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Breakdown of universal Lindemann criterion in the melting of Lennard-Jones polydisperse solids

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Abstract

It is commonly believed that melting occurs when mean square displacement (MSD) of a particle of crystalline solid exceeds a threshold value. This is known as the Lindemann criterion, first introduced in the year of 1910 by Lindemann. However, Chakravarty et al., demonstrated that this common wisdom is inadequate because the MSD at melting can be temperature dependent when pressure is also allowed to vary along the coexistence line of the phase diagram [Chakravarty C, Debenedetti P G and Stillinger F H 2007 J. Chem. Phys. 126 204508]. We show here by extensive molecular dynamics simulation of both two and three dimensional polydisperse Lennard-Jones solids that particles on the small and large limits of size distribution exhibit substantially different Lindemann ratio at melting. Despite all the dispersion in MSD, melting is found to be first order in both the dimensions at 5–10% dispersity in size. Sharpness of the transition is incommensurate with the different rate of growth of MSD. The increased MSD values of smaller particles play a role in the segregation of them prior to melting.

Lindemann ratio (scaled root mean square displacement) is found to be strongly dependent on size of particles - smaller particles have higher values than bigger ones near melting transition of LJ polydisperse solid. Underlying cause of breakdown of universal Lindemann criterion is related to greater tendency of partial segregation of smaller particles prior to melting.

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References

  1. Chakravarty C, Debenedetti P G and Stillinger F H 2007 Lindemann measures for the solid-liquid phase transition J. Chem. Phys. 126 204508

    Article  Google Scholar 

  2. Bagchi B 2012 In Molecular relaxation in liquids (New York: Oxford University Press)

  3. (a) Lindemann F A 1910 The calculation of molecular vibration frequencies Phys. Z. 11 609; (b) Alba-Simionesco C, Coasne B, Dosseh G, Dudziak G, Gubbins K E, Radhakrishnan R and Sliwinska-Bartkowiak M 2006 Effects of confinement on freezing and melting J. Phys.: Condens. Matt. 18 R15

  4. Gilvarry J J 1956 The Lindemann and Grüneisen Laws Phys. Rev. 102 308

    Article  CAS  Google Scholar 

  5. Chakravarty C 2002 Path integral simulations of quantum Lennard-Jones solids J. Chem. Phys. 116 8938

    Article  CAS  Google Scholar 

  6. (a) Agrawal R and Kofke D A 1995 Thermodynamic and structural properties of model systems at solid-fluid coexistence. II. Melting and sublimation of the Lennard-Jones system Mol. Phys. 85 43; (b) Kofke D A and Bolhuis P G 1999 Freezing of polydisperse hard spheres Phys. Rev. E 59 618; (c) Bolhuis P G and Kofke D A 1996 Monte Carlo study of freezing of polydisperse hard spheres Phys. Rev. E 54 634

  7. Stillinger F H and Weber T A 1980 Lindemann melting criterion and the Gaussian core model Phys. Rev. B 22 3790

    Article  CAS  Google Scholar 

  8. Sarkar S, Biswas R, Ray P P and Bagchi B 2015 Use of polydispersity index as control parameter to study melting/freezing of Lennard-Jones system: Comparison among predictions of bifurcation theory with Lindemann criterion, inherent structure analysis and Hansen-Verletrule J. Chem. Sci. 127 1715

    Article  CAS  Google Scholar 

  9. Sarkar S, Biswas R, Santra M and Bagchi B 2013 Solid-liquid transition in polydisperse Lennard-Jones systems Phys. Rev. E 88 022104

    Article  Google Scholar 

  10. Sarkar S and Bagchi B 2011 Inherent structures of phase-separating binary mixtures: Nucleation, spinodal decomposition, and pattern formation Phys. Rev. E 83 031506

    Article  Google Scholar 

  11. Wales D J 2003 In Energy landscapes (UK: Cambridge University Press)

  12. Press W, Flannery B, Teukolsky S and Vetterling W 1992 In Numerical recipes in Fortran 77: The art of scientific computing (UK: Cambridge University Press)

  13. Hansen J-P and Verlet L 1969 Phase transitions of the Lennard-Jones System Phys. Rev. 184 151

    Article  CAS  Google Scholar 

  14. Hansen J-P and McDonald I R 2006 Theory of simple liquids 3rd ed. (New York: Academic Press)

  15. Bosse J and Thakur J S 1987 Delocalization of small particles in a glassy matrix Phys. Rev. Lett. 59 998

    Article  CAS  Google Scholar 

  16. Williams S R, Snook I K and van Megen W 2001 Molecular dynamics study of the stability of the hard sphere glass Phys. Rev. E 64 021506

  17. Maeda K, Matsuoka W, Fuse T, Fukui K and Hirota S 2003 Solid-liquid phase transition of binary Lennard-Jones mixtures on molecular dynamics simulations J. Mol. Liq 102 1

    Article  CAS  Google Scholar 

  18. Hopkins A J and Woodcock L V 1990 Effects of Polydispersity on Osmotic Properties of Colloidal Suspensions J. Chem. Soc., Faraday Trans. 86 3419

    Article  CAS  Google Scholar 

  19. Abraham S E and Bagchi B 2008 Suppression of the rate of growth of dynamic heterogeneities and its relation to the local structure in a supercooled polydisperse liquid Phys. Rev. E 78 051501

