Study of Electrical Conductivity in Fiber Composite Dounia zed Mezdour 1,2, Salah Sahli 1 and Mohamed Tabellout 2 1 Laboratoire de Microsystèmes et Instrumentation, Département d'Electronique Faculté des Sciences de l'Ingénieur, Université Mentouri de Constantine Route d'Ain El Bey, Constantine 25000, Algérie 2 Laboratoire LPEC, CNRS UMR 6087, Université du Maine, Avenue olivier Messiaen, Le Mans, CEDEX 9

Introduction Composite materials are widely used as pressure- sensing elements, resistors, transducers, thermistors, piezoresistors, chemical sensors and as packaging materials for substrates in electronic applications. The investigation of their heat and electrical transport phenomena represent an important research topic. Electrical conductive fibers (e.g., carbon fibers) provide the possibility to establish in polymer materials electrical conductivity and good mechanical properties simultaneously. With the percolation theory, other non-percolation models have also been proposed. The structure- oriented models are the most promising ones. They are based on microstructural data such as fiber orientation, length and packing arrangement. objective: Evaluation of the electrical resistance of polymer/carbon fiber composites using the proposed approach, based on updating the connectivity of the resistor network formed by the conducting clusters.

Assumptions Each conducting fiber is represented by a group of L adjacent cubes lined up along one of the principal axes of the lattice Conduction occurs between two different fibers when they touch along a non-zero surface The sample is considered to percolate if it is spanned by a conducting path in a given direction chosen to calculate the resistance Conducting paths are not straight chains but meander randomly through the composite Percolation is detected by the height of the conducting cluster instead of its number at the two electrodes. Tunnelling resistance is negligible The electrical resistivity of the polymer is not considered in the resistivity of the composite when the probability of percolation exceeds 50 %, else the electrical resistance of the composite is taken to be the electrical resistance of the matrix.

Sample generation The fibers are introduced randomly starting with an empty lattice. An orientation is randomly choused for the fiber. A new fiber is rejected if it would overlap a fiber already introduced. For each fiber introduced, a cluster number is attributed to it. Touching fibers are found and the smaller cluster number of them is attributed to the two. Then, cluster length in the considered direction is calculated and the corresponding electrical resistance computed. A spanning cluster is found by testing cluster length. If this occurs, the volume fraction is kept at this stage called percolation threshold p c. The spatial distribution of conductive fillers is generated by repeating the above described procedure until the pre-determined volume percentage is reached. n n n

Electrical resistance computation The resistance of a fiber-to-fiber contact R ff is the sum of the constriction resistance R cr and the bulk resistance R b of the fiber : R ff = R c +R b = R cr + R b (1) R b =  f × L/ S (2)‏ R cr =  f /d (3) The total path resistance R s is deduced by adding bulk resistance R b to contact resistance R c of fibers in contact: R s =  (R c +R b ) (4)‏ The effective electrical resistance Re of the sample is the equivalent resistance of the conducting pathways assumed to be in parallel: 1/R e =  1/R s (5)‏ L is the fiber's length S is the fiber’s cross section  f is the intrinsic fiber resistivity d is the diameter of the contact spot.

FIGURE 1. Variation of composite electrical resistance versus volume fraction in polycarbonate/ milled PAN-based carbon fiber composite for fiber length L=4 and different sample sizes n:  n=30,  n=40, + n=50 Transition from insulating to conducting behavior  percolation threshold p c Finite size effect  decreasing p c with increasing n

FIGURE 2. Distributions of Percolation Thresholds p c : (a)for a Fiber Length L = 6 and Different Sample Size n:  n=20, O n=30, Δ n=40, + n=50, X n=60 (Finite size effect)‏ (b) for a Fiber Length L=20 and Sample Size n=100.

FIGURE 3. Log-Log Plot of Distributions Maxima L G max Versus Sample Sizes n for L= 6 The curve is a traight line having the slope (1/ υ), then υ = 1/slope = 0,68.

FIGURE 4. n -1/v p (Vol.%)‏ Plot of n -1/ν Versus for ν = 0,68 and L= 6. The obtained curve is a straight line. Following the equation e xtrapolation gives p c values of 8,67 % for L= 6.

FIGURE 5. Variation of composite electrical resistance versus volume fraction in polycarbonate/ milled PAN-based carbon fiber composite for sample size n=100 and different fiber lengths:  L=5,  L=10,  L=15,  L=20 Increasing L  decreasing p c Lower value of p c for L=20 and n=100

FIGURE 6. Effect of the matrix type on the electrical resistance of carbon fiber composite: n=100, L=20  Polycarbonate,  PET,  Nylon 6.6,  Polyurethane The high boundary of the electrical resistance is controlled by the electrical resisistivity of the matrix  PC >  PET >  Nylon >  Polyurethane

FIGURE 7. Variation of conductivity versus volume fraction in polycarbonate/ milled PAN-based carbon fiber composite (n=100, L=20):  Experimental result,  Simulation result Superposition of experimental and simulated electrical resistance levels before and after percolation. p c resides between 6 and 7 vol.%

Conclusions A simulation method was presented to model the dependence of the electrical resistance as a function of load and particle size for a carbon fiber-filled polymer. The evaluation of electrical composite resistance was based on the notion that the composite is the result of a series of a large number of resistors combined in series and parallel. The evolution of electrical resistance versus filler volume fraction exhibited a percolative behavior for different sample sizes and fiber lengths. Finite size scaling was noticed on the curves. Reducing percolation threshold with longer fibers was also verified. Lower percolation thresholds were obtained for L=20 and n=100 Although simplifications were introduced in the model, the latter adjusts very well the experimental data and allows prediction of the electrical resistance curve for other particles sizes. Percolation threshold resides between 6 and 7 vol% for simulated polycarbonate/PAN based Carbon fiber composite. This method enabled investigation of the effects of different parameters on the electrical properties of carbon fiber-filled polymers.