in the case of an electrical percolation, conductivity increases sharply (by several orders of magnitude) when the concentration of inclusions or filler exceeds a critical value.

With the same components, we can have both conductors or insulators composite by changing only the concentration or proportion of inclusions.

in the case of an electrical percolation, conductivity increases sharply (by several orders of magnitude) when the concentration of inclusions or filler exceeds a critical value.

With the same components, we can have both conductors or insulators composite by changing only the concentration or proportion of inclusions.

Percolation threshold is that minimum filler content in the polymer matrix after which there is no significant change in the electrical properties of the composites. It is also related with the percolation network formation within the matrix for the electrical conduction. Percolation threshold is an important phenomenon for the polymer matrix composites which shows that at which minimum weight % of the filler the conductivity of the polymer matrix composite increased.

I agree with Nadia Saidi. A good way to examine percolation is in terms of volume fractions, that of the polymer matrix continuous phase, and that of the inclusions (filer, discontinuous phase). A very intersting way to examine percolation mathematically is by plotting the logarithm of the transport variable of interest (electrical conductivity, thermal conductivity) as a function of the logarithm of the [- volume fraction of filler plus the volume fraction of filler at the threshold]. Similarly,

above the threshold value one would examine the logarithm of [filler volume fraction - threshold volume fraction]. When either or both of these behaviors exhibit straight lines, one can say they exhibit "scaling". The slopes of any straight lines are called "scaling exponents". They can be related to fractal dimensions and to geometrical dimensions. For example in my old paper with Mark Lelental [“Network formation in nanoparticulate tin oxide - gelatin thin films;” J. Texter and M. Lelental, Langmuir, 15, 654-661 (1999).] we found some interesting scaling behaviors that indicated the electrical conductivity of some nanoparticle-doped polymer films scaled in conformance with an theoretical expectatiion. Such phenomena in complex fluids are often seen at quite low percolation thresholds, and the general topic is fodder for colloid and polymer physicists. Practical appications, for example, include knowing how much carbon black you can dope polymer beads with, before making the beads significantly conductive. Such beads (toner particles) are the basis of imaging in xerographic printing and in most laserjet printing engines.

Thanks Nadia Saidi & John Texter

sorry to somewhat disagree with the wording of the question and some parts of the various answers. "Percolation" relates to a theory why something happens, in this case you are looking at: a critical concentration at which by adding a little bit more of a conductive filler, suddenly the conductivity increases by several orders of magnitude. When using "percolation theory" you assume that the conductive particles are statistically evenly distributed within the polymer matrix.

Hence one would start with a conclusion before having analyzed, and then based on this conclusion ("the particles are statistically evenly distributed"), one makes a theory how then all of a sudden conductivity jumps up.

I did not begin with the conclusion, I started with the question: "How are the conductive particles distributed in a polymer matrix below, at, and above critical concentration?"

The result is this paper:

https://www.researchgate.net/publication/229816762_Electrical_conductivity_in_heterogeneous_polymer_systems._V_(1)_Further_experimental_evidence_for_a_phase_transition_at_the_critical_volume_concentration

Please look at this and later publications I have made (available here in RG) which explain

- conductive particles (if smaller than about 1 µm) are not statistically evenly distributed, the become dispersed and are forming complex aligned structures

- I have elucidated the mechanism by which conductivity all of sudden jumps up

- and later, I also explained this with a new non-equilibrium thermodynamical theory

https://www.researchgate.net/publication/202290104_Critical_Shear_Rate_-_the_Instability_Reason_for_the_Creation_of_Dissipative_Structures_in_Polymers?ev=prf_pub (which is part 2 of the theory)

https://www.researchgate.net/publication/233979584_Dispersion_hypothesis_and_non-equilibrium_thermodynamics_key_elements_for_a_material_science_of_conductive_polymers._A_key_to_understanding_polymer_blends_or_other_multiphase_polymer_systems?ev=prf_pub (part 1)

Article Electrical conductivity in heterogeneous polymer systems. V ...

Article Critical Shear Rate - the Instability Reason for the Creatio...

Article Dispersion hypothesis and non-equilibrium thermodynamics: ke...

Unmesh Vibhute has contacted me directlwith some more precise question, and I think that the group here should get to know his question and my answer. He is dealing with PPS and Ni filler. I do not know the Ni particle size, nor how the composite was prepared.

Such composites do NOT behave as I was describing in the above cited literature. I am only describing systems where the conductive phase is smaller than 1 µm. Ni fillers are usually bigger than that and they behave in a way which can roughly be described by percolation theory.

