Heat Generation and Diffusion in an Assembly of Magnetic Nanoparticles: Application to Magnetic Hyperthermia

Abstract

We investigate the thermal generation and transport properties of an assembly of magnetic nanoparticles embedded in a solid or fluid matrix, subjected to an AC magnetic field. For this purpose, we first build the heat equation for the assembly using the effective thermal transport coefficients obtained within the effective medium approach. In the present calculation, the SAR is obtained from the (linear) dynamic response of the assembly to the AC magnetic field. We numerically solve the extended heat equation and, as a preliminary study, we obtain the space-time profile of the temperature and total power absorbed by the system.

1. Introduction

Magnetic hyperthermia (MH) is a process by which magnetic nanoparticles (or nanomagnets, NMs), subjected to a time-dependent magnetic field (AFM), convert electromagnetic energy into heat. This process offers an advantage over the traditional methods of cancer treatment by irradiation [1,2,3,4,5,6,7,8], since it avoids the side effects of the latter by a directed and localized heating. Moreover, MH has been shown to kill cancer cells faster than the traditional methods and this reduces the therapy administration time [9]. In order to turn this technique into an efficient application, we still have to fully understand and control the mechanisms by which heat is generated and diffused in the assembly of NMs. The efficiency of MH is usually assessed through the specific absorption rate (SAR) [10,11,12,13,14,15,16,17,18]. Calorimetric methods provide another more precise means for characterizing MH and assessing its efficiency by studying the temperature rise after the start of heating, upon varying the AMF characteristics and the sample’s properties [19,20,21]. This technique is quite relevant in that it reduces the risk of heating the surrounding healthy cells. Today, a very lively debate is underway about the device-independence of these experiments, especially the measurement of the initial time slope of the temperature elevation [22].

In bulk homogeneous systems, heating mechanisms are well understood and the temperature rise is well described by the heat equation built on Fick’s and Fourier’s laws. However, at the nanoscale or in heterogeneous media, such as an assembly of NMs embedded in a solid matrix or immersed in a liquid, it is not obvious how the heat equation can be extended. Indeed, at reduced dimensions, even the definition of temperature itself becomes questionable. For heterogeneous media, several physical quantities, such as electrical and thermal conductivities and dielectric functions, have been calculated with the help of the effective medium approach [23,24,25,26,27,28,29,30,31,32,33,34]. This is a standard homogenization approach that leads to a macroscopic characterization of the medium with fewer parameters than those needed for a full description of the original system. For example, the whole micro-structure with the dielectric variation of a sample is reduced to an effective permittivity, which one then tries to relate to that of the inclusions and the matrix. The textbook [35] provides an extensive review of the topic and the various models and approaches that have been developed for tackling this issue.

In the present work, we follow this approach to provide expressions for the effective thermal conductivity and heat capacity of the assembly (or ferrofluid) of NMs. Then, replacing the latter by an effective medium with its effective transport coefficients, we write the corresponding extended heat equation with a source given by the heat generated by the NMs under the effect of the AMF. We then solve the extended heat equation using numerical tools and compute the space-time profile of the temperature elevation and total absorbed power for a typical assembly of magnetite nanoparticles.

We would like to emphasize that the aim of this work is not to present an extensive study of the results that can be inferred by varying all relevant physical parameters of the problem. It is not either an immediate application of MH in biological media and therapeutic treatment of cancer tumors. Such issues have been addressed by a few groups (see Refs. [36,37,38] and references therein) focusing on the bioheat equation and heat transfer in living tissues. Here, we propose a general formalism for studying heat generation and transport in a heterogeneous system and present a few results to illustrate the possible effects and observables that can be studied by such an approach. More precisely, we consider the system setup presented in Figure 1 below [39]. It is an assembly of nanomagnets (NMs), e.g., magnetite, embedded in a solid matrix (polymer, silica, etc) or floating in a fluid (ferrofluid), the base fluid (bf), which could be water. This system is in contact with its environment that may be the container of the sample (a tube, a box), air or a tissue. In the sequel, the subscript “ff” will stand for ferrofluid; “bf” for base fluid and “Fe” for iron (or magnetic substance of the inclusions). In the equations below, the quantities $\rho ,c_v,\kappa ,\mathcal{P}$ and L refer to the whole sample (NM+host) and should then carry the subscript “ff”. However, to keep the equations simpler, we omit the subscript. We will introduce the density of the magnetic substance within the particles $m_{\mathrm{Fe}}/V_{\mathrm{Fe}}={\rho }_{\mathrm{Fe}}$ and the volume fraction of the ferrofluid $\varphi \equiv V_{\mathrm{Fe}}/V_{\mathrm{ff}}$.

