Stream broadening due to fluid shear across the wider transverse dimension of a free-flow zone electrophoresis channel

Abstract

While the pressure-gradient applied along the length of a free-flow zone electrophoresis (FFZE) chamber is known to produce a parabolic flow profile for the carrier electrolyte across the narrower channel dimension (typically the channel depth), additional fluid shear can arise across the channel width due to a variety of reasons. Most commonly, any variation in the pressure-drop or channel depth across this wider dimension can lead to a gradient in the liquid flow velocity along it, significantly altering the stream broadening and, thereby, the separation performance of the assay. This article assesses the effect of such fluid shear on stream broadening during the FFZE process by describing a mathematical framework for solving the relevant advection-diffusion equation based on the method-of-moments approach. A closed-form expression for the leading order term describing the additional contribution to the spatial stream variance has been derived considering a small linear gradient in the liquid velocity across the wider transverse dimension of the FFZE chamber. The noted analysis predicts this contribution to be governed by two Péclet numbers that are evaluated based on the axial pressure-driven flow and transverse electrophoretic solute velocities. More importantly, this contribution is shown to vary quadratically with the axial distance traversed by the analyte stream as opposed to the classical linear variation known for all other stream broadening contributions in FFZE systems. The results from the analytic theory have been validated with Monte Carlo simulations, which also establish a time and length scale over which the noted analytical results are applicable.

I. INTRODUCTION

The ability to resolve analyte streams in a free-flow zone electrophoresis (FFZE) assay critically depends on the extent of broadening these zones undergo during their transit through the separation chamber.^1–3 Such broadening arises both due to solute diffusion perpendicular to the flow direction of the sample stream and a variation in the analyte advection velocity across the channel cross section.^3–5 While the first contribution is relatively simple to estimate, the latter one can originate from multiple sources. Under ideal conditions, the spatial variance (σ²) of a solute stream in an FFZE assay may be expressed as^6,7

$$\begin{array}{ccc}\hfill {\sigma }^2& =\hfill & \frac{b^2}{\underset{\begin{array}{c}\text{injection}\\ \text{variance}\end{array}}{\underbrace{12}}}+\frac{2{\left({\mu }_{EOF}+{\mu }_{EP}\right)}^2{E}^{2}DL}{\underset{\begin{array}{c}\text{orientation}\\ \text{factor}\end{array}}{\underbrace{U_P^3}}}+\frac{2DL}{\underset{\begin{array}{c}\text{diffusive}\\ \text{broadening}\end{array}}{\underbrace{U_P}}}\hfill \\ \hfill & \hfill & +\underset{\begin{array}{c}\text{hydrodynamic}\\ \text{broadening}\end{array}}{\underbrace{\frac{1}{105}\frac{{\left({\mu }_{EOF}+{\mu }_{EP}\right)}^2{E}^2{d}^{2}L}{U_{P}D}}},\hfill \end{array}$$ (1)

where the symbols D, U_P, μ_EOF, and μ_EP refer to the analyte diffusion coefficient, spatially averaged axial pressure-driven velocity, electroosmotic mobility of the fluid, and electrophoretic mobility of the analyte species, respectively. In addition, the characters E, b, d, and L in this equation represent the transverse electric field, the width of the sample stream at the channel entrance, and the depth and length of the FFZE compartment, respectively. Equation (1) shows that the contribution to zone variance due to “diffusive broadening” perpendicular to sample flow simply scales with the solute diffusion coefficient (D) times the residence time (T = L/U_p) of the analyte molecules in the FFZE channel. On the other hand, the “hydrodynamic broadening” component induced by the parabolic pressure-driven flow profile of the carrier electrolyte is determined by a complex interplay between the electrophoretic and the pressure-driven transport of the analyte molecules in conjunction with their cross-streamline migration via diffusion. In addition, the overall zone variance also has contributions from the injection width and the nonorthogonal orientation of the solute stream with respect to the direction along which σ is measured. These two contributions, which have been referred to as the “injection variance” and “orientation factor,” respectively, in Eq. (1) may be shown to be additive to the diffusive and hydrodynamic broadening components included in the same mathematical expression.

While the sources for stream broadening noted above apply to an ideal FFZE assay, several other factors^8–10 tend to modify this zone variance from that predicted by Eq. (1) in an actual experimental setting. For example, Joule heating of the carrier electrolyte has been recently shown to moderately reduce the “hydrodynamic broadening” contribution due to the dominant effect of an increased cross-streamline diffusion of the analyte species.¹¹ In other situations, the presence of an unwanted transverse pressure-gradient in the system may be shown to significantly increase fluid shear in FFZE assays leading to additional zone dispersion.¹² Along the same lines, the spatial variance of the analyte stream can be expected to rise due to the existence of any gradients in the axial pressure-driven flow velocity across the wider transverse dimension of the channel. Such gradients can most commonly originate from an unwanted variation in the axial pressure-drop and/or channel depth across this dimension as well as due to the presence of pores at the channel walls.¹³ In addition, the presence of walls and/or other structures around the channel side-regions^14–17 can also produce such gradients whose effects on stream broadening have not been previously assessed in the literature. In this article, a mathematical framework for analyzing the effect of a small linear gradient in the liquid flow velocity across the wider transverse dimension of a FFZE chamber on zone dispersion has been described. The present framework relies on the method-of-moments approach for solving the relevant material balance equations yielding a closed-form expression for the stream position and zone variance under equilibrium conditions. The results from the analytic theory have been later validated with Monte Carlo simulations, which also establish a time and length scale over which the predicted stream width may be attained for a chosen set of operating conditions.

