1
|
Coupette F, Härtel A, Schilling T. Continuum percolation expressed in terms of density distributions. Phys Rev E 2020; 101:062126. [PMID: 32688611 DOI: 10.1103/physreve.101.062126] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2019] [Accepted: 04/15/2020] [Indexed: 06/11/2023]
Abstract
We present an approach to derive the connectivity properties of pairwise interacting n-body systems in thermal equilibrium. We formulate an integral equation that relates the pair connectedness to the distribution of nearest neighbors. For one-dimensional systems with nearest-neighbor interactions, the nearest-neighbor distribution is in turn related to the pair-correlation function g through a simple integral equation. As a consequence, for those systems, we arrive at an integral equation relating g to the pair connectedness, which is readily solved even analytically if g is specified analytically. We demonstrate the procedure for a variety of pair potentials including fully penetrable spheres as well as impenetrable spheres, the only two systems for which analytical results for the pair connectedness exist. However, the approach is not limited to nearest-neighbor interactions in one dimension. Hence, we also outline the treatment of external fields and long-range interactions and we illustrate how the formalism can applied to higher-dimensional systems using the three-dimensional ideal gas as an example.
Collapse
Affiliation(s)
- Fabian Coupette
- Institute of Physics, University of Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
| | - Andreas Härtel
- Institute of Physics, University of Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
| | - Tanja Schilling
- Institute of Physics, University of Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
| |
Collapse
|
2
|
Sapkota J, Martinez Garcia JC, Lattuada M. Reinterpretation of the mechanical reinforcement of polymer nanocomposites reinforced with cellulose nanorods. J Appl Polym Sci 2017. [DOI: 10.1002/app.45254] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023]
Affiliation(s)
- Janak Sapkota
- Chair of Polymer Processing; Department of Polymer Engineering and Science, Montanuniversitaet Leoben; Leoben 8700 Austria
| | | | - Marco Lattuada
- Department of Chemistry; University of Fribourg; Fribourg 1700 Switzerland
| |
Collapse
|
3
|
Meyer H, van der Schoot P, Schilling T. Percolation in suspensions of polydisperse hard rods: Quasi universality and finite-size effects. J Chem Phys 2015; 143:044901. [PMID: 26233158 DOI: 10.1063/1.4926946] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present a study of connectivity percolation in suspensions of hard spherocylinders by means of Monte Carlo simulation and connectedness percolation theory. We focus attention on polydispersity in the length, the diameter, and the connectedness criterion, and we invoke bimodal, Gaussian, and Weibull distributions for these. The main finding from our simulations is that the percolation threshold shows quasi universal behaviour, i.e., to a good approximation, it depends only on certain cumulants of the full size and connectivity distribution. Our connectedness percolation theory hinges on a Lee-Parsons type of closure recently put forward that improves upon the often-used second virial approximation [T. Schilling, M. Miller, and P. van der Schoot, e-print arXiv:1505.07660 (2015)]. The theory predicts exact universality. Theory and simulation agree quantitatively for aspect ratios in excess of 20, if we include the connectivity range in our definition of the aspect ratio of the particles. We further discuss the mechanism of cluster growth that, remarkably, differs between systems that are polydisperse in length and in width, and exhibits non-universal aspects.
Collapse
Affiliation(s)
- Hugues Meyer
- Research Unit for Physics and Materials Science, Université du Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Paul van der Schoot
- Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 3500 MB Eindhoven, The Netherlands
| | - Tanja Schilling
- Research Unit for Physics and Materials Science, Université du Luxembourg, L-1511 Luxembourg, Luxembourg
| |
Collapse
|
4
|
Chatterjee AP. Connectedness percolation in isotropic systems of monodisperse spherocylinders. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:375302. [PMID: 26325334 DOI: 10.1088/0953-8984/27/37/375302] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The dependence upon aspect ratio and connectedness range of the percolation threshold for monodisperse spherocylinders as calculated from an analogy between lattice and continuum percolation is compared with findings from: (i) recent Monte Carlo (MC) simulations, and: (ii) a recent elaboration of an integral equation-based approach that uses the connectedness Ornstein-Zernike equation. The results from each of these approaches are found to be in reasonable semi-quantitative agreement, especially for the case of particles that have a finite hard core diameter.
Collapse
Affiliation(s)
- Avik P Chatterjee
- Department of Chemistry, SUNY College of Environmental Science and Forestry, One Forestry Drive, Syracuse, NY 13210, USA
| |
Collapse
|
5
|
Chatterjee AP, Grimaldi C. Random geometric graph description of connectedness percolation in rod systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:032121. [PMID: 26465440 DOI: 10.1103/physreve.92.032121] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2015] [Indexed: 06/05/2023]
Abstract
The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The probability that an edge (or link) connects any randomly selected pair of nodes depends upon the rod volume fraction as well as the distribution over their sizes and shapes, and also upon quantities that characterize their state of dispersion (such as the orientational distribution function). We employ the observation that contributions from closed loops of connected rods are negligible in the limit of large aspect ratios to obtain percolation thresholds that are fully equivalent to those calculated within the second-virial approximation of the connectedness Ornstein-Zernike equation. Our formulation can account for effects due to interactions between the rods, and many-body features can be partially addressed by suitable choices for the edge probabilities.
