Constitutive relationship and governing physical properties for magnetophoresis

Significance

The motion of magnetic particles under magnetic field, referred to as magnetophoresis, is fundamental to nature. While several empirical observations have been gathered, the multitude and complexity of magnetic-field-induced interactions on magnetic particles have prevented the full understanding of this phenomenon. The magnetophoresis is central to several advanced technologies such as magnetic drug targeting, bioseparation, and imaging, yet the efficient targeting of magnetic particles is hampered for the lack of a theoretical framework. We present a fundamental approach utilizing a detailed force balance on magnetic particles to develop a constitutive relationship for magnetophoresis that includes contributions from magnetic diffusion, magnetic convection, and residual magnetization. The application of this theory will enable the predictive design of magnetic targeting applications.

Abstract

Magnetophoresis is an important physical process with application to drug delivery, biomedical imaging, separation, and mixing. Other than empirically, little is known about how the magnetic field and magnetic properties of a solution affect the flux of magnetic particles. A comprehensive explanation of these effects on the transport of magnetic particles has not been developed yet. Here we formulate a consistent, constitutive equation for the magnetophoretic flux of magnetic nanoparticles suspended in a medium exposed to a stationary magnetic field. The constitutive relationship accounts for contributions from magnetic diffusion, magnetic convection, residual magnetization, and electromagnetic drift. We discovered that the key physical properties governing the magnetophoresis are magnetic diffusion coefficient, magnetic velocity, and activity coefficient, which depend on relative magnetic energy and the molar magnetic susceptibility of particles. The constitutive equation also reveals previously unknown ballistic and diffusive limits for magnetophoresis wherein the paramagnetic particles either aggregate near the magnet or diffusive away from the magnet, respectively. In the diffusive limit, the particle concentration is linearly proportional to the relative magnetic energy of the suspension of paramagnetic particles. The region of the localization of paramagnetic particles near the magnet decreases with increasing the strength of the magnet. The dynamic accumulation of nanoparticles, measured as the thickness of the nanoparticle aggregate, near the magnet compares well with the theoretical prediction. The effect of convective mixing on the rate of magnetophoresis is also discussed for the magnetic targeting applications.

Magnetophoresis is a physical phenomenon concerning the motion of magnetic particles in response to an external magnetic field (1, 2). This phenomenon has been exploited in a broad range of applications involving magnetic seals and inks (3), catalysts (4), ferrofluids (5), contrast agents for magnetic resonance imaging (6, 7), carriers for targeted drug delivery (7–9), and magnetic hyperthermia (10). Most of these applications aim at directing paramagnetic nanoparticles to a target location using external magnets (a process referred to as a magnetic targeting (11)), where the objective is to obtain the desired concentration of magnetic nanoparticles within a neighborhood of the targeted location in specific time duration. The spatiotemporal evolution of the concentration of magnetic nanoparticles during magnetic targeting is due to an interplay of various factors such as size, shape, concentration, surface charge density, and diffusion coefficient of the particles, magnetic susceptibility of the solution, temperature, strength of magnetic field and its gradient. The effects of some of these factors such as size, concentration, and magnetic field on the magnetic targeting has been investigated experimentally (12–15). However, the findings of such experiments are specific to the factors studied and cannot be generalized to other systems. A fundamental theory capable of predicting would allow one to better comprehend and predict magnetic targeting in various applications. Since most of the theoretical models are based on the dynamics of single particle, they cannot be applied to magnetophoresis of a population of nanoparticles in a concentrated solution near the target where interparticle interactions, steric effects, and concentration-dependent magnetic field play a crucial role. The emerging applications of magnetic targeting in biomedical and materials science need a predictive theoretical model based on fundamental principles to guide the design of functional materials and systems.

