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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-Verlet rule

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Abstract

Using polydispersity index as an additional order parameter we investigate freezing/melting transition of Lennard-Jones polydisperse systems (with Gaussian polydispersity in size), especially to gain insight into the origin of the terminal polydispersity. The average inherent structure (IS) energy and root mean square displacement (RMSD) of the solid before melting both exhibit quite similar polydispersity dependence including a discontinuity at solid-liquid transition point. Lindemann ratio, obtained from RMSD, is found to be dependent on temperature. At a given number density, there exists a value of polydispersity index (δ P) above which no crystalline solid is stable. This transition value of polydispersity(termed as transition polydispersity, δ P ) is found to depend strongly on temperature, a feature missed in hard sphere model systems. Additionally, for a particular temperature when number density is increased, δ P shifts to higher values. This temperature and number density dependent value of δ P saturates surprisingly to a value which is found to be nearly the same for all temperatures, known as terminal polydispersity (δ TP). This value (δ TP∼ 0.11) is in excellent agreement with the experimental value of 0.12, but differs from hard sphere transition where this limiting value is only 0.048. Terminal polydispersity (δ TP) thus has a quasiuniversal character. Interestingly, the bifurcation diagram obtained from non-linear integral equation theories of freezing seems to provide an explanation of the existence of unique terminal polydispersity in polydisperse systems. Global bond orientational order parameter is calculated to obtain further insights into mechanism for melting.

The Lindemann criterion for melting, inherent structure analysis and Hansen Verlet rule of freezing are shown to be consistent with each other in providing a measure for terminal polydispersity of Lennard-Jones system. A two order parameter scaled phase diagram showing limits of stability for liquid and solid is also in good agreement.

