69 6.15

2.91 3.38 4.76 6.10 6.98 9.09

The individual CB particles have a mean diameter of 21 nm, and the aggregates have a DBPA number of 175 mL/100 g. The DBPA gives a measure of the degree of branching of the CB aggregates used to make our composites. The high degree of branching results in elongated and porous aggregates (see Figure 1 for a representative image of the carbon black used in this study). The aggregates have a total surface area of 242 m2/g (as a result of the individual 21 nm CB particles) but have an external surface area of 130 m2/g (a measure of the agglomerate surface area). Therefore, the CB used in fabricating our composites may be treated as anisotropic fillers. Carbon black was dispersed in the polymer through two methods. The first method of mixing was mechanical mixing at room temperature using a blender. The second method of mixing was dispersing carbon black in a PMMA solution with the help of an ultrasonic bath and a magnetic stirrer. The PMMA solvent was ethyl acetate, and the solid-to-solvent weight ratio was 1:6. The liquid dispersion was cast into a film, and then, the film was chopped into little pieces. In the next stage, the composite mixtures, prepared by either mechanical mixing or solution mixing, were compression-molded at 170 °C using 20 kN of pressure so as to form pellets of 31.7 mm in diameter and approximately 1 mm in thickness. Table 1 lists the carbon black concentrations used for making specimens by the two methods. The concentrations were chosen so as to ensure detection of the percolation threshold for each fabrication method. The concentrations of CB listed in Table 1 are given in three different units in order to provide an easy frame of reference for polymer researchers and materials scientists simultaneously. The unit phr is the concentration unit often used by engineers for convenience, for example, 1 phr means that for each 100 g of PMMA, 1 g of carbon black was used. The volume percent and weight percent were calculated using the densities of the PMMA and CB.

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Figure 2. SEM images of fractured cross sections of PMMA/CB specimens with different CB concentrations made by the mechanical mixing method: (a) pure PMMA, (b) 0.1 phr CB, (c) 1 phr CB, and (d) 10 phr CB.

The microstructures of the specimens were obtained using scanning electron microscopy (SEM) of the fractured surfaces. Pellet specimens were fractured at room temperature, and then, they were gold-coated before being examined in a Hitachi S-800 scanning electron microscope. The accelerating voltage used was 15 kV. Optical measurements could not be used because most of these specimens were optically dark. However, at the lowest concentration of 0.1 phr, it was possible to obtain transmission optical images because the specimen at that composition was transparent. In preparation for the electrical property measurements, the specimen surfaces were painted with a conductive silver paint (SPI Supplies). Impedance measurements were performed using three different instruments. The Solartron 1260 Impedance/Gain Phase Analyzer coupled with a 1296 Dielectric Interface was used to measure from 10-3 to +107 Hz. The QuadTech 1910 Inductance Analyzer was used to measure from 20 to 106 Hz. Although the frequency range of this instrument is narrower, the data for more conductive samples are better, especially at the high-frequency end. The Agilent E4991A RF Impedance/ Material Analyzer extends the frequency range from 106 Hz to 3 × 109 Hz. The dc resistivity data were estimated by fitting the impedance data with equivalent circuits. For comparison, the dc resistance of the samples was also measured directly using a Keithley 617 Programmable Electrometer.

III. Results A. Microstructure Evaluation. Figure 2 shows SEM images of the fractured surfaces of a pure PMMA specimen and three PMMA/CB composite specimens fabricated with the mechanical mixing method with different levels of carbon black filler. Figure 2a shows the pure PMMA fractured surface, which displays the normally expected ductile behavior of PMMA. Figure 2b, which displays the fracture surface of the 0.1 phr specimen, shows the formation of partially faceted structures. The most striking picture is the one with 1 phr CB (Figure 2c), which looks like a collection of crystal grains. Yet, the PMMA/CB composite was revealed by X-ray diffraction to be noncrystalline, which is expected from a noncrystalline PMMA and a noncrystalline carbon black filler. The formation of such a faceted structure can be understood from the fabrication process. During the mixing stage, which is dry blending at room temperature, carbon black particles, which are much smaller than the PMMA particles, coat the surfaces of the PMMA particles. In the compression molding stage, because of the lack of shearing forces, the particle structure of the PMMA is not destroyed. Rather, the particles are deformed into polyhedra when the faces of the PMMA particles coated with CB run together and may force some of the CB particles toward the triple junctions. Such a processing procedure results in a