The effects of the magnetic field on the mobility of magnetic species such as ions, biological molecules, and nanoparticles have been investigated theoretically (16–19) and experimentally (8, 20). In such magnetophoresis studies, the motion of magnetic nanoparticles due to three types of body forces such as paramagnetic gradient force, field gradient force, and the Lorentz force have been postulated (18, 21). Various methods to model magnetophoresis of nanoparticles by accounting these body forces have been reported in the literature. In one of the approaches, the magnetic forces were included as the external body forces in the Navier–Stokes equation, and the resulting velocity was added to the convective term of the mass flux expression (17, 18, 22). This method is mostly applied to study magnetohydrodynamics, where liquid such as ferrofluid is susceptible to the magnetic field. However, a dilute solution of nanoparticles cannot provide enough body force for the solution to move. In another approach, the velocity of magnetic nanoparticles is modeled as a product of mobility and a net magnetic force, which is then included in the convective term of the mass flux expression (19, 23–25). This model resulting from single-particle motion cannot capture the effect of paramagnetic gradient force due to concentration gradient on the flux of nanoparticles. The third approach is to model magnetophoresis as the migration term of the Nernst–Planck equations, where the migration is modeled as a magnetic-field-dependent hall and transversal mobilities (16). Such models are not effective in evaluating the magnetophoretic motion of uncharged nanoparticles. In addition to the issues with current approaches, these models cannot predict magnetophoresis in concentrated solutions accounting for magnetic dipole interactions, steric effects, and interaction between concentration and magnetic fields. The mathematical details regarding these models are given in SI Appendix, section S1.

A comprehensive theoretical model for magnetophoresis that overcomes the limitations of current models is required for various magnetic-targeting-based applications. Additionally, there are several outstanding questions in magnetophoresis that need to be addressed through the development of such a model: 1) what are the physical properties required to realize magnetic targeting in practical scenarios? 2) what are the length scales and timescales of targeting? 3) what are the primary driving forces for magnetophoresis and how to control them? and 4) how do interparticle interactions affect magnetic targeting? To address these fundamental knowledge gaps, here we develop a constitutive equation for magnetophoretic flux utilizing a well-established framework of concentrated solution theory (26). The physical properties such as magnetic diffusion coefficient, magnetic velocity, and activity coefficients are also related to the magnetic field and nanoparticle properties. The theoretical analysis of magnetophoresis in one-dimensional (1D) and three-dimensional (3D) system at equilibrium is discussed and scaling relationships between concentration and magnetic field are obtained from the analytical solution. The critical length scales and timescales of magnetic targeting are obtained from transient experiments and validated with simulations. Finally, the simulation results of the spatiotemporal distribution of concentrations are validated with representative experiments.

Theory

The constitutive relationship for magnetophoretic flux can be obtained from a detailed balance of driving and drag forces acting on magnetic nanoparticles in a solution in the presence of an external magnetic field. The magnetic nanoparticles can experience two primary driving forces: 1) magnetic forces due to the gradient in the magnetic energy density $\left(U_m\right)$, and 2) electrochemical forces due to the gradient in the electrochemical potential $\left(\mu \right)$ of the solution. Another driving force could be the Lorentz force $F_L=zc\left(\overrightarrow{\mathbf{v}}\times \overrightarrow{\mathbf{B}}\right)$, which is typically much smaller as compared to the magnetoelectrochemical forces for a dilute solution of neutral nanoparticles (22). The net driving force acting on nanoparticles is balanced by the drag force due to hydrodynamic interactions with solvent (26), such that

$$\underset{\text{Electrochemical}\hspace{0.17em}\text{Forces}}{\underbrace{-c\overrightarrow{\nabla }\mu }}\underset{\text{Magnetic}\hspace{0.17em}\text{Forces}}{\underbrace{-\overrightarrow{\nabla }{U}_m}}=\underset{\text{Drag}\hspace{0.17em}\text{Force}}{\underbrace{K_s\left(\overrightarrow{\mathbf{v}}-\overrightarrow{\mathbf{u}}\right)}},$$ [1]

