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Nonlinear Nonequilibrium Simulations of Magnetic Nanoparticles

  • Daniel B. ReevesEmail author
Chapter

Abstract

Magnetic nanoparticles are found in computer memory and in futuristic biomedical applications. General models for the particle dynamics are essential to understanding and predicting dynamical behaviors in diverse conditions. Many approaches have been used to varying degrees of success. Here we present the most general methods for modeling: nonlinear, nonequilibrium models that typically require computational solving. We maintain rigor throughout so that the expressions arise from as close to first principles calculations as possible. We present the intuitively simpler, conceptually satisfying models too but are clear on their ranges of validity. We are also explicit in the computational implementation where necessary. At the end, we summarize the state of the art and the interesting problems that remain unsolved.

Keywords

Magnetic Nanoparticles Stochastic Differential Equation Magnetic Particle Wiener Process Langevin Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The author would like to thank Dr. John Weaver, Dr. Martin Wybourne, Dr. Miles Blencowe, and Dr. Michael Martens for excellent feedback and help with early drafts. Professor Weaver in particular has been a constant motivation and driving force behind my study of magnetic nanoparticles and their wondrous and potentially life-saving applications, and this work would not be possible without him.

References

  1. 1.
    Felderhof BU, Jones RB (2003) Mean field theory of the nonlinear response of an interacting dipolar system with rotational diffusion to an oscillating field. J Phys Condens Matter 15(23):4011CrossRefGoogle Scholar
  2. 2.
    Shliomis MI, Stepanov VI (1994) Theory of the dynamic susceptibility of magnetic fluids. Adv Chem Phys 87:1–30Google Scholar
  3. 3.
    Miguel MC, Rub JM (1995) Relaxation dynamics in suspensions of ferromagnetic particles. Phys Rev E 51(3):2190CrossRefGoogle Scholar
  4. 4.
    Elfimova EA, Ivanov AO, Camp PJ (2013) Thermodynamics of ferrofluids in applied magnetic fields. Phys Rev E 88(4):042310CrossRefGoogle Scholar
  5. 5.
    Koh I, Josephson L (2009) Magnetic nanoparticle sensors. Sensors 9(10):8130–8145CrossRefGoogle Scholar
  6. 6.
    Zhang X, Reeves DB, Perreard IM, Kett WC, Griswold KE, Gimi B, Weaver JB (2013) Molecular sensing with magnetic nanoparticles using magnetic spectroscopy of nanoparticle Brownian motion. Biosens Bioelectron 50:441–446CrossRefGoogle Scholar
  7. 7.
    Dieckhoff J, Lak A, Schilling M, Ludwig F (2014) Protein detection with magnetic nanoparticles in a rotating magnetic field. J Appl Phys 115(2):024701CrossRefGoogle Scholar
  8. 8.
    Chung SH, Hoffmann A, Bader SD, Liu C, Kay B, Makowski L, Chen L (2004) Biological sensors based on Brownian relaxation of magnetic nanoparticles. Appl Phys Lett 85(14):2971–2973CrossRefGoogle Scholar
  9. 9.
    Sun C, Lee JSH, Zhang M (2008) Magnetic nanoparticles in MR imaging and drug delivery. Adv Drug Deliv Rev 60(11):1252–1265CrossRefGoogle Scholar
  10. 10.
    Pankhurst QA, Thanh NTK, Jones SK, Dobson J (2009) Progress in applications of magnetic nanoparticles in biomedicine. J Phys D Appl Phys 42(22):224001CrossRefGoogle Scholar
  11. 11.
    Buzea C, Pacheco II, Robbie K (2007) Nanomaterials and nanoparticles: sources and toxicity. Biointerphases 2(4):MR17–MR71CrossRefGoogle Scholar
  12. 12.
    Chemla YR, Grossman HL, Poon Y, McDermott R, Stevens R, Alper MD, Clarke J (2000) Ultrasensitive magnetic biosensor for homogeneous immunoassay. Proc Natl Acad Sci 97(26):14268–14272CrossRefGoogle Scholar
  13. 13.
    Weaver JB, Rauwerdink AM, Hansen EW (2009) Magnetic nanoparticle temperature estimation. Med Phys 36:1822CrossRefGoogle Scholar
  14. 14.
    Rauwerdink AM, Weaver JB (2010) Viscous effects on nanoparticle magnetization harmonics. J Magn Magn Mater 322(6):609–613CrossRefGoogle Scholar
  15. 15.
