Assessing Performance of Irregular Microvoids in Electrophoresis

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Industrial & Engineering Chemistry Research

Assessing Performance of Irregular Microvoids in Electrophoresis Separations

Jyothirmai J. Simhadri1, Holly A. Stretz1, Mario A. Oyanader2, and Pedro E. Arce1*

1. Department of Chemical Engineering, Tennessee Technological University, Cookeville, TN-USA 2. Department of Chemical Engineering, California Baptist University 8432 Magnolia Ave, Riverside, CA 92504

To whom correspondence should be addressed. Dr. Pedro E. Arce. Tel: +1 (931) 372 3189. Fax: +1 (931) 372 6352- Email: [email protected]

ACS Paragon Plus Environment


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Abstract

Within the last decade, novel gel materials with a modified internal morphology have been synthesized and are available for a wide range of applications including drug delivery, separation of biomolecules with relevance in clinical diagnostics, electrokinetic pumping, and sensors

1-3

. One way to achieve this internal modification is to embed nanoparticles into the

polymer matrix, so that when an electrical field is applied to the system, the presence of these particles can change the biomacromolecule transport through the gel, possibly altering the separation 4. As a result, gaining a very deep understanding of the role that these nanoparticles can have on the separation efficiency in terms of electrophoresis techniques becomes very important. To the best of our knowledge, this contribution is the first effort that examines the role of morphology (in the form of a axially-diverging microdomain) on the prediction of optimal separation times for two protein models when they are subjected to electrophoresis.

The

research conducted predicts that the optimal separation time of a short, straight pore yields the best resolution in this type of electrophoretic separation.

Keywords: Electrophoresis, Nanocomposite, Optimal Time, Separation, Poiseuille flow


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1.

Motivation and Introduction Electrophoresis is one of the most widely accepted high resolution techniques for the

separation of numerous substances such as proteins, nucleic acids, pharmaceuticals, metabolites, and peptides. It has been shown previously that electrophoresis in a gel (which can act as a molecular sieve5) gives much higher resolution than electrophoresis in a free solution 6. Separation of the biomolecules is important for the study of genes, development of new pharmaceuticals, clinical diagnostics, and tissue scaffolding

2, 7

. In gel-electrophoresis, the gel

matrix has pores or, in general, microvoids of different size distributions that either can be interconnected or be in isolation; furthermore, these pores or microvoids can also provide a tortuous path through which the biomolecule migrates in the presence of electric field8. Historically, the pore or domain morphology has been modified by changing the cross-link density or monomer concentration, which also affects the electrophoretic mobility of the biomolecules

(Stellwagen et al.)9. Synthesis of novel gel materials with modified internal

morphology has a wide range of applications including drug delivery, biomolecules separation and sensor applications, the latter of which have been the focus within the last decade. For example, Huang et al.10 has shown that the separation of Apolipoprotein and Complement C3, which have the same physicochemical properties, can be achieved by modifying the internal morphology of the gel with the addition of carbon nanotubes. In addition,Thompson et al.11 have modified polyacrylamide gel with gold nanoparticles and has found a potential improvement in separation of proteins. The different routes for the modification of gel morphology and different models associated with the evaluation of electrophoretic mobility for different internal morphologies have been discussed in detail in a review by Simhadri et al.3.



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Several contributions have been reported in the literature related to electrophoretic transport in microvoids of varying cross sections (such as rectangular, cylindrical, and annular) for applications in bioseparations and microfluidics. For example, in SDS-gel electrophoresis the gel media provides a differential medium for the lower molecular weight analyte resulting in broader, less distinct bands in a linear voltage gradient when compared to high molecular weight analyte for which the smaller pores are inaccessible12. This effect can be overcome by a nonlinear voltage gradient which can be achieved by modifying the cylindrical shape of the gel pore morphology to a diverging conical/wedge shape. At equivalent loading of protein samples, the resolution in diverging domains is much higher than uniform gels12. Boncinelli et al.13 used an agarose slab of steadily increasing thickness to generate a gradient of decreasing electric-field strength which improved the resolution of DNA fragments. Ross et al.14 used diverging microchannels to reduce the effects of dispersion as they created a self-sharpening front on the analyte/solute band as it traveled down the channel. Straight and diverging microchannels were compared across the initial injection width of the dye as the primary factor for determining the best separation resolution. These researchers found that diverging channels gave the best resolution when the injection width was relatively wide. On the other hand, when the injection width was small, then straight channels gave the best resolutions. This finding is important since the quality of samples may play a role in the separation efficiency. Ghosal et al.15 studied the transport of a polyelectrolyte through a nanopore with variable axial cross section to examine the effects of an applied electric field in the axial direction. Expressions for the electrophoretic velocity of the polyelectrolyte in a gel through the use of electrostatics and hydrodynamics within a continuum mechanics framework were determined. Four different modes of transport of the polyelectrolyte were identified as a function of the dimensionless height of the channel and