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Figure 3. Transmission optical micrographs of a transparent PMMA/CB composite containing 0.1 phr CB. Above this concentration, the composites become opaque.

Figure 4. SEM images of fractured cross sections of PMMA/CB composite specimens containing 5 phr CB made by the (a) mechanical mixing method and (b) solution mixing method.

microstructure with a highly ordered segregated network of CB particles. While many studies have obtained segregated network microstructures in the past, it appears that no previous studies have reported any faceted structures such as those we are reporting here. Finally, with 10 phr CB (Figure 2d), the edges of the grains are not as sharp as the grains formed with 1 phr CB. This is again evidence that the carbon black filler is located between the faceted PMMA particles. Excess carbon black particles will locate at the triple junctions of the PMMA faceted grains and on the faceted faces and serve as a cushion so that the grain edges are not as sharply defined in the 10 phr specimen as for the 1 phr specimen. While the majority of the PMMA/CB specimens we fabricated were optically dark, we were able to obtain transmission optical micrographs for the lowest CB specimen prepared (0.1 phr ) 0.067 vol %) because it was optically transparent. The transmission optical micrographs displayed in Figure 3 clearly show the faceted PMMA grains surrounded by CB filler at two different magnifications The images show that the carbon black particles are preferentially located at the triple-point junctions of the PMMA particles creating a 3D nanowire-like network. This type of segregated structure has important consequences on the electrical properties of the composites which result in the achievement of percolation at extremely low volume fractions of the CB filler and will be discussed more later. In contrast, the microstructure of PMMA/CB composites fabricated by dissolution of the PMMA followed by compression molding results in a more random distribution of the filler.

Figure 4 compares the microstructure of PMMA/CB composites of the same filler concentration (5 phr CB) made by the two different methods. While the ordered faceted structure is clear in the specimen made by our mechanical mixing method (see Figure 4a), it is absent in the composite made by the solution method (see Figure 4b). With the solution method, carbon black particles are more randomly dispersed within the PMMA matrix, and the composite specimens fracture in a similar fashion as that of the pure PMMA specimen displayed in Figure 2a. B. Electrical Measurements. Measurements were carried out using dc resistivity measurements and impedance spectroscopy measurements. The advantage of impedance spectroscopy is that it can contain a large amount of microstructural information in addition to providing conductivity and dielectric response information. Not only is impedance a complex number with a real part and an imaginary part, both of which are a function of frequency, but different functions can be used to present the data.44 Figure 5 displays complex plane impedance plots for the PMMA/carbon black composites made by the solution method, where the imaginary part of impedance is plotted against the real part of impedance. Semicircles are expected to be seen on the complex plane impedance plot, with frequency increasing from the right side of the semicircles to the left. A large semicircle indicates a large resistance and hence a more insulating specimen, whereas a small semicircle means a small resistance and a more conducting specimen. Because the conductivity value spans over 13 orders of magnitude when the

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Figure 6. The ac conductivity vs frequency for PMMA/CB composites made by the mechanical mixing method with various CB concentrations.

