Journal of Chromatography A, 1339 (2014) 219–223
Contents lists available at ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Short communication
Hydrodynamic chromatography and field flow fractionation in finite aspect ratio channels T.N. Shendruk ∗,1 , G.W. Slater ∗ University of Ottawa, Department of Physics, 150 Louis-Pasteur, Ottawa, ON K1N 6N5, Canada
a r t i c l e
i n f o
Article history: Received 3 October 2013 Received in revised form 11 February 2014 Accepted 1 March 2014 Available online 13 March 2014 Keywords: Hydrodynamic chromatography Field-flow fractionation Retention theory Rectangular channel
a b s t r a c t Hydrodynamic chromatography (HC) and field-flow fractionation (FFF) separation methods are often performed in 3D rectangular channels, though ideal retention theory assumes 2D systems. Devices are commonly designed with large aspect ratios; however, it can be unavoidable or desirable to design rectangular channels with small or even near-unity aspect ratios. To assess the significance of finite-aspect ratio effects and interpret experimental retention results, an ideal, analytical retention theory is needed. We derive a series solution for the ideal retention ratio of HC and FFF rectangular channels. Rather than limiting devices’ ability to resolve samples, our theory predicts that retention curves for normal-mode FFF are well approximated by the infinite plate solution and that the performance of HC is actually improved. These findings suggest that FFF devices need not be designed with large aspect ratios and that rectangular HC channels are optimal when the aspect ratio is unity. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Hydrodynamic chromatography (HC) [1] and field-flow fractionation (FFF) [2] are versatile and robust separation techniques. The basic retention theories describing these techniques depend on simplifying assumptions such as spherical solute particles [3], dilute concentrations [4], no-slip [5], negligible lift forces [6] and negligible wall-drag [7]. One prevalent assumption is that of a channel of finite height h but infinite width w. Since large aspect ratios cannot be cast in PDMS [8] or may be inconvenient for device design, an analytical retention theory that accounts for finite aspect ratios is desirable. Edge effects [9] and their impact on band broadening [10,11] have been investigated theoretically previously and careful experimental studies have been performed [12]. These studies have demonstrated that the axial dispersion of the bands increases by nearly an order of magnitude even when the aspect ratio remains large [13,10]. Despite this deterrent, on-chip FFF devices with aspect ratios ∼10 have been successfully developed in various laboratories [14,15,7].
∗ Corresponding authors. E-mail addresses: [email protected] (T.N. Shendruk), [email protected] (G.W. Slater). 1 Current address: The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom. Tel.: +44 1865 273952; fax: +44 1865 272400. http://dx.doi.org/10.1016/j.chroma.2014.03.002 0021-9673/© 2014 Elsevier B.V. All rights reserved.
This manuscript theoretically considers the impact of finite aspect ratios on the mean elution time of solutes. It presents a series solution for the ideal retention ratio of HC and FFF. Such an ideal model is expected to be accurate when wall-drag is negligible, which occurs when solutes are small compared to the channel height [7]. The model predicts that finite aspect ratios should not radically alter the retention ratio for small solutes and that the separation of peaks during HC benefits from the presence of lateral walls. This work concludes that large aspect ratios commonly utilized in FFF apparatuses are not necessary and that devices with modest aspect ratios are viable. Thus, although dispersion may increase, experimentalist employing microfluidic FFF devices with finite aspect ratios to study solutes much smaller than the channel height should expect retention curves similar to those produced by appratuses with much larger aspect ratios.
2. Theory Hydrodynamic chromatography separates solutes based on size by hydrodynamically transporting a solution through a channel [16,17]. As solutes are carried along by the flowing solvent their centres are excluded from the near-wall regions by steric repulsion. Larger solutes have a larger exclusion zone and are subject to the faster flowing velocity in the centre region of the channel. This causes larger solutes to elute more quickly than smaller solutes. HC has been performed in packed channels (packed-column HC), in
220
T.N. Shendruk, G.W. Slater / J. Chromatogr. A 1339 (2014) 219–223
Fig. 1. Solvent velocity field for a finite aspect ratio of w˜ = 10 and a square-duct w˜ = 1. In these figures, the velocity fields are normalized by the mean parabolic Poiseuille
˜ = ∞. The w ˜ = 10 geometry has a maximum solvent velocity of vmax ≈ 1.5v0,2D , while the square-duct (w) ˜ has a maximum of velocity v0,2D = h2 dP/dx / (12) for w
vmax = 0.88v0,2D .