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they also reported that the use of continuum hydrodynamics with water as the buffer, was valid for pores of sizes as low as tenths of a nanometer. Xuan et al.16 experimentally studied the effect of pressure on the electrophoretic motion and separation of polystyrene particles within a converging-diverging microchannel fabricated with poly (dimethylsiloxane) chips. They studied the dependence of the separation on electrical field strength, particle size, and morphology by using the ratio of the velocity of the particle at the throat of the device to that in a straight microchannel. Yariv et al.17studied the electrophoretic transport in rectangular channels of periodically varying cross section. They obtained expressions for the effective velocity and effective diffusivity within the channel based on the macrotransport theory. They found that for both small and large Peclet numbers their analysis yielded asymptotic results for the effective transport parameters. Yu et al.18,19 prepared gold nanotube membranes with controlled inner diameters and studied the electrophoretic transport of proteins in such membranes. They optimized the selectivity and flux of the proteins by studying the effect of a combination of parameters such as transmembrane potential, nanotube diameter, and also the pH. Berenzhkovskii et al.20,21 derived formulas to show the dependence of the transport coefficients (i.e., effective mobility and diffusivity) on driving force and geometric parameters of the tube based on mapping the particle random walk. Their results also show the potential impact of axially-varying shapes within a microvoid. The key focus of the paper is to numerically validate some of the trends found in the analytical approaches by relaxing some of the assumptions used by the researchers. A number of models containing regular geometrical pores with rectangular, cylindrical, spherical, and conical shapes have been developed for describing the structure of electrophoresis media22. Models with microvoids of rectangular shape are relevant for the current work. For



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example, Trinh et al.23 used a simple geometry of the capillary channels and found that the geometry could play a significant role in the study of transport properties relevant for the separation of biomacromolecules. In this contribution, the focus is on the role of Poiseuille flow and secondary electrical fields on the separation of bio-molecules in rectangular diverging microvoids. For the case of electrophoretic applications, this effort studies the application of the secondary orthogonal field as opposed to the usual axial applied field to enhance the separation. Previous contributions in the above mentioned aspect include, for example, Sauer et al.24, that studied (from a scalingmodeling point of view) the effect of an orthogonal field on the transport of solute in a Poiseuille flow regime using a straight capillary domain (α=0º) of rectangular geometry. Oyanader and Arce25 (as a follow up contribution) derived simpler expressions for the effective transport parameters by recognizing dependence on the orthogonal electrical field with geometrical scaling parameters of the system; this property leads to the non-requirement of numerical-assisted integrals. In the sections below, the description of the model formulation as well as the solution of the model by numerical-based approaches will be presented. Several illustrative results will be discussed. One of the objectives of this contribution is to assess the role of the divergent angle, α, in determining the optimal time of separations and, therefore, whether or not it improves the separation efficiency in conjunction with the orthogonal applied field. This piece of information is relevant in view of the discussion above and in regard to the recent separation potential of magnetic-aligned nanocomposite gels26. 2.

