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Mathematical Models for Polymer Crystallization Processes

  • Vincenzo Capasso
  • Martin Burger
  • Alessandra Micheletti
  • Claudia Salani
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 2)

Abstract

Polymer industry raises a large amount of relevant mathematical problems with respect to the quality of manufactured polymer parts. These include in particular questions about the crystallization kinetics of the polymer melt, in presence of a tem perature field.

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References

  1. 1.
    G. Alfonso, Polimeri cristallini, in Fondamenti di Scienza dei Polimeri, (M. Guaita et al. Eds.), Pacini, Pisa, 1998.Google Scholar
  2. 2.
    L. Ambrosio, N. Fusco, D. Paliara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press Oxford, 2000.Google Scholar
  3. 3.
    M. Avrami, Kinetics of phase change. Part I, J. Chem. Phys., 7, 1103–112 (1939).CrossRefGoogle Scholar
  4. 4.
    P. Brémaud, Point Processesand Queues, Martingale Dynamics, Springer-Verlag, New York, 1981.CrossRefGoogle Scholar
  5. 5.
    M. Burger, Iterative regularization of an identification problem arising in polymer crystallization, SFB Report 99-21 (University of Linz, 1999), and SIAM J. of Numerical Analysis (submitted).Google Scholar
  6. 6.
    M. Burger, Direct and Inverse Problems in Polymer Crystallization Processes, PhD-Thesis, University of Linz, 2000.Google Scholar
  7. 7.
    M. Burger, V. Capasso, Mathematical modelling and simulation of nonisothermal crystallization of polymers. Math. Models and Methods in Appl. Sciences, 11 (2001), 1029–1053.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. Burger, V. Capasso, G. Eder, Modelling crystallization of polymers in temperature fields Z. Angew. Math. Mech. 82 (2002), 51–63.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. Burger, V. Capasso, H.W. Engl, Inverse problems related to crystallization of polymers, Inverse Problems 15 (1999), 155–173.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M. Burger, V. Capasso, A. Micheletti, Optimal control of polymer morphologies, Quaderno n. 11/2002, Dip. di Matematica, Universita di Milano, 2002.Google Scholar
  11. 11.
    M. Burger, V. Capasso, C. Salani, Modelling multidimensional crystallization of polymers in interaction with heat fransfer, Nonlinear Analysis: Real World Application, 3 (2002), 139–160.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    V. Capasso, A. Micheletti, Stochastic geometry of birth-and-growth processes, Quadernon. 20/1997, Dip. di Matematica, Università di Milano, 1997.Google Scholar
  13. 13.
    V. Capasso, A. Micheletti, Birth-and-Growth stochastic processes modelling polymer crystallization Technical Report 14, Industrial Mathematics Institute, J. Kepler Universität, Linz, 1997.Google Scholar
  14. 14.
    V. Capasso, A. Micheletti, Local spherical contact distribution function and local mean densities for inhomogeneous random sets, Stachastics and Stoch. Rep., 71, (2000), 51–67.MathSciNetzbMATHGoogle Scholar
  15. 15.
    V. Capasso, A. Micheletti, The hazard function of an inhomogeneous birthand-growth process, Quaderno n.39/2001, Dip. di Matematica, Universita’ di Milano.Google Scholar
  16. 16.
    V. Capasso, A. Micheletti, M. Burger, Densities of n-facets of incomplete Johnson-Mehl tessellations generated by inhomogeneous birth-and-growth processes. Quaderno n.38/2001, Dip. di Matematica, Università di Milano.Google Scholar
  17. 17.
    V. Capasso, A. Micheletti, M. De Giosa, R. Mininni, Stochastic modelling and statistics of polymer crystallization processes, Surv. Math. Ind., 6, (1996), 109–132.zbMATHGoogle Scholar