In the general scheme for the measurement of the specific absorption rate (SAR, denoted as $\mathfrak{S}$), the sample is initially set to the same temperature as its environment, $T_0$. Then, the AC magnetic field (AMF) is switched on at $t=0$. The latter excites the NMs which then release heat within the sample, i.e., they heat up the matrix/fluid (ferrofluid). The heat thus generated then diffuses across the sample and the heat flux reaches the sample’s limits, or its interface with the environment, to which heat is transferred by conduction, convection or radiation. If we assume that the NMs are homogeneously distributed through the sample, the heat power by unit volume is $\mathcal{P}/V_{\mathrm{ff}}$ (that is in $\mathrm{W}/{\mathrm{m}}^3$).

2. Heat Equation

In general, the heat equation is an example of what is called IVP-BVP problem, i.e., a differential equation for a time- and space-dependent function f with both initial and boundary conditions, namely (e.g., for a solid medium of length L)

$$\left\{\begin{array}{ccc}\frac{\partial f}{\partial t}\left(x,t\right)-\kappa {\partial }_x^{2}f\left(x,t\right)\hfill & =\hfill & 0,\hfill \\ f\left(x,0\right)=T_0,\hfill \\ f\left(0,t\right)=\alpha ,f\left(L,t\right)\hfill & =\hfill & \beta .\hfill \end{array}\right.$$

(1)

$\alpha ,\beta$ are given constants.

In our context, the space-time evolution of the temperature $T\left(\mathit{r},t\right)$ within a given medium, measured at position $\mathit{r}$ and time t, may be described by the “extended” heat equation with source, namely

$$\rho c_v\frac{\partial T}{\partial t}\left(\mathit{r},t\right)+\rho c_v\mathit{v}\left(\mathit{r},t\right)·\nabla T\left(\mathit{r},t\right)-\kappa {\nabla }^{2}T\left(\mathit{r},t\right)=\mathcal{P}\left[T\left(\mathit{r},t\right)\right]/V_{\mathrm{ff}},$$

(2)

where $\rho \left(\mathrm{kg}/{\mathrm{m}}^3\right)$, $c_v\left(\mathrm{J}/\mathrm{K}/\mathrm{kg}\right)$, and $\kappa \left(\mathrm{W}/\mathrm{K}/\mathrm{m}\right)$ are, respectively, the density, the specific heat and the thermal conductivity of the medium (ferrofluid); $\mathit{v}\left(\mathit{r},t\right)$ is the local flow velocity (in m/s); and $\mathcal{P}\left(\mathrm{W}\right)$ is the heat power. Note that the power $\mathcal{P}$ depends on the temperature $T\left(\mathit{r},t\right)$. Indeed, in our context, $\mathcal{P}$ is generated by the NMs subjected to an AMF which drives its magnetization into an oscillatory motion with a characteristic time that depends on the temperature in the immediate vicinity of the NMs, namely the temperature $T\left(\mathit{r},t\right)$. More precisely, T enters $\mathcal{P}$ through the AC susceptibility and the relaxation time of the NMs [see Section 2.3].

In Equation (2) one implicitly assumes that the sample is insulated at its interface with the outer environment (air in the room, other parts of a body, etc). In reality, this is not rigorously true and the sample does exchange heat with its environment at room temperature $T_0$. One of the mechanisms for such an exchange is given by Newton’s law of cooling which states that the rate of temperature change as a result of the sample-environment exchange is (for the whole sample of volume $V_s$)

$$C\frac{\partial T}{\partial t}\left(\mathit{r},t\right)=L\left[T\left(\mathit{r},t\right)-T_0\right],$$

(3)

where L is the so called Newton (positive) coefficient measured in W/K and $C\left(\mathrm{J}/\mathrm{K}\right)$ the heat capacity of the sample. See discussion in Ref. [39] and references therein.