Figure 5. Complex plane impedance plots of PMMA/CB composites made by the solution mixing method. Parts a-c are different magnifications of the same figure.

composite changes from an insulator to a conductor, the semicircles must be viewed on different scales. On the scale of Figure 5a, a large semicircle can be seen. The diamond symbols, which form a short arc, represent the experimental data of the specimen containing 2 phr carbon black, while the solid line is the fitted curve. This specimen is very insulating, and thus, only a small part of the semicircle is observed. To extract out the dc resistance value, the equivalent circuit of a resistance, R, and a constant phase element (CPE) in parallel was used to fit the experimental data (see equivalent circuit model in Figure 5a). The CPE is used in place of a capacitor to account for slight deviations from a perfect semicircle. A CPE is defined by the following equation:

Z ) 1/[Y0(iω)R]

(1)

where Y0 is the CPE constant and R is the CPE power (0 e R e 1). If R ) 1, then the equation is identical to that of a capacitor and an ideal capacitor is obtained:

Z ) 1/(iωC)

(2)

From the intercept of the fitted semicircle with the real axis, the dc resistance can be obtained. It should be mentioned that it is not uncommon for impedance spectra to not show a perfect semicircle. The usage of the CPE is equivalent to what is known as a Cole-Cole plot in dielectric materials analysis.44, 45 Figure 5b shows a higher magnification of Figure 5a to reveal features that are hidden in the origin region for the 2 phr

specimen and others. At this magnification, a big semicircle featuring the 5 phr CB specimen and a small semicircle close to the origin can be observed for the 6.5 phr CB specimen. At still higher magnification (Figure 5c), yet another semicircle belonging to the 10 phr CB specimen is revealed. From the intercepts of the semicircles on the real axis, the dc resistance values of these specimens made by the solution method can be obtained and the dc resistivities can be calculated. The dc resistivity values of specimens made from the mechanical mixing method were obtained from similar complex plane impedance plots (not shown). The dc resistivity data will be further analyzed in the discussion section. To show frequency dependence, the real part of conductivity is explicitly plotted against frequency in Figure 6 for the composite specimens made by the mechanical mixing method. For pure PMMA and low-concentration (e 0.2 phr CB) composites, the ac conductivity is low and increases with increasing frequency. This is the typical behavior of an insulating material. As the carbon black concentration is increased, the ac conductivity shows a plateau region in the low-frequency region and then increases with frequency after a critical frequency, fc (marked by the dashed line in Figure 6), is reached. The slope of conductivity versus the frequency in the high-frequency region decreases with an increasing amount of carbon black used. At 3 phr CB concentration, the ac conductivity curve becomes flat. The curves for 5 phr CB, 7.5 phr CB, and 10 phr CB are not shown because they are also frequency-independent and virtually overlap with the 3 phr CB curve. The frequency-dependent conductivity for the specimens made by the solution mixing method show similar responses to those of the mechanical mixing method shown in Figure 6 except for the magnitudes. Another major difference is that all of the specimens, up to the highest CB concentration studied (10 phr), show a frequency dependence of conductivity. IV. Discussion According to classical percolation theory, σdc ) σm|p - pc|t above pc. That is, the electrical conductivity increases, or the resistivity decreases, by several orders of magnitude above the percolation threshold. In the meantime, the dielectric constant diverges around the percolation threshold according to (0) ) d|p - pc|-s. Figure 7 shows the resistivity as a function of filler concentration for the PMMA/carbon black composites made by the two different processing methods. The resistivity values were obtained from the ac impedance plots as discussed earlier in

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Figure 7. Comparison of the dc resistivity vs CB concentration for PMMA/CB composites made by the mechanical mixing and solution mixing methods.