    Article  Google Scholar 

  20. Murarka R K and Bagchi B 2003 Diffusion and viscosity in a supercooled polydisperse system Phys. Rev. E 67 051504

    Article  Google Scholar 

  21. Ingebrigtsen T S and Tanaka H 2015 Effect of size polydispersity on the nature of Lennard-Jones liquids J. Phys. Chem. B 119 11052

    Article  CAS  Google Scholar 

  22. Ingebrigtsen T S and Tanaka H 2016 Effect of energy polydispersity on the nature of Lennard-Jones liquids J. Phys. Chem. B 120 7704

    Article  CAS  Google Scholar 

  23. Sollich P and Wilding N B 2010 Crystalline phases of polydisperse spheres Phys. Rev. Lett. 104 118302

    Article  Google Scholar 

  24. Wilding N B, Sollich P and Fasolo M 2005 Finite-size scaling and particle-size cutoff effects in phase-separating polydisperse fluid Phys. Rev. Lett. 95 155701

    Article  Google Scholar 

  25. Auer S and Frenkel D 2001 Suppression of crystal nucleation in polydisperse colloids due to increase of the surface free energy Nature (London) 413 711

    Article  CAS  Google Scholar 

  26. Kawasaki T, Araki T and Tanaka H 2007 Correlation between dynamic heterogeneity and medium-range order in two-dimensional glass-forming liquids Phys. Rev. Lett. 99 215701

    Article  Google Scholar 

  27. Bartlett P and Warren P B 1999 Reentrant melting in polydispersed hard spheres Phys. Rev. Lett. 82 1979

    Article  CAS  Google Scholar 

  28. Pusey P 1987 The effect of polydispersity on the crystallization of hard spherical colloids J. Phys. 48 709

    Article  CAS  Google Scholar 

  29. Pusey P 1991 In Les Houches, session LI, liquids, freezing and glass transitions, NATO Advanced Study Institute, series B: Physics J P Hansen, D Levesque and J Zinn-Justin (Eds.) (Amsterdam: Elsevier) Ch. 10

  30. Chaudhuri P, Karmakar S, Dasgupta C, Krishnamurthy H R and Sood A K 2005 Equilibrium glassy phase in a polydisperse hard-sphere system Phys. Rev. Lett. 95 248301

    Article  Google Scholar 

  31. Watanabe K and Tanaka H 2008 Direct Observation of medium-range crystalline order in granular liquids near the glass transition Phys. Rev. Lett. 100 158002

    Article  Google Scholar 

  32. Abraham S E, Bhattacharrya S M and Bagchi B 2008 Energy landscape, antiplasticization, and polydispersity induced crossover of heterogeneity in supercooled polydisperse liquids Phys. Rev. Lett. 100 167801

    Article  Google Scholar 

  33. Steinhardt P J, Nelson D R and Ronchetti M 1983 Bond-orientational order in liquids and glasses Phys. Rev. B 28 784

    Article  CAS  Google Scholar 

  34. Hoang V V 2011 Atomic mechanism of glass-to-liquid transition in simple monatomic glasses Philos. Mag. 91 3443

    Article  Google Scholar 

  35. Stillinger F H 1995 A topographic view of supercooled liquids and glass formation Science 267 1935

    Article  CAS  Google Scholar 

  36. Nelson D R 2002 In Defects and geometry in condensed matter physics (UK: Cambridge University Press)

  37. Jin Z H, Gumbsch P, Lu K and Ma E 2001 Melting mechanisms at the limit of superheating Phys. Rev. Lett. 87 055703

    Article  CAS  Google Scholar 

  38. Fixman M 1969 Highly anharmonic crystal J. Chem. Phys. 51 3270

    Article  CAS  Google Scholar 

  39. Stoessel J P and Wolynes P G 1984 Linear excitations and the stability of the hard sphere glass J. Chem. Phys. 80 4502

    Article  CAS  Google Scholar 

  40. Singh Y, Stoessel J P and Wolynes P G 1985 Hard-sphere glass and the density-functional theory of aperiodic crystals Phys. Rev. Lett. 54 1059

    Article  CAS  Google Scholar 

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Acknowledgements

This paper is dedicated to the memory of Prof. Charusita Chakravarty who was a respected colleague, valued friend and highly admired person. We and Indian science shall miss her dearly.

We would like to thank Prof. Stuart A. Rice from the University of Chicago for scientific discussions. This work was supported in parts by grants from DST, BRNS and CSIR (India). B.B. thanks DST for support through J. C. Bose Fellowship. S. S. and C. J. contributed equally.

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  1. Solid State and Structural chemistry Unit, Indian Institute of Science, Bangalore, 560 012, India

    SARMISTHA SARKAR, CHANDRAMOHAN JANA & BIMAN BAGCHI

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  1. SARMISTHA SARKAR
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  2. CHANDRAMOHAN JANA
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Correspondence to BIMAN BAGCHI.

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Supplementary Information (SI)

In order to ascertain that segregation happens just before melting, we have shown two snapshots from equilibrated samples (as shown in Figures ?? and ??) deep in the solid phase of three dimensional polydisperse system. It is interesting to note that no obvious segregation pattern is found.

Special Issue on THEORETICAL CHEMISTRY/CHEMICAL DYNAMICS

Dedicated to the memory of the late Professor Charusita Chakravarty

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SARKAR, S., JANA, C. & BAGCHI, B. Breakdown of universal Lindemann criterion in the melting of Lennard-Jones polydisperse solids. J Chem Sci 129, 833–840 (2017). https://doi.org/10.1007/s12039-017-1245-y

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