Model Description In Figure 1(a) the schematic of the diverging microvoid domain used in the calculations

is shown. The domain considered here is a two-dimensional rectangular geometry with the length


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L along the x-direction, height H(x) along the y-direction, and has an axially varying crosssection. As noted in Figure 1(a), H(x=0) = H/2 the cross-section in the axial direction is controlled by the divergence angle, α. Poiseuille flow, a laminar flow of a Newtonian fluid, between two parallel walls driven by a pressure gradient, is considered in this problem. For the case of α=0°, the microvoid morphology of Figure 1(a) reduces to the limiting case of the flow of an incompressible, Newtonian fluid/buffer between two parallel plates; here, the velocity profile along the axial direction is just a function of the height of the domain (which means vy=0 and vx=vx(y)) for the case of fully developed flow. However, for cases where α≠0°, the microvoid domain has axially varying cross-section the velocity profile in both x and y directions become important. In other words, the 1D flow characteristic is lost to become a 2D flow system. In this paper, the effective diffusion coefficient and effective velocities, for the straight microvoid, and the microdomain with axially varying cross section i.e., α≠0°, are compared; moreover, optimal separation times of two model species are presented for various values of the divergence angle in order to assess the effect of the diverge of the microvoid domain on the separation efficiency.

2.1. Governing Equations In order to determine the velocity profiles for the system under study, as shown in Figure 1, the Navier-Stokes equations27 that govern the steady two-dimensional flow of a typical incompressible flow of a Newtonian buffer are used28: Hydrodynamics: The 2D Navier-Stokes equations are as follows:

 ∂2v  ∂2 vx ∂P  = − + µ 2x + ∂x ∂y 2   ∂x

  

(1)

 ∂2vy ∂2vy ∂v y   ∂v y ∂P    ρ v x + vy  = − ∂y + µ  ∂x 2 + ∂y 2 ∂ x ∂ y   

   

(2)

 ∂v ∂v ρ v x x + v y x ∂x ∂y 



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and the continuity equation for incompressible flow:

∇.v =

∂v x ∂v y + =0 ∂x ∂y

(3)

where vx and vy are velocity in the x and y-directions, respectively, ρ is the density of the fluid, µ is the Newtonian viscosity, and P is the pressure. As mentioned above, key assumptions include that the fluid is Newtonian and the flow is incompressible; the system is under steadystate; and the hydrodynamic flow regime is pressure driven within two stationary walls i.e., Poiseuille flow. Since the cross-section of the microvoid is varying axially, the pressure difference is calculated as a function of axial direction, ∆P(x), for a given constant volumetric flow rate and is given as:

∆P(x) = −

12Q µL H 3 h(x) 3

(4)

where Q is the volumetric flow rate, L is microvoid length, H is the height, h(x) is the equation of the line representing the walls of the microvoid. For the fluid flow, a value of the pressure corresponding to a given flow rate Q is specified at the channel inlet. At the outlet, a constant pressure (atmospheric) is specified (i.e., a pressure gradient is imposed as a function of x-direction which can be determined from the flow rate Q). No-slip boundary conditions are imposed on all the remaining walls. The following is the mathematical representation of the above conditions:

p = constant

for

x = 0, ∀y

(5)

p = atmospheric

for

x = L, ∀y

(6)

v x ( x, y = h( x)) = 0, v y ( x, y = h( x)) = 0, ∀x

(7)

v x ( x, y = − h( x)) = 0, v y ( x, y = − h( x)) = 0, ∀x

(8)


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Electric Potential: Non-charged walls of the microvoid domain are assumed and, therefore, the Electro-Osmotic Flow (EOF) is negligible29; the applied electric potential ϕ, can be calculated by simply solving the Laplace electric potential equation with constant conductivity2,24:

+

=0

(9)

The electric potential model governed by the Laplace equation31 is subject to the following boundary conditions: A fixed potential drop is applied on the boundaries in the transverse direction32.

Convective-Diffusive-Electromigrative Molar Species Transport:

The molar species

continuity equation for flow in a two-dimensional rectangular microdomain and under an applied electric field with the no EOF assumption is given by the following equation2:

+ +

+ +

= μ + μ +

+

(10)

, , is the concentration of the i-th species, μ , , are mobility, valence number, and molecular diffusivity, respectively. is the Faraday constant, is the electrical potential. The velocity field can be obtained from the solution of the Navier-Stokes equations specified earlier.