  18. 18.
    V. Capasso, C. Salani, Stochastic-birth-and-growth processes modelling crystallization of polymers with spatially heterogeneous parameters, Nonlinear Analysis: Real World Application, 1, (2000) pp. 485–498.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    N. A.C. Cressie, Statistics for Spatial Data, Wiley, New York, 1993.Google Scholar
  20. 20.
    G. Eder, Mathematical modelling of crystallization processes as occurring in polymer processing, Nonlinear Analysis 30 (1997), 3807–3815.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    G. Eder, H. Jameschitz-Krieql, Structure development during processing: crystallization, in Materials Science and Technology, Vol. 18 (H. Meijer, Ed.), Verlag Chemie, Weinheim, 1997.Google Scholar
  22. 22.
    H. W. Engl, O. Scherzer, Convergence rate results for iterative methods for solving nonlinear ill-posed problems, in: D. Colton, H.W. Engl, J. McLaughlin, A. Louis, W. Rundell, eds., Surveys on Solution Methods for Inverse Problems (Springer, Vienna, New York, 2000).Google Scholar
  23. 23.
    V.R. Evans, The laws of expanding circles and spheres in relation to the lateral growth rate of surface films and the grain-size of metals, Trans. Faraday Soc., 41 (1945), 365–374.CrossRefGoogle Scholar
  24. 24.
    L.H. Friedman, D.C. Chrzan, Scaling theory of the Hall-Petch relation for multilayers, Phys. Rev. Letters, 81 (1998), 2715–2719.CrossRefGoogle Scholar
  25. 25.
    A. Friedman, J.J.L. Velasquez, A free boundary problem associated with crystallization of polymers in a temperature field, Indiana Univ. Math. Journal, 50 (2001), 1609–1650.CrossRefzbMATHGoogle Scholar
  26. 26.
    R. V. Gamkrelidze, Principles of Optimal Control Theory Plenum Press, N. Y., London, 1976.Google Scholar
  27. 27.
    B.G. Ivanoff, E. Merzbach, A martingale characterization of the set-indexed Poisson Process, Stochastics and Stochastics Report 51 (1994), 69–82.MathSciNetzbMATHGoogle Scholar
  28. 28.
    H. Janeschitz-Kriegl, E. Ratajski, H. Wippel, The physics of athermal nuclei in polymer crystallization, Colloid & Polymer Science 277 (1999), 217–226.CrossRefGoogle Scholar
  29. 29.
    W.A. Johnson, R.F. Mehl, Reaction Kinetics in processes of nucleation and growth, Trans. A.I.M.M.E., 135, 416–458 (1939).Google Scholar
  30. 30.
    J.D. Kalbfleisch, R.L. Prentice, The Statistical Analysis of Failure Time Data, John Wiley and Sons, New York, 1980.zbMATHGoogle Scholar
  31. 31.
    A.N. Kolmogorov, On the statistical theory of the crystallization of metals, Bull. Acad.Sci USSR, Math. Ser. 1 (1937), 355–359.Google Scholar
  32. 32.
    D. Kröner, Numerical Schemes for Conservation Laws, Wiley & Teubner, Chichester, Stuttgart, 1997.zbMATHGoogle Scholar
  33. 33.
    G. Knowles, An Introduction to Applied Optimal Control, Academic Press, New York, 1981.zbMATHGoogle Scholar
  34. 34.
    G. Last, A. Brandt, Marked Point Processes on the Real Line. A Dynamic Approach, Springer, New York, 1995.Google Scholar
  35. 35.
    R. Le Veque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, Boston, Berlin, 1990.Google Scholar
  36. 36.
    G. Matheron, Random Sets and Integral Geometry, Wiley, New York, 1975.zbMATHGoogle Scholar
  37. 37.
    S. Mazzullo, M. Paolini, C. Verdi, Polymer crystallization and processing: free boundary problems and their numerical approximation, Math. Engineering in Industry 2 (1989), 219–232.zbMATHGoogle Scholar
  38. 38.
    J.L. Meijering, Interface area, edge length, and number of vertices in crystal aggregates with random nucleation, Philips Res. Rep., 8, 270–290 (1953).zbMATHGoogle Scholar
  39. 39.
