Percolation backbone structure analysis in electrically conductive carbon fiber reinforced cement composites

Composites: Part B 43 (2012) 3270–3275

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Composites: Part B journal homepage: www.elsevier.com/locate/compositesb

Percolation backbone structure analysis in electrically conductive carbon fiber reinforced cement composites Ning Xie a,b,⇑, Xianming Shi c,d, Decheng Feng a, Boqiang Kuang a, Hui Li b a

School of Transportation Science and Engineering, Harbin Institute of Technology, Harbin 150090, China School of Civil Engineering, Harbin Institute of Technology, Harbin 150001, China c Corrosion and Sustainable Infrastructure Laboratory, Western Transportation Institute, PO Box 174250, College of Engineering, Montana State University, Bozeman, MT 59717-4250, USA d Civil Engineering Department, 205 Cobleigh Hall, Montana State University, Bozeman, MT 59717-2220, USA b

a r t i c l e

i n f o

Article history: Received 7 October 2011 Received in revised form 22 December 2011 Accepted 24 February 2012 Available online 5 March 2012 Keywords: A. Carbon fiber A. Discontinuous reinforcement B. Electrical properties C. Computational modeling Percolation backbone analysis

a b s t r a c t A simple model was presented to quantitatively calculate the backbone density of carbon fillers in carbon/cement composites. In this model, a ‘‘structure factor’’, j, defined as a function of the aspect ratio of the carbon filler, was first introduced to calculate the backbone density. To obtain the actual backbone density, carbon fiber (CF) reinforced cement composites with different CF concentrations were prepared and their DC electrical conductivities were measured. It was found that, when the CF concentration slightly exceeded the percolation threshold, the electrically conductive critical exponent was neither a universal value nor a constant that increases with the CF concentration. The results also indicated that the backbone density of the CF decreased with increasing CF concentration. The mechanisms of backbone structure evolution with increasing CF concentration were presented. The experimental results showed that near the percolation threshold the backbone density is approximately 0.15, which agrees well with the simulation results. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Electrically Conductive Cement-based Composites (ECCCs) has been studied for decades, and its multifunctional properties can be applied in many fields. Since the ECCCs technology was first patented by the National Research Council of Canada in 1993 [1,2], it had been developed rapidly in recent years. In general, ECCC is composed of a concrete matrix and conductive fillers. In this system, the matrix is made of cementitious material with or without small amounts of silica fume, fly ash, or fine aggregates. The majority of conductive fillers are metallic fibers (MFs), carbon fiber (CF), carbon black (CB), graphite, or carbon nanotubes (CNTs) [1–7]. Existing research has demonstrated that the conductive fillers can play multiple roles in the intrinsically smart ECCCs as follows. (1) Electrically conductive phase, which changes the concrete from an electrical insulator to an electrical conductor [8–13]. Most of these fillers are MFs, CFs, graphite, CB, and CNTs used individually or in combination. (2) Piezoelectric element, which turns the concrete into a bulk piezoelectric material [14,15]. It can be used in health monitoring of bridges or construction parts, or in traffic monitoring by detecting the speed and weight of vehicles over ⇑ Corresponding author at: School of Transportation Science and Engineering, Harbin Institute of Technology, Harbin 150090, China. Tel./fax: +86 451 86282191. E-mail address: [email protected] (N. Xie). 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2012.02.032

the pavement. In this aspect, most of the work has been focused on CF/concrete or CNT/concrete composites [16,17]. (3) Thermistor, which accurately and efficiently measures the temperature of a structure [18]. In many cases, the performance, operations and safety of a structure are largely affected by temperature, the monitoring of which is thus important especially during the construction phase. (4) Electromagnetic shielding element, which enhances the capability of the concrete matrix to prevent electromagnetic interference [19–23]. (5) Corrosion protection component, which reduces the corrosion risk of steel bars by decreasing the chloride ion diffusivity in concrete while acting as secondary anode for the cathodic protection of reinforced concrete [24,25], or by acting as auxiliary anode for the electrochemical extraction of chlorides from reinforced concrete [26]. (6) Heating element, which is typically used for de-icing and snow melting during winter season in cold regions [27,28]. (7) Strengthening phase, which increase the mechanical properties of the concrete [29]. Thanks to the decreased production cost of CFs and CNTs and the increasing needs for multifunctional composites, ECCCs have found widespread applications in various industries. In all ECCC functions, the key parameter is its electrical conductivity, which is resulted from the addition of electrically conductive fillers. ECCC is an interesting type of percolation system and its transport properties have received much attention because of the research challenges. When the DC electrical conductivity (reff) is