Initial and Boundary Conditions: A mixture of two sample test solutes A and B (point particles), each having a fixed valence (z), mobility, diffusivity, and a given initial concentration are injected at the center of the inlet wall of microvoid. The mathematical representation of initial and boundary conditions are given below. The walls of the microvoid are assumed to be impermeable to the analyte/solute transport, and thus the no-flux boundary condition is applied and it yields:  =0 y = − h( x ) y = − h( x) y = − h( x )   ∂Ci ∂φ Di + v y Ci + µi Ci = 0  y = h ( x ) ∂y y = h ( x ) ∂y y = h ( x )  ACS Paragon Plus Environment Di

∂Ci ∂y

+ v y Ci

+ µi Ci

∂φ ∂y

(11)


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At the initial time, t=0, a Gaussian plug of solute is injected at the center of the microvoid

C0 C i (x, y, t = 0) = σ 0 2π

 x2 + y2 −  2σ 2 0 

e

   

(12)

Where, σ 0 is the initial injection standard deviation, and C0 is the initial concentration of solute.

3.

Model Solution The primary goal for the study of the electrophoretic system (with a diverging microvoid

domain) is to determine the effect of geometrical design parameters, as well as the operating parameters (such as the applied electric field), as these parameters govern the separation efficiency of the analytes. The most important parameter that measures the performance of the system is the resolution of peaks in a given chromatogram; this parameter helps to understand how the solute bands are placed apart so that separation can take place. Giddings et al.33 mathematically defined the resolution as a ratio of distance between the two adjacent solute bands to the function of band dispersion as: R=

x 2 − x1 4σ avg

(13)

In the above equation, x1 and x2 can be obtained from the first moments, whereas the average standard deviation of the two bands can be calculated from the second moments for the solute bands approximated as Gaussian distributions

33

(see Figure 1(b)). ypically, the desirable

resolution is 1.5, which corresponds to the “baseline” resolution. The resolution may be lower than desired due to, for example, the dispersion of solute/analyte in the microvoid, which, in general, could be a result of diffusion, microvoid geometry, and flow conditions, among other parameters.


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In order to evaluate the effective transport properties such as diffusivity and convective velocity, the statistical moment analysis is used to determine the location and shape of the Gaussian distribution. The nth moment about the mean is defined as follows 2,24,33. " , ! #"

≡

(14)

The first moment gives the position of the peak, which is related to the effective velocity for a particular given time, t, by the following equations: ∞

x c (t) =

∫ xc( x, t )dx

(15)

0 ∞

∫ c( x,0)dx 0

(16)

xc (t) = Veff t

The second moment gives the peak width, which is related to the effective diffusivity for a given time, t, by the following equations: ∞

∫[x − x

σ (t) = 2

c

(t )]2 c( x, t )dx

(17)

0 ∞

∫ c( x,0)dx 0

σ (t) = (2 Deff t )1/ 2

(18)

The concentration of the solute is plotted along the length of the microvoid as shown in Figure 1(b). From these effective parameters, the optimal separation time of two solutes, A and B, can be determined as follows24:

τ op

^

D eff ,i

 DeffA  DeffB +  DA DB   =   VˆeffA − VˆeffB    

2

^

V eff ,i ACS Paragon Plus Environment

(19)


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Where

DeffA; DeffB

and VeffA; VeffB

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are the effective diffusivity and effective velocity

coefficients of solutes A and B, respectively. This is referred to as the optimal separation time because it is the time in which two solute molecules are separated with a resolution of 2. The significance of this optimal separation time is that important information related to parameter values can be used (potentially) by experimentalists in the fabrication of polymer gels or in the optimization of parameter values for achieving optimal separation. For example, based on Equation 19, the magnitude of the applied electrical field that will give the lowest optimal separation time can be determined for a mixture of solutes having specific mobility by relating Deff and Veff to fundamental parameters of the system. In this work, in order to obtain the above mentioned effective transport parameters and eventually the optimal time of separation from the moment analysis, numerical simulations are conducted. The microdomain geometry is created and meshed using COMSOL Multiphysics v3.5a, a numerical solver based on finite element method. Triangular mesh is used for the simulation and refined mesh is used near the walls of the channel, as that is the place of larger variations. Under the assumptions of the model, the standard transient electrokinetic transport module in COMSOL is utilized for the evaluation of concentration profiles in the system; various parameter combinations describing the microdomain geometry (such as aspect ratio, and angle of divergence) and operating parameters such as those related to the orthogonal electric field are used.

4.