2.1. Extended Heat Equation

As discussed in the introduction, an example of the systems targeted by the present study is sketched in Figure 1. In the effective media approach, this heterogeneous system, with parameters ${\rho }_{\mathrm{NP}},{\kappa }_{\mathrm{NP}}$ (for the nanoparticles or NMs) and ${\rho }_{\mathrm{m}},{\kappa }_{\mathrm{m}}$ (for the matrix), is replaced by a homogeneous effective system with the effective parameters ${\rho }_{\mathrm{eff}},{\kappa }_{\mathrm{eff}}$ (the subscript “eff” is dropped in the sequel), as sketched in Figure 2.

Equation (2) may be extended to include the contribution from the interface, leading to the following more general equation (here L is measured in W/K/m³)

$$\rho c_v\frac{\partial T}{\partial t}\left(\mathit{r},t\right)+\rho c_v\mathit{v}\left(\mathit{r},t\right)·\nabla T\left(\mathit{r},t\right)-\kappa {\nabla }^{2}T\left(\mathit{r},t\right)=\frac{\mathcal{P}}{V_{\mathrm{ff}}}-L\left[T\left(\mathit{r},t\right)-T_0\right].$$

(4)

Dividing Equation (4) by $T_0$, we rewrite it in terms of $\theta \left(\mathit{r},t\right)\equiv \left(T\left(\mathit{r},t\right)-T_0\right)/T_0$, the relative temperature elevation, as follows:

$$\rho c_v\frac{\partial \theta }{\partial t}\left(\mathit{r},t\right)+\rho c_v\mathit{v}\left(\mathit{r},t\right)·\nabla \theta \left(\mathit{r},t\right)-\kappa {\nabla }^2\theta \left(\mathit{r},t\right)=q\left[\theta \right],$$

(5)

where we have defined the source as a function of $\theta$:

$$q\left[\theta \right]\equiv \frac{1}{T_0}\frac{\mathcal{P}\left[\theta \right]}{V_{\mathrm{ff}}}-L\theta .$$

(6)

Next, dividing by L, we note that $\rho c_v/L$ has the dimensions of time and accordingly introduce the characteristic time

$$t_s\equiv \frac{\rho c_v}{L}.$$

(7)

Then, we rewrite Equation (5) as follows:

$$\frac{\partial \theta }{\partial \tau }\left(\mathit{r},\tau \right)+\tilde{\mathit{v}}\left(\mathit{r},\tau \right)·\nabla \theta \left(\mathit{r},\tau \right)-\tilde{\kappa }{\nabla }^2\theta \left(\mathit{r},\tau \right)=\tilde{q}\left(\theta \right).$$

(8)

Here, we also introduce the (dimensionless) time $\tau =t/t_s$ and the new quantities

$$\tilde{v}\equiv t_{s}v,\tilde{\kappa }\equiv \frac{\kappa }{L},\tilde{q}\left(\theta \right)\equiv \frac{q}{L}=-\theta +\frac{1}{L{T}_0}\frac{\mathcal{P}\left[\theta \right]}{V_{\mathrm{ff}}}.$$

(9)

We may rewrite $\tilde{q}$ using the SAR $\mathfrak{S}=\mathcal{P}/m_{\mathrm{Fe}}$ (in W/kg) as follows:

$$\begin{array}{cc}\hfill \tilde{q}\left(\theta \right)& =-\theta +\frac{\mathfrak{S}}{L{T}_0}{\rho }_{\mathrm{Fe}}.\hfill \end{array}$$

(10)

The quantity $\frac{\mathfrak{S}}{L{T}_0}{\rho }_{\mathrm{Fe}}\varphi$ is the dimensionless SAR $\Xi \left[\theta \right]$ introduced in Ref. [39].

Consequently, we have the final (dimensionless) heat equation

$$\frac{\partial \theta }{\partial \tau }\left(\mathit{r},\tau \right)+\tilde{\mathit{v}}\left(\mathit{r},\tau \right)·\nabla \theta \left(\mathit{r},\tau \right)-\tilde{\kappa }{\nabla }^2\theta \left(\mathit{r},\tau \right)=-\theta +\Xi \left[\theta \right].$$

(11)

2.2. Physical Parameters

As discussed earlier, we may consider a system composed of a fluid (here, water) and magnetic inclusions (maghemite or magnetite nanoparticles). An important relevant quantity is then the mass of the whole magnetic material introduced into the fluid, which we denote by $m_{\mathrm{Fe}}$. In practice, we need the SAR $\mathfrak{S}$ which is the power $\mathcal{P}$ divided by the mass of the magnetic substance $m_{\mathrm{Fe}}$. In addition, we note that since $c_v$$,\rho ,\kappa$ and L refer to the ferrofluid, they depend on the volume fraction $\varphi$ of the inclusions, namely the number of NMs introduced in the ferrofluid volume ($V_{\mathrm{ff}}$), in addition to the corresponding parameters of the nanofluid components.