capillaries (open-tube HC), in a rectangular channels and between plates [18–22]. Field-flow fractionation separates by imposing an external, transverse force to the solutes. The competition between the potential energy drop across the channel and thermal energy establishes an exponential concentration distribution. As in HC, this distribution is carried through the channel by a flow profile. Since different solutes have different concentration distributions, they have different mean velocities and separation can be achieved. HC is the zero-force limit of FFF. The system is defined as follows: A solvent of viscosity flows in the xˆ -direction in response to a pressure gradient dP/dx. The external force F acts perpendicular to the flow, in the −ˆy-direction. The height of the rectangular channel in the yˆ -direction is h, which is the characteristic length scale and non-dimensionalizes all other lengths. The rectangular channel has a width w in the zˆ -direction ˜ = w/h. In zˆ , the channel and so is characterized by an aspect ratio w goes from −w/2 to w/2. The origin is at the centre of the accumulation wall (Fig. 1). The concentration distribution resulting from the potential energy drop Fh and thermal energy kB T is described by the retention parameter = kB T/Fh, which is essentially an inverse Péclet number. Assuming the force scales with particle size as F∼˜r ˛ (where r˜ = r/h is the normalized solute radius), the retention parameter has a solute size dependence that we write = ˜r −˛ , where the device retention parameter has no implicit solute size dependence [23]. In HC, = ∞. In FFF, the power ˛ differs for different external fields. For flow-, and thermal-based fields ˛ = 1 [24,25], while ˛ = 3 for buoyancy-based techniques [26]. The resulting concentration distribution for spherical solutes is
c=
⎧ ⎨ c0 e−(˜y−˜r )˜r ˛ / for r˜ ≤ y˜ ≤ 1 − r˜ and − w˜ + r˜ ≤ z˜ ≤ w˜ − r˜ ⎩
2
0
2
(1)
otherwise.
The mean transport of solutes through the channel is described by the retention ratio R, which is simply the mean of the velocity of the solutes V(y, z) normalized by the mean solvent velocity v, i.e. R = cV/cv. ˜ = ∞, the system is 2D and When the aspect ratio is infinite w the flow obeys the parallel-plate Poiseuille equation. The average velocity of the 2D flow is v = h2 (dP/dx)/(12) ≡ v0,2D . We choose v0,2D as the characteristic velocity for this study. Neglecting hydrodynamic friction [18], the solute particles move with a velocity given by Faxén’s law
V=
r2 2 1+ ∇ 6
where L( · ) is the Langevin function [23]. When → ∞, the HC parallel-plate retention ratio is R2D = 1 + bHC r˜ − cHC r˜ 2 ,
with bHC = 2 and cHC = 4 as is ideally expected for HC in a channel [17]. The retention ratio for HC and FFF in rectangular channels can be determined by accounting for the non-parabolic flow profile of the carrier solvent (Fig. 1). The fluid velocity profile in a rectangular channel is given by the Fourier sum
v v0,2D
=
(2)
∞ 48
(n)3
n=1
v
R2D =
6 [1 − 2˜r ] L r˜ ˛
[1 − 2˜r ] r˜ ˛ 2
+ 6˜r 1 −
4˜r 3
,
(3)
cosh(ny˜ ) cosh( n2w˜ )
v0,2D
=1−
∞ 192 tanh n=1
(n)5
sin(n˜z ),
(5)
nw˜ 2
˜ w
(6)
.
In order to calculate the retention ratio, Faxén’s law operates on the solvent velocity to find V
v0,2D
=
∞ 48 sin (n˜z )
(n)3
n=1
2 cosh (ny˜ ) (n˜r ) × 1− − 6 cosh n2w˜
.
(7)
The denominator of the retention ratio is the product of the average concentration and Eq. (6), while the numerator is determined by integrating the weighted solute speed term by term. This produces a lengthy, though straight-forward, solution for the rectangular retention ratio. It is convenient to define a set of functional coefficients (listed in Appendix A), which allows the solution for the rectangular retention ratio to be written succinctly as ∞
R3D = A
n=1
(Bn −Cn )(Dn −En ) (n)3 Fn
1−
∞
.
(8)
Gn
n=1
This series solution for the retention ratio in rectangular channels is the main result of this work. In the absence of an external field, the HC rectangular retention ratio becomes ∞
R3D =
48 1 − 2˜r
(Bn −Cn ) Hn (n)4
n=1
1− By substituting the Poiseuille equation into Faxén’s law and let˜ → ∞ in Eq. (1), the ideal parallel-plate retention ratio is found ting w to be
× 1−
represents summation over odd terms only [27]. A few where terms are adequate to produce satisfactorily accurate solutions [28]. The cross-sectional averaged solvent velocity is found to be
v.
(4)
∞
.
(9)
Gn
n=1
3. Results and discussion Since the rectangular retention ratio for FFF and HC are series solutions, their behaviour is unclear. Therefore, consider some
T.N. Shendruk, G.W. Slater / J. Chromatogr. A 1339 (2014) 219–223
Retention Ratio, R
1.4 1.2 1.0
Λ =10−9 w˜ = ∞ w˜ = 10 w˜ = 2 w˜ = 1
Λ =10−5
221
Λ =∞
0.8 0.6 0.4 0.2 0.0 0.00
0.05
0.10
0.00
0.05
0.10
0.00
0.05
0.10
0.15
Particle Radius, r˜ Fig. 2. Retention ratio (Eq. (8)) as a function of solute size r˜ for ˛ = 3 and various external field strengths −1 . Series solutions were truncated at ten terms.