Illustrative Results and Discussion This section includes graphical illustrations showing the numerical results obtained from

the simulation of diverging microvoid morphology as indicated in Figure 1. These illustrations show the variation of macroscopic effective transport parameters such as the effective diffusivity


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and the effective velocity with the chosen parameters of the system. Results will show that the parameters dictating the geometry of the pore (i.e. L/H, the aspect ratio, α, the angle of divergence) can play a significant role in predicting the diffusive-convective-electromigrative behavior of the system. This transport parameter set is very important in (eventually) predicting the separation performance of the solute molecules in the microvoid as shown, for example, in the optimal time of separation (see Equation 19). For the present analysis, where the solutes A and B have a valence of -1, a mobility ratio of 2 is considered. Other values of the parameters used for the orthogonal electric field are well within the range of experimental conditions relevant for finding the electrophoretic mobility in nanocomposite gels. Also, in the discussions, the morphological parameters of the microdomain such as the “shape parameter” which is indicated by the divergence angle, α and skew parameter which is indicated by the aspect ratio L/H will be also used.

4.1

Comparison of the Optimal Times with the Analytical Model (for α=0˚) In order to validate the numerical calculations, the model developed by Oyanader and

Arce25 was used since it shows the analytical result when considering the role of the dimensional ratio of the domain given by the aspect ratio, L/H. From Oyanader et al.25 expressions for the effective transport parameters to calculate the time of separation of two sample solutes having different electrophoretic mobility were used. In this section values for the optimal separation times, τop (defined by Equation 19), as a function of Ω, the non-dimensional electric field in the orthogonal direction for a Pe=2, 6 and 10 are shown in Figure 2. The (material) domain considered for the analysis is a straight microvoid of fixed aspect ratio of L/H=9 having an orthogonal electric field applied to the direction of flow. A solute mixture having a mobility ratio



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of µb/µa=2 is considered for the hypothetical solutes A and B. It can be observed that for a Pe=2, the minimum values of the time of separation for the numerical model coincide closely with the analytical model result at the omega value of 5 as shown in Figure 2(a). The optimal times of separation for Pe=6 and 10 are also in good agreement with the analytical model. The values of omega (Ω) for the minimum values of τop varied depending on the Pe. For example, the minimum value of optimal times occurred at Ω=6 for a Pe=6, and Ω=7 for a Pe=10 as shown in Figure 2(b) and Figure 2(c), respectively.

4.2

Effective Transport Parameters in a Diverging Microvoid (α≠0˚) In the previous section, the optimal times of separation were analyzed for the straight

microdomain, α=0°. Here in the first part of the section, the effect of the microvoid shape (given by the parameter α) for a fixed microvoid size parameter (given by L/H) of the domain on the effective transport parameters (obtained by numerical calculations) are analyzed. Later, the effect of the applied orthogonal electric field given by the parameter, Ω, is analyzed in addition to the above mentioned morphological parameter; finally the optimal times of separation of a solute mixture are reported. Figure 3 (a), (b), (c) illustrate the effect of morphological (shape) parameter, α, on effective diffusivity for varying Peclet numbers of 2, 6, and 10. It is shown that as α increases, the dispersion increases, since the axial cross section varies more as α increases. Figure 4 (a), (b), and (c) illustrate the effect of the same shape parameter on the effective velocity for varying Peclet, Pe. It can be seen from the plots that the dimensionless effective velocity is equal to one for a parallel wall domain with Ω=0 indicating the effective velocity is equal to the average


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ˆ velocity in the domain. Another observation is the value of Veff decreases as the shape parameter increases in value showing retardation in the velocity as the axial cross-section increases. The plots of effective transport parameters mentioned previously illustrate the effect of morphological parameters, such as shape parameter (α) of the micro-domain on the effective transport parameters, Deff, Veff for a fixed skew factor (L/H). The effects of electromigration given by the parameter Ω will be added next. Figure 5 (a), (b), (c) illustrate the effects of the morphological parameter, α, which is the divergence angle that controls the shape of the microvoid. This result is reported for a fixed mobility ratio of the model proteins, (µb/µa=2). The plot illustrates the optimal time of separation of two test protein species A & B (considered for the current analysis) as a function of the applied electrical field, Ω. The minimum value for τop as a function of the applied electrical field, Ω, is lower when α=0˚ but increases when α≠0˚ for a fixed Pe=2 (see Figure 5(a)). The qualitative trend observed is the same for all other Pe numbers such as 6 (see Figure 5(b)), and 10 (see Figure 5(c)). It can be seen that for a parallel wall domain the optimal time of separation τop is the lowest (for a given Pe) and it increases as the divergence angle, α, gets bigger. Furthermore, the optimal time of separation, which is given by an equation (see Equation (19)) relating it to the different forces acting on the species, is lowest for the straight channel case for a Pe of 2. This is a useful piece of information in view of the recent experiments