• Specific heat: The sample (ferrofluid) specific heat is given by

  $$c_v\left(\varphi \right)=\frac{\varphi {\rho }_{\mathrm{Fe}}{c}_v^{\mathrm{Fe}}+\left(1-\varphi \right){\rho }_{\mathrm{bf}}{c}_v^{\mathrm{bf}}}{\varphi {\rho }_{\mathrm{Fe}}+\left(1-\varphi \right){\rho }_{\mathrm{bf}}},$$

  (12)

  with

  $$\rho \left(\varphi \right)=\varphi {\rho }_{\mathrm{Fe}}+\left(1-\varphi \right){\rho }_{\mathrm{bf}}.$$

  (13)
• Thermal conductivity: There are several models for $\kappa \left(\varphi \right)$ of a mixture based on the effective-medium approach from which we choose that of Yu and Choi [40]

  $$\kappa \left(\varphi \right)={\kappa }_m\frac{{\kappa }_{\mathrm{pm}}+2+2\varphi \left({\kappa }_{\mathrm{pm}}-1\right){\left(1+\beta \right)}^3}{{\kappa }_{\mathrm{pm}}+2-\varphi \left({\kappa }_{\mathrm{pm}}-1\right){\left(1+\beta \right)}^3},$$

  (14)

  with $\beta$ being the ratio of the thickness of the shell surrounding the nanoparticle to the radius of the latter and ${\kappa }_{\mathrm{pm}}\equiv {\kappa }_{\mathrm{Fe}}/{\kappa }_m$ the relative thermal conductivity.
• Power vs. SAR: In Ref. [39] we already saw that the power $\mathcal{P}$ divided by the mass of the magnetic substance ($m_{\mathrm{Fe}}$) yields the SAR $\mathfrak{S}$ in W/kg. So

  $$\frac{\mathcal{P}}{m_{\mathrm{Fe}}}=\mathfrak{S}.$$

  (15)
  See discussion before Equation (11).

2.3. Specific Absorption Rate

From previous works [16,39], using the theory of linear magnetic response, the SAR can be written as

$$\Xi ={\Xi }_0\left(h,\omega \right)\Phi \left(\theta \right)$$

(16)

where

$${\Xi }_0\left(h\right)\equiv \frac{{\left({\mu }_{0}h\right)}^2{m}^2}{6k_B{l}_{\mathrm{ff}}{T}_0^2}\omega ,\Phi \left(\theta \right)\equiv \frac{\eta \left(\theta \right)}{\left(1+\theta \right)\left[1+{\eta }^2\left(\theta \right)\right]}.$$

(17)

Here

$$\begin{array}{cc}\hfill \eta \left(\theta \right)& =\frac{\sqrt{\pi }}{2}{\varpi }_0\sqrt{1+\theta }\frac{e^{\frac{{\sigma }_0}{1+\theta }}}{\sqrt{{\sigma }_0}},\hfill \\ \hfill {\varpi }_0& \equiv \omega {\tau }_0,{\sigma }_0\equiv \frac{KV}{k_B{T}_0}.\hfill \end{array}$$

(18)

We also introduce the following quantity

$${\eta }_0\left(\omega \right)=\frac{\sqrt{\pi }}{2}\frac{{\varpi }_0{e}^{{\sigma }_0}}{\sqrt{{\sigma }_0}}=\eta \left(T_0,\omega \right).$$

(19)

To plot the SAR as a function of temperature, we use the experimental data from Ref. [21] for magnetite ferrofluid: nanoparticles of iron oxide with a concentration of $8.6$ mg/cm³, a diameter of 9 nm, anisotropy constant $K=3\times {10}^4\mathrm{J}/{\mathrm{m}}^3$, a saturation magnetization $M_s=480\times {10}^3\mathrm{A}/\mathrm{m}$, and a density $\rho =5.2\mathrm{g}/{\mathrm{cm}}^3$. $T_0=318\mathrm{K}$, frequency $f=\omega /2\pi =194\times {10}^3\mathrm{Hz},$ and magnetic field $h=38.2\mathrm{kA}/\mathrm{m}$. Finally, from Ref. [39] we have ${\tau }_0=9.5\times {10}^{-10}\mathrm{s}$ and $l_{\mathrm{ff}}=0.330\mathrm{W}/\mathrm{K}/\mathrm{g}$. The plot of the corresponding SAR is shown in Figure 3 for an assembly with lognormal size distribution with three values of the standard deviation and the same mean diameter.

