Determination of the functioning parameters in asymmetrical flow field-flow fractionation with an exponential channel

Journal of Chromatography A, 1305 (2013) 213–220

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Determination of the functioning parameters in asymmetrical flow field-flow fractionation with an exponential channel P. Déjardin ∗ Institut Européen des Membranes, UMR 5635 (CNRS, UM2, ENSCM), Université Montpellier 2, 2 Place Eugène Bataillon, F-34095 Montpellier Cedex 5, France

a r t i c l e

i n f o

Article history: Received 11 March 2013 Received in revised form 13 June 2013 Accepted 15 June 2013 Available online 28 June 2013 Keywords: Flow field-flow-fractionation Exponential channel Rectangular channel High retention limit Determination of channel height

a b s t r a c t The flow conditions in normal mode asymmetric flow field-flow fractionation are determined to approach the high retention limit with the requirement d  l  w, where d is the particle diameter, l the characteristic length of the sample exponential distribution and w the channel height. The optimal entrance velocity is determined from the solute characteristics, the channel geometry (exponential to rectangular) and the membrane properties, according to a model providing the velocity fields all over the cell length. In addition, a method is proposed for in situ determination of the channel height. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Field-flow fractionation is a tool for the separation and characterization of macromolecules and particles [1,2]. Flow field-flow fractionation (FlFFF) is now mainly used in the asymmetrical configuration (AsFl-FFF or AF4) in various fields as food analysis [3], recovery of nanoparticles and proteins [4], drug delivery [5], fractionation of superferrimagnetic multicore nanoparticles [6], characterization of protein conjugate [7], analysis of starch [8,9] and liposomes [10]. It has been analyzed by several models [11–16] and a critical overview appeared recently [17]. Many publications [14–16,18] have been devoted to improve the interpretation of asymmetric flow field-flow fractionation (AsFl-FFF) data through a better description of the transverse velocity field in the domain near the wall where the solute is spread. These works are based on the assumption of a cross-flow velocity constant implying consequently the estimation of the axial transport velocity field. However taking into account the pressure variations in a recent model [19,20] led to an improved description of the flow rates through the whole system. Two cell designs were proposed to approach more accurate constancy of both velocity fields all over the length of the cell than in the usual cells. However the present manuscript proposes an analysis of data using the expression of both axial through the channel and transversal through the

∗ Tel.: +33 467149121; fax: +33 467149119. E-mail addresses: [email protected], [email protected] 0021-9673/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chroma.2013.06.037

membrane velocity fields where we consider the usual cell geometries, i.e. membrane with constant permeability and permeate compartment of constant large height compared to the sample channel one. The analysis will examine the two cases of channel constant breadth (rectangular) and exponential decreasing breadth in the direction of axial flow (exponential). Generally, the membranes are characterized by their molecular weight cut off, a useful parameter as they must retain the solutes, macromolecules or particles, in the sample channel. However, in the present work, the important characteristic is their resistance to the flow of solvent. The scope is to find the right balance between axial flow in the channel and transverse flow through the membrane to achieve an efficient separation, in the high retention limit. The model includes the assumption of the Poiseuille parabolic axial velocity profile. 2. Theory 2.1. Mean axial velocity and membrane (cross-flow) velocity as a function of distance The system under study is schematically represented in Fig. 1, with the flow rate in the channel qc (z) and the flow rate through the membrane qm (z). There is no flow through the impermeable upper wall of the channel of length L. The height of the channel w is very small with respect to the breadth, which is an exponential function of the distance (characteristic length s−1 ): b(z) = b(0)e−sz

(1)

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which becomes in the configuration of a spacer of constant breadth:

rq,s=0 =

1 cosh L∗ + ˇL∗ sinh L∗

(9)

hence ˇ(sL) = Fig. 1. Scheme of flow cell of length L with varying channel breadth with distance z to entrance. Cross-flow or membrane flow rate qm (z). Sample channel flow rate qc (z). qin = qc (0). qout = qc (L).

A characteristic length  was previously introduced [19,20] which, for channel and membrane of same constant breadth, is the length where the channel and the membrane have the same hydraulic resistance. It depends on the channel height w and a characteristic length of the membrane (+ support) m .



=

w3 m

1/2

 U = L∗2 +

,

L L =  ∗

(2b)

2

U cosh[U(1 − Z)] + (ˇL∗2 − (sL/2)) sinh[U(1 − Z)] sLZ/2 e U cosh U + (ˇL∗2 − (sL/2)) sinh U (3) ˇU cosh[U(1 − Z)]+(ˇ(sL/2)+1) sinh[U(1−Z)] sLZ/2 e ˇU cosh U + (ˇ(sL/2) + 1) sinh U (4)

In the configuration of a spacer of constant breadth, hence s = 0 and U = L*, these expressions become:

vc (Z) = vc (0)

cosh[L∗ (1 − Z)] + ˇL∗ sinh[L∗ (1 − Z)] cosh L∗ + ˇL∗ sinh L∗

vm (Z) = vm (0)

(5)

ˇL∗ cosh[L∗ (1 − Z)] + sinh[L∗ (1 − Z)] ˇL∗ cosh L∗ + sinh L∗

(10a)

(10b)

L∗ sinh L∗

The condition ˇ ≥ 0 provides the upper boundary of rq which corresponds to the same pressure at the exit of the channel and in the permeate compartment: Ue−sL/2 U cosh U − (sL/2) sinh U

rq,max (sL) =

(11a)