25

where an excellent possibility for improving

separation has been observed in magnetic-aligned nanocomposite gels. Both the model and the experiments are in qualitative agreement in predicting that the straight channels can give a resolution of 2 (which is the optimal spatial distance between the two solutes), and yields the lowest possible optimal time of separation compared to other cases of divergent angles.



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5.

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Summary and Concluding Remarks In this contribution, a differential model was developed for simulating the transport of

two test sample, point-like, protein molecules A and B that move within a nanocomposite gel. Such motion of the protein species is described by their electrophoretic mobility, diffusivity, valence, etc., as they travel within a single axially-diverging channel located in a nanocomposite gel matrix. Rectangular coordinates were used to develop the two-dimensional numerical model; the macroscopic transport properties, such as the effective diffusivity and effective velocity, were evaluated from the concentration profiles using the moment theory. The dependence of morphological aspects (i.e., shape and size) of the micro-channel and also the effect of orthogonal electric field have been analyzed. It can be concluded that for the same amount of convection (given by Pe) both the effective parameters are affected by changes in the angle of divergence and also by the aspect ratio. Hence, it can be said that the morphological parameters play a role in yielding high separation resolutions. Also, when an orthogonal field was applied, its effect was much more pronounced. From the analysis of optimal time of separation it can be concluded that shorter microvoids perform better separations for lower Peclet numbers such as 2, but longer microvoids are better when the Pe is increased; however, in both the cases the straight microvoids gave better optimal times compared to other cases. One important observation from this study is that the predictions of this model are in qualitative agreement with experimental findings for the case of magnetic-aligned nanocomposite gels26. A more quantitative study with direct comparisons between model predictions and experimental results will be the subject matter of a future contribution. Finally, the computational strategy presented in this contribution is suitable to study comparisons with predictions based on analytical results from the application of spatial averaging methods34. This aspect will be subject matter of a future communication.


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Acknowledgements: We are privileged to contribute to Professor D. Ramkrishna Festschrift honoring his substantial and notable contributions to the Chemical Engineering Profession. For those of us that had the honor of being his mentees, his continuous and tremendous beneficial impact on our professional careers has been priceless. Also, we are grateful for the assistance provided to one of us (JS) from the Center for Manufacturing Research (CMR), and from the Center for the Management, Utilization & Protection of Water Resources at Tennessee Technological University during the graduate studies. The authors are grateful to Dr. Paula ArceTrigatti (Tulane University) for helpful comments to improve the manuscript.

LIST OF FIGURES

Figure 1: (a) Schematic of a diverging microvoid formed by the inter-connection of several pores having different size distributions with the nanoparticles and gel material forming the microvoid walls; (b) Exit Concentration Chromatograms of the sample peaks represented as the usual Gaussian distribution. Figure 2: Comparison of predicted optimal times of separation of the solute mixture study in this contribution with those reported by Oyanader and Arce25. The excellent agreement among them indicates a validation of the results obtained by the method described in this study. Figure 3: Effect of morphological parameter such as the divergence angle, α, on effective diffusivity for a Pe of (a) 2 (b) 6, and (c) 10 in the presence of orthogonal electric field. The domain of the negative electrical field, Ω, is a “mirror image” of the positive side and it has been omitted. Figure 4: Effect of morphological parameter such as the divergence angle, α, on effective velocity for a Pe of (a) 2 (b) 6, and (c) 10 in the presence of orthogonal electric field. As in the case of the Deff, the negative side of the electrical field, Ω, is a mirror image and it has been omitted. Figure 5: Effect of morphological parameter such as the divergence angle, α, on the optimal times of separation for a Pe of (a) 2 (b) 6, and (c) 10 in the presence of orthogonal electric field.