As discussed in Ref. [39], $\theta$ is small for prototypical ferrofluids and, as such, we may expand $\Xi \left(\left[\theta \right],\varphi \right)$, appearing in Equation (11), in powers of $\theta$ and write to first order: $\Xi \simeq {\Xi }_0+{\Xi }_1\theta$. Then, the right-hand side of Equation (11) becomes ${\Xi }_0+\left({\Xi }_1-1\right)\theta$. Indeed, we expand the function $\Phi$ in powers of $\theta$:

$$\Phi \left(\theta \right)\simeq {\Phi }_0+{\Phi }_1\theta +\mathcal{O}\left({\theta }^3\right)$$

(20)

with [${\Phi }_n\left(\omega \right)={\Phi }_n\left(\omega ,T_0\right)$]

$$\begin{array}{cc}\hfill {\Phi }_0& =\frac{{\eta }_0}{1+{\eta }_0^2},\hfill \\ \hfill {\Phi }_1& =-\frac{{\eta }_0}{2{\left(1+{\eta }_0^2\right)}^2}\left[\left(1+2{\sigma }_0\right)+\left(3-2{\sigma }_0\right){\eta }_0^2\right].\hfill \end{array}$$

(21)

Therefore, the SAR expansion reads

$$\begin{array}{cc}\hfill \Xi & ={\Xi }_0\left(h,\omega \right)\frac{{\eta }_0}{1+{\eta }_0^2}-{\Xi }_0\left(h,\omega \right)\frac{{\eta }_0}{2{\left(1+{\eta }_0^2\right)}^2}\left[\left(1+2{\sigma }_0\right)+\left(3-2{\sigma }_0\right){\eta }_0^2\right]\times \theta \hfill \end{array}$$

(22)

and thereby the quantities ${\Xi }_0$ and ${\Xi }_1$ introduced above are explicitly given by

$$\left\{\begin{array}{ccc}{\Xi }_0\hfill & =\hfill & {\Xi }_0\left(h,\omega \right)\frac{{\eta }_0}{1+{\eta }_0^2}=\left(\frac{1}{l_{\mathrm{ff}}{T}_0}\right)\frac{{\left({\mu }_{0}h\right)}^2{m}^2}{6\left(k_B{T}_0\right)}\omega \frac{{\eta }_0}{1+{\eta }_0^2},\hfill \\ \\ {\Xi }_1\hfill & =\hfill & -{\Xi }_0\left(h,\omega ,T_0\right)\frac{\left(1+2{\sigma }_0\right)+\left(3-2{\sigma }_0\right){\eta }_0^2}{2\left(1+{\eta }_0^2\right)}.\hfill \end{array}\right.$$

(23)

Note that ${\Xi }_0$ is the (dimensionless) SAR evaluated at $T_0$, namely $\Xi \left(h,\omega ,T_0\right)$.

Therefore, we have the final heat equation

$${\partial }_{\tau }\theta \left(\mathit{r},t\right)+\tilde{\mathit{v}}\left(\varphi \right)·\nabla \theta \left(\mathit{r},t\right)-\tilde{\kappa }\left(\varphi \right){\nabla }^2\theta \left(\mathit{r},t\right)=\Xi \left(h,\omega ,T_0\right)+\left({\Xi }_1-1\right)\theta \left(\mathit{r},t\right)$$