1. A strong field characterized by = 10−9 . 2. A moderate field with = 10−5 . 3. No field, which is HC. For each of these, the ideal parallel-plate retention ratiois shown ˜ = 10, 2, 1 for the as well as that for the finite aspect ratios w range 0 < r˜ ≤ 0.15. Only solutes that are small compared to the channel height are shown because the ideal parallel-plate retention theory has been shown to be accurate for r˜ 0.05 but wall-induced lag becomes significant for larger solutes [7]. When the external field is large ( = 10−9 ), the normal-mode of FFF (for which the retention ratio drops rapidly from nearunity to a global minimum) is relatively unaffected with respect to the parallel-plate theory (Fig. 2a). This behaviour is seen for all aspect ratios and Fig. 2a demonstrates that normal-mode FFF is well described by the 2D parallel-plate retention ratio regardless of aspect ratio when the external field is strong. Similarly when the field strength is moderate ( = 10−5 ) the normal-mode region is shifted up from the parallel-plate retention ratio but the slope does not change drastically. For both fields, the position of the minimum (the steric-inversion point [29]) does not shift visibly from the size predicted by parallel-plate retention theory (Fig. 2a and b). Past the steric-inversion point is steric-mode FFF. Even for the ˜ = 10, the rectangular retention relatively small aspect ratio of w ratio is observed to be well approximated by the parallel-plate retention ratio (Fig. 2a and b). This suggests that, apparatuses need not be designed with astronomical aspect ratios in order to apply Eq. (3) for either normal- or steric-mode FFF. Smaller aspect ratios than commonly used could be utilized and parallel-plate retention theory would remain accurate. Other values of the force scaling exponent ˛ are physically relevant for FFF (not shown). The retention curves for these are qualitatively similar to Fig. 2. The primary difference is that the steric-inversion point is shifted to smaller solute sizes for smaller ˛. Operational-mode diagrams have demonstrated that this is a ˜ [23]. property of FFF and not a consequence of finite aspect ratios w As with ˛ = 3, the steric-inversion point does not shift to different ˜ decreases. solute sizes as w Retention curves for aspect ratios smaller than unity were con˜ 1 channels, which follows the trend seen in Fig. 2. Howthan in w ever, the ultimate consequence of this is that the total drop in R through normal-mode FFF is reduced. In fact, the retention ratio quite rapidly approaches Eq. (4) even for strong fields (small ). Next, consider true HC (Fig. 2c). Unlike the finite curves, there is no negative-slope region representing normal-mode FFF. The rectangular retention ratio increases to greater values than the 2D parallel-plate prediction. This suggests that peak separation in HC may benefit from finite aspect ratios (although the reader is reminded that dispersion should be expected to increase [10–12]). It is clear from Fig. 2c that Eq. (4) with the coefficients bHC = 2 and cHC = 4 is not adequate for HC in rectangular channels with finite aspect ratios. Fig. 2c suggests that it is reasonable to continue assuming a quadratic form for the HC retention ratio. We fit Eq. (4) to the calculated retention ratio and obtain bHC and cHC as ˜ by minimizing 2 (Fig. 3). For infifunctions of the aspect ratio w nite aspect ratios, bHC = 2 and cHC = 4 as expected. As the channel width approaches the height, the constants rise to roughly bHC ≈ 4 and cHC ≈ 8. By comparing the HC retention ratio to that for an open-tube cylindrical capillary of diameter d, we will now question the validity of assuming that the HC retention is quadratic with respect to particle size (Eq. (4)). Briefly, in a cylindrical tube the concentration distribution is homogeneous beyond steric-exclusion, the flow
2.2 b HC/2 2.0
HC fitting constants
example parameters. For convenience, the retention curves are plotted in Fig. 2 for the power ˛ = 3. We consider three field strengths:
c HC/4
1.8 1.6 1.4 1.2 1.0 0.0
0.2
0.4
0.6
0.8
1.0
Inverse Aspect Ratio, w˜ −1 Fig. 3. The parabolic fitting constants for hydrodynamic chromatography from Eq. ˜ normalized by their parallel-plate values. (4) as functions of the aspect ratio w, Errorbars represent uncertainty on the fit parameters.