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measured as a function of the volume concentration (p) of the conductive fillers, reff can increase by several orders of magnitudes when p exceeds a critical value, known as the percolation threshold. As such, the studies of the percolation phenomena in ECCC carbon network systems have been focused on the percolation threshold. It was found that the percolation threshold was widely distributed with different carbon fillers [30]. Furthermore, it has been accepted that the percolation threshold decreases with increasing aspect ratio and decreasing particle size of the fillers [8,31,32–34]. In a carbon/cement composite, the carbon fillers can be divided into two different parts, the isolated clusters and the infinite cluster. The infinite cluster carries the electrical current, whereas the isolated clusters do not contribute to transport of electrons. The infinite cluster can be divided into two different structures: the ‘‘backbone’’ demonstrating the real path that carries the current and the ‘‘dangling ends’’, which can be removed from the infinite cluster when a voltage is applied to the system because they do not carry any current [35]. In a percolation system, the geometry of the cluster resembles that of the infinite cluster which is very ‘‘weak’’ at the percolation threshold [36]. As such, the removal of a few bonds can break the infinite cluster into finite clusters that disconnects the edges of the system from each other. Therefore, the backbone analysis is crucial for understanding, improving, and utilizing the DC transport in ECCCs as it sheds light on the real current path. There are existing studies that focused on the backbone structure analysis in percolation systems. Most of them, however, were purely based on computational results since the backbone is very hard to be detected in actual materials even with advanced technologies such as Computed Tomography (CT) [32]. Although ECCC has been widely studied in the past decades, to the best of our knowledge, there was little research on the percolation backbone structure of the fillers in ECC carbon network systems, which is important for the understanding of such disordered systems. In this study, the backbone density was quantitatively calculated and analyzed in CF/cement composite with the calculation of percolation critical exponent t and ‘‘structure factor’’ j.

2. Material and methods The cement based materials fabricated were made from normal Portland cement pastes (#42.5) from Yatai Corp. in China, and the water/cement ratio was 0.35. No admixtures or additives other than CFs were added into the cement paste. The dosage of CFs were 0.1%, 0.25%, 0.5%, 0.75%, 1.0%, 1.25%, 1.5%, and 1.75%, respectively, by weight of cement. The CFs are isotropic with diameter of about 8 lm and length of 9 mm, featuring an electrical resistivity of 1.8  103 X cm as obtained by Sinosteel Jilin Carbon Co. Ltd. Fig. 1 shows the Scanning Electron Microscopy (SEM) image of the CFs. Before mixing the CFs and cement, the CFs were soaked in ethanol and ultrasonically vibrated for 1 h followed by drying in an oven at 80 °C for 2 h. A 1-l rotary mixer was used for mixing. The CFs and cement were mixed with low rotary speed for 120 s and high rotary speed for 300 s before they were poured in steel molds with the dimension of 40  40  160 mm3. The specimens were demolded after 24 h and cured at room temperature for 28 days under a relative humidity of 95%. The electrical conductivity was measured with a four-probe method, i.e., two current contacts and two voltage contacts were used as probes. Fig. 2 illustrates the configuration of the specimens for the electrical conductivity tests. Four copper meshes were embedded in each paste specimen as the electrical contacts for the tests. Three specimens of each mix design were tested to ensure data reliability.

Fig. 1. SEM image of the CFs.

3. Results and discussion 3.1. Modeling considerations One of the most important factors of a carbon/cement network composite is the DC electrical conductivity. When the electrical conductivity (reff) measured as a function of the volume (or weight) concentration (p) of the conductive fillers, it shows a typical ‘‘S’’ shape due to the percolation phenomenon [37]. It follows a power law behavior with the form [36]:

reff ¼ rc ðp  pc Þt p > pc reff ¼ rc ðpc  pÞs p < pc

ð1Þ

where reff is the effective electrical conductivity of the conductor– insulator composite, rc is the electrical conductivity of the conducting fillers, p is the weight or volume content of the conductor, pc is the percolation threshold, and t and s are the electrical conductivity critical exponents above and below the percolation threshold, respectively. In this equation, the values of rc and pc are constants, reff was experimentally measured. In previous studies, according to numerical and experimental results, the values of the conductivity critical exponents in bond and site percolation lattices were considered to be universal such that t  1.3–1.4, s  0.5 (in two dimensions) and t  1.6–2.0, s  0.6 (in three dimensions) based on the renormalization group theory [36,37], and in practical applications, they were generally considered belonging to the same universality class. However, some experimental and numerical results have indicated that the lattice percolation problems and the practical application problems may belong to different universality classes. Although the percolation phenomenon has been studied for decades, the non-universality of the critical conductivity exponents observed experimentally has remained difficult to explain. Regarding the non-universality of critical conductivity exponents, Kogut and Straley (KS) [38], first claimed that if the low-conductance bonds in percolation networks were characterized by an anomalous conductivity distribution, the universality of the conductivity exponents would be broken. In their model (KS model) derived from the mean field theory, by assigning to each neighboring pair in a regular lattice, a bond with finite conductivity g with probability l and zero conductivity with probability 1  l, the resulting bond conductivity distribution function becomes:

qðgÞ ¼ lhðgÞ þ ð1  lÞdðgÞ

ð2Þ

where d(g) is the Dirac delta function and h(g) is the distribution function of the finite bond conductivity. If h(g) has a power law divergence for small g of the form:

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Fig. 2. Configuration of electrical conductivity measurement.

lim hðgÞ / g a

ð3Þ

g!0

where a 6 1, then the universality is lost for sufficiently large values of exponent a. Based on the KS model, two models were introduced by Halparin [39] and Balberg [40], known as ‘‘Random Void’’ model (Swiss– Cheese Model) and ‘‘Tunneling Model’’ corresponding to granular materials and carbon/polymer composites, respectively. Theses two models were widely used for describing the non-universality of the percolation system. In both random void and tunneling models, however, the conductive fillers are treated as spherical; thus, these two models are insufficient in describing the percolation system with non-spherical fillers such as CF filled concrete or CNT network composites. It is well known, in geological considerations, since the conductivities can be considered as in series and in parallel, the effective conductivities are given by the average conductance values. One extreme case is that all the resistance in a percolation system is in parallel, where the equivalent effective conductivity can be written as:

reff /

X

!1 Ri

i

/

X

!1

r1 i

ð4Þ

i

If all fillers are geometrically identical, then the reff is proportional to the second sum. Another extreme case is that all the resistances are in series, where the equivalent effective conductivity can be written as:

reff /

X

ri

ð5Þ

  MD g ¼ g 0 exp j MB

ð6Þ

where g0 is a constant, MD and MB are dangling ends and backbone masses belonging to two randomly selected nodes, and j is the ‘‘structure factor’’ that represents the geometry and topology structure of the conductor. Fig. 3 gives the assumed possible geometrical shapes of carbon fillers in a cement–matrix carbon network composite. According to this assumption, the value of j can be given as:

1      a k fi log  b  þ 1

j¼P 

ð7Þ

i

where f is the volume (or weight) fraction of the conductive fillers, i is the index of the filler types, a/b represents the aspect ratio of the filler, and k is 1, 0, and 1 corresponding to a > b, a = b, and a < b, respectively. Due to the fractal nature of the fillers, the size dependence can be neglected in this model. In a carbon/cement network composite, the value of j will reach its highest value, namely 1, when the individual geometrical shape of the original filler is spherical (a = b). This is because the sphere is the simplest geometrical shape and the value of j decreases with increasing geometrical complexity of the conductor. In Eq. (6), the tunneling conductance parts should be considered as a part of the backbone. In order to find the conduction distribution function of the network H (MD/MB), the distribution function of the dangling ends and the backbone must be presented. In a recent

i

There is a same geometrical restriction in these two cases that the individual elements must all be congruent, namely, all in the same size and shape; otherwise, it would be necessary to include geometrical factors to describe them [38]. In a real percolation network, the resistance of the whole system is a combination of parallel and series; therefore, the critical path needs to be analyzed. A study presented a new model to describe a possible nonuniversal behavior in a conductor–insulator composite system [41]. In the model, the backbone mass MB and the dangling ends mass MD are presented as key parameters. The backbone or dangling ends density is defined as the portion of total backbone or dangling ends that belong to the percolation infinite clusters, respectively. The conductance between two randomly selected nodes, g, is given by:

Fig. 3. Possible geometrical shapes of carbon fillers in a cement–matrix carbon network composite.