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Figure 1: (a) Schematic of a diverging microvoid formed by the inter-connection of several pores having different size distributions with the nanoparticles and gel material forming the microvoid walls; (b) Exit Concentration Chromatograms of the sample peaks represented as the usual Gaussian distribution.


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Figure 2: Comparison of predicted optimal times of separation of the solute mixture study in this contribution with those reported by Oyanader and Arce25. The excellent agreement among them indicates a validation of the results obtained by the method described in this study.



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Figure 3: Effect of morphological parameter such as the divergence angle, α, on effective diffusivity for a Pe of (a) 2 (b) 6, and (c) 10 in the presence of orthogonal electric field. The domain of the negative electrical field, Ω, it is a “mirror image” of the positive side and it has been omitted.


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Figure 4: Effect of morphological parameter such as the divergence angle, α, on effective velocity for a Pe of (a) 2 (b) 6, and (c) 10 in the presence of orthogonal electric field. As in the case of the Deff, the negative side of the electrical field, Ω, is a mirror image and it has been omitted.



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Figure 5: Effect of morphological parameter such as the divergence angle, α, on the optimal times of separation for a Pe of (a) 2 (b) 6, and (c) 10 in the presence of orthogonal electric field.


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REFERENCES 1.

Watson, J. D.; Baker, T. A.; Bell, S. P.; Gann, A.; Levine, M.; Losick, R., Molecular

biology of gene. Cold Spring Harbor Lab: San Francisco, CA, 2004. 2.

Masliyah, J. H.; Bhattacharjee, S., Electrokinetic and Colloid Transport Phenomena.

John Wiley and Sons Inc: Hoboken, N.J., 2006. 3.

Simhadri, J. J.; Stretz, H. A.; Oyanader, M.; Arce, P. E., Role of Nanocomposite

Hydrogel Morphology in the Electrophoretic Separation of Biomolecules: A Review. Ind. Eng. Chem. Res 2010, 49, 11866-11877. 4.

Matos, M. A.; White, L. R.; Tilton, R. D., Enhanced mixing in polyacrylamide gels

containing embedded silica nanoparticles as internal electroosmotic pumps. Colloids and Surfaces B: Biointerfaces 2008, 61, 262-269. 5.

In gel-electrophoresis there are other separation regimes in addtion to the one controls by

"sieving". Sinhadri et al., 2010 [3] reviwed these potential regimes and the technical conditions behind them. 6.

Vesterberg, O., A short history of electrophoretic methods. Electrophoresis 1993, 14,

1243-1249. 7.

Probstein, R., An Introduction to Physicochemical Hydrodynamics. John Wiley: New

York, 1991. 8.

Simhadri, J. J. Assessing the role of the gel microvoid scale, geometry, and shape on the

optimal time of separation in electrophoresis. Tennessee Technological University, 2012. 9.

Stellawagen, N. C.; Stellawagen, E., Effect of the matrix on DNA electrophoretic

mobility Journal of Chromatography A 2009, 1216, (10), 1917-1929.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

10.

Page 24 of 28

Huang, G.; Zhang, Y.; Ouyang, J.; Baeyens, R. W.; Delanghe, J., Application of carbon

nanotube-matrix assistant native polyacrylamide gel electrophoresis to the separation of apolipoprotein A-I and complement C3. Anal. Chim. Acta 2006, 557, 137-145. 11.

Thompson, J. W.; Stretz, H. A.; Arce, P. E., Preliminary Observations of the Role of

Material Morphology on Protein-Electrophoretic Transport in Gold Nanocomposite Hydrogels. Ind. Eng. Chem. Res 2010, 49, (23), 12104–12110. 12.

Dennison, C.; Lindner, W. A.; Phillips, N. C. K., Nonuniform field gel electrophoresis.

Analytical Biochemistry 1982, 120, 12-18. 13.

Boncinelli, E.; Simeone, A.; Falco, A. D.; Fidanza, V.; Volpe, A. L., An agarose gel

resolving a wide range of DNA fragment lengths. Analytical Biochemistry 1983, 134, 40-43. 14.