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T.N. Shendruk, G.W. Slater / J. Chromatogr. A 1339 (2014) 219–223
exceedingly small, the parabolic form is exceptionally accurate and the coefficients can be approximated by using b HC = 4 ;
cHC =
˜ 2 + 4w ˜ + 12 16w . ˜ +1 ˜ 2 + 2w w
(12)
These accurately reproduce the HC retention curves in Fig. 4 (dashed lines) and can be compared to the fitting parameters in Fig. 3. 4. Conclusions
Fig. 4. Hydrodynamic chromatography retention ratio as a function of particle radius normalized by hydraulic diameter for a number of geometries (solid lines). Dashed lines show the parabolic approximation of Eq. (12) in Eq. (11). The inset focuses on the small particle size behaviour.
obeys the Hagen–Poiseuille equation, and applying Faxén’s law (Eq. (2)) estimates the solute velocity. These facts can be used to find that the ideal retention ratio is R=1+4
28 r − 3 d
r 2 d
(10)
,
which is shown in Fig. 4. To compare a cylindrical capillary and channel, an a rectangular ˜ must be ascribed effective hydraulic diameter dH ≡ 2h/ 1 + 1/w to the latter [30]. Writing the presumed quadratic form of the HC retention ratio (Eq. (4)) in terms of hydraulic diameter allows us to compare between different channel geometries:
r r R = 1 + b HC − cHC dH dH
2 .
(11)
A comparison can now be made between cylindrical and rectangular channels (Fig. 4). Again, this ideal theory, which neglects hydrodynamic and other interactions with the channel walls, is expected to be accurate for solutes that are small compared to the characteristic size of the channel. From Eq. (10) the coefficients for a cylindrical capillary are b HC = 4 and cHC = 28/3, while Eq. (4) gives the coefficients of an = 16. Although infinite aspect ratio channel to be b HC = 4 and cHC b HC = 4 for both geometries, the two retention curves are seen to be rather different in Fig. 4. Surprisingly, the retention curves for rectangular channels fall between those of the cylindrical and the plate ˜ > 2 but for aspect ratios closer to unity, the geometries only for w retention ratios are greater than the cylindrical values. The reten˜ ≈ 2 channel behaves most like the cylindrical tion ratio of the w capillary (Fig. 4). These two retention ratios climb rather rapidly as a function of r/dH . On the other hand, the retention ratio of larger aspect ratio channels climb more slowly at any given r/dH (Fig. 4). Regardless of geometry, the retention curves for the rectangular channels in Fig. 4 appear to follow the presumed quadratic form [1,18,31], which is indeed accurate for most particle sizes. However, an investigation of the tiniest particle sizes reveals that the quadratic form breaks down (Fig. 4; inset). Therefore, we conclude that the commonly presumed form of Eq. (4) or Eq. (11) is an approximation for HC retention in rectangular channels. For particles that are small compared to the channel height but are not
In this manuscript, we have presented series solutions for the retention ratios applicable to hydrodynamic chromatography and field-flow fractionation in channels with rectangular cross sections that complement theoretical predictions of increased disperion [11]. The resulting ideal retention theory neglects hydrodynamic interactions with the channel walls and other complications and so is expected to be most accurate for colloids that are much smaller than the channel height. It predicts that normal-mode FFF is relatively unperturbed from the parallel-plate retention ratio. The steric-inversion point at which the retention behaviour transitions from normal-mode to steric-mode FFF is observed to remain essentially fixed regardless of the channel aspect ratio. In the steric-mode regime, the slope of the retention curve increases as the aspect ratio approaches unity, suggesting improved peak separation is possible. This increase is most substantial when the external force is zero. In this hydrodynamic chromatography limit, decreasing the aspect ratio increases the slope. A parabolic retention ratio is found to well approximate the retention ratio. Such rectangular geometries are convenient for the materials commonly used to cast microfluidic channels. Therefore, the fact that retention curves do not change dramatically should be encouraging to separation scientists who wish to construct microfluidic separation devices out of such materials. Acknowledgements We gratefully acknowledge support through an NSERC Discovery Grant to G.W.S. and the NSERC-CGS program to T.N.S. Appendix A. Coefficients The functional coefficients used in Eq. (8) and Eq. (9) are given by
A (˜r , ) ≡ 48 1 + coth Bn (˜r ) ≡ 1 − ˜ ≡ Cn (˜r , w)
(n)2 2 r˜ 6
2 n
1 ˜ − 2˜r w
En (˜r , ) ≡ e−(1−2˜r )˜r
˜ ≡ Gn (w)
˛ /
n 2 r˜ ˛
[1 − 2˜r ] r˜ ˛ 2
(13a) (13b)
Dn (˜r , ) ≡ sin (n˜r ) +
Fn (˜r , ) ≡
×
˜ − 2˜r ] /2 sinh n [w
˜ cosh nw/2
n cos (n˜r ) r˜ ˛ { sin (n [1 − r˜ ]) +
+1
(13c)
(13d) n cos (n [1 − r˜ ]) } r˜ ˛ (13e)
(13f)
˜ 192 tanh nw/2 ˜ w (n)5
Hn (˜r ) ≡ cos (n˜r ) − cos (n [1 − r˜ ]) .