Unmesh Vibhute found different values for surface and volume resistivity. My answer to his question why this is the case, was as follows:

My explanation for the differences:

1) surface resistivity and volume resistivity are measured by different techniques (at least you should do so), 2-point or ring electrode for surface R, 4-point-probe for volume R

2) for the 3 highly resistive samples (pure, 10 and 30), the differences are negligible, it is not easy to really precisely measure high R values (contact area differences, humidity, temperature differences, very sensitive)

3) for Ni-50, the difference is due to

a) at lower R values, only 4-point-probe gives you reliable results because it eliminates contact points polarisation

b) the higher surface resistivity reflects that you have a pure polymer layer which you contact, so you can not measure the "true" resistivity which is somewhat lowered by the insulating polymer surface layer.

@ Benhard Wessling: Thank you sir for your valuable suggestions.

How to calculate the 't' exponent from calculated values of electrical conductivity to fit the power law ?

"For a phase transition at the critical volume concentration."

Mr. Berhard Wessling's experimental observations are phenomenal and correct with three(3) keywords as far as I experienced:

• Nano- or quantum-size effect (based on Einstein's Dissertation 190
• Critical volume fraction
• "I did not begin with the conclusion, 'the particles are statistically evenly distributed.' I started with the question: How are the conductive particles distributed in a polymer matrix below, at, and above critical concentration?"

The topic appears to be closely related to "Brownian motion and diffusion" defined and explained in Albert Einstein's dissertation:

• Albert Einstein's original PhD dissertation in German

https://www.youtube.com/watch?v=TB9Jeeu6Z_4

• Is it true that Albert Einstein's Ph.D thesis is only 6 pages?

https://www.quora.com/Is-it-true-that-Albert-Einsteins-Ph-D-thesis-is-only-6-pages

• On Einstein's Doctoral Thesis

Norbert Straumann

https://arxiv.org/abs/physics/0504201

• Albert Einstein’s Dissertation

https://pdfs.semanticscholar.org/cff4/fbf0e5cd4140acb95cdb73da16936f1d1691.pdf

• Investigations on the Theory of the Brownian Movement

https://www.pitt.edu/~jdnorton/lectures/Rotman_Summer_School_2013/Einstein_1905_docs/Einstein_Dissertation_English.pdf

Thanks from Benjamin

very nice and funny comment (I hope you did not try to compare me or my work with Einstein).

I have a minor comment: Einstein was dealing with the question of the size of *molecules*, I am dealing with nanoparticles, which are at least one order of magnitude bigger than molecules.

And on the nanoparticle level, again other laws are directing the motions and behaviour, different from the molecular level and different as well from the micron-size or even bulk level.

The concept of double percolation was first proposed by

Sumita et al, in immiscible blends that were filled with carbon black. Two requirements are needed to realize the double percolation structure in

composites. One requires co-continuous blend morphologies,the other needs selective filling of only one of the blend phases (double percolation).

Studies on the selective localization of multi-walled carbon

nanotubes in blends of poly(vinylidene fluoride) and

polycaprolactone(DOI: 10.3144/expresspolymlett.2015.8)

Vafa Fakhri, I don't know to whom or to what your comment was directed, but I have 2 comments:

1) what you (and Sumita) call "percolation" and "double percolation" is not *percolation*, but a phase transition (from dispersed to flocculated status of the carbon black filler), and as I have outlined and shown in my article (link below) that this phenomenon is one of the many experimental evidence supporting my view that "percolation" is the absolutely wrong term for the sudden occurence of conductivity in carbon black filled composites

2) why do you say "Sumita has first proposed this" (double percolation which is a double phase transition)? His article is from 1992:

Article Double percolation effect on the electrical conductivity of ...

my article is from 1991:

Article Electrical conductivity in heterogeneous polymer systems. V ...

and Sumita is even not mentioning a reference to my publication (apart from that he anyway only has 1 literature reference ...)

and in fact, I published it much earlier: 1989, because I had huge problems getting my paper published (because the referees all were strong believers of "percolation theory" which simply can not be applied here); I had therefore filed a British patent application: GB-OS 2 214 511 date of filing: Jan. 23. 1989. date of publication:

Sept. 6. 1989

Only 2 years later I succeeded in getting it published in that Engineering research publication

Dear Bernhard Wessling

thanks for your very good and comprehensive comment.

I must read more and more about this topic.

your papers will be very useful.