Skip to main content
SpringerLink
Log in
Book cover

Trends in PDE Constrained Optimization pp 471–486Cite as

Birkhäuser

Modeling, Analysis and Optimization of Particle Growth, Nucleation and Ripening by the Way of Nonlinear Hyperbolic Integro-Partial Differential Equations

  • Chapter
  • First Online:

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 165))

Abstract

We consider the processes of particle nucleation, growth, precipitation and ripening via modeling by nonlinear 1-D hyperbolic partial integro differential equations. The goal of this contribution is to provide a concise predictive forward modeling of the processes including appropriate goal functions and to establish a mathematical theory for the open-loop optimization in this context. Beyond deriving optimality conditions in the synthesis process, we present the application of a fully implicit method for Ostwald ripening of ZnO quantum dots which preserves its numeric stability even with respect to the inherent high sensitivities and wide disparity of scales. FIMOR represents an appropriate method that can be integrated to subordinate optimization studies which enables its future application in the context of continuous particle syntheses and microreaction technology (MRT).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. D. Armbruster, P. Degond, C. Ringhofer, A model for the dynamics of large queuing networks and supply chains. SIAM J. Appl. Math. 66, 896–920 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Armbruster, D.E. Marthaler, C.A. Ringhofer, K.G. Kempf, T.C. Jo, A continuum model for a re-entrant factory. Oper. Res. 54, 933–950 (2006)

    Article  MATH  Google Scholar 

  3. R.M. Colombo, M. Herty, M. Mercier, Control of the continuity equation with a non local flow. ESAIM Control Optim. Calc. Var. 17, 353–379 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. J.M. Coron, M. Kawski, Z. Wang, Analysis of a conservation law modeling a highly re-entrant manufacturing system. Discrete Contin. Dyn. Syst. Ser. B 14, 1337–1359 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Gradl, Experimentelle und theoretische Untersuchungen der Bildungskinetik diffusions- sowie reaktionslimitierter Systeme am Beispiel der Nanopartikelfällung von Bariumsulfat und Zinkoxid (Cuvillier Verlag, Göttingen, 2010)

    Google Scholar 

  6. M. Gröschel, Optimization of particel synthesis, Ph.D. thesis, FAU Department of mathematics, 2013

    Google Scholar 

  7. M. Gugat, M. Herty, A. Klar, G. Leugering, Optimal control for traffic flow networks. J. Optim. Theory Appl. 126, 589–616 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Herty, A. Klar, B. Piccoli, Existence of solutions for supply chain models based on partial differential equations. SIAM J. Math. Anal. 39, 160 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. C. Jagadish, S.J. Pearton (Eds.), Zinc Oxide Bulk, Thin Films and Nanostructures: Processing, Properties, and Applications, 1st edn. (Elsevier, Amsterdam/London, 2006)

    Google Scholar 

  10. I. Lifshitz, V. Slyozov, The kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Solids 19, 35–50 (1961)

    Article  Google Scholar 

  11. A. Mersmann, Calculation of interfacial tensions. J. Cryst. Growth 102, 841–847 (1990)

    Article  Google Scholar 

  12. A. Mersmann, General prediction of statistically mean growth rates of a crystal collective. J. Cryst. Growth 147, 181–193 (1995)

    Article  Google Scholar 

  13. E.L. Petersen, M.W. Crofton, Measurements of high-temperature silane pyrolysis using SiH4 IR emission and SiH2 laser absorption. J. Phys. Chem. A 107, 10988–10995 (2003)

    Article  Google Scholar 

  14. D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering (Elsevier, Burlington, 2000)

    Google Scholar 

  15. D. Segets, M.A.J. Hartig, J. Gradl, W. Peukert, A population balance model of quantum dot formation: oriented growth and ripening of ZnO. Chem. Eng. Sci. 70, 4–13 (2012)