(13g) (13h)
T.N. Shendruk, G.W. Slater / J. Chromatogr. A 1339 (2014) 219–223
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Contents lists available at ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Short communication
Hydrodynamic chromatography and field flow fractionation in finite aspect ratio channels T.N. Shendruk ∗,1 , G.W. Slater ∗ University of Ottawa, Department of Physics, 150 Louis-Pasteur, Ottawa, ON K1N 6N5, Canada
a r t i c l e
i n f o
Article history: Received 3 October 2013 Received in revised form 11 February 2014 Accepted 1 March 2014 Available online 13 March 2014 Keywords: Hydrodynamic chromatography Field-flow fractionation Retention theory Rectangular channel
a b s t r a c t Hydrodynamic chromatography (HC) and field-flow fractionation (FFF) separation methods are often performed in 3D rectangular channels, though ideal retention theory assumes 2D systems. Devices are commonly designed with large aspect ratios; however, it can be unavoidable or desirable to design rectangular channels with small or even near-unity aspect ratios. To assess the significance of finite-aspect ratio effects and interpret experimental retention results, an ideal, analytical retention theory is needed. We derive a series solution for the ideal retention ratio of HC and FFF rectangular channels. Rather than limiting devices’ ability to resolve samples, our theory predicts that retention curves for normal-mode FFF are well approximated by the infinite plate solution and that the performance of HC is actually improved. These findings suggest that FFF devices need not be designed with large aspect ratios and that rectangular HC channels are optimal when the aspect ratio is unity. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Hydrodynamic chromatography (HC) [1] and field-flow fractionation (FFF) [2] are versatile and robust separation techniques. The basic retention theories describing these techniques depend on simplifying assumptions such as spherical solute particles [3], dilute concentrations [4], no-slip [5], negligible lift forces [6] and negligible wall-drag [7]. One prevalent assumption is that of a channel of finite height h but infinite width w. Since large aspect ratios cannot be cast in PDMS [8] or may be inconvenient for device design, an analytical retention theory that accounts for finite aspect ratios is desirable. Edge effects [9] and their impact on band broadening [10,11] have been investigated theoretically previously and careful experimental studies have been performed [12]. These studies have demonstrated that the axial dispersion of the bands increases by nearly an order of magnitude even when the aspect ratio remains large [13,10]. Despite this deterrent, on-chip FFF devices with aspect ratios ∼10 have been successfully developed in various laboratories [14,15,7].
∗ Corresponding authors. E-mail addresses: [email protected] (T.N. Shendruk), [email protected] (G.W. Slater). 1 Current address: The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom. Tel.: +44 1865 273952; fax: +44 1865 272400. http://dx.doi.org/10.1016/j.chroma.2014.03.002 0021-9673/© 2014 Elsevier B.V. All rights reserved.
This manuscript theoretically considers the impact of finite aspect ratios on the mean elution time of solutes. It presents a series solution for the ideal retention ratio of HC and FFF. Such an ideal model is expected to be accurate when wall-drag is negligible, which occurs when solutes are small compared to the channel height [7]. The model predicts that finite aspect ratios should not radically alter the retention ratio for small solutes and that the separation of peaks during HC benefits from the presence of lateral walls. This work concludes that large aspect ratios commonly utilized in FFF apparatuses are not necessary and that devices with modest aspect ratios are viable. Thus, although dispersion may increase, experimentalist employing microfluidic FFF devices with finite aspect ratios to study solutes much smaller than the channel height should expect retention curves similar to those produced by appratuses with much larger aspect ratios.
2. Theory Hydrodynamic chromatography separates solutes based on size by hydrodynamically transporting a solution through a channel [16,17]. As solutes are carried along by the flowing solvent their centres are excluded from the near-wall regions by steric repulsion. Larger solutes have a larger exclusion zone and are subject to the faster flowing velocity in the centre region of the channel. This causes larger solutes to elute more quickly than smaller solutes. HC has been performed in packed channels (packed-column HC), in
220
T.N. Shendruk, G.W. Slater / J. Chromatogr. A 1339 (2014) 219–223
Fig. 1. Solvent velocity field for a finite aspect ratio of w˜ = 10 and a square-duct w˜ = 1. In these figures, the velocity fields are normalized by the mean parabolic Poiseuille
˜ = ∞. The w ˜ = 10 geometry has a maximum solvent velocity of vmax ≈ 1.5v0,2D , while the square-duct (w) ˜ has a maximum of velocity v0,2D = h2 dP/dx / (12) for w
vmax = 0.88v0,2D .