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0.0

study [41], the dangling ends and backbone distribution function was considered as exponential distribution, and given by:

    MD g2 MD gM D H exp  / MB MB MB written as:

ð9Þ

with Eqs. (3), (6), and the KS model, the critical exponent t can be obtained as:



MB t ¼ tun þ j MD 0



-1.0 -1.5 -2.0 -2.5 -3.0 -3.5

1

1

1 MD B    C 1 ¼ tun þ @P  A  a k MB fi log  b  þ 1

-4.0

0.0

0.2

ð10Þ

i

where tun is the ‘‘universal value’’ of a percolation system based on the Effective Medium Theory (EMT) [38]. The lower the t value is, the higher the effective electrical Dconductivity of the system. In this E MB equation, the backbone density M varies from zero to infinity. D When the backbone density goes to zero, i.e. no backbone in the system, the t value goes to infinity, and the electrical conductivity goes to zero according to Eq. (1), in this extreme case, the system is an insulator. When the backbone density goes to infinity, i.e., no dangling ends in the system, the t value goes to zero, according to Eq. (1), the electrical conductivity goes to rc. In this extreme case, the system is a pure conductor. Compared with spherical fillers, the fiber fillers decrease the percolation threshold in carbon/cement composites [42–44]. In addition, unlike the spherical fillers, CF fillers tend to tangle each other and form more backbones in the infinite cluster due to its high aspect ratio, which decreases the critical exponent of the system. This phenomenon has been proven by Wen and Chung [9]. They partially replaced CF by CB and found that the electrical conductivity of CF (a > b) filled cement was higher than those filled with CB (a = b) even with same filler’s weight fraction, indicating that the CF/cement led to a higher effectiveness in enhancing electrical conductivity than the CB/cement. 3.2. Experimental validation In a carbon/cement composite system, since the concentration of the carbon fillers exceeds the percolation threshold, the addition of carbon fillers will increase the amount of carbon fillers that belong to the infinite cluster in the network, thus increasing electrical conductivity of the composite. However, the backbone density is not the same as the whole quantity of the carbon fillers. The evolution of electrical conductivity critical exponent t demonstrates the variations of the backbone density as a function of CF concentration in the area near the percolation threshold. Fig. 4 illustrates the electrical conductivity of the CF/cement composite as a function of the weight fraction of CF, based on the experimental results from this work. As shown in Fig. 4, the electrical conductivity increased by about three orders of magnitude as the weight fraction of the CF increased from zero to 0.5 wt.%. Fig. 5 shows a plot of percolation critical exponent t as a function of weight fraction of CF. The critical exponent t was calculated by t ¼ logðreff =rc Þ= logðp  pc Þ according to Eq. (1), and in this study, the electrical conductivity increased three orders of magnitude with the CF content increased from 0 to 0.5 wt.% and keeps a relative stable value once the CF content exceeds 0.5 wt.%. As such, the percolation threshold was determined as 0.25%, which is the mid-point from 0 to 0.5%. As can be seen in this figure, the value of t is not a universal value. It increased with the weight fraction

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Weight fraction of CF, % Fig. 4. Electrical conductivity as a function of weight fraction of CFs.

1.80 1.75

Critical Exponent t

MB MD



MB , MD

1.70 1.65 1.60 1.55 1.50 0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Weight Fraction of CF, % Fig. 5. Electrical conductivity exponent t as a function of weight fraction of CFs.

of CF and approached a constant once the weight fraction reached a critical value. Fig. 6 presents the plot of the backbone densityDas Ea function of MB the weight fraction of CF. The backbone density M was calcuD lated according to Eq. (10). In this case, t was calculated by t ¼ logðreff =rc Þ= logðp  pc Þ according to Eq. (1), tun was set as the universal value of 1. As illustrated in this figure, the backbone

0.160 0.155

MB / M D

g¼



ð8Þ Log σ, (Ω⋅cm)-1

where g is the average value of

-0.5

0.150 0.145 0.140 0.135 0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Weight fraction of CF, % Fig. 6. Backbone density as a function of weight fraction of CFs.

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Fig. 7. Schematic representation of the backbone structure variation mechanisms in the CF infinite cluster with addition of CFs above the percolation threshold. (a) Original infinite cluster; (b) the increased fiber locates on the loop and connected part of the previous dangling ends; (c and d) the increased fiber locates on the loop but does not connect any previous dangling ends; (e) the increased fiber belongs to the infinite cluster but neither locates on the loop nor connects any previous dangling ends; (f) the increased fiber is isolated and does not belong to the infinite cluster.

density decreased with the increasing CF concentration near the percolation threshold in CF/cement composites, and the value of D E MB , the ratio between the backbones and the dangling ends, M D