Figure 5. The resistivities range from 1016 (ohm cm) for pure PMMA down to less than 102 (ohm cm) for all specimens containing more than 5 phr CB (i.e., a change in resistivity of 14 orders of magnitude). This tremendous resistivity range for the PMMA/CB composites was corroborated by dc measurements on the more conducting specimens. The concentration where the largest (and steepest) decrease of resistivity occurs is the percolation threshold. The percolation threshold of the composites made by mechanical mixing followed by compression molding is about 0.26 vol %. This is the lowest percolation threshold the authors are aware of for PMMA/carbon black composites. On the other hand, solution mixing followed by compression molding results in a composite with a much higher (∼2.7 vol % CB) percolation threshold, although this is also quite a low percolation value. While it is tempting to try to obtain the values of the superscript t in the above classical percolation conductivity equation, a lot of work by a number of investigators has demonstrated that this type of interpretation is not valid for CB-containing composites. This view has recently been highlighted by Balberg et al.,42 where it was remarked that, because the CB particles are interconnected electrically even if they are not in direct physical contact (i.e., tunneling across thin layers of the polymer matrix), then classical percolation is not relevant. This issue will become clearer when we discuss the ac conductivity curves. The reason that the specimens made by the mechanical mixing method have a much lower percolation threshold than the specimens made by the solution mixing method can be understood from the drastically different microstructures (see Figure 4). For the mechanical mixing method, the specimens have a segregated microstructure as clearly displayed in the SEM fracture surface image displayed in Figure 4a and the optical transmission images depicted in Figure 3a and b. Carbon black is seen to be localized between the PMMA grains, at triple junctions and along grain boundaries for the mechanically mixed specimens.46 This is in contrast with composite specimens made by the solution mixing method, where the carbon black is distributed more randomly throughout the composite.47 This is also suggested in Figure 4b where the fractured surface is typical of that of a more ductile specimen, as expected from a more homogeneous polymer microstructure. Figure 8 displays 2D schematic drawings showing why a lower level of carbon black is needed to reach percolation for the segregated structure. It should be clear from the figure that less carbon black is needed to coat the surfaces of the faceted PMMA grains (Figure 8a) as compared to the higher amount needed to make an interconnected path throughout the bulk of

Figure 8. Schematic of the microstructure of the composites prepared by the (a) mechanical mixing method and (b) solution mixing method.

the PMMA amorphous matrix (Figure 8b) when the CB is more randomly distributed. Because percolation occurs when electrical interconnectivity of the filler particles is achieved across the composite, it would be expected that the composites prepared using the mechanical mixing method, where a large volume of material is excluded from the mixing with CB (as depicted in the schematic in Figure 8a), would achieve percolation at lower filler concentrations than those made by the solution mixing method (shown in Figure 8b). The validity of these models was demonstrated in the resistivity curves displayed in Figure 7 where the mechanical mixing method resulted in a lower percolation volume fraction than the solution mixing method. For the solution mixing method, where the carbon black is more evenly distributed, the fact that the PMMA is dissolved before mixing with the CB precludes exclusion of a large volume of PMMA. Despite this, our results indicate that even the composites made by the solution method undergo percolation at a relatively low volume fraction, suggesting that the highly branched structure of the carbon black used to make our composites also allows tunneling across the polymer in this case. Furthermore, because CDX975 has porous aggregates (as depicted in Figure 1), it provides the conditions needed for achieving a much shorter tunneling distance than would be expected if only solid CB particles were used at a given CB loading level. Experiments which prove that porous carbon black aggregates result in lower percolation volume fractions were also conducted by Verhelst et al. a number of years ago.48 As expected from the discussion presented above, the percolation threshold for composites fabricated using the mechanical mixing method (pc ) 0.26 vol % CB) is lower than the percolation threshold for those composites prepared using the dissolution method (pc ) 2.7 vol % CB). Both of the methods we used to fabricate PMMA/CB composite specimens resulted in lower percolation values than solution-prepared specimens reported by Alexander (∼ 8% CB).49 Differences may be associated with different starting PMMA particle sizes