capillaries (open-tube HC), in a rectangular channels and between plates [18–22]. Field-flow fractionation separates by imposing an external, transverse force to the solutes. The competition between the potential energy drop across the channel and thermal energy establishes an exponential concentration distribution. As in HC, this distribution is carried through the channel by a flow profile. Since different solutes have different concentration distributions, they have different mean velocities and separation can be achieved. HC is the zero-force limit of FFF. The system is defined as follows: A solvent of viscosity flows in the xˆ -direction in response to a pressure gradient dP/dx. The external force F acts perpendicular to the flow, in the −ˆy-direction. The height of the rectangular channel in the yˆ -direction is h, which is the characteristic length scale and non-dimensionalizes all other lengths. The rectangular channel has a width w in the zˆ -direction ˜ = w/h. In zˆ , the channel and so is characterized by an aspect ratio w goes from −w/2 to w/2. The origin is at the centre of the accumulation wall (Fig. 1). The concentration distribution resulting from the potential energy drop Fh and thermal energy kB T is described by the retention parameter = kB T/Fh, which is essentially an inverse Péclet number. Assuming the force scales with particle size as F∼˜r ˛ (where r˜ = r/h is the normalized solute radius), the retention parameter has a solute size dependence that we write = ˜r −˛ , where the device retention parameter has no implicit solute size dependence [23]. In HC, = ∞. In FFF, the power ˛ differs for different external fields. For flow-, and thermal-based fields ˛ = 1 [24,25], while ˛ = 3 for buoyancy-based techniques [26]. The resulting concentration distribution for spherical solutes is
c=
⎧ ⎨ c0 e−(˜y−˜r )˜r ˛ / for r˜ ≤ y˜ ≤ 1 − r˜ and − w˜ + r˜ ≤ z˜ ≤ w˜ − r˜ ⎩
2
0
2
(1)
otherwise.
The mean transport of solutes through the channel is described by the retention ratio R, which is simply the mean of the velocity of the solutes V(y, z) normalized by the mean solvent velocity v, i.e. R = cV/cv. ˜ = ∞, the system is 2D and When the aspect ratio is infinite w the flow obeys the parallel-plate Poiseuille equation. The average velocity of the 2D flow is v = h2 (dP/dx)/(12) ≡ v0,2D . We choose v0,2D as the characteristic velocity for this study. Neglecting hydrodynamic friction [18], the solute particles move with a velocity given by Faxén’s law
V=
r2 2 1+ ∇ 6
where L( · ) is the Langevin function [23]. When → ∞, the HC parallel-plate retention ratio is R2D = 1 + bHC r˜ − cHC r˜ 2 ,
with bHC = 2 and cHC = 4 as is ideally expected for HC in a channel [17]. The retention ratio for HC and FFF in rectangular channels can be determined by accounting for the non-parabolic flow profile of the carrier solvent (Fig. 1). The fluid velocity profile in a rectangular channel is given by the Fourier sum
v v0,2D
=
(2)
∞ 48
(n)3
n=1
v
R2D =
6 [1 − 2˜r ] L r˜ ˛
[1 − 2˜r ] r˜ ˛ 2
+ 6˜r 1 −
4˜r 3
,
(3)
cosh(ny˜ ) cosh( n2w˜ )
v0,2D
=1−
∞ 192 tanh n=1
(n)5
sin(n˜z ),
(5)
nw˜ 2
˜ w
(6)
.
In order to calculate the retention ratio, Faxén’s law operates on the solvent velocity to find V
v0,2D
=
∞ 48 sin (n˜z )
(n)3
n=1
2 cosh (ny˜ ) (n˜r ) × 1− − 6 cosh n2w˜
.
(7)
The denominator of the retention ratio is the product of the average concentration and Eq. (6), while the numerator is determined by integrating the weighted solute speed term by term. This produces a lengthy, though straight-forward, solution for the rectangular retention ratio. It is convenient to define a set of functional coefficients (listed in Appendix A), which allows the solution for the rectangular retention ratio to be written succinctly as ∞
R3D = A
n=1
(Bn −Cn )(Dn −En ) (n)3 Fn
1−
∞
.
(8)
Gn
n=1
This series solution for the retention ratio in rectangular channels is the main result of this work. In the absence of an external field, the HC rectangular retention ratio becomes ∞
R3D =
48 1 − 2˜r
(Bn −Cn ) Hn (n)4
n=1
1− By substituting the Poiseuille equation into Faxén’s law and let˜ → ∞ in Eq. (1), the ideal parallel-plate retention ratio is found ting w to be
× 1−
represents summation over odd terms only [27]. A few where terms are adequate to produce satisfactorily accurate solutions [28]. The cross-sectional averaged solvent velocity is found to be
v.
(4)
∞
.
(9)
Gn
n=1
3. Results and discussion Since the rectangular retention ratio for FFF and HC are series solutions, their behaviour is unclear. Therefore, consider some
T.N. Shendruk, G.W. Slater / J. Chromatogr. A 1339 (2014) 219–223
Retention Ratio, R
1.4 1.2 1.0
Λ =10−9 w˜ = ∞ w˜ = 10 w˜ = 2 w˜ = 1
Λ =10−5
221
Λ =∞
0.8 0.6 0.4 0.2 0.0 0.00
0.05
0.10
0.00
0.05
0.10
0.00
0.05
0.10
0.15
Particle Radius, r˜ Fig. 2. Retention ratio (Eq. (8)) as a function of solute size r˜ for ˛ = 3 and various external field strengths −1 . Series solutions were truncated at ten terms.
