We present a generalized connectedness percolation theory reduced to a compact form for a large class of anisotropic particle mixtures with variable degrees of connectivity. Even though allowing for an infinite number of components, we derive a compact yet exact expression for the mean cluster size of connected particles. We apply our theory to rodlike particles taken as a model for carbon nanotubes and find that the percolation threshold is sensitive to polydispersity in length, diameter, and the level of connectivity, which may explain large variations in the experimental values for the electrical percolation threshold in carbon-nanotube composites. The calculated connectedness percolation threshold depends only on a few moments of the full distribution function. If the distribution function factorizes, then the percolation threshold is raised by the presence of thicker rods, whereas it is lowered by any length polydispersity relative to the one with the same average length and diameter. We show that for a given average length, a length distribution that is strongly skewed to shorter lengths produces the lowest threshold relative to the equivalent monodisperse one. However, if the lengths and diameters of the particles are linearly correlated, polydispersity raises the percolation threshold and more so for a more skewed distribution toward smaller lengths. The effect of connectivity polydispersity is studied by considering nonadditive mixtures of conductive and insulating particles, and we present tentative predictions for the percolation threshold of graphene sheets modeled as perfectly rigid, disklike particles.
REFERENCES
1.
K. S.
Novoselov
, A. K.
Geim
, S. V.
Morozov
, D.
Jiang
, Y.
Zhang
, S. V.
Dubonos
, I. V.
Grigorieva
, and A. A.
Firsov
, Science
306
, 666
(2004
).2.
M.
Moniruzzaman
and K. I.
Winey
, Macromolecules
39
, 5194
(2006
).3.
S.
Stankovich
, D. A.
Dikin
, G. H. B.
Dommett
, K. M.
Kohlhaas
, E. J.
Zimney
, E. A.
Stach
, R. D.
Piner
, S. T.
Nguyen
, and R. S.
Ruoff
, Nature (London)
442
, 282
(2006
).4.
D.
Stauffer
and A.
Aharony
, Introduction to Percolation Theory
(Taylor & Francis
, London
, 1992
).5.
N.
Grossiord
, J.
Loos
, O.
Regev
, and C. E.
Koning
, Chem. Mater.
18
, 1089
(2006
).6.
E. J.
Garboczi
, K. A.
Snyder
, J. F.
Douglas
, and M. F.
Thorpe
, Phys. Rev. E
52
, 819
(1995
).7.
I.
Balberg
, C. H.
Anderson
, S.
Alexander
, and N.
Wagner
, Phys. Rev. B
30
, 3933
(1984
).8.
I.
Balberg
, Phys. Rev. B
33
, 3618
(1986
).9.
A. L. R.
Bug
, S. A.
Safran
, and I.
Webman
, Phys. Rev. B
33
, 4716
(1986
).10.
A. Celzard E.
McRae
, C.
Deleuze
, M.
Dufort
, G.
Furdin
, and J. F.
Mareche
, Phys. Rev. B
53
, 6209
(1996
).11.
K.
Leung
and D.
Chandler
, J. Stat. Phys.
63
, 837
(1991
).12.
A. V.
Kyrylyuk
and P. van der
Schoot
, Proc. Natl. Acad. Sci. U.S.A.
105
, 8221
(2008
).13.
Y. R.
Hernandez
, A.
Gryson
, F. M.
Blighe
, M.
Cadek
, V.
Nicolosi
, W. J.
Blau
, Y. K.
Gunko
, and J. N.
Coleman
, Scr. Mater.
58
, 69
(2008
).14.
P.
Sollich
, J. Phys.: Condens. Matter
14
, R79
(2002
).15.
K.
Shundyak
, R. van
Roij
, and P.
van der Schoot
, J. Chem. Phys.
122
, 094912
(2005
).16.
As produced graphene sheets are conductive but if they are first oxidized and later reduced in the processing of the nanocomposites, their conductivity can be strongly reduced. See also Ref. 57.
17.
H.
Deng
, R.
Zhang
, E.
Bilotti
, J.
