Topological melting in networks of granular materials. (English) Zbl 1417.82007
Summary: Granular materials represent a vast category of particle conglomerates with many areas of industrial applications. Here we represent these materials by graphs which capture their topological organization and ordering. Then, using the communicability function – a topological descriptor representing the thermal Green function of a network of harmonic oscillators – we prove the existence of a universal topological melting transition in these graphs. This transition resembles the melting process occurring in solids. We show here that crystalline-like granular materials melts at lower temperatures and display a sharper transition between solid to liquid phases than the random spatial graphs, which represent amorphous granular materials. In addition, we show the evolution mechanism of melting in these granular materials. In the particular case of crystalline materials the process starts by melting a central core of the crystal which then growth until the whole material is in the liquid phase. We provide experimental confirmation from published literature about this process.
MSC:
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
| 82D20 | Statistical mechanics of solids |
| 82D15 | Statistical mechanics of liquids |
| 05C80 | Random graphs (graph-theoretic aspects) |
Keywords:
granular materials; melting; phase transition; graph spectrum; communicability function; matrix functionsReferences:
| [1] | S. Alexander, Amorphous solids: their structure, lattice dynamics and elasticity. Phys. Rep. 296, 65-236 (1998) · doi:10.1016/S0370-1573(97)00069-0 |
| [2] | V. Alexiades, Mathematical Modeling of Melting and Freezing Processes (CRC Press, Boca Raton, 1992) |
| [3] | R. Bandyopadhyay, D. Liang, J.L. Harden, R.L. Leheny, Slow dynamics, aging, and glassy rheology in soft and living matter. Solid State Commun. 139, 589-598 (2006) · doi:10.1016/j.ssc.2006.06.023 |
| [4] | J. Borge-Holthoefer, Y. Moreno, A. Arenas, Modeling abnormal priming in Alzheimer’s patients with a free association network. PLoS ONE 6, e22651 (2011) · doi:10.1371/journal.pone.0022651 |
| [5] | R.W. Cahn, Melting and the surface. Nature 323, 668-669 (1986) · doi:10.1038/323668a0 |
| [6] | S.N. Dorogovtsev, A.V. Goltsev, J.F. Mendes, Critical phenomena in complex networks. Rev. Modern Phys. 80, 1275 (2008) · doi:10.1103/RevModPhys.80.1275 |
| [7] | E. Estrada, The Structure of Complex Networks: Theory and Applications (Oxford University Press, Oxford!, 2012) · Zbl 1267.05001 |
| [8] | E. Estrada, N. Hatano, Statistical-mechanical approach to subgraph centrality in complex networks. Chem. Phys. Lett. 439, 247-251 (2007) · doi:10.1016/j.cplett.2007.03.098 |
| [9] | E. Estrada, N. Hatano, Communicability in complex networks. Phys. Rev. E 77, 036111 (2008) · doi:10.1103/PhysRevE.77.036111 |
| [10] | E. Estrada, N. Hatano, M. Benzi, The physics of communicability in complex networks. Phys. Rep. 514, 89-119 (2012) · doi:10.1016/j.physrep.2012.01.006 |
| [11] | E. Estrada, D.J. Higham, Network properties revealed through matrix functions. SIAM Rev. 52, 696-714 (2010) · Zbl 1214.05077 · doi:10.1137/090761070 |
| [12] | K.R. Gabriel, R.R. Sokal, A new statistical approach to geographic variation analysis. Syst. Biol. 18, 259-278 (1969) |
| [13] | M. Hiraiwa, M.A. Ghanem, S.P. Wallen, A. Khanolkar, A.A. Maznev, N. Boechler, Complex contact-based dynamics of microsphere monolayers revealed by resonant attenuation of surface acoustic waves. Phys. Rev. Lett. 116, 198001 (2016) · doi:10.1103/PhysRevLett.116.198001 |
| [14] | Z. Jin, P. Gumbsch, K. Lu, E. Ma, Melting mechanisms at the limit of superheating. Phys. Rev. Lett. 87, 055703 (2001) · doi:10.1103/PhysRevLett.87.055703 |
| [15] | F.A. Lindemann, The calculation of molecular eigen-frequencies. Phys. Z. (West Germany) 11, 609-612 (1910) · JFM 41.0996.04 |
| [16] | Y.-Y. Liu, E. Csóka, H. Zhou, M. Pósfai, Core percolation on complex networks. Phys. Rev. Lett. 109, 205703 (2012) · doi:10.1103/PhysRevLett.109.205703 |
| [17] | M.Z. Miskin, H.M. Jaeger, Adapting granular materials through artificial evolution. Nat. Mater. 12, 326 (2013) · doi:10.1038/nmat3543 |
| [18] | J. Nagler, T. Tiessen, H.W. Gutch, Continuous percolation with discontinuities. Phys. Rev. X 2, 031009 (2012) |
| [19] | L. Papadopoulos, M.A. Porter, K.E. Daniels, D.S. Bassett, Network analysis of particles and grains. J. Complex Net. 6, 485-565 (2018) · doi:10.1093/comnet/cny005 |
| [20] | S.R. Phillpot, S. Yip, D. Wolf, How do crystals melt? Comput. Phys. 3, 20-31 (1989) · doi:10.1063/1.4822877 |
| [21] | M.A. Porter, P.G. Kevrekidis, C. Daraio, Granular crystals: nonlinear dynamics meets materials engineering. Phys. Today 68, 44-50 (2015) · doi:10.1063/PT.3.2981 |
| [22] | D.L. Powers, Graph partitioning by eigenvectors. Linear Algebra Its Appl. 101, 121-133 (1988) · Zbl 0666.05056 · doi:10.1016/0024-3795(88)90147-4 |
| [23] | B. Rudra, Y. Jiang, Y. Li, J. Shim, A class of diatomic 2-d soft granular crystals undergoing pattern transformations. Soft Matter 13, 5824-5831 (2017) · doi:10.1039/C7SM01430A |
| [24] | J.H. Smith, Some properties of the spectrum of a graph, in Combinatorial Structures and their Applications, eds. by R. Guy, H. Hanani, N. Sauer, J. Schonhcim (Gordon and Breach, New York, 1970), pp. 403-406 · Zbl 0249.05136 |
| [25] | J.C. Urschel, L.T. Zikatanov, Spectral bisection of graphs and connectedness. Linear Algebra Its Appl. 449, 1-16 (2014) · Zbl 1286.05101 · doi:10.1016/j.laa.2014.02.007 |
| [26] | S.R. Vippagunta, H.G. Brittain, D.J. Grant, Crystalline solids. Adv Drug Deliv. Rev. 48, 3-26 (2001) · doi:10.1016/S0169-409X(01)00097-7 |
| [27] | D.M. Walker, A. Tordesillas, Topological evolution in dense granular materials: a complex networks perspective. Int. J. Solids Struct. 47, 624-639 (2010) · Zbl 1183.74043 · doi:10.1016/j.ijsolstr.2009.10.025 |
| [28] | Z. Wang, F. Wang, Y. Peng, Y. Han, Direct observation of liquid nucleus growth in homogeneous melting of colloidal crystals. Nat. Commun. 6, 6942 (2015) · doi:10.1038/ncomms7942 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