II. MATHEMATICAL FORMULATION

To examine the effect of a small fluid shear across the channel width on solutal transport in an FFZE system, the flow of an analyte stream between two parallel plates separated by a distance d (see Fig. 1) has been considered. The sample flow is driven in this system by a pressure-gradient applied along the axial direction (z-coordinate) in the presence of an electric field (E) imposed across the width of the separation chamber (along the x-coordinate). The locations of the parallel plates are chosen to be y = ±d/2 to simplify the analysis producing liquid flow and analyte concentration profiles that are both symmetric with respect to the y-coordinate. Furthermore, the small fluid shear across the channel width has been accounted for by assuming a pressure-driven velocity profile u_p = (3U/2)(1 − 4y²/d²)(1 + εx/d) with ε(x/d) ≪ 1 and U being the depth-averaged value of u_p at x = 0. Analyte migration along the x–coordinate is assumed to be purely electrokinetic, yielding a stream velocity across the channel width given by u_t = μE, where μ and E refer to the net electrokinetic mobility (algebraic sum of the electrophoretic and electroosmotic mobilities) of the solute molecules and transverse electric field in the system, respectively. In this situation, the advection-diffusion equation governing the concentration profile of the sample species (C) may be written as

$$\begin{array}{ccc}\hfill \mu E\frac{\partial C}{\partial x}+\frac{3U}{2}\left(1-\frac{4y^2}{d^2}\right)\left(1+\epsilon \frac{x}{d}\right)\frac{\partial C}{\partial z}=D\left(\frac{{\partial }^{2}C}{\partial x^2}+\frac{{\partial }^{2}C}{\partial y^2}+\frac{{\partial }^{2}C}{\partial z^2}\right)& \hfill & \hfill \\ \hfill & \hfill & \hfill \end{array}$$ (2)

with D denoting the diffusion coefficient of the analyte species. The above equation may be reduced to a dimensionless form upon normalizing all length scales with respect to d, i.e., x^*, y^*, z^* = x/d, y/d, z/d, and the sample concentration by its value (C₀) at the channel inlet (z^* = 0), i.e., C^* = C/C₀, yielding

$$\begin{array}{ccc}\hfill P{e}_x\frac{\partial C^{*}}{\partial x^{*}}+\frac{3}{2}P{e}_z\left(1-4y^{*2}\right)\left(1+\epsilon x^{*}\right)\frac{\partial C^{*}}{\partial z^{*}}=\frac{{\partial }^2{C}^{*}}{\partial x^{*2}}+\frac{{\partial }^2{C}^{*}}{\partial y^{*2}}+\frac{{\partial }^2{C}^{*}}{\partial z^{*2}}.& \hfill & \hfill \\ \hfill & \hfill & \hfill \end{array}$$ (3)

The quantities Pe_x = μEd/D and Pe_z = Ud/D here denote the Péclet numbers in the x- and z-directions, respectively, which provide a measure for the rate of advection relative to diffusion along the width and length of the separation compartment. The analyte concentration in this system is subjected to the boundary conditions ∂C^*/∂y^* = 0 at y^* = ±1/2 and C^*, ∂C^*/∂x^* = 0 as x^* → ±∞. In addition to these constraints, the amount of analyte flowing per unit time through any x–y plane is set to a constant M in the present analysis which can be equated to the integral $\left(3C_{0}U{d}^2/2\right){\int }_{-1/2}^{1/2}{\int }_{-\infty }^{\infty }\left(1-4y^{*2}\right)\left(1+\epsilon x^{*}\right)C^{*}d{x}^{*}d{y}^{*}$ to satisfy the material balance in the system. Now multiplying Eq. (3) with $x^{*p}$ followed by integrating it along the x^*-coordinate from −∞ to ∞, it is possible to show that⁵

$$\begin{array}{ccc}\hfill \begin{array}{c}-pP{e}_x{\varphi }_{p-1}+P{e}_{z}g\frac{\partial {\varphi }_p}{\partial z^{*}}+P{e}_z\epsilon g\frac{\partial {\varphi }_{p+1}}{\partial z^{*}}=p\left(p-1\right){\varphi }_{p-2}+\frac{{\partial }^2{\varphi }_p}{\partial y^{*2}}+\frac{{\partial }^2{\varphi }_p}{\partial z^{*2}}.\\ \text{Boundary}\hspace{0.17em}\text{conditions}:{\left.\frac{\partial {\varphi }_p}{\partial y^{*}}\right|}_{y^{*}=\pm 1/2}=0,{\int }_{-1/2}^{1/2}\left({\varphi }_0+\epsilon {\varphi }_1\right)gd{y}^{*}=\frac{M}{U{C}_0{d}^2}=\delta ,\end{array}& \hfill & \hfill \end{array}$$ (4)

where ${\varphi }_p={\int }_{-\infty }^{\infty }{x}^{*p}{C}^{*}d{x}^{*}$ and $g=3\left(1-4y^{*2}\right)/2$. Further integrating Eq. (4) along the y^*–coordinate over the region between the parallel plates and defining $m_p={\int }_{-1/2}^{1/2}{\varphi }_{p}d{y}^{*}$, one can obtain

$$\begin{array}{ccc}\hfill & \hfill & -pP{e}_x{m}_{p-1}+P{e}_z{\int }_{-1/2}^{1/2}\left(g\frac{\partial {\varphi }_p}{\partial z^{*}}\right)d{y}^{*}+\epsilon P{e}_z{\int }_{-1/2}^{1/2}\left(g\frac{\partial {\varphi }_{p+1}}{\partial z^{*}}\right)d{y}^{*}\hfill \\ \hfill & \hfill & =p\left(p-1\right)m_{p-2}+\frac{d^2{m}_p}{d{z}^{*2}}.\hfill \end{array}$$ (5)

Note that the quantity m_p in this formulation represents the pth moment of C^* along the x^*–coordinate after spatially averaging over the y^* domain with m₁ quantifying the normalized x^*–position of the center of mass for the analyte stream and $m_2/m_0-m_1^2/m_0^2$ equaling its normalized spatial variance along the x-axis. The leading order effect of the fluid shear component ∂u_p/∂x on solutal transport may be assessed analytically in this situation by assuming ${\varphi }_p={\sum }_{i=0}^{\infty }{\epsilon }^i{\varphi }_{pi}$ and $m_p={\sum }_{i=0}^{\infty }{\epsilon }^i{m}_{pi}$ followed by solving for $\varphi$_p1 and m_p1 in the series solutions. Note that the quantities $\varphi$_p0 and m_p0 in the current analysis correspond to the solutions of $\varphi$_p and m_p when ε = 0, i.e., in the absence of any gradient in u_p along the x^*–coordinate.