Poly(methyl methacrylate)/Carbon Black Composites as well as the carbon black type used (i.e., with a different average particle size, different degree of branching, and different agglomerate size and porosity48) or the fact that we used compression molding to make the PMMA/CB mixtures more dense and they did not. It should also be remarked that the nature of the carbon black we used to make our PMMA/CB composites (which is highly branched with a large DBPA of 175 mL/100 g and was depicted in Figure 1) may lend itself to forming continuous CB networks upon compression molding of the PMMA into faceted particles. Percolation at such a low volume fraction as 0.26 vol % must be a result of CB compaction into 3D nanowire-like networks that become electrically conducting even if not all CB particles are in direct physical contact. The formation of the 3D interconnected networks is suggested in the transmission optical micrographs of the 0.1 phr CB specimen displayed in Figure 3 and the fractured SEM cross sections depicted in Figures 2b-d and 4a. The achievement of percolation at such low volume fractions in PMMA/CB is very exciting because our results rival those obtained for single-wall carbon nanotubes in PMMA, where percolation was found to occur at volume fractions ranging from 0.17 to 0.5 vol %.50,51 It should be mentioned that additional results obtained in our laboratory52 reveal that the faceting behavior obtained by compression molding of mechanically mixed specimens described here is not unique to PMMA composites. In fact, similar results, that is, a large difference in electrical response as a result of the different microstructures achieved, were also obtained for acrylonitrile-butadiene-styrene (ABS)/CB specimens made by manual mixing and solution mixing methods. The percolation volume fraction for the manually mixed ABS/CB specimens was even lower52 (0.0025%CB) than that for the PMMA/CB composites. It is believed that the primary reason for this is that the size of the starting ABS material that was used to make the ABS/CB composites reported in ref 52 was substantially larger than the PMMA particles used here and therefore resulted in a larger polymer/filler size ratio, which caused a lower percolation threshold, in accord with recent Monte Carlo simulations by He and Ekere.53 To get a better grasp on the tunneling behavior, the ac conductivity of the PMMA/CB composites needs to be analyzed in more detail. As it is seen in Figure 6, the more-conducting specimens show a frequency-independent conductivity below fc (marked by the dashed line), and conductivity increases with frequency above fc, according to a power law, σ(p,ω) ≈ ωx. The frequency dependence of conductivity has been explained on the basis of percolation in a fractal structure38 and tunneling across extremely thin insulating barriers.42 In the polymer-CB composites, carbon black exists in clusters of different sizes. Below the percolation threshold, all clusters are finite. The formation of the first infinite cluster denotes the onset of percolation. This is where the ac conductivity will first display a frequency-independent portion (related to direct interconnectivity) and a frequency-dependent portion (related to tunneling). This is first observed in Figure 6 at a concentration of 0.4 phr. However, as the filler concentration is increased above the concentration needed to achieve the first infinite cluster, more and more of the finite clusters join the infinite cluster so that the tunneling portion becomes less and less important (and fc moves to a higher frequency) until only a frequency-independent ac conductivity is obtained (e.g., for 2-3 phr CB in Figure 6). Further increases in the filler concentration above 3 phr have a minimal effect on the resultant conductivity where the conductivity appears to have reached a plateau (also observed in the

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Figure 9. Critical frequency fc vs (p-pc) for the PMMA/CB composites made by the mechanical mixing method.