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References

  1. Sarkar S, Biswas R, Santra M and Bagchi B 2013 Phys. Rev. E 88 022104

    Article  Google Scholar 

  2. Abraham S E, Bhattacharrya S M and Bagchi B 2008 Phys. Rev. Lett. 100 167801

    Article  Google Scholar 

  3. Bagchi B, Cerjan C, Mohanty U and Rice S A 1984 Phys. Rev. B 29 2857

    Article  Google Scholar 

  4. Phan S -E, Russel W B, Zhu J and Chaikin P M 1998 J. Chem. Phys. 108 9789

    Article  CAS  Google Scholar 

  5. Lindemann F A 1910 Phys. Z. 11 609

    CAS  Google Scholar 

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

  7. Hansen J -P and Verlet L 1969 Phys. Rev. 184 151

    Article  CAS  Google Scholar 

  8. Kirkwood J G and Monroe E 1941 J. Chem. Phys. 9 514

    Article  CAS  Google Scholar 

  9. Kozak J J, Rice S A and Weeks J D 1971 Physica 54 573

    Article  Google Scholar 

  10. Lovett R 1977 J. Chem. Phys. 66 1225

    Article  CAS  Google Scholar 

  11. Munakata T 1977 J. Phys. Soc. Jpn. 43 1723

    Article  Google Scholar 

  12. Munakata T 1978 J. Phys. Soc. Jpn. 45 749

    Article  CAS  Google Scholar 

  13. Lacks D J and Wienhoff J R 1999 J. Chem. Phys. 111 398

    Article  CAS  Google Scholar 

  14. Kofke D A and Bolhuis P G 1999 Phys. Rev. E 59 618

    Article  CAS  Google Scholar 

  15. Williams S R, Snook I K and Megen van W 2001 Phys. Rev. E 64 021506

    Article  CAS  Google Scholar 

  16. Auer S and Frenkel D 2001 Nature 413 711

    Article  CAS  Google Scholar 

  17. Hansen J-P and McDonald I R 2006 In Theory of Simple Liquids 3rd Edition (San Diego: Academic Press)

    Google Scholar 

  18. Maeda K, Matsuoka W, Fuse T, Fukui K and Hirota S 2003 J. Mol. Liq. 102 1

    Article  CAS  Google Scholar 

  19. Bolhuis P G and Kofke D A 1996 Phys. Rev. E 54 634

    Article  CAS  Google Scholar 

  20. Bagchi B, Cerjan C and Rice S A 1983 J. Chem. Phys. 79 5595

    Article  CAS  Google Scholar 

  21. Bagchi B, Cerjan C and Rice S A 1983 J. Chem. Phys. 79 6222

    Article  CAS  Google Scholar 

  22. Bagchi B, Cerjan C and Rice S A 1983 Phys. Rev. B 28 6411

    Article  Google Scholar 

  23. Radloff P L, Bagchi B, Cerjan C and Rice S A 1984 J. Chem. Phys. 81 1406

    Article  CAS  Google Scholar 

  24. Lovett R and Buff F P 1980 J. Chem. Phys. 72 2425

    Article  CAS  Google Scholar 

  25. Sarkar S and Bagchi B 2011 Phys. Rev. E 83 031506

    Article  Google Scholar 

  26. Chakraborty S N and Chakravarty C 2007 Phys. Rev. E 76 011201

    Article  Google Scholar 

  27. Singh M, Agarwal M, Dhabal D and Chakravarty C 2012 J. Chem. Phys. 137 024508

    Article  Google Scholar 

  28. Ingebrigtsen T S, Schrøder T B and Dyre J C 2012 Phys. Rev. X 2 011011

    Google Scholar 

  29. Fantoni R, Gazzillo D, Giacometti A and Sollich P 2006 J. Chem. Phys. 125 164504

    Article  Google Scholar 

  30. Tokuyama M and Terada Y 2005 J. Chem. Phys. B 109 21357

    Article  CAS  Google Scholar 

  31. Bartlett P and Warren P B 1999 Phys. Rev. Lett. 82 1979

    Article  CAS  Google Scholar 

  32. Ramakrishnan T V and Yussouff M 1979 Phys. Rev. B 19 2775

    Article  CAS  Google Scholar 

  33. Ramakrishnan T V 1982 Phys. Rev. Lett. 48 541

    Article  CAS  Google Scholar 

  34. Chaudhuri P, Karmakar S, Dasgupta C, Krishnamurthy H R and Sood A K 2005 Phys. Rev. Lett. 95 248301

    Article  Google Scholar 

  35. Singh Y, Stoessel J P and Wolynes P G 1985 Phys. Rev. Lett. 54 1059

    Article  CAS  Google Scholar 

  36. Corti D S, Debenedetti P G, Sastry S and Stillinger F H 1997 Phys. Rev. E 55 5522

    Article  CAS  Google Scholar 

  37. Haymet A D J and Oxtoby D W 1981 J. Chem. Phys. 74 2559

    Article  Google Scholar 

  38. Haymet A D J and Oxtoby D W 1986 J. Chem. Phys. 84 1769

    Article  CAS  Google Scholar 

  39. Oxtoby D W and Haymet A D J 1982 J. Chem. Phys. 76 6262

    Article  CAS  Google Scholar 

  40. Ramakrishnan T V and Sengupta S 1993 Density wave theory of freezing and of interfaces. In: Indo-US workshop on Interfaces, 1993, pp. 33–49

  41. Underwood S M, Taylor J R and Megen van W 1994 Langmuir 10 3550

    Article  CAS  Google Scholar 

  42. Pusey P 1987 J. Phys. 48 709

    Article  CAS  Google Scholar 

  43. 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

  44. Chakravarty C, Debenedetti P G and Stillinger F H 2007 J. Chem. Phys. 126 204508

    Article  Google Scholar 

  45. Agrawal R and Kofke D A 1995 Mol. Phys. 85 43

    Article  CAS  Google Scholar 

  46. Schneider T, Brout R, Thomas H and Feder J 1970 Phys. Rev. Lett. 25 1423

    Article  CAS  Google Scholar 

  47. Schneider T 1971 Phys. Rev. A 3 2145

    Article  Google Scholar 

  48. Yang A J M, Fleming P D and Gibbs J H 1976 J. Chem. Phys. 64 3732

    Article  CAS  Google Scholar 

  49. Löwen H, Palberg T and Simon R 1993 Phys. Rev. Lett. 70 1557

    Article  Google Scholar 

  50. Löwen H and Hoffmann G P 1999 Phys. Rev. E 60 3009

    Article  Google Scholar 

  51. Stoessel J P and Wolynes P G 1984 J. Chem. Phys. 80 4502

    Article  CAS  Google Scholar 

  52. Langer J S 2013 Phys. Rev. E 88 012122

    Article  CAS  Google Scholar 

  53. Leocmach M, Russo J and Tanaka H 2013 J. Chem. Phys. 138 12A536

    Article  Google Scholar 

  54. Lovett R, Mou C Y and Buff F P 1976 J. Chem. Phys. 65 570

    Article  CAS  Google Scholar 

  55. 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)

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

  57. Landau L D and Lifshitz E M 1981 In Theory of Elasticity Second revised and enlarger edition (Oxford: Pergamon press)

    Google Scholar 

  58. Jarić M V and Mohanty U 1988 Phys. Rev. B 37 4441

    Article  Google Scholar 

  59. Lipkin M D, Rice S A and Mohanty U 1985 J. Chem. Phys. 82 472

    Article  CAS  Google Scholar 

  60. Steinhardt P J, Nelson D R and Ronchetti M 1983 Phys. Rev. B 28 784

    Article  CAS  Google Scholar 

  61. Xu X and Rice S A 2008 Phys. Rev. E 78 011602

    Article  Google Scholar 

  62. Xu X and Rice S A 2011 Phys. Rev. E 83 021120

    Article  Google Scholar 

Download references

Acknowledgements

We would like to thank Prof. Stuart A. Rice, Prof. Charusita Chakravarty, Prof. Govardhan Reddy for scientific discussions. This work was supported in parts by grants from DST and CSIR (India). B.B. thanks DST for support through J.C. Bose Fellowship.

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Authors and Affiliations

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

    SARMISTHA SARKAR, RAJIB BISWAS & BIMAN BAGCHI

  2. Department of Physics, Jadavpur University, Kolkata, 700 032, India

    SARMISTHA SARKAR & PARTHA PRATIM RAY

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  1. SARMISTHA SARKAR
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  2. RAJIB BISWAS
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SARKAR, S., BISWAS, R., RAY, P.P. et al. 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-Verlet rule. J Chem Sci 127, 1715–1728 (2015). https://doi.org/10.1007/s12039-015-0937-4

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  • DOI: https://doi.org/10.1007/s12039-015-0937-4

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