    Weaver JB, Rauwerdink KM, Rauwerdink AM, Perreard IM (2013) Magnetic spectroscopy of nanoparticle Brownian motion measurement of microenvironment matrix rigidity. Biomed Eng 58(6):547–550CrossRefGoogle Scholar
  16. 16.
    Ginder JM, Davis LC, Elie LD (1996) Rheology of magnetorheological fluids: models and measurements. Int J Modern Phys B 10(23n24):3293–3303CrossRefGoogle Scholar
  17. 17.
    Steimel JP, Aragones JL, Alexander-Katz A (2014) Artificial tribotactic microscopic walkers: Walking based on friction gradients. Phys Rev Lett 113(17):178101CrossRefGoogle Scholar
  18. 18.
    Haun JB, Yoon TJ, Lee H, Weissleder R (2010) Magnetic nanoparticle biosensors. Wiley Interdiscip Rev Nanomed Nanobiotechnol 2(3):291–304CrossRefGoogle Scholar
  19. 19.
    Gleich B, Weizenecker J (2005) Tomographic imaging using the nonlinear response of magnetic particles. Nature 435(7046):1214–1217CrossRefGoogle Scholar
  20. 20.
    Pablico-Lansigan MH, Situ SF, Samia ACS (2013) Magnetic particle imaging: advancements and perspectives for real-time in vivo monitoring and image-guided therapy. Nanoscale 5.10:4040–4055Google Scholar
  21. 21.
    Ferguson R, Khandhar A, Saritas E, Croft L, Goodwill P, Halkola A, Borgert J, Rahmer J, Conolly S, Krishnan K (2015) Magnetic particle imaging with tailored iron oxide nanoparticle tracers. IEEE Trans Med Imaging 34.5:1077–1084Google Scholar
  22. 22.
    Weizenecker J, Gleich B, Rahmer J, Dahnke H, Borgert J (2009) Three-dimensional real-time in vivo magnetic particle imaging. Phys Med Biol 54(5):L1CrossRefGoogle Scholar
  23. 23.
    Konkle JJ, Goodwill PW, Carrasco-Zevallos OM, Conolly SM (2013) Projection reconstruction magnetic particle imaging. IEEE Trans Med Imaging 32(2):338–347CrossRefGoogle Scholar
  24. 24.
    Weizenecker J, Gleich B, Rahmer J, Borgert J (2012) Micro-magnetic simulation study on the magnetic particle imaging performance of anisotropic mono-domain particles. Phys Med Biol 57(22):7317CrossRefGoogle Scholar
  25. 25.
    Zanini L-F, Dempsey NM, Givord D, Reyne G, Dumas-Bouchiat F (2011) Autonomous micro-magnet based systems for highly efficient magnetic separation. Appl Phys Lett 99(23):232504CrossRefGoogle Scholar
  26. 26.
    Veiseh O, Gunn JW, Zhang M (2010) Design and fabrication of magnetic nanoparticles for targeted drug delivery and imaging. Adv Drug Deliv Rev 62(3):284–304CrossRefGoogle Scholar
  27. 27.
    Hergt R, Dutz S, Müller R, Zeisberger M (2006) Magnetic particle hyperthermia: nanoparticle magnetism and materials development for cancer therapy. J Phys Condens Matter 18(38):S2919CrossRefGoogle Scholar
  28. 28.
    Giustini AJ, Petryk AA, Cassim SM, Tate JA, Baker I, Hoopes PJ (2010) Magnetic nanoparticle hyperthermia in cancer treatment. Nano Life 1(01n02):17–32CrossRefGoogle Scholar
  29. 29.
    Weaver JB (2010) Hot nanoparticles light up cancer: targeting magnetic nanoparticles to tumours then applying an alternating magnetic field can improve the contrast for infrared thermal imaging. Nat Nanotechnol 5(9):630CrossRefGoogle Scholar
  30. 30.
    Hergt R, Dutz S, Zeisberger M (2010) Validity limits of the Néel relaxation model of magnetic nanoparticles for hyperthermia. Nanotechnology 21(1):015706CrossRefGoogle Scholar
  31. 31.
    Carrey J, Mehdaoui B, Respaud M (2011) Simple models for dynamic hysteresis loop calculations of magnetic single-domain nanoparticles: application to magnetic hyperthermia optimization. J Appl Phys 109(8):083921CrossRefGoogle Scholar
  32. 32.