FIG. 1.

(a) Schematic of the FFZE fractionation process as described in this article. (b) Top view of a FFZE device relevant to the mathematical analysis presented in this work.

Besides seeking the leading order analytic solution to σ² in the presence of a small fluid shear across the width of an FFZE chamber, Monte Carlo simulations were also performed to validate these results as well as establish the length and time scales over which the predictions of Eqs. (4) and (5) are applicable. These simulations involved the tracking of 10⁶ pointlike particles between two parallel plates which migrated based on a velocity field given by $\overrightarrow{u}=u_t{ê}_x+u_p{ê}_z$ coupled to a random walk diffusion model, where ${ê}_x$ and ${ê}_z$ denote the unit vectors along the x- and z-coordinates, respectively. The noted migration was simulated at each time step (of size Δt) through random particle displacement along the x-, y-, and z-coordinates by an amount governed by a Gaussian distribution with mean value $\overrightarrow{u}\mathrm{\Delta }t$ and standard deviation $\sqrt{2D\mathrm{\Delta }t}$. In addition, the effects of the no-slip and zero flux boundary conditions were captured by appropriately reflecting the molecules off the channel walls. To simplify the analysis, a parallel plate system separated by a unit distance was chosen in the simulations with the domain extending 2000 units along the direction of the pressure-driven flow (0 ≤ z^* ≤ 2000). The effect of the channel sidewalls on stream dispersion was eliminated by leaving the separation chamber unbounded along the x-direction. Moreover, the spatial variance of the injected stream was chosen to be at least 10⁴ times smaller than its steady state value at the channel outlet to render this contribution to stream broadening negligible relative to other sources. Furthermore, the spatial extent of the separation chamber along the z–direction was chosen to be at least 8 times larger than the product of the axial stream velocity (U) and the characteristic diffusion time scale (d²/4D) in the system, i.e., L ≥ 2Ud²/D, to ensure that the stream width at the channel outlet reached its equilibrium value. The initial distribution of the particles (i.e., at t = 0) was determined by a Gaussian distribution around the center of mass for the sample stream predicted at steady state with a spatial variance equal to that of the injected stream. The particles exiting the simulation domain at z^* = 2000 were conserved by reintroducing them back into the FFZE compartment through its inlet at z^* = 0 with their starting x and y coordinates determined by the distribution chosen for the initial condition. The initial z-positions $\left(z_0^{*}\right)$ for these reintroduced particles were set equal to the distance they had migrated beyond the outlet, i.e., $z_0^{*}=z^{*}-L$ in the previous time step. The Monte Carlo simulation described here was run over a time period equal to 5 times the larger of the advection and diffusion time scales in the system, i.e., L/U and d²/(4D), respectively, ensuring that the stream variance attained its steady state value. In-house written computer codes run using the MATLAB package were used to realize the simulations described above.

III. RESULTS AND DISCUSSION

As has been noted previously, the solution to the ε⁰ order terms, i.e., $\varphi$_p0 and m_p0, in the present analysis corresponds to the case of an ideal FFZE assay and therefore may be borrowed for p = 0, 1, and 2 from the published literature⁷ as

$$\begin{array}{ccc}\hfill {\varphi }_{00}& =\hfill & m_{00}=\delta ,{\varphi }_{10}=\frac{P{e}_x\delta }{P{e}_z}{z}^{*}+P{e}_x\delta \left(\frac{y^{*2}}{4}-\frac{y^{*4}}{2}-\frac{7}{480}\right),\hfill \\ \hfill m_{10}& =\hfill & \frac{P{e}_x\delta }{P{e}_z}{z}^{*},\hfill \\ \hfill {\varphi }_{20}& =\hfill & \frac{P{e}_x^2\delta }{P{e}_z^2}{z}^{*2}+\left[\frac{P{e}_x^2\delta }{P{e}_z}\left(\frac{y^{*2}}{2}-y^{*4}-\frac{11}{560}\right)+\frac{2\delta }{P{e}_z}\left(1+\frac{P{e}_x^2}{P{e}_z^2}\right)\right]z^{*}\hfill \\ \hfill & \hfill & +\frac{{\delta }^3}{12}+f\left(y^{*}\right),\hfill \\ \hfill m_{20}& =\hfill & \frac{P{e}_x^2\delta }{P{e}_z^2}{z}^{*2}+\left[\frac{P{e}_x^2\delta }{105P{e}_z}+\frac{2\delta }{P{e}_z}\left(1+\frac{P{e}_x^2}{P{e}_z^2}\right)\right]z^{*}+\frac{{\delta }^3}{12},\hfill \end{array}$$

where the function f(y^*) equals

$$\begin{array}{ccc}\hfill & \hfill & P{e}_x^2\delta \left(\frac{3y^{*8}}{28}-\frac{7y^{*6}}{60}+\frac{103y^{*4}}{3360}-\frac{y^{*2}}{6720}-\frac{253}{1612800}\right)\hfill \\ \hfill & \hfill & +\delta \left(1+\frac{P{e}_x^2}{P{e}_z^2}\right)\left(\frac{y^{*2}}{2}-y^{*4}-\frac{7}{240}\right).\hfill \end{array}$$ (6)