where $R$ is the gas constant, $T$ is the temperature, $c_s$ is the concentration of the solvent, $c$ is the concentration of nanoparticles, $D$ is the mass diffusion coefficient of the nanoparticles, $\overrightarrow{\mathbf{v}}$ is the velocity of nanoparticles, $\overrightarrow{\mathbf{u}}$ is the velocity of the solvent molecules, $K_s$ is the friction factor defined as $\frac{RTc\hspace{0.17em}{c}_s}{D\left(c+c_s\right)}$, $\mu$ is the electrochemical potential, and $U_m$ is the magnetic energy density of the nanoparticle solution. The electrochemical potential is given as $\mu =RT\hspace{0.17em}\ln \left(\gamma c\right)+zF\varphi$, where $\gamma$ is the activity coefficient that depends on the magnetic dipole–dipole interactions between nanoparticles, $\varphi$ is the streaming potential due to motion of charged nanoparticles, $z$ is the average charge on nanoparticles, and $F$ is Faraday’s constant. The magnetic energy density in Eq. 1 is defined as $U_m=-\frac{c{\chi }_n\hspace{0.17em}\overrightarrow{\mathbf{B}}\cdot \overrightarrow{\mathbf{B}}}{2{\mu }_0\left(1+c{\chi }_n\right)}$, where ${\chi }_n$ is the molar magnetic susceptibility of nanoparticles, $\overrightarrow{\mathbf{B}}$ is the external magnetic flux density, ${\mu }_0$ is the magnetic permeability of vacuum, and $\left(1+c{\chi }_n\right)$ is the relative permeability of the nanoparticle solution. The negative sign in the expression for $U_m$ is due to the work done by the nanoparticles to align their dipoles parallel to the magnetic field.

The principle of detailed force balance, similar to Eq. 1, has been applied previously to derive the Nernst–Planck equation for ion transport in concentrated solution (27), Maxwell–Stefan equation for multicomponent transport, and thermal diffusion equations (Soret and Dufour effects); see chapters. 12 and 13 of ref. (26). In the case of magnetophoresis, the electrochemical and magnetic forces are balanced by the drag force. The electrochemical forces include the diffusion force in the direction opposite to the concentration gradient and the migration force along the direction of the electric potential gradient for a positive charge. The magnetic forces consist of the paramagnetic gradient force in the direction of the susceptibility gradient and the magnetic gradient force in the direction of the magnetic field gradient. The detailed expressions for these four driving forces and their derivations are given in SI Appendix, section S2. The required constitutive relation for the magnetophoretic flux can be obtained by rearranging terms in Eq. 1:

$$\overrightarrow{\mathbf{N}}=\underset{\text{Diffusion}}{\underbrace{-D_e\overrightarrow{\nabla }c}}-\underset{\text{Migration}}{\underbrace{z{u}_{e}Fc\overrightarrow{\nabla }\varphi }}-\underset{\begin{array}{l}\text{Residual}\\ \text{Magnetization}\end{array}}{\underbrace{D_{e}c\overrightarrow{\nabla }\ln \gamma }}+\underset{\begin{array}{l}\text{Magnetic}\\ \text{Diffusion}\end{array}}{\underbrace{D_m\overrightarrow{\nabla }c}}+\underset{\begin{array}{l}\text{Magnetic}\\ \text{Convection}\end{array}}{\underbrace{{\overrightarrow{\mathbf{v}}}_{m}c}}+\underset{\begin{array}{l}\text{Hydrodynamic}\\ \text{Convection}\end{array}}{\underbrace{\overrightarrow{\mathbf{u}}c}},$$ [2]

where $D_e$ is the effective diffusion coefficient of nanoparticles, which is dependent on their concentration, $u_e=D_e/RT$ is the effective mobility of nanoparticles in the presence of surface charges, $D_m$ is the magnetic diffusion coefficient of nanoparticles, and ${\overrightarrow{\mathbf{v}}}_m$ is the magnetic velocity of nanoparticles. The physical parameters of Eq. 2 are defined as

$$D_e=\left(\frac{c+c_s}{\frac{c}{D_0}+\frac{c_s}{D}}\right),\hspace{0.17em}\hspace{0.17em}{D}_m=\frac{D_e{\lambda }_m}{{\left(1+c{\chi }_n\right)}^2},\hspace{0.17em}{\overrightarrow{\mathbf{v}}}_m=\frac{D_e\overrightarrow{\nabla }{\lambda }_m}{\left(1+c{\chi }_n\right)}$$ [3]

where $D_0$ is the self-diffusion coefficient of nanoparticles that can be determined from empirical correlation (28), and ${\lambda }_m$ is the relative magnetic energy of nanoparticles, which is the ratio of magnetic energy of nanoparticles in vacuum to the thermal energy of solution

$${\lambda }_m=\frac{{\chi }_n\overrightarrow{\mathbf{B}}\cdot \overrightarrow{\mathbf{B}}}{2{\mu }_{0}RT}\hspace{0.17em}.$$ [4]