(24)

with

$$\left\{\begin{array}{ccc}\tilde{v}\left(\varphi \right)\hfill & =\hfill & \frac{v}{L}\left[\varphi {\rho }_{\mathrm{Fe}}{c}_v^{\mathrm{Fe}}+\left(1-\varphi \right){\rho }_{\mathrm{bf}}{c}_v^{\mathrm{bf}}\right],\hfill \\ \\ \tilde{\kappa }\left(\varphi \right)\hfill & =\hfill & \frac{{\kappa }_m}{L}\frac{{\kappa }_{\mathrm{pm}}+2+2\varphi \left({\kappa }_{\mathrm{pm}}-1\right){\left(1+\beta \right)}^3}{{\kappa }_{\mathrm{pm}}+2-\varphi \left({\kappa }_{\mathrm{pm}}-1\right){\left(1+\beta \right)}^3},\hfill \\ \\ {\Xi }_0\left(h,\omega ,T_0\right)\hfill & =\hfill & \frac{{\left({\mu }_{0}h\right)}^2{m}^2}{6k_B{l}_{\mathrm{ff}}{T}_0^2}\omega \times \frac{{\eta }_0}{1+{\eta }_0^2},\hfill \\ \\ {\Xi }_1\hfill & =\hfill & -{\Xi }_0\left(h,\omega ,T_0\right)\frac{\left(1+2{\sigma }_0\right)+\left(3-2{\sigma }_0\right){\eta }_0^2}{2\left(1+{\eta }_0^2\right)}.\hfill \end{array}\right.$$

(25)

In the present study we ignore the shell, i.e., $\beta =0$. We also assume that the particles are fixed within the host and thereby ignore convection effects in this study.

3. Results and Discussion

We have used numerical tools to solve Equation (11) or (24) with the physical parameters of magnetite. For the dependent variable $\theta \left(\mathit{r},t\right)$, i.e., the relative temperature elevation, we use the initial/boundary conditions,

$$\theta \left(\mathit{r},0\right)=0,\theta \left(0,t\right)=\theta \left(L,t\right)=0.$$

(26)

A more realistic situation of three dimensions is supposed to mimic the fact that the ferrofluid is embedded in a medium that is supposed to be kept at a constant temperature $T_0$ (∼$42–45°\mathrm{C}$); recall that $\theta =\left(T-T_0\right)/T_0$. This is, for instance, the situation of a suspension of magnetic nanoparticles injected in a tumor, where the boundary conditions above prevent overheating of the surrounding area.

In this preliminary study, we only consider the case of a two-dimensional assembly of NMs, as shown in Figure 4, which may be either organized or randomly distributed in the matrix. In the present work, this does not matter since we do not include dipolar interactions between the NMs. The effect of the latter on the SAR was studied in Refs. [16,17,18] and references therein.

More precisely, we consider assemblies of magnetite particles as studied experimentally in Ref. [21] and, later, theoretically in Ref. [39]. In

Table 1. Physical parameters from Refs. [21,39].

Substance	${\mathit{c}}_{\mathit{v}}\left(\mathbf{J}/\mathbf{K}/\mathbf{kg}\right)$	$\mathit{\kappa }\left(\mathbf{W}/\mathbf{K}/\mathbf{m}\right)$	$\mathit{\rho }\left(\mathbf{kg}/{\mathbf{m}}^3\right)$	$\mathit{m}\left(\mathbf{kg}\right)$	$\mathit{V}\left({\mathbf{m}}^3\right)$	$\mathit{K}\left(\mathbf{J}/{\mathbf{m}}^3\right)$	${\mathit{M}}_{\mathit{s}}\left(\mathbf{A}/\mathbf{m}\right)$
Magnetite	619	5.1	5200	$1.032\times {10}^{-5}$	$1.985\times {10}^{-9}$	$3.0\times {10}^4$	$4.8\times {10}^5$
Water	4181	0.598	1000	$1.198\times {10}^{-3}$	$1.198\times {10}^{-6}$

, we collect all the relevant data used in the following figures.

The intensity of the AC magnetic field is $38.2\mathrm{kA}/\mathrm{m}$ and its frequency is $123\mathrm{kHz}$. Newton’s coefficient was obtained in Ref. [39] by fitting the experimental data from Ref. [21]: $l_{\mathrm{ff}}=330\left(\mathrm{W}/\mathrm{K}/\mathrm{kg}\right)$. The NMs are lognormal distributed in size with mean diameter $D_m=9\mathrm{nm}$ and standard deviation $\delta =0.25$. Shah et al. [21] used $1.2\mathrm{mL}$ of magnetite ferrofluid (magnetite in water) and a concentration ${\rho }_{\mathrm{Fe}}\simeq 8.6\mathrm{mg}/\mathrm{mL}\simeq 8.6{\mathrm{kg}/\mathrm{m}}^3$ of magnetite in the ferrofluid. The rest of the data related to the magnetization dynamics are given in Section 2.2. of Ref. [39]. In particular, the relaxation time related to the Brownian motion of NMs in the fluid is given and used to define the effective relaxation time.