    Lee J-H, Jang J-T, Choi J-S, Moon SH, Noh S-H, Kim J-W, Kim J-G, Kim I-S, Park KI, Cheon J (2011) Exchange-coupled magnetic nanoparticles for efficient heat induction. Nat Nanotechnol 6(7):418–422CrossRefGoogle Scholar
  33. 33.
    Branquinho LC, Carrião MS, Costa AS, Zufelato N, Sousa MH, Miotto R, Ivkov R, Bakuzis AF (2013) Effect of magnetic dipolar interactions on nanoparticle heating efficiency: implications for cancer hyperthermia. Sci Rep 3:2887Google Scholar
  34. 34.
    Mehdaoui B, Tan RP, Meffre A, Carrey J, Lachaize S, Chaudret B, Respaud M (2013) Increase of magnetic hyperthermia efficiency due to dipolar interactions in low-anisotropy magnetic nanoparticles: theoretical and experimental results. Phys Rev B 87(17):174419CrossRefGoogle Scholar
  35. 35.
    Dennis CL, Ivkov R (2013) Physics of heat generation using magnetic nanoparticles for hyperthermia. Int J Hyperthermia 29(8):715–729CrossRefGoogle Scholar
  36. 36.
    Denisov SI, Lyutyy TV, Pedchenko BO, Babych HV (2014) Eddy current effects in the magnetization dynamics of ferromagnetic metal nanoparticles. J Appl Phys 116(4):043911CrossRefGoogle Scholar
  37. 37.
    Nándori I, Rácz J (2012) Magnetic particle hyperthermia: power losses under circularly polarized field in anisotropic nanoparticles. Phys Rev E 86(6):061404CrossRefGoogle Scholar
  38. 38.
    Mehdaoui B, Meffre A, Carrey J, Lachaize S, Lacroix L-M, Gougeon M, Chaudret B, Respaud M (2011) Optimal size of nanoparticles for magnetic hyperthermia: a combined theoretical and experimental study. Adv Funct Mater 21(23):4573–4581CrossRefGoogle Scholar
  39. 39.
    Goya GF, Berquo TS, Fonseca FC, Morales MP (2003) Static and dynamic magnetic properties of spherical magnetite nanoparticles. J Appl Phys 94:3520CrossRefGoogle Scholar
  40. 40.
    Khandhar AP, Ferguson RM, Krishnan KM (2011) Monodispersed magnetite nanoparticles optimized for magnetic fluid hyperthermia: implications in biological systems. J Appl Phys 109:07B310CrossRefGoogle Scholar
  41. 41.
    Martinez-Boubeta C, Simeonidis K, Makridis A, Angelakeris M, Iglesias O, Guardia P, Cabot A, Yedra L, Estradé S, Peiró F (2013) Learning from nature to improve the heat generation of iron-oxide nanoparticles for magnetic hyperthermia applications. Sci Rep 3:1652Google Scholar
  42. 42.
    Reeves DB, Weaver JB (2014) Nonlinear simulations to optimize magnetic nanoparticle hyperthermia. Appl Phys Lett 104(10):102403CrossRefGoogle Scholar
  43. 43.
    Reeves DB, Weaver JB (2014) Approaches for modeling magnetic nanoparticle dynamics. Crit Rev Biomed Eng 42(1):85–93CrossRefGoogle Scholar
  44. 44.
    Kesserwan H, Manfredi G, Bigot J-Y, Hervieux P-A (2011) Magnetization reversal in isolated and interacting single-domain nanoparticles. Phys Rev B 84(17):172407CrossRefGoogle Scholar
  45. 45.
    Mayergoyz ID, Bertotti G, Serpico C (2009) Nonlinear magnetization dynamics in nanosystems. Elsevier, AmsterdamGoogle Scholar
  46. 46.
    d’Aquino M, Serpico C, Miano G, Forestiere C (2009) A novel formulation for the numerical computation of magnetization modes in complex micromagnetic systems. J Comput Phys 228(17):6130–6149CrossRefGoogle Scholar
  47. 47.
    Cullity BD, Graham CD (2011) Introduction to magnetic materials. Wiley, SomersetGoogle Scholar
  48. 48.
    Néel L (1949) Théorie du tranage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites. Ann Géophys 5(2):99–136Google Scholar
  49. 49.
    Reeves DB, Weaver JB (2015) Comparisons of characteristic timescales and approximate models for Brownian magnetic nanoparticle rotations. J Appl Phys 117.23:233905.CrossRefGoogle Scholar
  50. 50.
    Fannin PC, Charles SW (1994) On the calculation of the Néel relaxation time in uniaxial single-domain ferromagnetic particles. J Phys D Appl Phys 27(2):185CrossRefGoogle Scholar
  51. 51.
