Skip to main content
SpringerLink
Log in

Nonlinear normal and anomalous response of non-interacting electric and magnetic dipoles subjected to strong AC and DC bias fields

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The perturbation theory approach via the Smoluchowski equation to the nonlinear dielectric relaxation of noninteracting permanent electric dipoles (Coffey and Paranjape, Proc R Ir Acad A 78:17, 1978) and the analogous Brownian magnetic relaxation of ferrofluids where Néel relaxation is ignored is revisited for the particular case of a strong dc bias field superimposed on a strong ac field. Unlike weak ac and strong bias dc fields, a frequency-dependent dc term now appears in the response as well as additional nonlinear terms at the fundamental and second harmonic frequencies. These may be experimentally observable particularly in the ferrofluid application. The corresponding results for the dc term for anomalous relaxation based on the fractional Smoluchowski equation are also given.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Debye, P.: Polar Molecules. Chemical Catalog Co., New York (1929). Reprinted Dover, New York (1954)

    MATH  Google Scholar 

  2. Coffey, W.T., Paranjape, B.V.: Dielectric and Kerr effect relaxation in alternating electric fields. Proc. R. Ir. Acad. Sect. A 78, 17 (1978)

    MathSciNet  Google Scholar 

  3. Déjardin, J.L., Kalmykov, Y.P.: Nonlinear dielectric relaxation of polar molecules in a strong ac electric field: steady state response. Phys. Rev. E 61, 1211 (2000)

    Article  Google Scholar 

  4. Déjardin, J.L., Kalmykov, Y.P.: Steady state response of the nonlinear dielectric relaxation and birefringence in strong superimposed ac and dc bias electric fields: polar and polarizable molecules. J. Chem. Phys. 112, 2916 (2000)

    Article  Google Scholar 

  5. Coffey, W.T., Kalmykov, Y.P.: The Langevin Equation, 3rd edn. World Scientific, Singapore (2012)

    Book  MATH  Google Scholar 

  6. Déjardin, J.L., Kalmykov, Y.P., Déjardin, P.M.: Birefringence and dielectric relaxation in strong electric fields. Adv. Chem. Phys. 117, 271 (2001)

    Google Scholar 

  7. Déjardin, J.L., Debiais, G., Ouadiou, A.: On the nonlinear behavior of dielectric relaxation in alternating fields. II. Analytic expressions of the nonlinear susceptibilities. J. Chem. Phys. 98, 8149 (1993)

    Article  Google Scholar 

  8. De Smet, K., Hellemans, L., Rouleau, J.F., et al.: Rotational relaxation of rigid dipolar molecules in nonlinear dielectric spectra. Phys. Rev. E 57, 1384 (1998)

    Article  Google Scholar 

  9. Jadżyn, J., Kędziora, P., Hellemans, L.: Frequency dependence of the nonlinear dielectric effect in diluted dipolar solutions. Phys. Lett. A 251, 49 (1999)

    Article  Google Scholar 

  10. Jadżyn, J., Kędziora, P., Hellemans, L., et al.: Nonlinear dielectric relaxation in non-interacting dipolar systems. Chem. Phys. Lett. 289, 541 (1999)

    Google Scholar 

  11. Fannin, P.C., Scaife, B.K.P., Charles, S.W.: A study of the complex ac susceptibility of magnetic fluids subjected to a constant polarizing magnetic field. J. Magn. Magn. Mater. 85, 54 (1990)

    Article  Google Scholar 

  12. Fannin, P.C., Giannitsis, A.T.: Investigation of the field dependence of magnetic fluids exhibiting aggregation. J. Mol. Liq. 114, 89 (2004)

    Article  Google Scholar 

  13. Coffey, W.T., Kalmykov, Y.P.: Thermal fluctuations of magnetic nanoparticles: fifty years after brown. J. Appl. Phys. 112, 121301 (2012)

    Article  Google Scholar 

  14. Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284 (1940)

    Article  MATH  MathSciNet  Google Scholar 

  15. Brown, W.F.: Thermal fluctuations of a single-domain particle. Phys. Rev. 130, 1677 (1963)

  16. Fannin, P.C., Charles, S.W., Mac Oireachtaigh, C., et al.: Investigation of possible hysteresis effects arising from frequency- and field-dependent complex susceptibility measurements of magnetic fluids. J. Magn. Magn. Mater. 302, 1 (2006)

    Article  Google Scholar 

  17. Coffey, W.T., Crothers, D.S.F., Kalmykov, Y.P., Déjardin, P.M.: Nonlinear response of permanent dipoles in a uniaxial potential to alternating fields. Phys. Rev. E 71, 062102 (2005)

    Article  Google Scholar 

  18. Coffey, W.T., Crothers, D.S.F., Kalmykov, Y.P.: Nonlinear response of permanent dipoles in a mean-field potential to alternating fields. J. Non-Cryst. Solids 352, 4710 (2006)