Besides the mathematical expressions included in Eq. (6), the present analysis also requires a solution to $\varphi$₃₀ which has been determined by solving the differential equation

$$-3P{e}_x{\varphi }_{20}+P{e}_{z}g\frac{\partial {\varphi }_{30}}{\partial z^{*}}=6{\varphi }_{10}+\frac{{\partial }^2{\varphi }_{30}}{\partial y^{*2}}+\frac{{\partial }^2{\varphi }_{30}}{\partial z^{*2}},$$ (7)

yielding a solution

$$\begin{array}{ccc}\hfill {\varphi }_{30}& =\hfill & \frac{P{e}_x^3\delta }{P{e}_z^3}{z}^{*3}+\left[\right.\frac{P{e}_x^3\delta }{P{e}_z^2}\left(\frac{3y^{*2}}{4}-\frac{3y^{*4}}{2}-\frac{17}{1120}\right)\hfill \\ \hfill & \hfill & +\frac{6P{e}_x\delta }{P{e}_z^2}\left(1+\frac{P{e}_x^2}{P{e}_z^2}\right)\left]\right.z^{*2}+h_1{z}^{*}+h_2,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill h_1& =\hfill & \frac{P{e}_x^3\delta }{P{e}_z}\left(\frac{9y^{*8}}{28}-\frac{7y^{*6}}{20}+\frac{87y^{*4}}{1120}+\frac{3y^{*2}}{448}-\frac{21881}{41395200}\right)\hfill \\ \hfill & \hfill & +\frac{P{e}_x{\delta }^3}{4P{e}_z}+\frac{2P{e}_x^3\delta }{35P{e}_z^3}+\frac{P{e}_x\delta }{P{e}_z}\left(1+\frac{P{e}_x^2}{P{e}_z^2}\right)\left(3y^{*2}-6y^{*4}-\frac{33}{280}\right)\hfill \\ \hfill & \hfill & +\frac{12P{e}_x\delta }{P{e}_z^3}\left(1+\frac{P{e}_x^2}{P{e}_z^2}\right).\hfill \end{array}$$ (8)

Interestingly, it turns out that the quantity m₂₁, and subsequently σ², only depends on the derivative ∂$\varphi$₃₀/∂z^* which is unaffected by the function h₂. As a result, the evaluation of h₂ has been omitted in the present work. Nevertheless, a mathematical expression for m₃₀ has been derived by integrating $\varphi$₃₀ over the y^*-coordinate and setting $m_{30}={\int }_{-1/2}^{1/2}{h}_{2}d{y}^{*}$ at z^* = 0, yielding

$$\begin{array}{ccc}\hfill m_{30}& =\hfill & \frac{P{e}_x^3\delta }{P{e}_z^3}{z}^{*3}+\left[\frac{P{e}_x^3\delta }{35P{e}_z^2}+\frac{6P{e}_x\delta }{P{e}_z^2}\left(1+\frac{P{e}_x^2}{P{e}_z^2}\right)\right]z^{*2}\hfill \\ \hfill & \hfill & +\frac{P{e}_x\delta }{P{e}_z}\left[\frac{29P{e}_x^2}{80850}+\left(\frac{2}{35}+\frac{12}{P{e}_z^2}\right)\left(1+\frac{P{e}_x^2}{P{e}_z^2}\right)+\frac{2P{e}_x^2}{35P{e}_z^2}+\frac{{\delta }^2}{4}\right]z^{*}\hfill \\ \hfill & \hfill & +{\int }_{-1/2}^{1/2}{h}_{2}d{y}^{*}.\hfill \end{array}$$ (9)

Proceeding further, the mathematical expression governing $\varphi$₀₁ may be determined by matching the ε order terms in Eq. (4) for p = 0, leading to the equation

$$P{e}_{z}g\frac{\partial {\varphi }_{01}}{\partial z^{*}}+P{e}_{z}g\frac{\partial {\varphi }_{10}}{\partial z^{*}}=\frac{{\partial }^2{\varphi }_{01}}{\partial y^{*2}}+\frac{{\partial }^2{\varphi }_{01}}{\partial z^{*2}}.$$ (10)

Equation (10) may be shown to yield an analytic solution of the form $\varphi$₀₁ = −(Pe_xδ/Pe_z)z^* + Pe_xδ/210, which also implies m₀₁ = $\varphi$₀₁ as $\varphi$₀₁ does not have any dependence on y^*. In this situation, the leading order effect of ∂u_p/∂x on m₀ may be expressed as m₀ ≈ m₀₀ + εm₀₁ = δ + ε[Pe_xδ/210 − (Pe_xδ/Pe_z)z^*] which indicates a decrease in the amount of analyte molecules at any channel cross section, i.e., x–y plane, with increasing z^* for ε > 0 (see Fig. 2). This decrease occurs as the analyte stream is subjected to higher pressure-driven velocities as it migrates laterally to larger values of x^* during its transit through the FFZE chamber. The noted acceleration causes the analyte stream to dilute with increasing z^* in order to then preserve mass balance in the system. Expectedly, the magnitude of this dilution is greater for larger values of Pe_x/Pe_z as the analyte stream is rapidly migrated to regions with higher pressure-driven velocities under these conditions. Note that because the analysis presented here is only valid for small variations in the axial pressure-driven velocity, i.e., εx^* ≪ 1, the quantity εPe_xz^*/Pe_z must also be much less than unity as the lateral migration distance of the analyte stream may be approximated as m₁ ≈ Pe_xz^*/Pe_z to the leading order. This constraint ensures that m₀ will always assume positive values as is required based on its definition in this work. Equation (10) also shows that if the mass flux at any x–y plane is set equal to a value independent of ε as in the present analysis, i.e., δ, the amount of solute molecules at the channel entrance, i.e., m₀ at z^* = 0, will be influenced by ∂u_p/∂x and will therefore have an ε order contribution to it. The term εPe_xδ/210 quantifies this contribution which is the additional material that needs to be introduced into the system to maintain the desired flux. Now, matching the ε order terms in the equation governing $\varphi$₁, it is possible to show that