resistivity displayed in Figure 7). This is a result of the fact that most of the CB aggregates are connected with the infinite cluster, and the portion contributed by tunneling is very small to nonexistent. Thus, in the case of the mechanically mixed specimens, CB concentrations above 3 phr or 1.9 vol % CB show frequency-independent conductivity. 3 phr CB or 1.9 vol % CB may seem to be a low concentration to have high interconnectivity, but for the segregated structure where the carbon black particles are confined to the boundary regions between the faceted PMMA grains and the triple junctions (see Figures 2b-d, 3a,b, and 8a), there is a high local CB concentration. Thus, it is possible that the majority of the finite clusters disappear (i.e., become part of an infinite cluster) above this concentration. Evidence that this occurs has been obtained using ultrasmall-angle X-ray scattering imaging where a transition in the microstructure of PMMA/CB specimens is observed from 3D nanowire-like networks to 2D conducting sheets.46 From Figure 6, we can also determine the critical frequency, fc, at which the ac conductivity changes from frequencydependent behavior (related to tunneling) to frequencyindependent behavior (related to direct interconnectivity). When the ac conductivity is frequency-independent, it means that the contribution of tunneling is less important than the physical interconnection of the CB particles present. Therefore, as the carbon black concentration increases, physical interconnection dominates and the critical frequency fc increases with the carbon black concentration. The critical frequency, fc, above which tunneling will be detected even if the dc conductivity is small, is found by drawing two tangent lines from the frequencyindependent region and the frequency-dependent region. Figure 9 shows the critical frequency as a function of (p - pc) plotted in a log-log scale for the specimens made by the mechanical mixing method. In the equation above, p represents the volume fraction of filler at a given composition, while pc represents the volume fraction at which percolation occurs for a given matrixfiller system. A linear relationship is found with a slope of 1.6, which means fc ∝ |p - pc|1.6. The exponent of 1.6 for our PMMA/CB composites is in agreement with the value of 1.5 found by Connor et al. for poly(ethylene terephthalate)/CB composites.38 This value is also in agreement with the fractal dimension measured by Balberg et al.,42 who carried out conductive atomic force microscopy experiments and measured the fractional area that contributed the current carrying portions of the microstructure. Even though our electrical measurements were conducted in 3-D, Balberg’s 2-D interpretation may also be used to explain our results because, stereologically, area

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TABLE 2: The x and y Exponents of the PMMA/CB Composites Made by the Mechanical Mixing Method CB phr

CB vol %

x

y

x+y

0 0.1 0.2 0.4 0.6 0.8 1.0 2.0

0 0.065 0.13 0.26 0.39 0.52 0.65 1.29

0.85 0.87 0.87 0.71 0.50 0.30 0.24 0.22

0.04 0.04 0.04 0.28 0.29 0.45 0.54 0.52

0.89 0.91 0.91 0.99 0.79 0.75 0.78 0.74

TABLE 3: The x and y Exponents of the PMMA/CB Composites Made by the Solution Mixing Method CB phr

CB vol %

x

y

x+y

1.0 2.0 3.5 5.0 6.5 10

0.65 1.29 2.24 3.17 4.09 6.15

0.89 0.89 0.91 0.82 0.75 0.51

0.04 0.04 0.04 0.16 0.16 0.25

0.93 0.93 0.95 0.98 0.91 0.76

fraction is equivalent to volume fraction when a material is randomly oriented.54,55 We can now move on to calculate the frequency-dependent power law dependences of the ac conductivity. It can be seen in Figure 6 that the slope of the frequency-dependent region, which is the x exponent from the equation σ(p,ω) ≈ ωx, decreases with an increasing carbon black concentration, until the curve becomes flat at a CB concentration well above the percolation threshold. This is in contrast with the literature report that the slope is independent of carbon black concentration.38 It is speculated that previous investigators were unable to obtain this information accurately because of the limited frequency range of their measurement equipment. We also find that the slope of the decreasing dielectric constant with frequency (not shown) is dependent on the carbon black concentration. This slope gives the y exponent in the equation (p,ω) ≈ ω-y. Tables 2 and 3 list the x and y exponents obtained by analyzing the frequency-dependent plots for all of the specimens measured. According to the theory, close to the percolation threshold, x ) y ) 0.5 for two-dimensional systems, whereas x ) 0.72 and y ) 0.28 for three-dimensional systems.37 Our data in Tables 2 and 3 indicate that the x and y exponents are not constant, but that they change with the carbon black concentration. The x exponent decreases with the carbon black concentration and the y exponent increases with the carbon black concentration for both the mechanical mixing method and the solution mixing method. The sum of (x + y) reaches a maximum value that is very close to 1 for the 0.4 phr or 0.26 vol % CB specimen made by the mechanical mixing method. This is the composition around which percolation occurs and the resistivity changes sharply (see Figure 7). At this carbon concentration, the x value of 0.71 and y value of 0.28 are in excellent agreement with the theoretical prediction (x ) 0.72 and y ) 0.28) for threedimensional systems. For the solution mixing method, the sum of (x + y) approaches 1 between 3.5 phr (2.24 vol %) CB and 5 phr (3.17 vol %) CB concentrations. This happens to be the range where the resistivity changes dramatically for the solution mixing method (see Figure 7). The x and y exponents for these two concentrations (3.5 phr, x ) 0.91 and y ) 0.04 and 5 phr, x ) 0.82 and y ) 0.16) for the solution mixing method are closer to the three-dimensional percolation model numbers (x ) 0.72 and y ) 0.28) than the two-dimensional percolation model numbers (x ) 0.5 and y ) 0.5). In summary, both the mechanically mixed and the solution mixed PMMA/CB specimens behave as three-dimensional