Ross, D.; Ivory, C. F.; Locascio, L. E.; Cott, K. E. V., Peak compression and resolution

for electrophoretic separations in diverging microchannels. Electrophoresis 2004, 25, 3694– 3704. 15.

Ghosal, S., Lubrication theory for electro-osmotic flow in a microfluidic channel of

slowly vayring cross-section and wall charge. Journal of Fluid Mechanics 2002, 459, 103-128. 16.

Xuan, X.; Li, D., Particle motions in low-Reynolds number pressure-driven flows

through converging–diverging microchannels J.Micromechanics and Microengineering 2006, 16, (1). 17.

Yariv, E.; Dorfman, K. D., Electrophoretic transport through channels of periodically

varying cross section. Physics of Fluids 2007, 19, (3). 18.

Yu, S.; Lee, S. B.; Kang, M.; Martin, C. R., Size-Based Protein Separations in

Poly(ethylene glycol)-Derivatized Gold Nanotubule Membranes. Nano Letters 2001, 1, 495-498.


Page 25 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


19.

Yu, S.; Lee, S. B.; Martin, C. R., Electrophoretic Protein Transport in Gold Nanotube

Membranes. Anal. Chem 2003, 75, 1239-1244. 20.

Berezhkovskii, A. M.; Dagdug, L., Analytical treatment of biased diffusion in tubes with

periodic dead ends. J. Chem Phys 2011, 134, 124109. 21.

Berezhkovskii, A. M.; Dagdug, L.; Mahknovskii, Y. A.; Zitserman, V. Y., Drift diffusion

in a tube of periodically varying diameter. Driving force induced intermittency. J. Chem Phys

2010, 132, 221104. 22.

Righetti, P. G. In On the Pore Size and Shape of Hydrophilic Gels for Electrophoretic

Analysis, Electrophoresis; Berlin, 1981. 23.

Trinh, S.; Locke, B. R.; Arce, P., Diffusive-Convective and Diffusive-Electroconvective

Transport in Non-Uniform Channels with Application to Macromolecular Separations. Separation and Purification Technology 1999, 15, 255-269. 24.

Sauer, S. G.; Locke, B. R.; Arce, P. E., Effects of Axial and Orthogonal Applied Electric

Fields on Solute Transport in Poiseuille Flows. An Area Averaging Approach Ind Eng Chem Res

1995, 34, 886-894. 25.

Oyanader, M.; Arce, P. E., Role of geomterical dimensions in electrophoresis

applications with orthogonal fields. Electrophoresis 2005, 26, 2857. 26.

Thompson, J. W. Role of Material Morphology on Separation of Proteins in

Nanocomposite Hydrogels, PhD Dissertation, Tennessee Technological University, 2012. 27.

Bird, R. B.; Stewart, W.; Lightfoot, E. N., Transport Phenomena. John Wiley: New

York, NY, 2002.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

28.

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For this study, the walls of the capillary are assumed having either no charge or a

negligible value so that the Electrosmotic Flow (EOF) can be safely neglected (see Section 2.1.2). Model is similar to that of Sauer et al. [24] and Oyanader and Arce [25]. 29.

The role of the EOF flow on fluids and the geometrical impact on them are discussed in

Pascal et al., 2012 [30]. The influence on the optimal time of separation will be a subject matter of an up-coming communication. 30.

Pascal, J. A., Oyanader, M. and Arce, P. E. “Effect of Capillary Geometry on Predicting

Electroosmotic Volumetric Flowrates in Porous or Fibrous Media,” Journal of Colloid and Interface Science, 2012, 378 (1), 241-250. 31.

Since no electroosmotic (EOF) effect is considered due to that the microvoids are

assumed with no charges present on the walls, the Poisson equation reduces to the Laplace equation in the the geometrical domain under analysis. 32.

This is the same condition used by Sauer et al.[24].

33.

Giddings, C. J., Unified Separation Science. John Wiley & sons: New York, 1991.

34.

Pascal, J. A. ,

“Up-Scaling Electrical-based Separations by Electrokinetic

Hydrodynamics (EKHD) and the Role of Scaling in Learning Transport Phenomena Concepts,” PhD Dissertation, Tennessee Technological University, 2011.


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Assessing Performance of Irregular Microvoids in Electrophoresis

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