$$-P{e}_x{\varphi }_{01}+P{e}_{z}g\frac{\partial {\varphi }_{11}}{\partial z^{*}}+P{e}_{z}g\frac{\partial {\varphi }_{20}}{\partial z^{*}}=\frac{{\partial }^2{\varphi }_{11}}{\partial y^{*2}}+\frac{{\partial }^2{\varphi }_{11}}{\partial z^{*2}},$$ (11)

which again admits an analytic solution given by

$${\varphi }_{11}=-\frac{3P{e}_x^2\delta }{2P{e}_z^2}{z}^{*2}+\left[\frac{P{e}_x^2\delta }{P{e}_z}\left(\frac{y^{*4}}{2}-\frac{y^{*2}}{4}+\frac{7}{480}\right)-\frac{5P{e}_x^2\delta }{P{e}_z^3}-\frac{2\delta }{P{e}_z}\right]z^{*}+q_1,$$

where

$$\begin{array}{ccc}\hfill q_1& =\hfill & P{e}_x^2\delta \left(\frac{3y^{*8}}{56}-\frac{3y^{*6}}{40}+\frac{227y^{*4}}{6720}-\frac{83y^{*2}}{13440}+\frac{109}{460800}\right)\hfill \\ \hfill & \hfill & +\frac{P{e}_x^2\delta }{P{e}_z^2}\left(\frac{3y^{*4}}{2}-\frac{3y^{*2}}{4}+\frac{7}{160}\right).\hfill \end{array}$$ (12)

In obtaining the function q₁, the quantity m₁₁ was assumed to equal zero at the channel entrance (z^* = 0) which corresponded to setting the integral ${\int }_{-1/2}^{1/2}{q}_{1}d{y}^{*}=0$ in the present analysis. Subsequently, Eq. (12) may be integrated over the y^*-coordinate to obtain $m_{11}=-3P{e}_x^2\delta z^{*2}/\left(2P{e}_z^2\right)-\left(2\delta /P{e}_z+5P{e}_x^2\delta /P{e}_z^3\right)z^{*}$, which then provides an estimate for the leading order effect of ∂u_p/∂x on the normalized lateral equilibrium position of the analyte stream as

$$\begin{array}{ccc}\hfill \frac{m_1}{m_0}& \approx \hfill & \frac{m_{10}}{m_{00}}+\epsilon \left(\frac{m_{11}}{m_{00}}-\frac{m_{10}{m}_{01}}{m_{00}^2}\right)\hfill \\ \hfill & =\hfill & \frac{P{e}_x}{P{e}_z}{z}^{*}-\epsilon \left[\frac{P{e}_x^2}{2P{e}_z^2}{z}^{*2}+\left(\frac{P{e}_x^2}{210P{e}_z}+\frac{2}{P{e}_z}+\frac{5P{e}_x^2}{P{e}_z^3}\right)z^{*}\right].\hfill \end{array}$$ (13)

Equation (13) predicts the fluid shear component ∂u_p/∂x to reduce the lateral drift in the stream position for positive values of ε by an amount which varies quadratically with z^* to the leading order. This reduction occurs as the analyte stream is subjected to higher pressure-driven velocities in the present system compared to the case when ε = 0 which decreases the residence time of the solute molecules in the FFZE chamber and thereby their lateral migration distance in it. The noted quadratic dependence also implies an increasingly dominating effect of ∂u_p/∂x on m₁/m₀ for long separation channels [see Fig. 3(a)]; however, Eq. (13) strictly remains valid for εPe_xz^*/Pe_z ≪ 1 as has been noted previously. To understand the physical origin for the ε order terms in Eq. (13), the motion of an individual particle migrating through the FFZE chamber considered here can be insightful. The trajectory of this entity ($x_t^{*}$) can be shown to be governed by the equation dz^*/dx^* = (Pe_z/Pe_x) (1 + εx^*) after ignoring its diffusive behavior, which yields the following solution for the lateral migration distance to the leading order in ε:

$$x_t^{*}=\underset{m_{10}}{\underbrace{\frac{P{e}_x}{P{e}_z}{z}^{*}}}-\epsilon \underset{m_{11}}{\underbrace{\frac{P{e}_x^2}{2P{e}_z^2}{z}^{*2}}}.$$ (14)

As may be seen, Eq. (14) captures the stream trajectory in the absence of any fluid shear component ∂u_p/∂x as well as the ε order term with the quadratic dependence on z^*. In this situation, the remaining contribution to m₁/m₀ in Eq. (13), i.e., $-\epsilon \left[2/P{e}_z+5P{e}_x^2/P{e}_z^3+P{e}_x^2/\left(210P{e}_z\right)\right]z^{*}$, may be attributed to that arising from the nonuniform distribution of the analyte molecules around the center of mass of the sample stream. Now, for a typical microfluidic FFZE system, the quantities Pe_x and Pe_z tend to be on the order of 20 and 1000, respectively, as when μ = 2 × 10⁻⁴ V cm⁻² s⁻¹, E = 200 V cm⁻¹, d = 20 μm, U = 2 cm s⁻¹, and D = 4 × 10⁻⁶ cm² s⁻¹. Under these conditions, the quadratic term dominates the leading order contribution from ∂u_p/∂x on m₁/m₀ for z ≫ 10d ≈ 0.2 mm. Remember that for the operating parameters chosen above, the present analysis remains valid for z ≪ 10²d/ε ≈ 20 cm assuming ε = 0.01, suggesting Eq. (14) to well describe the lateral stream migration for a majority of the FFZE channel. For the choice of ε, Pe_x, and Pe_z as noted above, the quadratic term in Eq. (14) corrects the quantity m₁/m₀ by about 10% in the presence of a fluid shear component ∂u_p/∂x when z = 2 cm [see Fig. 3(b)].