Eq. 2 shows the magnetophoretic flux of nanoparticles as a sum of six different fluxes due to mass diffusion, residual magnetization, migration, magnetic diffusion, magnetic convection, and hydrodynamic convection. The mass and magnetic diffusion are two different processes that are based on the Brownian motion of nanoparticles with a characteristic diffusion coefficient $D_e$. While the mass diffusion occurs in the direction of decreasing chemical potential (or decreasing concentration), the magnetic diffusion acts along the direction of decreasing magnetic energy density (or increasing concentration). The magnetic diffusion flux can be described as the product of diffusion coefficient $\left(D_e\right)$ and gradient of relative magnetic energy density at constant magnetic flux density ${\left(-\nabla U_m/RT\right)}_{\overrightarrow{\mathbf{B}}}$,

$${\mathbf{N}}_m=-D_e{\left(\frac{\overrightarrow{\nabla }{U}_m}{RT}\right)}_{\overrightarrow{\mathbf{B}}}=\frac{D_e{\lambda }_m}{{\left(1+c{\chi }_n\right)}^2}\overrightarrow{\nabla }c.$$ [5]

The magnetic diffusion flux can also be written in terms of concentration gradient, as described above, to yield an apparent diffusion coefficient, which is referred to as a magnetic diffusion coefficient. The magnetic convection flux is described as a product of concentration and magnetic velocity of nanoparticles. The magnetic velocity is proportional to the magnetic flux gradient, which is similar to the description of electrophoretic velocity in terms of zeta potential and electric field ( SI Appendix, section S5). The effect of magnetic diffusion and magnetic convection is to oppose the mass diffusion of nanoparticles at equilibrium. The residual magnetization flux (RMF) represents resistance to nanoparticles’ diffusion due to higher interparticle dipole–dipole interactions. The contribution from the RMF can be substantial near the magnet, where the concentration and hence the interactions are higher. Since the activity coefficient decreases with increasing concentration and dipole–dipole interactions, the gradient of activity coefficient will, therefore, act along the direction of the concentration gradient resulting in the RMF of nanoparticles. The activity coefficient for magnetic dipole–dipole interaction can be obtained by relating magnetic energy density with magnetic potential as follows:

$$\gamma \left(c\right)=\frac{1}{{\gamma }^{\theta }c}\exp \left({\displaystyle \int \frac{1}{RTc}\overrightarrow{\nabla }{U}_m\cdot d\overrightarrow{\mathbf{r}}}\right).$$ [6]

The secondary reference state is required to identify proportionality constant ${\gamma }^{\theta }$ such that $\gamma \to 1$ as $c\to 0$ (26). The secondary reference state may also vary with magnetic flux density. The magnetic potential, along with the electrochemical potential, can be used to define magnetoelectrochemical potential, whose gradient directly produces all of the driving forces acting on nanoparticles. Such a formulation of activity coefficient can be applied to a wide range of concentrations as magnetic dipole–dipole interactions have a short range. This is not the case with electrophoresis, where the long-range Coulombic interactions are dominant. The magnetic dipole–dipole interactions between nanoparticles and the external wall can also be treated in a similar manner by implementing wall-specific activity coefficient in the boundary condition, which is discussed in SI Appendix, section S6.

The magnetophoretic motion of charged nanoparticles can also result in a streaming potential $\left(\varphi \right)$, whose gradient can impose a local electric field resulting in the migration flux in Eq. 2. This is an electrokinetic phenomenon (for details see chap. 9 of ref. 26). induced by magnetophoresis. The migration flux due to gradient in streaming potential is usually very small as compared to magnetic diffusion and convection fluxes.