Collapse
Affiliation(s)
- Avik P Chatterjee
- Department of Chemistry, SUNY College of Environmental Science and Forestry, One Forestry Drive, Syracuse, New York 13210, USA
| | - Claudio Grimaldi
- Laboratory of Physics of Complex Matter, Ecole Polytechnique Fédérale de Lausanne, Station 3, CH-1015 Lausanne, Switzerland
| |
Collapse
|
6
|
Chatterjee AP, Grimaldi C. Tunneling conductivity in anisotropic nanofiber composites: a percolation-based model. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:145302. [PMID: 25788468 DOI: 10.1088/0953-8984/27/14/145302] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The critical path approximation ('CPA') is integrated with a lattice-based approach to percolation to provide a model for conductivity in nanofiber-based composites. Our treatment incorporates a recent estimate for the anisotropy in tunneling-based conductance as a function of the relative angle between the axes of elongated nanoparticles. The conductivity is examined as a function of the volume fraction, degree of clustering, and of the mean value and standard deviation of the orientational order parameter. Results from our calculations suggest that the conductivity can depend strongly upon the standard deviation in the orientational order parameter even when all the other variables (including the mean value of the order parameter 〈S〉) are held invariant.
Collapse
Affiliation(s)
- Avik P Chatterjee
- Department of Chemistry, SUNY College of Environmental Science and Forestry, One Forestry Drive, Syracuse, New York 13210, USA
| | | |
Collapse
|
7
|
Nigro B, Grimaldi C, Miller MA, Ryser P, Schilling T. Depletion-interaction effects on the tunneling conductivity of nanorod suspensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042140. [PMID: 24229148 DOI: 10.1103/physreve.88.042140] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/08/2013] [Revised: 09/25/2013] [Indexed: 06/02/2023]
Abstract
We study by simulation and theory how the addition of insulating spherical particles affects the conductivity of fluids of conducting rods, modeled by spherocylinders. The electrical connections are implemented as tunneling processes, leading to a more detailed and realistic description than a discontinuous percolation approach. We find that the spheres enhance the tunneling conductivity for a given concentration of rods and that the enhancement increases with rod concentration into the regime where the conducting network is well established. By reformulating the network of rods using a critical path analysis, we quantify the effect of depletion-induced attraction between the rods due to the spheres. Furthermore, we show that our conductivity data are quantitatively reproduced by an effective-medium approximation, which explicitly relates the system tunneling conductance to the structure of the rod-sphere fluid.
Collapse
Affiliation(s)
- B Nigro
- LPM, Ecole Polytechnique Fédérale de Lausanne, Station 17, Case Postale 1015 Lausanne, Switzerland
| | | | | | | | | |
Collapse
|
8
|
Loveday SM, Su J, Rao MA, Anema SG, Singh H. Effect of Calcium on the Morphology and Functionality of Whey Protein Nanofibrils. Biomacromolecules 2011; 12:3780-8. [DOI: 10.1021/bm201013b] [Citation(s) in RCA: 66] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Affiliation(s)
- Simon M. Loveday
- Riddet Institute, Massey University, Private Bag 11 222, New Zealand
| | - Jiahong Su
- Riddet Institute, Massey University, Private Bag 11 222, New Zealand
| | - M. Anandha Rao
- Department of Food Science, Cornell University, Geneva,
New York 14456, United States
| | | | - Harjinder Singh
- Riddet Institute, Massey University, Private Bag 11 222, New Zealand
| |
Collapse
|
9
|
Sarkisov L. Theory of pair connectedness in templated quenched-annealed systems. J Chem Phys 2008; 128:044707. [DOI: 10.1063/1.2823734] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
10
|
Chatterjee AP. Integral equation theory for athermal solutions of linear polymers. J Chem Phys 2004; 121:11432-9. [PMID: 15634103 DOI: 10.1063/1.1814977] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
An integral equation model is developed for athermal solutions of flexible linear polymers with particular reference to good solvent conditions. Results from scaling theory are used in formulating form factors for describing the single chain structure, and the impact of solvent quality on the chain fractal dimension is accounted for. Calculations are performed within the stringlike implementation of the polymer reference interaction site model with blobs (as opposed to complete chains) treated as the constituent structural units for semidilute solutions. Results are presented for the second virial coefficient between polymer coils and the osmotic compressibility as functions of the chain length and polymer volume fraction, respectively. Findings from this model agree with results from scaling theory and experimental measurements, as well as with an earlier investigation in which self-avoiding chains were described using Gaussian form factors with a chain length and concentration-dependent effective statistical segment length. The volume fractions at the threshold for connectedness percolation are evaluated within a coarse-grained closure relation for the connectedness Ornstein-Zernike equation. Results from these calculations are consistent with the usual interpretation of the semidilute crossover concentration for model solutions of both ideal and swollen polymer coils.
Collapse
Affiliation(s)
- Avik P Chatterjee
- Department of Chemistry, SUNY-ESF, 121 Edwin C. Jahn Laboratory, Syracuse, NY 13210, USA
| |
Collapse
|
11
|
Wang X, Chatterjee AP. Depletion interactions induced by flexible polymers in solutions of rod-like macromolecules. J Chem Phys 2003. [DOI: 10.1063/1.1625368] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
12
|
Wang X, Chatterjee AP. Connectedness percolation in athermal mixtures of flexible and rigid macromolecules: Analytic theory. J Chem Phys 2003. [DOI: 10.1063/1.1575201] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
13
|
|
14
|
Kondrat G. Influence of temperature on percolation in a simple model of flexible chains adsorption. J Chem Phys 2002. [DOI: 10.1063/1.1505866] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
15
|
Wang X, Chatterjee AP. Continuum percolation in athermal mixtures of flexible and rigid macromolecules. J Chem Phys 2002. [DOI: 10.1063/1.1423619] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
16
|
Wang X, Chatterjee AP. An integral equation study of percolation in systems of flexible and rigid macromolecules. J Chem Phys 2001. [DOI: 10.1063/1.1372761] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
|