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Thermodynamics of Interacting Magnetic Nanoparticles This is a repository copy of Thermodynamics of interacting magnetic nanoparticles. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/168248/ Version: Accepted Version Article: Torche, P., Munoz-Menendez, C., Serantes, D. et al. (6 more authors) (2020) Thermodynamics of interacting magnetic nanoparticles. Physical Review B. 224429. ISSN 2469-9969 https://doi.org/10.1103/PhysRevB.101.224429 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Thermodynamics of interacting magnetic nanoparticles P. Torche1, C. Munoz-Menendez2, D. Serantes2, D. Baldomir2, K. L. Livesey3, O. Chubykalo-Fesenko4, S. Ruta5, R. Chantrell5, and O. Hovorka1∗ 1School of Engineering and Physical Sciences, University of Southampton, Southampton SO16 7QF, UK 2Instituto de Investigaci´ons Tecnol´oxicas and Departamento de F´ısica Aplicada, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain 3UCCS Biofrontiers Center and Department of Physics, University of Colorado at Colorado Springs, Colorado Springs, Colorado 80918, USA 4Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, ES-28049 Madrid, Spain and 5Department of Physics, University of York, York YO10 5DD, UK (Dated: June 5, 2020) We apply the concepts of stochastic thermodynamics combined with the transition state theory to develop a framework for evaluating local heat distributions across the assemblies of interacting mag- netic nanoparticles (MP) subject to time-varying external magnetic fields. We show that additivity of entropy production in the particle state-space allows separating the entropy contributions and evaluating the heat produced by the individual MPs despite interactions. Using MP chains as a model system for convenience, without losing generality, we show that the presence of dipolar in- teractions leads to significant heat distributions across the chains. Our study also suggests that the typically used hysteresis loops cannot be used as a measure of energy dissipation at the local particle level within MP clusters, aggregates or assemblies, and explicit evaluation of entropy production based on appropriate theory, such as developed here, becomes necessary. I. INTRODUCTION assumption of ‘local equilibrium’ becomes problematic [15]. Similarly, the frequently used linear response the- Magnetic nanoparticles (MP) are prototypical examples ory is limited only to dynamical processes close to equi- of out-of-equilibrium thermodynamic systems. Their librium [16, 17]. Thermodynamics of hysteresis based on behaviour is often manifested by memory effects and superimposed collections of bistable elements [18, 19], or hysteresis, which are exploited in a variety of applica- the more general stochastic approach to non-equilibrium tions ranging from information storage to environmen- thermodynamics [20–22] were also proposed, but so far tally friendly refrigeration and medicine [1–5]. A fully remain unused for systematic studies of systems of inter- consistent non-equilibrium thermodynamics of MPs that acting MPs. would allow advancing these applications still remains to In this work we fill the gap and apply the non- be developed, especially for systems with non-negligible equilibrium stochastic thermodynamics combined with inter-particle interactions. the standard transition state theory [21, 22] to systems For example, the complementary approach to cancer of interacting MPs. We show that it is possible to as- therapy based on MP hyperthermia, where MPs inter- sociate the expressions for entropy production with the nalised inside cancerous cells and subject to alternating individual MPs within an interacting assembly, which in magnetic fields heat the tumours to cytotoxic temper- turn allows computing the single particle heat distribu- atures, requires relating the heat production to physi- tion across the assembly. Although the formalism is gen- cal properties of MPs and time varying fields [6]. It eral, to simplify the computations we consider chains of also requires understanding the effects of inter-particle identical MPs with aligned uniaxial anisotropy and cou- interactions, which become significant when MPs aggre- pled by dipolar interactions. This configuration allows gate during the internalisation inside cells and most fre- obtaining analytical expressions for the energy barriers quently lead to the reduction of the heating efficiency and thermal transitions rates, and applying the