Loos
, and A. A. J. M.
Peijs
, J. Appl. Polym. Sci.
113
, 742
(2009
).18.
J. P.
Hansen
and I.
MacDonald
, Theory of Simple Liquids
, 2nd ed. (Academic
, London
, 1986
).19.
R. H. J.
Otten
and P. van der
Schoot
, Phys. Rev. Lett.
103
, 225704
(2009
).20.
A.
Coniglio
, U. De
Angelis
, and A.
Forlani
, J. Phys. A
10
, 1123
(1977
).21.
M. C.
Hermant
, B.
Klumperman
, A. V.
Kyrylyuk
, P. van der
Schoot
, and C. E.
Koning
, Soft Matter
5
, 878
(2009
).22.
S. I.
White
, R. M.
Mutiso
, P. M.
Vora
, D.
Jahnke
, S.
Hsu
, J. M.
Kikkawa
, J.
Li
, J. E.
Fischer
, and K. I.
Winey
Adv. Funct. Mater.
20
, 2709
(2010
).23.
G.
Stell
, Physica A
231
, 1
(1996
).24.
R.
Fantoni
, D.
Gazzillo
, and A.
Giacometti
, J. Chem. Phys.
122
, 034901
(2005
).25.
A. P.
Chatterjee
, J. Phys.: Condens. Matter
20
, 255250
(2008
).26.
S.
Torquato
, Random Heterogeneous Materials: Microstructure and Macroscopic Properties
(Springer
, New York
, 2002
).27.
In fact, P and C+ can be expressed in terms of a sum of graphs and describe average probabilities of having the particles connected in such a graph, which is not an actual cluster.
28.
G. J.
Vroege
and H. N. W.
Lekkerkerker
, Rep. Prog. Phys.
55
, 1241
(1992
).29.
T.
DeSimone
, S.
Demoulini
, and R. M.
Stratt
, J. Chem. Phys.
85
, 391
(1986
).30.
J. P.
Straley
, Phys. Rev. A
8
, 2181
(1973
).31.
L.
Onsager
, Ann. N. Y. Acad. Sci.
51
, 627
(1949
).32.
G.
Ambrosetti
, C.
Grimaldi
, I.
Balberg
, T.
Maeder
, A.
Danani
, and P.
Ryser
, Phys. Rev. B
81
, 155434
(2010
).33.
M. C.
Hermant
, P. van der
Schoot
, B.
Klumperman
, and C. E.
Koning
, ACS Nano
4
, 2242
(2010
).34.
F.
Dalmas
, R.
Dendievel
, L.
Chazeau
, J.-Y.
Cavaillé
, and C.
Gauthier
, Acta Mater.
54
, 2923
(2006
).35.
A. P.
Chatterjee
, J. Chem. Phys.
132
, 224905
(2010
).36.
In probability theory,
$\sqrt{s} = \sqrt{{\rm Var}(L_i)}/\langle L_i\rangle _i$
is called the coefficient of variation (or variation coefficient) of the distribution of Li. See also Ref. 45.37.
C.
Lu
and J.
Liu
, J. Phys. Chem. B
110
, 20254
(2006
).38.
C. L.
Cheung
, A.
Kurtz
, H.
Park
, and C. M.
Lieber
, J. Phys. Chem. B
106
, 2429
(2002
).39.
T.
Yamada
, T.
Namai
, K.
Hata
, D. N.
Futaba
, K.
Mizuno
, J.
Fan
, M.
Yudasaka
, M.
Yumura
, and S.
Iijima
, Nat. Nanotechnol.
1
, 131
(2006
).40.
S.
Wang
, Z.
Liang
, B.
Wang
, and C.
Zhang
, Nanotechnology
17
, 634
(2006
).41.
J.
Li
and J.-K.
Kim
, Compos. Sci. Technol.
67
, 2114
(2007
).42.
A. F.
Holloway
, D. A.
Craven
, L.
Xiao
, J. Del
Campo
, and G. G.
Wildgoose
, J. Phys. Chem. C
112
, 13729
(2008
).43.