    Martsenyuk MA, Raikher YL, Shliomis MI (1974) Kinetics of magnetization of suspensions of ferromagnetic particles. Sov J Theor Phys 38(2):413–416Google Scholar
  52. 52.
    Deissler RJ, Wu Y, Martens MA (2013) Dependence of Brownian and Néel relaxation times on magnetic field strength. Med Phys 41(1):012301CrossRefGoogle Scholar
  53. 53.
    Reif F (2009) Fundamentals of statistical and thermal physics. Waveland Press, Long GroveGoogle Scholar
  54. 54.
    Abramowitz M, Stegun IA (1964) Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Number 55. Courier Corporation, New YorkGoogle Scholar
  55. 55.
    Stoner EC, Wohlfarth EP (1948) A mechanism of magnetic hysteresis in heterogeneous alloys. Philos Trans R Soc Lond A 240:599–642CrossRefGoogle Scholar
  56. 56.
    Poperechny IS, Raikher YL, Stepanov VI (2010) Dynamic magnetic hysteresis in single-domain particles with uniaxial anisotropy. Phys Rev B 82(17):174423CrossRefGoogle Scholar
  57. 57.
    Langevin P (1908) On the theory of Brownian motion. Correspondences of the Royal Academy of Science, ParisGoogle Scholar
  58. 58.
    Gardiner CW (1986) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer Ser Synergetics 13:149–168Google Scholar
  59. 59.
    Coffey W, Kalmykov YP, Waldron JT (2004) The Langevin equation: with applications to stochastic problems in physics, chemistry, and electrical engineering, vol 14. World Scientific, River EdgeGoogle Scholar
  60. 60.
    Shreve SE (2004) Stochastic calculus for finance II: continuous-time models, vol 11. Springer, New YorkGoogle Scholar
  61. 61.
    Øksendal B (1998) Stochastic differential equations: an introduction with applications. Springer, BerlinCrossRefGoogle Scholar
  62. 62.
    Chandrasekhar S (1943) Stochastic problems in physics and astronomy. Rev Mod Phys 15(1):1CrossRefGoogle Scholar
  63. 63.
    Lemons DS (2002) An introduction to stochastic processes in physics. JHU Press, BaltimoreGoogle Scholar
  64. 64.
    Van Kampen NG (1992) Stochastic processes in physics and chemistry, vol 1. Elsevier, AmsterdamGoogle Scholar
  65. 65.
    Wong E, Zakai M (1965) On the convergence of ordinary integrals to stochastic integrals. Ann Math Stat 36:1560–1564CrossRefGoogle Scholar
  66. 66.
    Scholz W, Schrefl T, Fidler J (2001) Micromagnetic simulation of thermally activated switching in fine particles. J Magn Magn Mater 233(3):296–304CrossRefGoogle Scholar
  67. 67.
    Reeves DB, Weaver JB (2012) Simulations of magnetic nanoparticle Brownian motion. J Appl Phys 112(12):124311CrossRefGoogle Scholar
  68. 68.
    Raible M, Engel A (2004) Langevin equation for the rotation of a magnetic particle. Appl Organ Chem 18(10):536–541CrossRefGoogle Scholar
  69. 69.
    Reynolds O (1883) An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Proc R Soc Lond 35(224–226):84–99CrossRefGoogle Scholar
  70. 70.
    Kubo R (1966) The fluctuation-dissipation theorem. Rep Prog Phys 29(1):255CrossRefGoogle Scholar
  71. 71.
    Einstein A (1956) Investigations on the theory of the Brownian movement. Dover, New YorkGoogle Scholar
  72. 72.
    Gilbert TL (2004) A phenomenological theory of damping in ferromagnetic materials. IEEE Trans Magn 40(6):3443–3449CrossRefGoogle Scholar
  73. 73.
    Landi GT (2014) Role of dipolar interaction in magnetic hyperthermia. Phys Rev B 89(1):014403CrossRefGoogle Scholar
  74. 74.
    Lakshmanan M (2011) The fascinating world of the Landau-Lifshitz-Gilbert equation: an overview. Philos Trans R Soc A 369(1939):1280–1300CrossRefGoogle Scholar
  75. 75.
    Kloeden PE, Platen E (1992) Numerical solution of stochastic differential equations. Springer, BerlinGoogle Scholar
  76. 76.
    Gard TC (1988) Introduction to stochastic differential equations. M. Dekker, New YorkGoogle Scholar
  77. 77.
