Magnetokinetics

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Magnetokinetics

Magnetokinetics is a branch of physics and chemistry that investigates the magnetically induced motion of magnetized particles, fluids, or interfaces, encompassing a range of interfacial phenomena driven by magnetic fields or generating them through mechanical means. Key effects include magnetophoresis (motion of dispersed particles in a fluid under a magnetic field), magnetoosmosis (bulk fluid motion through a porous medium or capillary due to an applied magnetic field), streaming magnetization (magnetic potential generated by fluid flow over a magnetized surface), and sedimentation magnetization (magnetic field from settling magnetized particles). These phenomena originate from the magnetic double layer (MDL) at solid-liquid interfaces, where fixed surface magnetizations are balanced by mobile counter-magnets, leading to tangential forces under external fields. Magnetokinetics plays a crucial role in ferrofluid science, with theoretical foundations linking magnetostatics, fluid dynamics, and transport processes, and has been quantitatively described since the 20th century. Modern extensions incorporate nonlinear effects, AC fields, and nanoscale considerations for advanced applications.

History

• The discovery of magnetokinetic phenomena dates back to the early 20th century when researchers observed magnetophoretic motion in magnetic colloids, marking the first documented magnetokinetic effect.
  • Experiments demonstrated particle movement towards regions of higher magnetic field strength, attributing it to magnetic forces on magnetized interfaces.
• In the 1940s, the development of ferrofluids by NASA scientists like Stephen Papell led to practical demonstrations of magnetoosmosis.
• Theoretical advancements began in the 1950s with models of magnetic susceptibility and the introduction of magnetic zeta potential analogs.
  • Subsequent work in the 1960s and 1970s refined equations for thin magnetic layers.
  • Models for diffuse magnetic layers were developed in the late 20th century.
• The late 20th and early 21st centuries saw widespread application in nanotechnology, biomedicine, and energy technologies.
  • Induced-charge magnetokinetics (ICMK) concepts emerged in the 2000s.

Theoretical Basis

Magnetokinetics couples the equations for magnetic potential with the Navier–Stokes equations and particle transport.

• Magnetostatic equation in the MDL:

$\nabla ^{2}\phi =-{\frac {\mu _{0}M}{\mu }}$

where $\phi$ is the magnetic potential and $M=\sum m_{i}c_{i}$ is the magnetization density.

• Analogous magnetoosmotic velocity (thin MDL limit, $\kappa a\gg 1$):

$v_{\mathrm {MO} }=-{\frac {\mu \xi }{\eta }}\mathbf {B}$

• Magnetophoretic velocity (analogous Hückel–Onsager limit for thick MDL, $\kappa a\ll 1$):

$v_{\mathrm {MP} }={\frac {2\mu \xi }{3\eta }}\mathbf {B}$

Henry's function $f(\kappa a)$ interpolates between 1 (Hückel) and 1.5 (Smoluchowski).

• Streaming magnetization coefficient:

$\left({\frac {\Delta \phi }{\Delta P}}\right)_{I=0}=-{\frac {\mu \xi }{\eta \sigma _{m}}}$

• Full system for nonlinear and transient magnetokinetics:

${\frac {\partial c_{i}}{\partial t}}+\nabla \cdot \left(-D_{i}\nabla c_{i}-{\frac {m_{i}D_{i}\mu _{0}}{kT}}c_{i}\nabla \phi +c_{i}\mathbf {v} \right)=0$

$\rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+\eta \nabla ^{2}\mathbf {v} +\mathbf {M} \cdot \nabla \mathbf {B}$

• Induced-charge magnetoosmosis (ICMO) slip velocity on polarizable surfaces:

$\mathbf {v} _{\mathrm {slip} }=-{\frac {\mu }{\eta }}\langle \mathbf {B} _{t}\rangle \Delta \mathbf {B} _{t}$

• Analogous Dukhin number (surface vs. bulk conduction):

$\mathrm {Du} ={\frac {K_{m}^{s}}{K_{m}^{b}a}}$

Applications

• Soil and groundwater remediation (magnetokinetic remediation) – removal of magnetic contaminants from low-permeability soils.
• Microfluidics – magnetoosmotic and magnetophoretic pumping in lab-on-a-chip devices; magnetic separation for biomolecules.
• Biomedical – magnetic drug targeting, MRI contrast agents, cell sorting via magnetophoresis.
• Geophysics – magnetic surveys for mineral exploration and fracture mapping.
• Energy – magnetic fluid power generation and magnetocaloric refrigeration.
• Industrial – magnetic levitation, ferrofluid seals, nanoparticle assembly in manufacturing.

Discipline	Relevant Mainstream Object/Equation	Role in Magnetokinetics
Fluid Dynamics	$\rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+\eta \nabla ^{2}\mathbf {v} +\mathbf {M} \cdot \nabla \mathbf {B}$	Navier–Stokes with body force from magnetic layer
Magnetism	$\nabla ^{2}\phi =-{\frac {\mu _{0}M}{\mu }}$	Equation for potential in the MDL
Colloid Science	$v_{\mathrm {MP} }={\frac {\mu \xi }{\eta }}\mathbf {B} \cdot f(\kappa a)$	Henry’s equation for particle mobility
Environmental Engineering	$v_{\mathrm {MO} }=-{\frac {\mu \xi }{\eta }}\mathbf {B}$	Analogous magnetoosmotic flow
Microfluidics	$\mu _{\mathrm {mp} }={\frac {\mu \xi }{\eta }}$	Magnetophoretic mobility in separations
Geophysics	$C_{s}={\frac {\Delta \phi }{\Delta P}}=-{\frac {\mu \xi }{\eta \sigma _{m}}}$	Streaming magnetization for subsurface imaging
Biomedical Engineering	$J_{\mathrm {drug} }=-\mu _{\mathrm {MO} }C_{\mathrm {drug} }B$	Magnetic flux for drug delivery
Nanotechnology	$\mathbf {F} _{\mathrm {MDEP} }=2\pi \mu _{m}r^{3}\mathrm {Re} [K(\omega )]\nabla B^{2}$	Magnetodielectrophoretic force for nanoparticle manipulation
Chemical Engineering	$\mathbf {J} _{i}=-D_{i}\nabla c_{i}-{\frac {m_{i}D_{i}\mu _{0}c_{i}}{kT}}\nabla \phi +c_{i}\mathbf {v}$	Flux equation for magnetic particle transport
Nonlinear Magnetokinetics	$\mathrm {Du} ={\frac {K_{m}^{s}}{K_{m}^{b}a}},\ \ \mathrm {Wi} ={\frac {\beta Ba}{2D}}$	Dimensionless numbers governing surface effects