MB AB þ BC þ CD þ DE þ EA þ x00 y00 ¼ 0 00 MD AA þ AA þ BB0 þ CC 0 þ CC 00 þ DD0 þ EE0 þ EE00 þ xx00 þ yy00 ð13Þ

ranged from 0.16 to 0.14 with CF concentrations from 0.5 wt.% to 1.75 wt.%. This agrees well with the simulation results of the square lattice percolation system [45]. In a previous study, near the percolation threshold, most of the conductive fillers belonged to the dangling ends, not to the backbone [46], in other words, most of the weight fraction p made no contribution to the conductivity r. Fig. 7 is the schematic representation of backbone structure variation mechanisms with the addition of CF fillers (xy) above the percolation threshold in the CF infinite cluster. Note that the locations of the nodes A, B, C, D, E and the ends A0 ; A00 ; B0 ; C 0 ; C 00 ; D0 ; E0 ; E00 are not shown in Fig. 7b–f for a clearer demonstration, but are exactly the same as shown in Fig. 7a. Fig. 7a shows the original infinite cluster of the CF in a CF/cement composite. As can be seen in the figure, the loop ABCDE belongs to the backbone MB, and AA0 ; AA00 ; BB0 ; CC 0 ; CC 00 ; DD0 ; EE0 and EE00 belong to the dangling ends MD in the infinite cluster. As such, the ratio of the backbone and the dangling ends is:

under this condition, the variation of the backbone density depended on the length ratio of xx00 þ yy00 and x00 y00 in Fig. 7c and xx000 þ yy000 and x000 y000 in Fig. 7d. In Fig. 7e, the newly added fiber xy belonged to the infinite cluster but was neither located on the loop nor connected any previous dangling ends. In Fig. 7f, the newly added fiber xy was isolated and did not belong to the infinite cluster, therefore, the backbone density changed to:

MB AB þ BC þ CD þ DE þ EA ¼ M D AA0 þ AA00 þ BB0 þ CC 0 þ CC 00 þ DD0 þ EE0 þ EE00

ð11Þ

Fig. 7b–f represents the backbone structure variations with increasing CF concentration on the basis of Fig. 7a. As shown in Fig. 7b, the newly added fiber xy was located on the loop and connected parts of the previous dangling end BB0 and EE0 , therefore, the backbone density changed to:

MB AB þ BC þ CD þ DE þ EA þ xy ¼ M D AA0 þ AA00 þ CC 0 þ CC 00 þ DD0 þ EE00 þ xx0 þ yy0 þ B0 y0 þ E0 x0 ð12Þ In Fig. 7c and d, the newly added fiber xy was located on the loop but did not connect any previous dangling ends, therefore, the backbone density changed to:

MB AB þ BC þ CD þ DE þ EA ¼ MD AA0 þ AA00 þ BB0 þ CC 0 þ CC 00 þ DD0 þ EE0 þ EE00 þ xy

ð14Þ

Under this condition, the backbone density decreased with the increase in CF concentration. Due to the complicated nature of the backbone with links and blobs, the conductivity critical exponent t varied with different backbone density derived from different filler structures. In this work, t showed a constant value once the backbone density reached its saturation value, and the calculated backbone density was in the range of 0.14–0.16 with the CF content of 0.5–1.75 wt.%, which means that most of the CFs contributed to the dangling ends rather than the backbone. Combined with the data shown in Figs. 3–5, it can be deduced that Fig. 7c–e were the dominant mechanisms of the backbone variations in the infinite cluster of CF/cement composite near the percolation threshold. 4. Conclusions The backbone density of electrically conductive CF/cement composites was quantitatively calculated by a simple model that was developed by using the percolation electrical conductivity critical exponent t and a ‘‘structure factor’’ j as parameters. In this model, the ‘‘structure factor’’ j was defined as a function of the aspect ratio of the carbon fillers. It reached its highest value,

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namely 1, with spherical shape fillers, and decreased with the increasing of aspect ratio of the fillers. Meanwhile, the nonuniversality of the electrical conductivity critical exponent t was quantitatively analyzed as well. Electrically conductive CF/cement composites were prepared to calculate the backbone density of CF in a real CF/cement composite. According to the percolation theory, it was found that in a CF/cement composite system, the critical exponent is neither a universal value nor a constant with the fillers’ concentration slightly above the percolation threshold. It increased slightly with the increasing of CF concentrations. Based on the model, the quantitative calculation results showed that the backbone density decreased from 0.16 to 0.14 with fiber concentration increasing from 0.5 wt.% to 1.75 wt.%. The backbone structure analysis shows that the backbone is very ‘‘weak’’, and the dangling ends and isolated clusters are the dominant mechanisms with increasing CF concentration near the percolation threshold. Acknowledgements The authors would like to thank Riga Zhahe and Jing Zhong at Harbin Institute of Technology for their help with this work. Ning Xie acknowledges Prof. Wenzhu Shao for his help and support. This work was financially supported by the Project (HIT.NSRIF.2009100) of Natural Scientific Research Innovation Foundation in Harbin Institute of Technology and Project (HIT.KLOF.2009105) of Key Laboratory Opening Funding of Special Materials Lab on Transportation Safety. References [1] [2] [3] [4] [5]