    A. Micheletti, The surface density of a random Johnson-Mehl tessellation, Quaderno n. 17/2001, Dip. di Matematica, Università di Milano, 2001.Google Scholar
  40. 40.
    A. Micheletti, M. Burger, Stochastic and deterministic simulation of nisothermal crystallization of polymers, J. Math. Chem., 30 (2001), 169–193.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    A. Micheletti, V. Capasso, The stochastic geometry of polymer crystallization processes, Stoch. Anal. Appl. 15 (1997), 355–373.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    A. Micheletti, V. Capasso, G. Eder The density of the n-facets of an incomplete Johnson-Mehl tessellation. Preprint of the Institute for Industrial Mathematics, Johannes-Kepler University of Linz (Austria), 1997.Google Scholar
  43. 43.
    J. Moller, Random Johnson-Mehl tessellations, Adv. Appl. Prob., 24, 814–844 (1992).CrossRefGoogle Scholar
  44. 44.
    H. Niessner, Stability of Lax-Wendroff methods extended to deal with source terms, ZAMM 77 (1997), Suppl. 2, S637–S638.zbMATHGoogle Scholar
  45. 45.
    T. Ohta, Y. Enomoto, R. Kato, Domain growth with time dependent front velocity in one dimension, 1990, preprint.Google Scholar
  46. 46.
    S. Ohta, T. Ohta, K. Kawasaki, Domain growth in systems with multiple degenerate ground states, Physica, 140A, 478–505 (1987).MathSciNetGoogle Scholar
  47. 47.
    E. Ratajski, H. Janeschitz-Kriegl, How to determine high growth speeds in polymer cryst allization, Colloid Polym. Sci. 274 (1996), 938–951.CrossRefGoogle Scholar
  48. 48.
    M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations, Springer, New York, 1993.zbMATHGoogle Scholar
  49. 49.
    S.M. Ross, Simulation. 2nd ed. Academic Press, San Diego, 1997.zbMATHGoogle Scholar
  50. 50.
    C. Salani, Crystallization of polymers with thermal heterogeneities, ECMI Thesis, Linz (1997).Google Scholar
  51. 51.
    C. Salani, On the mathematics of polymer crystallization processes: stochastic and deterministic models., Ph.D. Thesis, University of Milano, Italy, 2000.Google Scholar
  52. 52.
    G.E.W. Schulze, T.R. Naujeck, A growing 2D spherulite and calculus of variations, Colloid & Polymer Science 269 (1991), 689–703.CrossRefGoogle Scholar
  53. 53.
    J.A. Sethian, Level Set Methods. Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge Univ. Press, Cambridge, 1996.zbMATHGoogle Scholar
  54. 54.
    J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, Berlin, Heidelberg, 1983.CrossRefzbMATHGoogle Scholar
  55. 55.
    D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Application, John Wiley & Sons, New York, 1995.Google Scholar
  56. 56.
    P. Supaphol, J.E. Spruiell, Thermal properties and isothermal crystallization of syndiodactic polypropylenes: differential scanning calorimetry and overall crystallization kinetics, Journ. Appl. Polym. Sci. 75 (2000), 44–59.CrossRefGoogle Scholar
  57. 57.
    J.E. Taylor, J.W. Cahn, C.A. Handwerker, Geometric models of crystal growth, Acta metall. mater. 40 (1992), 1443–1475.CrossRefGoogle Scholar
  58. 58.
    D.W. Van Krevelen, Properties of Polymers, 5th ed., Elsevier, Amsterdam, 1990.Google Scholar
  59. 59.
    Y. Zhang, B. Tobarrok, Modifications to the Lax-Wendroff scheme for hyperbolic systems with source terms, Int. J. Numer. Methods Eng. 44 (1999), 27–40.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Vincenzo Capasso
    • 1
  • Martin Burger
    • 2
  • Alessandra Micheletti
    • 1
  • Claudia Salani
    • 1
  1. 1.MIRIAM - Milan Research Centre for Industrial and Applied Mathematics and Department of MathematicsUniversity of MilanoMilanoItaly
  2. 2.Industrial Mathematics InstituteJohannes Kepler UniversitätLinzAustria

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