Experimental Methods

Materials.

Iron (III) oxide (Fe₂O₃) magnetic nanoparticle solution (1mg ml⁻¹ Fe in water, Sigma-Aldrich) was used to prepare the samples. The magnetization of the magnetic nanoparticle solution was >45 emu g⁻¹, at room temperature, under 4,500 Oe. The average particle size of the nanoparticle was 10 nm, as measured using the transmission electron microscope. The solid density of nanoparticles was 0.996 g cm⁻³ at 25 °C. Two diluted samples at concentrations of 0.5 and 0.25 µM were prepared from the stock solution of 2.2 mM concentration.

Experimental Setup and Measurement of Nanoparticle Aggregation.

Fig. 1A shows the experimental setup for measuring the dynamic change in the height of the aggregated nanoparticles near the magnet. A glass tube (0.5-cm diameter and 17.7-cm length) was initially filled with 3.5 mL of the sample solutions under vacuum and installed horizontally on a stand. The tip of the glass tube was placed near the edge of the 1.05-T magnet (7.4-cm diameter and 5-cm height). The glass tube was exposed to a white light beam and time-lapsed images were taken from the camera (Canon EOS Rebel T6i with Canon zoom lens EF-S 55–250 mm, 5×). The images of agglomerated nanoparticles at a fixed focus were taken at every hour for 24 h using EOS Utility software. The time-lapsed images were then processed for the calculation of the height of the agglomerated nanoparticles using MATLAB R2018 (The MathWorks Inc.). The images were first calibrated by counting the pixels between the points A and B of known distance to measure the height of the nanoparticle aggregate in the glass tube. The number of pixels from A to C was then converted to distance using the calibration.

Fig. 1.

(A) Experimental setup to measure the aggregation of Fe₂O₃ nanoparticle in a solution of concentrations 0.25 and 0.5 µM under 1.05-T magnet. (B) Schematic representation of nanoparticle aggregation near the magnet. The thickness (height) of the concentrated nanoparticle aggregate was measured from the base to the tip of the aggregate.

Simulation of Magnetic Field and Magnetophoresis.

Maxwell’s equations for magnetostatics were solved for 1D and two-dimensional (2D) systems in COMSOL Multiphysics to obtain the distribution of magnetic potential and magnetic flux density. The parameters used to solve Maxwell’s equations were initial concentration of paramagnetic nanoparticles of 1 µM, and molar magnetic susceptibility $\left({\chi }_n\right)$ of 63,070 m³/mol. The distribution of nanoparticles was obtained by solving the continuity equation using magnetophoretic flux expression (2) in COMSOL Multiphysics. Since molar magnetic susceptibility of paramagnetic nanoparticles is much larger than that of water (solvent), the variation in the local concentration of nanoparticles due to magnetophoresis had a minimal effect on the relative permeability of the solution and hence the magnetic flux density.

Results and Discussion

Fig. 2 provides fundamental insight into the key physical properties that govern magnetophoresis, namely, effective mass diffusion coefficient, magnetic diffusion coefficient, magnetic velocity, and activity coefficient, which are dependent on the magnetic field and concentration of nanoparticles. Fig. 2A shows a sharp decrease in the effective mass diffusion coefficient with the increasing mole fraction of nanoparticles. The effective diffusion coefficient approaches the self-diffusion limit for higher mole fractions of nanoparticles, which also suggests that nanoparticles will be less mobile near the magnetic target. Fig. 2B shows the variation in the magnetic diffusion coefficient relative to the effective mass diffusion coefficient as a function of mole fraction of nanoparticles and magnitude of magnetic flux density. The magnetic diffusion coefficient increases quadratically with increasing magnitude of magnetic flux density and decreases with the increasing mole fraction of nanoparticles. Fig. 2C shows the ratio of magnetic velocity to the effective mass diffusion coefficient increases linearly with the magnitude of field gradient and decreases with increasing the mole fraction of nanoparticles. It can be seen in Fig. 2 B and C that the magnetic field greatly affects the magnetic diffusion coefficient and velocity, which in turn increases the effectiveness of magnetophoresis. Fig. 2D shows a rapid decrease in the activity coefficient of magnetic nanoparticles with increasing its mole fraction, which is obtained from Eq. 5. From fitting the data in Fig. 2D, the activity coefficient model can be approximated as $\ln \left(\gamma \right)=-\beta c$, where β is the interaction parameter. We note that this interaction parameter is similar to the proportionality constant in the Debye–Hückel limiting law. The activity coefficient is independent of the magnetic flux density because the magnetic dipoles of paramagnetic particles are permanent and do not change with the magnetic field. The long-range interactions between magnetic dipoles of nanoparticles cause the activity coefficient to decrease substantially with increasing nanoparticle concentration, which also indicates increasing residual (or remanent) magnetization in nanoparticles near the magnetic target.