    Article  Google Scholar 

  19. Titov, S.V., El Mrabti, H., Déjardin, P.M., Kalmykov, YuP: Nonlinear magnetization relaxation of superparamagnetic nanoparticles in superimposed ac and dc magnetic bias fields. Phys. Rev. B 82, 100413 (2010)

    Article  Google Scholar 

  20. Cole, K.S., Cole, R.H.: Dispersion and absorption in dielectrics. I. Alternating current characteristics. J. Chem. Phys. 9, 341 (1941)

  21. Davidson, D.W., Cole, R.H.: Dielectric relaxation in glycerol, propylene glycol, and n-propanol. J. Chem. Phys. 19, 1484 (1951)

    Article  Google Scholar 

  22. Havriliak, S., Negami, S.: A complex plane representation of dielectric and mechanical relaxation processes in some polymers. J. Polym. Sci. Part A-1 14, 99 (1966); Polymer 8, 161 (1967).

  23. Nigmatullin, R.R., Ryabov, Ya. A.: Cole-Davidson dielectric relaxation as a self-similar relaxation process. Fiz. Tverd. Tela (St. Petersburg) 39, 101 (1997). [Phys. Solid State 39, 87 (1997)].

  24. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  25. Novikov, V.V., Privalko, V.P.: Temporal fractal model for the anomalous dielectric relaxation of inhomogeneous media with chaotic structure. Phys. Rev. E 64, 031504 (2001)

    Article  Google Scholar 

  26. Hilfer, R.: H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems. Phys. Rev. E 65, 061510 (2002)

    Article  Google Scholar 

  27. Coffey, W.T., Kalmykov, Y.P., Titov, S.V.: Anomalous dielectric relaxation in the context of the Debye model of noninertial rotational diffusion. J. Chem. Phys. 116, 6422 (2002)

    Article  Google Scholar 

  28. Gudowska-Nowak, E., Bochenek, K., Jurlewicz, A., Weron, K.: Hopping models of charge transfer in a complex environment: coupled memory continuous-time random walk approach. Phys. Rev. E 72, 061101 (2005)

    Article  Google Scholar 

  29. Coffey, W.T., Kalmykov, Y.P., Titov, S.V.: Fractional rotational Brownian motion and anomalous dielectric relaxation in dipole systems. Adv. Chem. Phys. 133B, 285 (2006)

    Google Scholar 

  30. Goychuk, I.: Anomalous relaxation and dielectric response. Phys. Rev. E 76, 040102 (2007)

    Article  Google Scholar 

  31. Coffey, W.T., Kalmykov, Y.P., Titov, S.V.: Anomalous nonlinear dielectric and Kerr effect relaxation steady state responses in superimposed ac and dc electric fields. J. Chem. Phys. 126, 084502 (2007)

    Article  Google Scholar 

  32. Uchaikin, V.V., Sibatov, R.T.: Fractional Kinetics in Solids: Anomalous Charge Transport in Semiconductors, Dielectrics and Nanosystems. World Scientific Publishing Company, Singapore (2012)

    Google Scholar 

  33. Khamzin, A.A., Nigmatullin, R.R., Popov, I.I.: Log-periodic corrections to the Cole-Cole expression in dielectric relaxation. Theor. Math. Phys. 173, 1604 (2012); Physica A, 392, 136 (2013).

  34. Fröhlich, H.: Theory of Dielectrics, 2nd edn. Oxford University Press, Oxford (1958)

    MATH  Google Scholar 

  35. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  36. West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, New York (2003)

    Book  Google Scholar 

  37. Uchaikin, V.V.: Fractional Derivatives for Physicists and Engineers, Vol. 1, Background and Theory, Vol. 2, Application. Springer, Berlin (2013)

    Book  Google Scholar 

Download references

Acknowledgments

W. T. Coffey thanks Ambassade de France en Irlande for a research visit to Perpignan. N. Wei acknowledges the Government of Ireland Scholarship Award. We thank Dr. S. V. Titov for helpful conversations.

Author information

Authors and Affiliations

  1. Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland

    W. T. Coffey & N. Wei

  2. Laboratoire de Mathématiques et de Physique (EA 4217), Université de Perpignan Via Domitia, 66860 , Perpignan, France

    Y. P. Kalmykov

Authors
  1. W. T. Coffey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. P. Kalmykov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Wei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Coffey, W.T., Kalmykov, Y.P. & Wei, N. Nonlinear normal and anomalous response of non-interacting electric and magnetic dipoles subjected to strong AC and DC bias fields. Nonlinear Dyn 80, 1861–1867 (2015). https://doi.org/10.1007/s11071-014-1488-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1488-9

Keywords

Search

Navigation