FIG. 2.

Concentration distribution along the axis of the FFZE chamber estimated based on the analytic theory described in the article. ε was set equal to 0.01 for obtaining these results.

FIG. 3.

(a) Magnitude of the various terms in the ε order contribution to m₁/m₀ as predicted by Eq. (13) for Pe_z = 2000 and Pe_x = 10 as a function of the axial position in the FFZE chamber. (b) Variation in the ratio of the ε order contribution to m₁/m₀ over m₁₀/m₀₀, i.e., magnitude of m₁/m₀ when ε = 0, with the axial position in the FFZE chamber. ε was set equal to 0.01 for obtaining these results.

The influence of the fluid shear component ∂u_p/∂x on zone broadening may be similarly assessed by matching the ε order terms in the equation governing $\varphi$₂ to arrive at

$$-2P{e}_x{\varphi }_{11}+P{e}_{z}g\frac{\partial {\varphi }_{21}}{\partial z^{*}}+P{e}_{z}g\frac{\partial {\varphi }_{30}}{\partial z^{*}}=2{\varphi }_{01}+\frac{{\partial }^2{\varphi }_{21}}{\partial y^{*2}}+\frac{{\partial }^2{\varphi }_{21}}{\partial z^{*2}},$$ (15)

which yields a solution of the form

$$\begin{array}{ccc}\hfill {\varphi }_{21}& =\hfill & -\frac{2P{e}_x^3\delta }{P{e}_z^3}{z}^{*3}-\left[\right.\frac{P{e}_x^3\delta }{P{e}_z^2}\left(\frac{3y^{*2}}{4}-\frac{3y^{*4}}{2}-\frac{17}{1120}\right)\hfill \\ \hfill & \hfill & +\frac{17P{e}_x^3\delta }{P{e}_z^4}+\frac{9P{e}_x\delta }{P{e}_z^2}\left]\right.z^{*2}+r_1{z}^{*}+r_2,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill r_1\left(y^{*}\right)& =\hfill & \frac{P{e}_x^3\delta }{P{e}_z}\left(\frac{y^{*4}}{24}-\frac{y^{*6}}{30}-\frac{7y^{*2}}{480}+\frac{107}{201600}\right)\hfill \\ \hfill & \hfill & +\frac{P{e}_x\delta }{P{e}_z}\left(3y^{*4}-\frac{3y^{*2}}{2}+\frac{23}{336}\right)+\frac{P{e}_x^3\delta }{P{e}_z^3}\left(11y^{*4}-\frac{11y^{*2}}{2}+\frac{57}{560}\right)\hfill \\ \hfill & \hfill & -\frac{P{e}_x{\delta }^3}{4P{e}_z}-\frac{30P{e}_x\delta }{P{e}_z^3}-\frac{46P{e}_x^3\delta }{P{e}_z^5}.\hfill \end{array}$$ (16)

It must be noted that because the current analysis assumes m₂₁ = γ at the channel inlet, i.e., z^* = 0, the integral ${\int }_{-1/2}^{1/2}{r}_{2}d{y}^{*}=\gamma$. Now, as the spatial variance (σ²) of the analyte stream only depends on this integral across the channel gap rather than the exact functional form of r₂, a solution to the noted function (r₂) has not been sought in the present work. Equation (16) may be subsequently integrated over y^* to then arrive at an expression for m₂₁ given by

$$\begin{array}{ccc}\hfill m_{21}& =\hfill & -\frac{2P{e}_x^3\delta }{P{e}_z^3}{z}^{*3}-\left(\frac{P{e}_x^3\delta }{35P{e}_z^2}+\frac{9P{e}_x\delta }{P{e}_z^2}+\frac{17P{e}_x^3\delta }{P{e}_z^4}\right)z^{*2}\hfill \\ \hfill & \hfill & -\left(\right.\frac{P{e}_x^3\delta }{4200P{e}_z}+\frac{2P{e}_x\delta }{105P{e}_z}+\frac{23P{e}_x^3\delta }{105P{e}_z^3}+\frac{30P{e}_x\delta }{P{e}_z^3}\hfill \\ \hfill & \hfill & +\frac{46P{e}_x^3\delta }{P{e}_z^5}+\frac{P{e}_x{\delta }^3}{4P{e}_z}\left)\right.z^{*}+\gamma ,\hfill \end{array}$$ (17)

which then allows the estimation of the normalized spatial stream variance (σ^*2 = σ²/d²) as

$$\begin{array}{ccc}\hfill {\sigma }^{*2}& =\hfill & \frac{m_2}{m_0}-{\left(\frac{m_1}{m_0}\right)}^2\approx \frac{m_{20}}{m_{00}}-\frac{m_{10}^2}{m_{00}^2}\hfill \\ \hfill & \hfill & +\epsilon \left(\frac{m_{21}}{m_{00}}-\frac{m_{01}{m}_{20}}{m_{00}^2}-2\frac{m_{10}{m}_{11}}{m_{00}^2}+2\frac{m_{01}{m}_{10}^2}{m_{00}^3}\right)\hfill \\ \hfill & =\hfill & \left[\frac{P{e}_x^2}{105P{e}_z}+\frac{2}{P{e}_z}\left(1+\frac{P{e}_x^2}{P{e}_z^2}\right)\right]z^{*}+\frac{{\delta }^2}{12}\hfill \\ \hfill & \hfill & -\epsilon \left[\left(\frac{P{e}_x^3}{70P{e}_z^2}+\frac{3P{e}_x}{P{e}_z^2}+\frac{5P{e}_x^3}{P{e}_z^4}\right)z^{*2}\right.\hfill \\ \hfill & \hfill & +\left.\frac{P{e}_x}{P{e}_z}\left(\frac{P{e}_x^2}{3528}+\frac{1}{35}+\frac{8P{e}_x^2}{35P{e}_z^2}+\frac{30}{P{e}_z^2}+\frac{46P{e}_x^2}{P{e}_z^4}+\frac{{\delta }^2}{6}\right)z^{*}+\frac{P{e}_x{\delta }^2}{2520}-\frac{\gamma }{\delta }\right].\hfill \\ \hfill & \hfill & \hfill \end{array}$$ (18)