As discussed earlier, heat Equation (11) can be solved with free boundary conditions or with boundaries at fixed temperatures. The first case would apply in a situation where, for instance, we were interested in the spatial extent of heating. The second case with fixed boundaries would apply to a situation where the assembly is confined in some space brought into contact with a surrounding that is kept at a fixed temperature.

Next, the heat equation can also be applied to an assembly of NMs organized or not on a solid matrix, as shown in Figure 4. In the following, all results will be shown for an assembly of NMs that are randomly distributed in space.

As already stressed in the Introduction, this work is not an exhaustive study of the effect of the numerous physical parameters of the problem at hand, namely those related with NM assembly (size, shape, distributions, and concentration) and external excitation (amplitude and frequency). The main objective instead is to propose a formalism and a toolbox for investigating the heat generation and transport in assemblies of NMs with the possibility to vary all the relevant parameters. In the present study, we only show a sample of results to illustrate how our formalism can be used and what physical observables can be obtained. By way of example, in the following, we discuss the calculation of the total heat injected into the assembly at time $\tau$, which may be defined as ($V_s$ is the volume of the sample):

$$\begin{array}{cc}\hfill \Delta Q\left(\tau \right)& \equiv Q\left(t\right)-Q\left(0\right)=k_{\mathrm{B}}{T}_0\times \frac{1}{V_s}\underset{V_s}{\int }d\mathit{r}\theta \left(\mathit{r},t\right).\hfill \end{array}$$

(27)

In Figure 5, we compare the two situations with fixed boundary conditions (BCs) and free BC. We show both the $2D$ (relative) temperature field and $\Delta Q$. The plot of the latter clearly shows that the maximum heat is reached much more quickly in an assembly with fixed temperature $T_0$ (i.e., $\theta =0$) at the boundaries. As is corroborated by the $2D$ temperature field, in the case with fixed BC, the heat cannot propagate freely but is confined in a smaller region. However, this depends on the heat capacity and thermal conductivity of the system, together with other physical parameters such as the concentration and spatial arrangements of the NMs. This is shown in the sequel.

Indeed, in Figure 6, we show the results of $\Delta Q\left(t\right)$ for a monodisperse assembly of NMs with a diameter D and two size-distributed assemblies with mean diameter smaller or larger than D. It is seen that globally the generated heat increases with the mean diameter, and this is due to the fact that the SAR (the heat source) also increases. The plots also show that the initial slope of $\Delta Q\left(t\right)$ depends on the size distribution of the assembly, and this is related to the fact that the diffusion of heat depends on the size and spatial distributions of the assembly.

In Figure 7, we compare assemblies with different values of the thermal conductivity of the host (${\kappa }_m$) (see Equations (14) and (25)). The first observation for such randomly distributed assemblies is that the result strongly depends on the time interval, and for long times we see that in a host with higher conductivity, the maximum heat is reached much faster.

4. Conclusions

In this work, we have presented a preliminary study of heat generation and transport in an assembly of magnetic nanoparticles subjected to a time-dependent magnetic field. For this purpose we have proposed an extended heat equation based on the effective medium approach which allows us to obtain the effective thermal transport coefficients for a heterogeneous medium such as an assembly of nanomagnets. Then, we numerically solved the extended heat equation and discussed the space-time profile of the temperature elevation for boundaries with fixed temperature or with free boundaries.

The present methodology may form the basis for studying the heat generation and transport of an assembly of nanomagnets. It makes it possible to compute, in a self-contained manner, the specific absorption rate and the space-time profile of the temperature elevation as functions of the various magnetic materials parameters, such as the concentration of the assembly, its spatial organization and size distribution, the convection coefficient, and the characteristics of the external excitation.

A comparison with experiments and a fit for the relevant physical parameters were performed in Ref. [39] for the time profile of the temperature elevation in magnetic ferrofluids. Such a comparison for the spatial profile requires measurements of the local temperature within the assembly [41,42,43].