Marko- [7–9]. Moreover, it is currently unclear if it is the net vian probabilistic master-equation formalism frequently heating effect of the entire MP aggregate or the local employed for modelling the long-timescale thermal fluc- heat distribution across it that leads to tumour death tuations in MP systems. [10–14]. Answering these questions, and analogous ques- tions pertaining to other applications, requires develop- Solving numerically the full master-equation we quan- ing a consistent non-equilibrium thermodynamic descrip- tify the temporal evolution of the particle states, and tion of systems of interacting MPs. through the developed thermodynamics formalism eval- The challenge is that due to the small size of MPs, ther- uate the individual particle heat inside the chains. Our mal fluctuations become dominant and applying the stan- calculations suggests that dipolar interactions lead to sig- dard non-equilibrium thermodynamics founded on the nificant heat distribution across chains, often varying by as much as 50% - 100% of the mean heat practically mea- surable from the hysteresis loop area of an entire MP sys- tem, which is a significant finding. We also show that due ∗ [email protected] to the presence of interactions, the area of the hysteresis 2 Z-chains pression: 0 1 7 M V 1 r ~ d s 1 Hi = − 3 3 (−sˆj + 3(ˆsj · rˆij)ˆrij) (2) 4πa rij 2 j<i r 6 X where rij = Rij/a corresponds to particle-to-particle dis- 3 2 tances normalised by the particle diameter a. The dis- 5 tance between the neighbouring particles will be denoted 3 1 2 3 4 by the symbol r. The dimensionless particle moment vec- (a)X-chains (b) torss ˆi follow from the definition of an intrinsic magnetic moment of a MP: FIG. 1. (a) Illustration of chains of N = 3 particles in ‘Z’ ~mi = VMssˆi (3) configuration such thatr ˆij k kˆ k H~ , and in ‘X’ configuration ˆ ~ where Ms is the saturation magnetisation. withr ˆij ⊥ k k H. Inter-particle distance is denoted by r, and The minima of energy Eq. (1) correspond to stable ~ H is the applied field vector. (b) A sketch of a network of spin configurations defining the states α of the system. possible single-particle transitions between the 23 states α of Thermally activated dynamics is described by the transi- a 3-particle chain. Dashed, dash-dotted, and dotted lines cor- respond to transitions of the right, middle, and left particle. tion state theory as a hopping process between the states, with transition rates from any state β to α determined by the standard Arrhenius form [24]: sub-loops corresponding to the individual MPs does not ∆Eαβ represent the particle heat, and performing explicit cal- wαβ = f0 exp − (4) k T culations based on evaluating the entropy production is B necessary. where ∆Eαβ are the energy barriers separating the states The article is organised as follows. Section II intro- α and β, the constant f0 is the attempt frequency, kB is duces the model of interacting MPs including the ther- the Boltzmann constant, and T is temperature. Find- mally activated dynamics described by the standard tran- ing the energy barriers in many-particle systems is gen- sition state theory. Thermodynamics of particle chains erally a difficult optimisation problem. Here we restrict is discussed in Section III, based on the definitions of our considerations to co-linear particle chains and assume the first law of thermodynamics and entropy balance. single-particle transitions, as illustrated in Fig. 1, which Section IV offers similar discussion related to thermody- allows expressing the energy barriers analytically as [25]: namics of individual particles within a chain. Thermody- 2 ~ ~ d namic implications of the practically relevant cyclic pro- |H + Hi | cesses produced by alternating external magnetic fields ∆Eαβ = KV 1 ± (5) HK ! are discussed in Section V. The discussion of the dipolar effects and conclusions are given in Sections VI and VII. where the dipolar field index i implies that the states α and β are related by the switched particle i (Fig. 1(b)), and HK = 2K/µ0Ms. The ‘±’ sign in Eq. (5) II. MODEL OF THERMALLY ACTIVATED refers to energy barriers for particle i switching down PARTICLE CHAINS (+) or up (−), respectively. Detailed discussion of the algorithms used to track down the network of single To use a specific well established model of MPs, we con- particle transitions and computing the energy barriers sider chains of N interacting identical particles, each in more general cases
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