Fig. 2.

Dependence of key physical properties of magnetophoresis on the concentration of nanoparticles and magnetic flux density. (A) The effect of a mole fraction of nanoparticles on their effective mass diffusion coefficient. (B) Variation in the magnetic diffusion coefficient with the increasing mole fraction of nanoparticles and the magnitude of magnetic flux density. (C) Increase in magnetic velocity of nanoparticles with increasing mole fraction and gradient of relative magnetic energy. (D) Effect of mole fraction on nanoparticles on the activity coefficient.

The critical insights into the process of magnetophoresis can be obtained by considering the equilibrium of an ideal solution of noninteracting $\left(\gamma =1\right)$, charge-neutral $\left(z=0\right)$ paramagnetic nanoparticles under stagnant conditions $\left(\overrightarrow{\mathbf{u}}=0\right)$ exposed to a stationary magnetic field, such that

$$\left(1-\frac{D_m}{D_e}\right)\overrightarrow{\nabla }c=\frac{{\overrightarrow{\mathbf{v}}}_m}{D_e}c.$$ [7]

Since the magnetic diffusion and convection counteract the mass diffusion, two physically different transport limits in magnetophoresis can be realized: 1) Diffusive limit; when $\frac{D_m}{D_e}<1$, Eq. 7 yields a smooth concentration profile such that the net diffusion is in the same direction as the magnetic convection, and 2) Ballistic limit; when $\frac{D_m}{D_e}\ge 1$, the net diffusion and magnetic convection are in the same direction such that the net flux is never zero, which also implies the magnetophoresis is unimpeded by the diffusional resistance resulting in stacking (or aggregation) of nanoparticles near the magnet. The surface in Fig. 2B shows a white line $\left(\frac{D_m}{D_e}=1\right)$ separating regions of diffusive and ballistic limits.

Additionally, there can be two different limiting cases depending on the preferential mode of magnetophoretic transport, such as magnetic diffusion or magnetic convection. In the neighborhood of a strong magnet where the magnetic flux density is uniform $\left(\overrightarrow{\nabla }\cdot \overrightarrow{\mathbf{B}}\approx 0\right)$, the flux due to magnetic convection can be negligible. In such cases, the primary mode of transport is magnetic diffusion. The other limiting case can be realized in a highly concentrated solution of nanoparticles where $\overrightarrow{\nabla }c\approx \overrightarrow{0}$, such that the diffusion is negligible and the primary mode of transport in magnetic convection.

The distribution of concentration of nanoparticles in the diffusive limit is given by the solution of Eq. 7, written as

$$1+c\left(\overrightarrow{\mathbf{r}}\right){\chi }_n={\lambda }_m\left(\overrightarrow{\mathbf{r}}\right)+K,$$ [8]

where $\overrightarrow{\mathbf{r}}$ is the position vector and $K$ is the integration constant, which can be evaluated by conserving the total number of nanoparticles. A similar solution can be obtained from the variational principle, which is shown in SI Appendix, section S4. Eq. 8 is the fundamental scaling relationship for magnetophoresis, which shows the relative magnetic permeability $\left(1+c{\chi }_n\right)$ varies linearly with the relative magnetic energy $\left({\lambda }_m\right)$ of the solution (see Fig. 3A for β = 0). The effect of interparticle interaction β on the concentration distribution for a nonideal system can be obtained from the solution of the continuity Eq. 7 with the RMF term

$$\left(1-\frac{D_m}{D_e}-\beta c\right)\overrightarrow{\nabla }c=\frac{{\overrightarrow{\mathbf{v}}}_m}{D_e}c.$$ [9]