    Rogge H, Erbe M, Buzug TM, Lüdtke-Buzug K (2013) Simulation of the magnetization dynamics of diluted ferrofluids in medical applications. Biomed Eng 58(6):601–609CrossRefGoogle Scholar
  78. 78.
    Reeves DB, Weaver JB (2015) Comparisons of characteristic timescales and approximate models for Brownian magnetic nanoparticle rotations. J Appl Phys 117(23):233905CrossRefGoogle Scholar
  79. 79.
    Serpico C, Bertotti G, d’Aquino M, Ragusa C, Ansalone P, Mayergoyz ID (2008) Path integral approach to stochastic magnetization dynamics in uniaxial ferromagnetic nanoparticles. IEEE Trans Magn 44(11):3157–3160CrossRefGoogle Scholar
  80. 80.
    Aron C, Barci DG, Cugliandolo LF, Arenas ZG, Lozano GS (2014) Magnetization dynamics: path-integral formalism for the stochastic Landau-Lifshitz-Gilbert equation. arXiv:1402.1200Google Scholar
  81. 81.
    Brown WF Jr (1963) Thermal fluctuations of a single-domain particle. J Appl Phys 34(4):1319–1320CrossRefGoogle Scholar
  82. 82.
    Debye P (1929) Polar molecules. Dover, New YorkGoogle Scholar
  83. 83.
    Raikher YL, Shliomis MI (1994) The effective field method in the orientational kinetics of magnetic fluids. Adv Chem Phys 87:595–751Google Scholar
  84. 84.
    Cohen A (1991) A padé approximant to the inverse Langevin function. Rheol Acta 30(3):270–273CrossRefGoogle Scholar
  85. 85.
    Martens MA, Deissler RJ, Wu Y, Bauer L, Yao Z, Brown R, Griswold M (2013) Modeling the Brownian relaxation of nanoparticle ferrofluids: comparison with experiment. Med Phys 40(2):2303CrossRefGoogle Scholar
  86. 86.
    Haase C, Nowak U (2012) Role of dipole-dipole interactions for hyperthermia heating of magnetic nanoparticle ensembles. Phys Rev B 85(4):045435CrossRefGoogle Scholar
  87. 87.
    Song N-N, Yang H-T, Liu H-L, Ren X, Ding H-F, Zhang X-Q, Cheng Z-H (2013) Exceeding natural resonance frequency limit of monodisperse Fe3o4 nanoparticles via superparamagnetic relaxation. Sci Rep 3:3161Google Scholar
  88. 88.
    Fannin PC, Marin CN (2006) Determination of the Landau-Lifshitz damping parameter by means of complex susceptibility measurements. J Magn Magn Mater 299(2):425–429CrossRefGoogle Scholar
  89. 89.
    Jiles DC, Atherton DL (1986) Theory of ferromagnetic hysteresis. J Magn Magn Mater 61(1):48–60CrossRefGoogle Scholar
  90. 90.
    Raghunathan A, Melikhov Y, Snyder JE, Jiles DC (2009) Generalized form of anhysteretic magnetization function for Jiles-Atherton theory of hysteresis. Appl Phys Lett 95(17):172510CrossRefGoogle Scholar
  91. 91.
    Mamiya H, Jeyadevan B (2011) Hyperthermic effects of dissipative structures of magnetic nanoparticles in large alternating magnetic fields. Sci Rep 1:157Google Scholar
  92. 92.
    Reeves DB, Weaver JB (2014) Magnetic nanoparticle sensing: decoupling the magnetization from the excitation field. J Phys D Appl Phys 47(4):045002CrossRefGoogle Scholar
  93. 93.
    Titov SV, Déjardin P-M, El Mrabti H, Kalmykov YP (2010) Nonlinear magnetization relaxation of superparamagnetic nanoparticles in superimposed AC and DC magnetic bias fields. Phys Rev B 82(10):100413CrossRefGoogle Scholar
  94. 94.
    Rosensweig RE (2002) Heating magnetic fluid with alternating magnetic field. J Magn Magn Mater 252:370–374CrossRefGoogle Scholar
  95. 95.
    Raikher YL, Stepanov VI (1995) Stochastic resonance and phase shifts in superparamagnetic particles. Phys Rev B 52(5):3493CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Physics and AstronomyDartmouth CollegeHanoverUSA
  2. 2.Vaccine and Infectious Disease DivisionFred Hutchinson Cancer Research CenterSeattleUSA

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