1. A strong field characterized by = 10−9 . 2. A moderate field with = 10−5 . 3. No field, which is HC. For each of these, the ideal parallel-plate retention ratiois shown ˜ = 10, 2, 1 for the as well as that for the finite aspect ratios w range 0 < r˜ ≤ 0.15. Only solutes that are small compared to the channel height are shown because the ideal parallel-plate retention theory has been shown to be accurate for r˜ 0.05 but wall-induced lag becomes significant for larger solutes [7]. When the external field is large ( = 10−9 ), the normal-mode of FFF (for which the retention ratio drops rapidly from nearunity to a global minimum) is relatively unaffected with respect to the parallel-plate theory (Fig. 2a). This behaviour is seen for all aspect ratios and Fig. 2a demonstrates that normal-mode FFF is well described by the 2D parallel-plate retention ratio regardless of aspect ratio when the external field is strong. Similarly when the field strength is moderate ( = 10−5 ) the normal-mode region is shifted up from the parallel-plate retention ratio but the slope does not change drastically. For both fields, the position of the minimum (the steric-inversion point [29]) does not shift visibly from the size predicted by parallel-plate retention theory (Fig. 2a and b). Past the steric-inversion point is steric-mode FFF. Even for the ˜ = 10, the rectangular retention relatively small aspect ratio of w ratio is observed to be well approximated by the parallel-plate retention ratio (Fig. 2a and b). This suggests that, apparatuses need not be designed with astronomical aspect ratios in order to apply Eq. (3) for either normal- or steric-mode FFF. Smaller aspect ratios than commonly used could be utilized and parallel-plate retention theory would remain accurate. Other values of the force scaling exponent ˛ are physically relevant for FFF (not shown). The retention curves for these are qualitatively similar to Fig. 2. The primary difference is that the steric-inversion point is shifted to smaller solute sizes for smaller ˛. Operational-mode diagrams have demonstrated that this is a ˜ [23]. property of FFF and not a consequence of finite aspect ratios w As with ˛ = 3, the steric-inversion point does not shift to different ˜ decreases. solute sizes as w Retention curves for aspect ratios smaller than unity were con˜ 1 channels, which follows the trend seen in Fig. 2. Howthan in w ever, the ultimate consequence of this is that the total drop in R through normal-mode FFF is reduced. In fact, the retention ratio quite rapidly approaches Eq. (4) even for strong fields (small ). Next, consider true HC (Fig. 2c). Unlike the finite curves, there is no negative-slope region representing normal-mode FFF. The rectangular retention ratio increases to greater values than the 2D parallel-plate prediction. This suggests that peak separation in HC may benefit from finite aspect ratios (although the reader is reminded that dispersion should be expected to increase [10–12]). It is clear from Fig. 2c that Eq. (4) with the coefficients bHC = 2 and cHC = 4 is not adequate for HC in rectangular channels with finite aspect ratios. Fig. 2c suggests that it is reasonable to continue assuming a quadratic form for the HC retention ratio. We fit Eq. (4) to the calculated retention ratio and obtain bHC and cHC as ˜ by minimizing 2 (Fig. 3). For infifunctions of the aspect ratio w nite aspect ratios, bHC = 2 and cHC = 4 as expected. As the channel width approaches the height, the constants rise to roughly bHC ≈ 4 and cHC ≈ 8. By comparing the HC retention ratio to that for an open-tube cylindrical capillary of diameter d, we will now question the validity of assuming that the HC retention is quadratic with respect to particle size (Eq. (4)). Briefly, in a cylindrical tube the concentration distribution is homogeneous beyond steric-exclusion, the flow
2.2 b HC/2 2.0
HC fitting constants
example parameters. For convenience, the retention curves are plotted in Fig. 2 for the power ˛ = 3. We consider three field strengths:
c HC/4
1.8 1.6 1.4 1.2 1.0 0.0
0.2
0.4
0.6
0.8
1.0
Inverse Aspect Ratio, w˜ −1 Fig. 3. The parabolic fitting constants for hydrodynamic chromatography from Eq. ˜ normalized by their parallel-plate values. (4) as functions of the aspect ratio w, Errorbars represent uncertainty on the fit parameters.