A. S.
Berdinsky
, P. S.
Alegaonkar
, H. C.
Lee
, J. S.
Jung
, J. H.
Han
, J. B.
Yoo
, D.
Fink
, and L. T.
Chadderton
, NANO
2
, 59
(2007
).44.
A. N. G.
Parra-Vasquez
, I.
Stepanek
, V. A.
Davis
, V. C.
Moore
, E. H.
Haroz
, J.
Shaver
, R. H.
Hauge
, R. E.
Smalley
, and M.
Pasquali
, Macromolecules
40
, 4043
(2007
).45.
R. V.
Hogg
and A. T.
Craig
, Introduction to Mathematical Statistics
, 4th ed. (Macmillan
, New York
, 1978
).46.
The skewness
$\gamma \equiv \langle (L_i-\langle L_i\rangle _i)^3\rangle _i/\langle (L_i-\langle L_i\rangle _i)^2\rangle _i^{3/2}$
need not go to zero in the monodisperse limit, because both the numerator and the denominator go to zero and the speed of convergence of both determines the monodisperse value of γ.47.
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
, edited by M.
Abramowitz
, I. A.
Stegun
, and Irene
(Dover Publications
, New York
, 1972
).48.
Á.
Kukovecz
, T.
Kanyó
, and Z.
Kónya
, I.
Kiricsi
, Carbon
43
, 994
(2005
).49.
A.
Lucas
, C.
Zakri
, M.
Maugey
, M.
Pasquali
, P. van der
Schoot
, and P.
Poulin
, J. Phys. Chem. C
113
, 20599
(2009
).50.
G.-H.
Jeong
, S.
Suzuki
, Y.
Kobayashi
, A.
Yamazaki
, H.
Yoshimura
, and Y.
Homma
, J. Appl. Phys.
98
, 124311
(2005
).51.
M. S.
Arnold
, A. A.
Green
, J. F.
Hulvat
, S. I.
Stupp
, and M. C.
Hersam
, Nat. Nanotechnol.
1
, 60
(2006
).52.
B.
Yu
, P.-X.
Hou
, F.
Li
, B.
Liu
, C.
Liu
, and H.-M.
Cheng
, Carbon
48
, 2941
(2010
).53.
This requires the conductive layer to be above its glass temperature in the preparatory phase in the production stages of the composite. This in practice is the case. See Ref. 21.
54.
E. T.
Thostenson
and T.-W.
Chou
, J. Phys. D: Appl. Phys.
35
, L77
(2002
).55.
R.
Fantoni
, D.
Gazzillo
, A.
Giacometti
, M. A.
Miller
, and G.
Pastore
, J. Chem. Phys.
127
, 234507
(2005
).56.
J.
Phillips
and M.
Schmidt
, Phys. Rev. B
81
, 041401
(2010
).57.
E.
Tkalya
, M.
Ghislandi
, A.
Alekseev
, C.
Koning
, and J.
Loos
, J. Mater. Chem.
20
, 3035
(2010
).58.
H.-B.
Zhang
, W.-G.
Zheng
, Q.
Yan
, Y.
Yang
, J.-W.
Wang
, Z.-H.
Lu
, G.-Y.
Ji
, and Z.-Z.
Yu
, Polymer
51
1191
–1196
(2010
).59.
A. V.
Kyrylyuk
, M. C.
Hermant
, T.
Schilling
, B.
Klumperman
, C. E.
Koning
, and P. van der
Schoot
, “Controlling electrical percolation in multi-component carbon nanotube dispersions
,” Nature Nanotechnology
(to be published).60.
T. L.
Hill
, J. Chem. Phys.
23
, 617
(1955
).61.
For a Gaussian distribution centered around 〈Lk〉k this is not strictly the case because by definition Lk > 0; however, for s ≪ 1 it becomes approximately true with Lshort = 0 and Llong = 2〈Lk〉k.
© 2011 American Institute of Physics.
2011
American Institute of Physics
You do not currently have access to this content.