Equation (18) shows that the fluid shear component ∂u_p/∂x again introduces a leading order correction to the stream variance that is both subtractive and varies quadratically with the axial coordinate (z^*) as in the case of the lateral stream position. The noted decrease in stream width (for ε > 0) mostly occurs as the analyte molecules residing along the edge located at x^* > m₁/m₀ are accelerated in the axial direction to a greater extent compared to those populating the opposite edge. While the δ²/12 − ε(Pe_xδ²/2520 − γ/δ) term in Eq. (18) may be identified as the contribution from the injection width, the physical origin for the other contributions is not as evident. In particular, it is important to point out that the residence time of the analyte stream (t) in the FFZE device considered here depends on its lateral migration distance and therefore is a function of Pe_x in addition to depending on Pe_z. In this situation, it becomes more appropriate to assess the stream broadening in terms of the Taylor-Aris dispersion coefficient (K) in the system which relates to the stream variance as K = D(dσ²/dt)/2 ≈ DPe_z(1 + εPe_xz^*/Pe_z) (dσ^*2/dz^*)/2. Upon incorporating this information into Eq. (18) and making the relevant rearrangement, it is then possible to express

$$\begin{array}{ccc}\hfill \frac{K}{D}& =\hfill & 1+\frac{P{e}_x^2}{210}+\frac{P{e}_x^2}{P{e}_z^2}-\epsilon \left[\left(\frac{P{e}_x^3}{105P{e}_z}+\frac{2P{e}_x}{P{e}_z}+\frac{4P{e}_x^3}{P{e}_z^3}\right)z^{*}\right.\hfill \\ \hfill & \hfill & +\left.\left(\frac{P{e}_x^3}{7056}+\frac{P{e}_x}{70}+\frac{4P{e}_x^3}{35P{e}_z^2}+\frac{P{e}_x{\delta }^2}{12}+\frac{15P{e}_x}{P{e}_z^2}+\frac{23P{e}_x^3}{P{e}_z^4}\right)\right].\hfill \\ \hfill & \hfill & \hfill \end{array}$$ (19)

As may be seen from the above equation, the leading order correction to the Taylor-Aris dispersion coefficient in the presence of a fluid shear component ∂u_p/∂x is again subtractive consistent with the predictions of Eq. (18). Moreover, this corrective term varies linearly with the axial distance (z^*) but is never greater in magnitude than the ε⁰ order term due to the constraint εPe_xz^*/Pe_z ≪ 1 in the present analysis. The magnitudes of each of the terms in the ε order contribution to K/D as predicted by Eq. (19) have been compared in Figs. 4(a) and 4(b) which shows the $P{e}_x^3/\left(105P{e}_z\right)$ term to dominate the coefficient multiplying z^* under experimental conditions commonly applicable to FFZE assays. The ε order contribution independent of z^* on the other hand is seen to be dominated by the $P{e}_x^3/7056$ term under similar situations. Also, for a choice of Pe_x = 20 and Pe_z = 1000, the ε order term in Eq. (19) is seen to be dominated by the contribution varying linearly with z^* for z ≫ 12d ≈ 0.24 mm [see Fig. 4(c)]. Moreover, the ε order term in Eq. (19) is predicted to correct K/D by over 22% in the presence of a fluid shear component ∂u_p/∂x at z^* = 1000 and this correction increases monotonically for larger values of Pe_x [see Fig. 4(d)]. While this correction is subtractive for ε > 0, it adds to the overall stream broadening when ε assumes negative values. The noted increase/decrease in the Taylor-Aris dispersivity due to the fluid shear component ∂u_p/∂x can be further shown to be comparable to the contributions arising from other nonidealities typically encountered in FFZE systems. For example, the presence of a transverse pressure-driven backflow of magnitude equal to 20% of the transverse electrophoretic flow in an FFZE device may be shown to increase K/D by about 56% for a given value of Pe_x and Pe_z.¹² On the other hand, for Pe_z = 1000 and electric field strengths corresponding to a value of Pe_x = 20, Joule heating is expected to alter stream dispersion by less than 0.1%.¹¹ While making these comparisons, it is important to remember however that the extents to which the various nonidealities influence the magnitude of K/D in an FFZE system are not completely independent of each other. In this situation, an accurate estimate of the contribution to K/D from the fluid shear component ∂u_p/∂x in the presence of other nonidealities would require an extensive modeling effort that is beyond the scope of this study. Nevertheless, the analysis included in this manuscript offers a leading order estimate for this contribution in the absence or presence of other nonidealities that influence stream dispersion in FFZE assays.

FIG. 4.

(a) Magnitude of the various terms (before multiplying with ε) constituting the coefficient of z^* in the ε order contribution to K/D as predicted by Eq. (19). (b) Magnitude of the various terms (before multiplying with ε) independent of z^* in the ε order contribution to K/D as predicted by Eq. (19). (c) Magnitude of the terms constituting the ε order contribution to K/D in Eq. (19) when Pe_x = 20. (d) Ratio of the various ε order terms to the contribution independent of ε in Eq. (19) when z^* = 1000. For all of the subfigures above, Pe_z = 1000 and ε = 0.01.

