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Critical behaviour of a three-dimensional hardcore cylinders composite system

Critical behaviour of a three-dimensional hardcore cylinders composite system. J. Silva 1,2 , R. Simoes 2,3 , S. Lanceros- Mendez 1 1- Center of Physics, University of Minho - 4710-057 Braga, Portugal.

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Critical behaviour of a three-dimensional hardcore cylinders composite system

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  1. Critical behaviour of a three-dimensional hardcore cylinders composite system J. Silva1,2, R. Simoes2,3, S. Lanceros-Mendez1 1- Centerof Physics, University of Minho - 4710-057 Braga, Portugal. 2- Institute for Polymers and Composites IPC/I3N, University of Minho - Campus de Azurem, 4800-058 Guimares, Portugal. 3- School of Technology, Polytechnic Institute of Cavado and Ave - 4750-810 Barcelos, Portugal.

  2. Critical behaviour of a three-dimensional hardcore cylinders composite system • Introduction • Description of a composite as a network. • Results and Discussion. • The size of the giant component • The mean size of the finite components • Predicting the correlation length critical exponent (ν ) • Finite size scaling analysis • Scaling at the criticality • The Alexander-Orbach conjuncture • How to determine the critical exponents experimentally • Conclusions

  3. Critical behaviour of a three-dimensional hardcore cylinders composite system 1 – Introduction • The percolation problem of 2D random sticks was first addressed by Pike et. al [Phys. Rev. B 10, 1421 (1974)], with the authors focusing on the determination of the point of emergence of the giant component that spans the system. • Later, Balberg et. al [Phys. Rev. Lett. 51, 1605 (1983)] derived the percolation critical exponents for the 2D random sticks problem, demonstrating that the 2D continuum percolation problem belongs to the same universality class as the lattice percolation, i.e., the continuum and the lattice percolation problems share the same critical exponents. • Remarkably, while for the 2D soft-core problem the determination of the critical exponents has been reported, few studies [Compos. Sci. Technol. 46, 379 (1993), Compos. Sci. Technol. 56, 911 (1996)] are focused on the determination of the critical exponents for 3D hard-core caped cylinders with an isotropic distribution.

  4. Critical behaviour of a three-dimensional hardcore cylinders composite system 1 – Introduction The critical exponents are universal, as they depend just on the system dimension, the symmetry of the order parameter and on the range of interactions

  5. Critical behaviour of a three-dimensional hardcore cylinders composite system 2 – Description of a composite as network • Cubes were filled with cylinders with aspect ratios of 100 and δmax = 1 and the generated volume fractions ranged from 10-4 to 4.1 × 10-3, corresponding to a number of cylinders from a few hundred to ∼ 104 • For each data point (set of material parameters) of the results shown, ∼ 104 different microstructures were simulated and all the respective graph properties were averaged.

  6. Critical behaviour of a three-dimensional hardcore cylinders composite system 3 – Results and discussion: The size of the giant component

  7. Critical behaviour of a three-dimensional hardcore cylinders composite system 3 – Results and discussion: The mean size of the finite components

  8. Critical behaviour of a three-dimensional hardcore cylinders composite system 3 – Results and discussion: Predicting the correlation length critical exponent (ν ) In a second order phase transition the width of the transition, ∆, should scale with system size as: • ∆ can be determined by taking the value of the FWHM of a Gaussian using two methods: • By fitting a Gaussian to <s> vsΦ: • ν = 0.514 ± 0.016 • By fitting a Gaussian to dS/dΦvsΦ: • ν = 0.502 ± 0.022

  9. Critical behaviour of a three-dimensional hardcore cylinders composite system 3 – Results and discussion: Finite size scaling analysis

  10. Critical behaviour of a three-dimensional hardcore cylinders composite system 3 – Results and discussion: Finite size scaling analysis

  11. Critical behaviour of a three-dimensional hardcore cylinders composite system 3 – Results and discussion: Scaling at the criticality

  12. Critical behaviour of a three-dimensional hardcore cylinders composite system 3 – Results and discussion: Scaling at the criticality

  13. Critical behaviour of a three-dimensional hardcore cylinders composite system 4 – The Alexander-Orbach conjuncture Compos. Sci. Technol69, 1486 (2009) The Alexander-Orbach conjuncture is based on the diffusion on fractals and predicts that the dimensionality of the quantized vibrational states on a fractal is close to 4/3 for d ≥ 2, independent of the system dimensionality.

  14. Critical behaviour of a three-dimensional hardcore cylinders composite system 5 – How to determine the critical exponents experimentally Using a conducting –tip atomic force microscopy it is possible (using the principal of Delesse) to calculate the β critical exponent using a finite size scaling analysis. For that, several composites with different volumes but with same volume fraction must be analyzed. PRL 102, 116601 (2009)

  15. Critical behaviour of a three-dimensional hardcore cylinders composite system 6 – Conclusions • The critical exponents for a hard-core 3D cylinder system with short-range interactions has been obtained, making use of the network theory, and these are related through the common hyperscaling for a 3D system • In contrast to the 2D stick system and the 3D hard-core cylinder system, the determined critical exponents do not belong to the same universality class as the lattice percolation. Instead, the correlation length critical exponent has a typical mean field value, the γ critical exponent has a value that is close to the mean field one, and β = 0.3 • Using the Alexander-Orbach conjuncture, it is found that the relation between the conductivity and the correlation length critical exponents for a 3D system is obeyed.

  16. Thank you • The authors are in debt to Alberto Proença for providing the computational resources (SEARCH) used in this work; • The numerical calculations were performed on the Advanced Research Computing Services with HTC/HPC clusters, SEARCH cluster, financed through the Programa de ReequipamentoCientíficoNacional, FCT-Fundaçãopara a CiênciaeTecnologia, CONCREEQ/443/EEI/2005 and by the Program POCI 2010 funded by FEDER; • This work is also funded by FEDER funds through the ”ProgramaOperacionalFactores de Competitividade COMPETE” and by the FCT project references: PTDC/CTM/69316/2006, NANO/NMed- -SD/0156/2007, PTDC/EME-PME/108859-2008, and PTDC/CTM-NAN/112574/2009; • The authors also thank support from the COST Action MP1003-”European Scientific Network for Artificial Muscles” and the COST action MP0902-”Composites of Inorganic Nanotubes and Polymers (COINAPO)” • Jaime Silva acknowledges FCT grant SFRH/BD/60623/2009.

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