    Article  Google Scholar 

  16. D.H. Segets, J. Gradl, R.K. Taylor, V. Vassilev, W. Peukert, N.S. Distribution, Analysis of optical absorbance spectra for the determination of ZnO nanoparticle size distribution, solubility, and surface energy. ACS nano 3, 1703–1710 (2009)

    Article  Google Scholar 

  17. D.H. Segets, L.M. Tomalino, J. Gradl, W. Peukert, Real-time monitoring of the nucleation and growth of ZnO nanoparticles using an optical hyper-rayleigh scattering method. J. Phys. Chem. C 113, 11995–12001 (2009)

    Article  Google Scholar 

  18. P. Shang, Z. Wang, Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system. J. Diff. Equ. 250, 949–982 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. D.V. Talapin, A.L. Rogach, M. Haase, H. Weller, Evolution of an ensemble of nanoparticles in a colloidal solution: theoretical study. J. Phys. Chem. B 105, 12278–12285 (2001)

    Article  Google Scholar 

  20. R. Viswanatha, S. Sapra, B. Satpati, P.V. Satyam, B.N. Dev, D.D. Sarma, Understanding the quantum size effects in ZnO nanocrystals. J. Mater. Chem. 14, 661 (2004)

    Article  Google Scholar 

  21. C. Wagner, Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald-Reifung). Zeitschrift für Elektrochemie Berichte der Bunsengesellschaft für physikalische Chemie 65, 581–591 (1961)

    Google Scholar 

  22. Z.L. Wang, Zinc oxide nanostructures: growth, properties and applications. J. Phys. Condens. Matter 16, 829–858 (2004)

    Article  Google Scholar 

  23. M. Wulkow, Modeling and simulation of crystallization processes using parsival. Chem. Eng. Sci. 56, 2575–2588 (2001)

    Article  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the work of the PhD-candidates A. Keimer, L. Pflug, J. Semmler and M. Walther, in particular, with respect to the numerical solutions and R. Wagner for the studies on Si formation (experiment and population balance modeling), Doris Segets and Martin Hartig for the studies with ZnO quantum dots. The authors would like to thank the German Research Council (DFG) for their financial support within the priority programs SPP 1679 (PE427/25) and SPP 1253 (LE595/23), for support within the DFG-Cluster of Excellence “Engineering of Advanced Materials” (www.eam.uni-erlangen.de) at the University of Erlangen-Nuremberg as well as for funding within the framework of the Elite Network of Bavaria: Identification, Optimization and Control with Applications in Modern Technologies.

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics 2, Friedrich-Alexander University Erlangen-Nürnberg, Cauerstrasse 11, 91058, Erlangen, Germany

    Michael Gröschel & Günter Leugering

  2. Institute of Particle Technology, Friedrich-Alexander University Erlangen-Nürnberg, Cauerstrasse 4, 91058, Erlangen, Germany

    Wolfgang Peukert

Authors
  1. Michael Gröschel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wolfgang Peukert
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Günter Leugering
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Günter Leugering .

Editor information

Editors and Affiliations

  1. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Günter Leugering

  2. Institut für Dynamik komplexer technisch, Max-Planck-Institut, Magdeburg, Germany

    Peter Benner

  3. Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Dortmund, Germany

    Sebastian Engell

  4. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Andreas Griewank

  5. Mathematisches Institut, Universität Basel, Basel, Switzerland

    Helmut Harbrecht

  6. Fachbereich Mathematik Optimierung und Approximation, Universität Hamburg, Hamburg, Germany

    Michael Hinze

  7. Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

    Rolf Rannacher

  8. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Stefan Ulbrich

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Gröschel, M., Peukert, W., Leugering, G. (2014). Modeling, Analysis and Optimization of Particle Growth, Nucleation and Ripening by the Way of Nonlinear Hyperbolic Integro-Partial Differential Equations. In: Leugering, G., et al. Trends in PDE Constrained Optimization. International Series of Numerical Mathematics, vol 165. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-05083-6_30

Download citation

Publish with us

Policies and ethics

Search

Navigation