It must be further noted that the local Péclet number based on the flow speed in the z-direction around the solute stream varies with x^* and thereby the lateral stream position due to the dependence of u_p on this spatial coordinate. The effect of this variation in Pe_z however may be factored out by defining a modified Péclet number ${\overline{Pe}}_z=P{e}_z\left(1+\epsilon x^{*}/2\right)\approx P{e}_z+\epsilon P{e}_x{z}^{*}/2$ obtained by time averaging its magnitude over the period spent by the analyte stream in the FFZE chamber. Upon re-expressing K/D in terms of ${\overline{Pe}}_z$, it is then possible to show

$$\begin{array}{ccc}\hfill \frac{K}{D}& =\hfill & 1+\frac{P{e}_x^2}{210}+\frac{P{e}_x^2}{{\overline{Pe}}_z^2}-\epsilon \left[\left(\frac{P{e}_x^2}{105{\overline{Pe}}_z}+\frac{2P{e}_x}{{\overline{Pe}}_z}+\frac{3P{e}_x^3}{{\overline{Pe}}_z^3}\right)z^{*}\right.\hfill \\ \hfill & \hfill & +\left.\left(\frac{P{e}_x^3}{7056}+\frac{P{e}_x}{70}+\frac{8P{e}_x^3}{70{\overline{Pe}}_z^2}+\frac{P{e}_x{\delta }^2}{12}+\frac{15P{e}_x}{{\overline{Pe}}_z^2}+\frac{23P{e}_x^3}{{\overline{Pe}}_z^4}\right)\right].\hfill \\ \hfill & \hfill & \hfill \end{array}$$ (20)

The analytic results presented above were further validated with Monte Carlo simulations which provided a more comprehensive description of the sample concentration profile in the FFZE chamber with a fluid shear component ∂u_p/∂x (see Fig. 5). These simulations showed that for the initial condition chosen in the study, the solute distribution at the channel entrance remains asymmetric around its center of mass [see Figs. 5(c) and 5(d)] but becomes increasingly symmetric going downstream [see Fig. 5(a)]. In addition, the time scale needed for attaining the steady state value of σ² was determined by diffusion across the channel depth, i.e., d²/4D [see Fig. 6(a)]. Interestingly, however, the Taylor-Aris dispersion limit provided a reasonable estimate for the stream variance (σ²) throughout the FFZE chamber [see Fig. 6(b)].

FIG. 5.

(a) Spatial distribution of analyte molecules in an FFZE chamber at steady state as estimated based on a Monte Carlo simulation performed for a choice of Pe_x = 40 and Pe_z = 100. Steady state distribution of analyte molecules along the width of the FFZE chamber at (b) z^* = 50, (c) z^* = 20, and (d) z^* = 10. Note that the symmetry in the molecular distribution improves downstream of the channel inlet, i.e., with increase in z^*. The vertical dotted lines in subfigures (b)–(d) denote the mean x^*-position for the distribution.

FIG. 6.

(a) Temporal evolution of the normalized stream variance as predicted by the Monte Carlo simulations and analytic theory for a choice of Pe_x = 40 and Pe_z = 100. (b) Spatial evolution of the stream variance downstream of the sample inlet, i.e., z = 0, at a normalized time of t^* = 1.2 as predicted by the Monte Carlo simulations and analytic theory again for a choice of Pe_x = 40 and Pe_z = 100.

Finally, it must be pointed out that while the mathematical formulation described here has been applied to evaluating the spatial variance of a sample stream in an FFZE system in the presence of small steady fluid shear component ∂u_p/∂x, it can be readily extended to calculating the higher order moments of the analyte concentration profile, e.g., skewness, kurtosis, etc. Nevertheless, the present analysis strictly remains valid only in situations when the transit time for the analyte molecules in the separation chamber is much larger than the characteristic diffusion time scale, i.e., $L/U\gg d^2/4D$. Moreover, the mathematical model described here ignores the influence of any other nonideality (e.g., Joule heating, electromigration effects, etc.) on sample transport in the FFZE system and hence on the hydrodynamic dispersion component of the analyte stream. Furthermore, additional stream broadening introduced by abrupt changes in the channel cross section at the entrance and exit of the separation chamber has not been accounted for in the present analysis. Fortunately, for a majority of microfluidic FFZE systems, these effects tend to be insignificant, rendering the current mathematical model highly useful in assessing the performance of free-flow zone electrophoretic separations.

IV. CONCLUSIONS

In summary, the present work outlines a useful theoretical framework for quantifying stream dispersion in FFZE systems with a small and steady fluid shear component ∂u_p/∂x. It predicts that this additional fluid shear can shift the center of mass of the sample zone either way of its equilibrium position relative to that realized under ideal conditions depending on the sign of ε, i.e., the correction is positive for ε < 0 and vice versa. This equilibrium position is also predicted to vary quadratically with the axial coordinate of the analyte zone, making the modification increasingly more significant going downstream of the FFZE chamber. In addition, the hydrodynamic dispersion contribution to the zone width is seen to be significantly corrected by such additional fluid shear again depending on the sign of ε, following the same trend noted for the equilibrium position of the stream. More importantly, the leading order contribution to the zone variance from ∂u_p/∂x is shown to vary quadratically with the axial distance traversed by the analyte stream as opposed to the classical linear variation known for all other stream broadening contributions in FFZE systems. The analysis presented in this work, however, is strictly valid under conditions in which εx^* ≪ 1 when analytic expressions for $\varphi$_i may be obtained based on the solutions to the same variable in the absence of any ∂u_p/∂x in an FFZE system. In this situation, the above analysis remains valid in the entire FFZE chamber of width W provided εW/(2d) ≪ 1.