Fig. 3A shows the increase in concentration with increasing interaction parameter β from 0 to 10. The interaction parameter β dictates the strength of interparticle magnetic interactions. For larger values of β, the activity coefficient of nanoparticles decreases rapidly with a small increase in nanoparticle concentration near the magnet. Further, this results in a higher RMF and higher concentration of nanoparticles near the magnet than the case without interparticle interactions.

Fig. 3.

(A) Dimensionless concentration versus relative magnetic energy. (B) Concentration profile of noninteracting nanoparticles (γ =1) for 1D system. (C) Variation of relative magnetic energy with the length of the medium to achieve ballistic limit. (D) Concentration profile of noninteracting nanoparticles in the 3D system.

The spatial distribution of magnetic flux density and the relative magnetic energy can be obtained from Maxwell’s equations for magnetostatics (in the absence of charge accumulation or current). From the spatial distribution of relative magnetic energy ${\lambda }_m\left(\overrightarrow{\mathbf{r}}\right)$, the spatial distribution of nanoparticle concentration $c\left(\overrightarrow{\mathbf{r}}\right)$ can then be obtained using Eq. 7, where the integration constant is determined from the conservation of the total number of nanoparticles in the system. Fig. 3B shows steady-state distributions of nanoparticles in a 1D channel of length 10 cm with an initial (or total) concentration of 1 µM at varying magnetic strength of 0, 0.04, 0.08, and 0.12 T. The nanoparticle concentration near the magnet increases with increasing the magnetic strength from 0 to 0.12 T, after which the nanoparticles start aggregating for magnetic strength >0.15 T. The initiation of aggregation is dictated by the condition for ballistic limit $\frac{D_m}{D_e}\ge 1$, which yields a relationship between relative magnetic energy, initial concentration of nanoparticles, molar magnetic susceptibility of nanoparticles, and length of the domain (SI Appendix). Fig. 3C shows such a relationship of $\frac{{\lambda }_m}{c_0{\chi }_n}$ vs. length of the domain (L), which separates the regions of ballistic and diffusive limits for magnetophoresis in 1D. It also shows the magnetic strength required to pull all of the nanoparticles near the magnet increases with increasing the length of the domain and/or increasing the initial concentration of nanoparticles. The operation of magnetophoresis under ballistic limits is essential for several magnetic targeting applications, such as drug delivery, where the majority of drug-loaded nanoparticles need to be localized near the magnetic target with a minimum diffuse layer. In contrast to the smooth concentration distribution in diffusive limits, the distribution of nanoparticles under the ballistic limit is uniform, which can be obtained from the packing density of nanoparticles. For example, the maximum achievable concentration of nanoparticles under the ballistic limit is $c_b=\frac{\sqrt{2}}{N_A{d}_p^3}=2.3\hspace{0.17em}\text{mM}$ when spherical nanoparticles of diameter $d_p=10\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{nm}$ are hexagonally packed.

The scaling relationship Eq. 8 can also yield the distribution of nanoparticles in the 3D system using the distribution of relative magnetic energy obtained from Maxwell’s relation. Fig. 3D show the distribution of nanoparticles around 0.12-T magnet. The localization of nanoparticles around the magnet can be identified as a region containing 90% of the total number of nanoparticles (see the region under the white curve in Fig. 3D). This region is referred hereto as a target zone, whose dimension decreases with increasing the strength of the magnet.