Xie P, Gu P, Fu Y, Beaudoin J. US patent 5447564; 1995. Pye G, Myers R, Arnott, Beaudoin J, Tumidajski PJ. US patent 6503318 B2; 2003. Wen SH, Chung DDL. Cem Concr Res 2004;34:329. Chen B, Wu KR, Yao W. Cem Concr Compos 2004;26:291. Li H, Xiao HG, Ou JP. Cem Concr Compos 2006;28:824–8.

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46]

3275

Cao JY, Chung DDL. Cem Concr Res 2003;33:1737. Konsta-Gdoutos MS, Metaxa ZS, Shah SP. Cem Concr Res 2010;40:1052. Wen SH, Chung DDL. Carbon 2007;45:263. Wen SH, Chung DDL. Carbon 2007;45:505. Chung DDL. Adv Cem Res 2004;16:167. Chung DDL. Composites: Part B 2000;31:511. Chung DDL. Carbon 2001;39:1119. Chugh R, Chung DDL. Carbon 2002;40:2285. Sun MQ, Liu QP, Li ZQ, Hu YZ. Cem Concr Res 2000;30:1593. Han BG, Yu X, Kwon E. Nanotechnology 2009;20:445501. Gong Hongyu, Zhang Yujun, Quan Jing, Che Songwei. Curr Appl Phys 2011;11:653. Wen Sihai, Chung DDL. Carbon 2006;44:1496. Wen SH, Chung DDL. Cem Concr Res 1999;29:1989. Fu X, Chung DDL. Carbon 1998;36:459. Cao J, Chung DDL. Carbon 2003;41:2433. Cao J, Chung DDL. Cem Concr Res 2003;33:1737. Chung DDL. J Mater Sci 2004;39:2645. Zhang Y, Zhi R, Zhu F. J Mater Res 1995;9:284 [in Chinese]. Hou JY, Chung DDL. Corros Sci 2000;42:1489. Xu J, Yao W. J Wuhan Univ Technol – Mater Sci Ed 2010;25:883. Tumidajski PJ, Xie P, Arnott M, Beaudoin JJ. Cem Concr Res 2003;33:1807. Yehia S, Tuan CY. ACI Mater J 1999;96:382. Yehia S, Tuan CY, Ferdon D, Chen B. ACI Mater J 2000;97:172. Konsta-Gdoutos MS, Metaxa ZS, Shah SP. Cem Concr Compos 2010;32:110. Xie P, Gu P, Beaudoin J. J Mater Sci 1996;31:4093. Balberg I. Carbon 2002;40:139. Wang X, Wang Y, Jin Z. J Mater Sci 2002;37:223. Chen B, Wu K, Yao W. Cem Concr Compos 2004;26:291. Shao WZ, Xie N, Li YC, Zhen L, Feng LC. Trans Non-Ferro Metal Soc China SP 2005;15:297. Hunt A, Ewing R. 2nd ed. Berlin Heidelberg: Springer; 2009 . Stauffer D, Aharony A. 2nd ed. Talor & Francis Inc.; 1984. Nan CW. Prog Mater Sci 1993;37:1. Kogut PM, Straley JP. J Phys C – Solid State Phys 1978;12:2152. Halperin BI. Phys Rev Lett 1985;54:2391. Balberg I. Phys Rev Lett 1987;59:1305. Shao WZ, Xie N, Zhen L, Feng LC. J Phys: Conduct Matter 2008;20:395235. Zhang QH, Rastogi S, Chen DJ, Lippits D, Lemstra PJ. Carbon 2006;44:778. Chen CC, Chou YC. Phys Rev B 1985;54:2529. Cornette V, Pastor AJR, Nieto F. Physica A 2003;327:71. Xie N. PhD thesis, Harbin institute of technology, Harbin, China; 2009. Ki DY, Woo KY, Lee SB. Phys Rev E 2000;62:821.