percolation systems, although the specimens prepared via the mechanical mixing method result in a much lower percolation threshold than those made by the solution method, as a result of the large excluded volume. This conclusion agrees with the formation of a 3D CB-nanowire-like network for the mechanically mixed specimens, as demonstrated by the images in Figure 3. The fact that percolation is achieved at relatively low volume fractions for the solution-prepared specimens also suggests that some volume exclusion may also be occurring in these specimens, albeit on a more random and finer scale than for the mechanically mixed specimens. If the CB aggregates were truly randomly distributed, the amount of CB filler needed to achieve percolation would have to be much higher than what our experiments have demonstrated. The schematic presented in Figure 8b can be used to speculate on how the size of the CB aggregates can determine when percolation may occur in the solution-made specimens. We believe that the highly porous and aggregated structure of the carbon black used in this study was a major contributor to this effect. More detailed studies focusing on the effect of the size and porosity of CB aggregates on the resultant conductivity will appear in future applications. Detailed quantification of the nanowire-like networks observed in similarly prepared mechanically mixed specimens will be available in the literature shortly.46 V. Conclusions This paper was used to demonstrate that fabricating PMMA/ CB specimens using the same starting materials but different processing methods can result in different microstructures with substantially different electrical conductivities. We demonstrated that, by mechanical mixing of the PMMA and CB precursor materials, followed by compression molding, a highly ordered faceted structure is obtained, which leads to a very low percolation threshold (0.26 vol %), where the CB is interconnected along the PMMA grain boundaries and forms a self-assembled 3-D nanowire-like structure. Evidence for this nanowire network was provided by transmission optical micrographs for the PMMA/CB specimen with the lowest CB concentration fabricated, which was transparent to visible light. In contrast, a higher percolation threshold than the mechanical mixing method (2.7% vs 0.26%) was obtained for the solution mixing method specimens. This difference is believed to be a result of the carbon black aggregates being distributed more randomly throughout the bulk in the solution prepared specimens. The mechanical mixing method allows the electrical response of the PMMA/CB specimens we fabricated and characterized to be comparable to that of single-wall nanotube/ PMMA composites. Acknowledgment. The authors thank the National Science Foundation (DMR-0076153 and DMR-0604211) for support of this work and Ms. Yolande Berta for acquiring the TEM image of the carbon black used in this study. References and Notes (1) Ma, C. C.; Hu, A. T.; Chen, D. K. Polym. Compos. 1993, 1, 9399. (2) Lu, G.; Li, X.; Jiang, H.; Mao, X. J. Appl. Polym. Sci. 1996, 62, 2193-2199. (3) Yonezawa, M. Jpn. Kokai Tokkyo Koho, Japanese Patent, JP 08039734, 1996. (4) Myagawa, K.; Shimizu, M.; Inoe, M. Jpn. Kokai Tokkyo Koho, Japanese Patent, JP 06305084, 1994. (5) Khazai, B.; Nichols, G. M. U.S. Patent 5902518, 1999. (6) Albert, K. J.; Lewis, N. S.; Schauer, C. L.; Sotzing, G. A.; Stitzel, S. E.; Vaid, T. P.; Walt, D. R. Chem. ReV. 2000, 100, 2595-2626.

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