Fig. 4A shows the decrease in the length of the target zone by increasing the strength of the magnet in the 1D system. The target zone decreases gradually until 0.08 T and then sharply from 0.08 to 0.15 T, which after attaining the ballistic limit, suddenly collapses to a length scale ∼$\frac{c_{0}L}{c_b}=4.26\times {10}^{-3}\hspace{0.17em}\text{cm}$. The time required to attain the steady-state target zone is also important in several applications, where the dynamics of magnetophoresis determines the response of nanoparticles at the magnetic target. Fig. 4B shows the experimental and computational evaluation of the ballistic dynamics of the aggregation of Fe₂O₃ nanoparticles in a solution of 0.25 and 0.5 µM concentrations using 1.05-T magnet. The experimental details are given in Experimental Methods, and the computational methods are provided in SI Appendix, section S3. The rate of nanoparticle aggregation (or capture) increases with increasing the concentration of nanoparticles, which is due to an increase in magnetic convection in Eq. 7. The time required to attain a steady state is usually very long (>24 h), even for a magnet as strong as 1.05 T. This timescale of magnetophoresis increases with decreasing the magnetic flux density. Fig. 4C shows the dynamic profiles of the concentration of nanoparticles over 24 h (in the diffusive limit) using a 0.12-T magnet. The rate of magnetophoresis can be increased by inducing convective mixing in the solution. The effect of convective mixing on the dynamics of magnetophoresis is evaluated by multiplying an enhancement factor $f$ to the effective diffusion coefficient $D_e$ in the flux expression (2). Fig. 4D shows the increase in the concentration of nanoparticles near the magnet (x = 0) with increasing the enhancement factor $f$ from 1 (stagnant case) to 100.

Fig. 4.

(A) Decrease in the size of a steady-state target zone with increasing strength of the magnet. (B) Experimentally measured and computationally predicted height of the nanoparticle aggregate over 24 h in 0.25 and 0.5 μM Fe₂O₃ nanoparticle solution exposed to 1.05-T magnet in the ballistic limit. The photos of the nanoparticle aggregate are marked at 10 and 18 h. (C) Dynamic change in the concentration profile of nanoparticles over 24 h using 0.12-T magnet (diffusive limit). (D) Effect of convective mixing on the rate of magnetophoresis. Convective enhancement of the effective diffusion increases the concentration of nanoparticles near the magnet.

Conclusions

This article presents the constitutive relationship and governing physical properties for magnetophoresis. The constitutive relationship for a magnetophoretic flux of magnetic nanoparticles includes three primary contributions: magnetic diffusion, magnetic convection, and residual magnetization, which act in the direction opposite to the mass diffusion of paramagnetic particles away from the magnet. This constitutive relationship is obtained from a detailed force balance. The theory is valid even for a highly concentrated solution of charged/uncharged nanoparticles with dipole–dipole interactions. The magnetophoretic flux due to magnetic convection is dominant, and it is 2–3 orders of magnitude higher than the magnetic diffusion flux. The RMF term accounts for interparticle dipole–dipole interactions, which increases with increasing particle concentrations. The governing physical properties for magnetophoresis are magnetic diffusion coefficient, magnetic velocity, and activity coefficient that are dependent on the relative magnetic energy, the gradient of relative magnetic energy, and the magnetic energy density of the solution, respectively. The magnetoelectrochemical potential is also defined, whose gradient provides all of the driving forces acting on paramagnetic particles.

Two distinct transport limits for magnetophoresis are identified from the continuity equation, namely, diffusive limit and ballistic limit. The diffusive limit is realized when the magnetic diffusion coefficient is less than the mass diffusion coefficient, in which case the concentration profiles are smooth, and particles do not aggregate near the magnet. The most important region is the ballistic limit when the magnetic diffusion coefficient is greater than the mass diffusion coefficient. In the ballistic limit, the magnetic velocity of particles is high enough to cross the diffusional barrier and allows particles to aggregate near the magnet. This necessary condition for the ballistic limit also provides the minimum strength of the magnet required to capture all of the magnetic particles from the solution. However, the timescale of capture can be several hours based on the spatial extent and concentration of nanoparticle solution, and the strength of the magnet. The rate and hence the timescale of magnetophoresis can be increased by active mixing in the solution. Finally, the transport equations developed herein can be used to optimize the magnetic targeting in various applications such as drug delivery, bioseparations, and imaging.

Data Availability.

All study data are included in the article and SI Appendix.