Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Electrophoresis

Field: Separation Science

The motion of dispersed particles relative to a fluid under the influence of a spatially uniform electric field.

Sequence of Expressions

Define the electrophoretic velocity vepv_{ep} and the applied electric field EE. The migration coefficient μ\mu is defined as the ratio of the electrophoretic velocity to the applied electric field:\n\nμ=vepE\mu = \frac{v_{ep}}{E}
Define the diffusion coefficient DD for a spherical particle of radius rr in a medium of viscosity η\eta at absolute temperature TT by the relation:\nD=kBT6πηrD = \frac{k_B T}{6\pi \eta r} \nwhere kBk_B is the Boltzmann constant.
Define the charge-to-size ratio Γ\Gamma for an analyte molecule as the ratio of its net charge qq (in Coulombs) to its effective mass mm (in kilograms): \nΓ=qm\Gamma = \frac{q}{m} \nThis ratio Γ\Gamma is the primary determinant of the electrophoretic mobility μ\mu in a given medium.
Define the zeta potential ζ\zeta at the capillary wall and the applied electric field EE. The electroosmotic flow velocity veofv_{eof} is given by the Helmholtz-Smoluchowski equation, which simplifies to:\n\nveof=ϵζ Eηv_{eof} = \frac{\epsilon \zeta \ E}{\eta}
Let H+H^+ be the concentration of protons and AA be the analyte. The net charge qq of the analyte, which depends on the buffer pH\text{pH}, is modeled by the Henderson-Hasselbalch relationship applied to the dissociation constant pKa\text{p}K_a:\n\nq=qmax+(qmin−qmax)11+10(pH−pKa)q = q_{max} + (q_{min} - q_{max}) \frac{1}{1 + 10^{(\text{pH} - \text{p}K_a)}}
Consider the separation velocity vv of a macromolecule of radius rr and charge qq within a porous gel matrix characterized by a sieving coefficient κ(r)\kappa(r). The effective mobility μeff\mu_{eff} is then defined by:\n\nμeff=μfree⋅κ(r)\mu_{eff} = \mu_{free} \cdot \kappa(r)
Let μep\mu_{ep} be the electrophoretic mobility and μeof\mu_{eof} be the electroosmotic mobility. The total observed velocity vobsv_{obs} of an analyte under an applied electric field EE is the superposition of these two components:\n\nvobs=vep+veof=E(μep+μeof)v_{obs} = v_{ep} + v_{eof} = E (\mu_{ep} + \mu_{eof})
Let C(x,t)C(\mathbf{x}, t) be the concentration of particles at position x\mathbf{x} and time tt, and J\mathbf{J} be the particle flux vector. \n\nFick's First Law (Flux): \nJ=−D∇C\mathbf{J} = -D \nabla C \n\nFick's Second Law (Diffusion Equation): \n∂C∂t=D∇2C\frac{\partial C}{\partial t} = D \nabla^2 C \nwhere DD is the diffusion coefficient and ∇2\nabla^2 is the Laplacian operator.
Let μ\mu be the electrophoretic mobility, qq be the net charge, mm be the effective mass, and E\mathbf{E} be the electric field vector. The mobility is defined by the relationship: \nμ=qm⋅κ\mu = \frac{q}{m} \cdot \kappa \nwhere κ\kappa is a proportionality constant incorporating the medium's resistance and the field strength, such that the migration velocity v\mathbf{v} is given by v=μE\mathbf{v} = \mu \mathbf{E}.
Let E(x,t)\mathbf{E}(\mathbf{x}, t) be the electric field vector at position x\mathbf{x} and time tt. The electric field gradient is mathematically represented by the gradient tensor ∇E\nabla \mathbf{E}, whose components are the partial derivatives of the field components: \n∇E=[∂Ei∂xj]i,j=13\nabla \mathbf{E} = \left[ \frac{\partial E_i}{\partial x_j} \right]_{i, j=1}^3 \nThis tensor quantifies how the electric field varies spatially.