222
T.N. Shendruk, G.W. Slater / J. Chromatogr. A 1339 (2014) 219–223
exceedingly small, the parabolic form is exceptionally accurate and the coefficients can be approximated by using b HC = 4 ;
cHC =
˜ 2 + 4w ˜ + 12 16w . ˜ +1 ˜ 2 + 2w w
(12)
These accurately reproduce the HC retention curves in Fig. 4 (dashed lines) and can be compared to the fitting parameters in Fig. 3. 4. Conclusions
Fig. 4. Hydrodynamic chromatography retention ratio as a function of particle radius normalized by hydraulic diameter for a number of geometries (solid lines). Dashed lines show the parabolic approximation of Eq. (12) in Eq. (11). The inset focuses on the small particle size behaviour.
obeys the Hagen–Poiseuille equation, and applying Faxén’s law (Eq. (2)) estimates the solute velocity. These facts can be used to find that the ideal retention ratio is R=1+4
28 r − 3 d
r 2 d
(10)
,
which is shown in Fig. 4. To compare a cylindrical capillary and channel, an a rectangular ˜ must be ascribed effective hydraulic diameter dH ≡ 2h/ 1 + 1/w to the latter [30]. Writing the presumed quadratic form of the HC retention ratio (Eq. (4)) in terms of hydraulic diameter allows us to compare between different channel geometries:
r r R = 1 + b HC − cHC dH dH
2 .
(11)
A comparison can now be made between cylindrical and rectangular channels (Fig. 4). Again, this ideal theory, which neglects hydrodynamic and other interactions with the channel walls, is expected to be accurate for solutes that are small compared to the characteristic size of the channel. From Eq. (10) the coefficients for a cylindrical capillary are b HC = 4 and cHC = 28/3, while Eq. (4) gives the coefficients of an = 16. Although infinite aspect ratio channel to be b HC = 4 and cHC b HC = 4 for both geometries, the two retention curves are seen to be rather different in Fig. 4. Surprisingly, the retention curves for rectangular channels fall between those of the cylindrical and the plate ˜ > 2 but for aspect ratios closer to unity, the geometries only for w retention ratios are greater than the cylindrical values. The reten˜ ≈ 2 channel behaves most like the cylindrical tion ratio of the w capillary (Fig. 4). These two retention ratios climb rather rapidly as a function of r/dH . On the other hand, the retention ratio of larger aspect ratio channels climb more slowly at any given r/dH (Fig. 4). Regardless of geometry, the retention curves for the rectangular channels in Fig. 4 appear to follow the presumed quadratic form [1,18,31], which is indeed accurate for most particle sizes. However, an investigation of the tiniest particle sizes reveals that the quadratic form breaks down (Fig. 4; inset). Therefore, we conclude that the commonly presumed form of Eq. (4) or Eq. (11) is an approximation for HC retention in rectangular channels. For particles that are small compared to the channel height but are not
In this manuscript, we have presented series solutions for the retention ratios applicable to hydrodynamic chromatography and field-flow fractionation in channels with rectangular cross sections that complement theoretical predictions of increased disperion [11]. The resulting ideal retention theory neglects hydrodynamic interactions with the channel walls and other complications and so is expected to be most accurate for colloids that are much smaller than the channel height. It predicts that normal-mode FFF is relatively unperturbed from the parallel-plate retention ratio. The steric-inversion point at which the retention behaviour transitions from normal-mode to steric-mode FFF is observed to remain essentially fixed regardless of the channel aspect ratio. In the steric-mode regime, the slope of the retention curve increases as the aspect ratio approaches unity, suggesting improved peak separation is possible. This increase is most substantial when the external force is zero. In this hydrodynamic chromatography limit, decreasing the aspect ratio increases the slope. A parabolic retention ratio is found to well approximate the retention ratio. Such rectangular geometries are convenient for the materials commonly used to cast microfluidic channels. Therefore, the fact that retention curves do not change dramatically should be encouraging to separation scientists who wish to construct microfluidic separation devices out of such materials. Acknowledgements We gratefully acknowledge support through an NSERC Discovery Grant to G.W.S. and the NSERC-CGS program to T.N.S. Appendix A. Coefficients The functional coefficients used in Eq. (8) and Eq. (9) are given by
A (˜r , ) ≡ 48 1 + coth Bn (˜r ) ≡ 1 − ˜ ≡ Cn (˜r , w)
(n)2 2 r˜ 6
2 n
1 ˜ − 2˜r w
En (˜r , ) ≡ e−(1−2˜r )˜r
˜ ≡ Gn (w)
˛ /
n 2 r˜ ˛
[1 − 2˜r ] r˜ ˛ 2
(13a) (13b)
Dn (˜r , ) ≡ sin (n˜r ) +
Fn (˜r , ) ≡
×
˜ − 2˜r ] /2 sinh n [w
˜ cosh nw/2
n cos (n˜r ) r˜ ˛ { sin (n [1 − r˜ ]) +
+1
(13c)
(13d) n cos (n [1 − r˜ ]) } r˜ ˛ (13e)
(13f)
˜ 192 tanh nw/2 ˜ w (n)5
Hn (˜r ) ≡ cos (n˜r ) − cos (n [1 − r˜ ]) .
(13g) (13h)
T.N. Shendruk, G.W. Slater / J. Chromatogr. A 1339 (2014) 219–223
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