Continuum Percolation Study of Carbon Nano Tube Composites Via Size Distribution Effects

Continuum Percolation study of carbon nano tube composites via Size distribution effects.

A.Afaghi ;S.Asiaei ;M.Baniasadi

Mechanical engineering department;University of Tehran.

The three-dimensional continuum percolation problem of hard-core and soft-core (permeable) objects was an area of active research in the 1980s[1]. Among the considered geometrical objects a very important category is the case of permeable sticks with the form of capped cylinders [2]. Advancements in capabilities of theories and numerical studies has lead to recent developments of polymer reinforced nanocomposites which overcome the need for having certain combination of electrical and mechanical properties [4]. While great strides have been made in exploiting the properties of carbon nanotubes (CNT) [5, 6] several publications document the progress made in fabrication and characterization of CNT nanocomposites [7-10]. In relation to percolation , it is difficult to draw definite conclusions about electrical conductivity from these published studies because the reported levels of CNT loading required to achieve a percolation concentration (i.e. an appreciable increase in electrical conductivity) vary widely, ranging from less than 1 to over 10% [3].

In general the main parameters affecting percolation are geometry (aspect ratio) and the state of orientation of sticks. However there are other production factors that will change expected percolation concentration such as "size distribution" .The laser counting carried out on the suspension of the particles reveals that the size distribution is asymmetrically extended on the side of the higher-than average values [11]. This fact itself tends to reduce the threshold since it has been shown that with percolating objects having large aspect ratios, the critical concentration diminishes as the size distribution is widened [12 ,13]. On the effects of polydispersed particles, Charlaix, Guyon, Riviers [14] pecifically noted that a ''larger weight'' was given to larger than average objects and thus critical concentration is maximum when the objects are of fixed size, otherwise it is the larger objects which determine the threshold.

Here we intend to investigate main approaches available in literature for predicting the required concentration of CNTs and ascertain the results with a numerical method specifically with respect to size distribution effects.

Moreover in previous works [3] the centers of mass of the cylinders were placed randomly within a unitscaled cell, insuring that no more than half of the cylinder extended beyond the boundary and orientations were generated by taking the center of mass as the origin of a unit sphere and generating a point randomly on the surface, using the method described in[15].This method insures the random isotropic distribution of sticks and prohibits the classical mistake as reported in [16].However this would lead to some difficulty since one must take the cell large enough to ascertain accuracy which will be reduced by restricting the sticks to be created with their centers only within the cell. But we have produced sticks fully random and no restriction was set to creating stick centers within the sample. So that much smaller samples can be used with little difficulty in computational efforts. In brief sticks are produced by generating N stick starting point in a completely random way and then the ending point is generated in such a way that it sweeps the perimeter of a sphere to obey a fully isotropic distribution. When a stick intersects the boundary only the fractional volume which is whitin the sample is considered. As pointed out in [16] the permeable stick assumption has to be carefully examined when applied to real composite materials. Particularly in numerical simulation this assumption will be ended in an overestimate of the threshold as some regions of space is occupied by more than one stick and we know that this is not the case. Here we will study the two assumptions and the level of affectability of the two by size distribution in our simulations.

The final part of this Paper is devoted to examining the results of varying size distribution (SD) on percolation compared with recent results that has been reported on the subject, attempting to explain the deviations observed with considering the fact. In another part of the study we examine the SD effect on the Bc (average number of objects bonded to a given object) and some concluding remarks will be given there in relation to the concepts of invariant excluded volume theory.

Here we present results of our study briefly. In relation to Bc behavior it is seen that in every nominal size of reinforcing nanotube a preferred distribution (Here Length Distribution only) exist that will be ended in a best conditioned percolating network with the stronger degree of connectivity.

Fig(1):mean number of connections per stick: we will have the best network connectivity at DI about 4 then the curve tends to the constant size sample

Fig (2): The best percolating network is formed at a DI about 3.The curve tends to the constant size sample.

Same trend is present with (Critical volume concentration needed for the onset of percolation). What we see is that all distributed length samples percolate at lower values compared with constant length samples, but considering

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