In the limit of high retention (characteristic length of the solute exponential distribution from the wall much smaller than the channel height), elution time is independent of the flow rate but dependent on the ratio of channel exit over entrance flow rates. Indeed, an increase of axial flow rate thus of axial velocity is counterbalanced by the transverse displacement of the solute towards the wall (membrane) induced by the simultaneous increase of the (cross-flow) membrane velocity (Fig. 2). The solute velocity averaged over its exponential distribution from the wall (characteristic length D/vm , assumption of half infinite space: see details on approximations in Appendix A or [15]) is:

qc,out vc (1) −sL rq = = = rv e−sL e qc,in vc (0)

(7)

6D vc (Z) w vm (Z)

(12)

The elution time over the cell length is:



1

t= 0

Let rq be the ratio of channel exit flow rate qc,out over channel entrance flow rate qc,in .

(11b)

2.3. Elution time in the high retention limit over the full length of the cell

Vs (Z) =

2.2. Ratio rq of channel exit flow rate over entrance flow rate. Connection to parameter ˇ

1 cosh L∗

Such a domain corresponds to a positive membrane velocity (i.e. from sample channel to permeate compartment): the condition ˇ > 0 is equivalent to the condition vm (1) > 0. Additional resistance at the exit of permeate compartment would allow to increase rq above rq ,max by increasing pressure in that compartment and inducing inverse flow through the membrane. Closing of that exit corresponds indeed to the absolute limit rq = 1, for instance used recently to determine the void-time [21]. It can be also verified that the upper limit (Eqs. (11)) is the unity when L* → 0 (impermeable membrane).

(6)

ˇ is the ratio of the resistance of the circuit after the channel exit over the channel resistance of a channel of constant breadth b(L). It is linked to the split of the entrance flow rate into the two exit flow rates out of the channel and permeate compartments.

From Eq. (3), we deduce:

rq−1 − cosh L∗

rq,max (0) =

 sL 2 1/2

vm (Z)=vm (0)

L∗2 sinh U

(2a)

For sake of simplicity, we will assume the pressure in the permeate compartment to be constant as in previous works [19,21]. In the writing of equations, the three variables L*, U and sL may appear; however it should be kept in mind that there are only two independent variables. The solutions for the mean axial velocity vc and membrane (cross-flow) velocity vm were given previously (Eqs. (A11)–(A12) in Annex of [19]) as a function of the reduced distance Z = z/L.

vc (Z)=vc (0)

ˇ(0) =

Ue−sL/2 rq−1 − (U cosh U − (sL/2) sinh U)

wL LdZ = 6D Vs (Z)

 0

1

vm (Z)dZ vc (Z)

(13)

Let us consider the general case of a breadth b varying with distance. The conservation of the mass of the incompressible fluid leads to the relation between membrane transverse velocity vm and axial channel velocity vc :

vm (Z) = −

w L

 dv

c

dZ

+



1 db vc (Z) b dZ

(14)

which, put in Eq. (13), leads to:



qc (1) sL U rq = exp − = 2 qc (0) U cosh U + (ˇL∗2 − (sL/2)) sinh U



 (8)

t=

w2 6D



ln

qc,in qc,out

(15)

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215

decreasing function of the distance to the entrance (see Appendix B), let us rewrite the condition (Eq. (17)) as: D D  vm (1) < vm (0)  w d or

(20)

D vm (1) D  vm (1)  w vm (0) d

(21)

Using the expressions of the velocities, we obtain the

vc (0) domain between the lower and upper boundaries as vc (0)lb  vc (0)  vc (0)ub (see Appendix C). The boundaries are chosen very far as corresponding to l = d or w. DL e−sL e−sL DL  vc (0)  ∗ ∗ 2 wd G(L , sL, rq ) w F(L , sL, rq )

(22)

where Ue−sL/2 − (U cosh U − (sL/2) sinh U)rq sinh U

F(L∗ , sL, rq ) = Fig. 2. Illustration of two parabolic velocity fields Vc (Vc2 = 2 × Vc1 ) near the wall as a function of distance x to the membrane wall. Increase of applied pressure induces the simultaneous increase of axial velocity field and membrane (cross-flow) velocity (vm2 = 2 × vm1 ), thus inducing no change in solute axial transport as its mean distance to the wall is decreased by the same factor. D is the solute diffusion coefficient.

 ∗

∗

G(L , sL, rq ) = F(L , sL, rq ) cosh U

 +

where qc,in and qc,out are the entrance flow rate and exit flow rate of the sample channel, respectively. In the derivation of Eq. (15), no specific function b(z) was assumed. Moreover, if the sample initial position is not at position Z = 0 but at Z = Z0 , then qc,in must be changed to qc (Z0 ). This relation is valid provided the exponential profile is established in the domain where the axial velocity varies linearly with distance x to the wall, say Vc (x) = x.  is the wall shear rate linked, for the Poiseuille parabolic profile, to the mean velocity vc by:  =6

vc

(16)

w

v∗c (0) =

or

(18)

Such a condition is generally very easily satisfied. However, if attractive interactions by the wall are described by an adsorption kinetic constant ka (dimension of velocity), the convection time over L must be much smaller than the reaction time. L D/vm  ka D/vm

or

ka 

6D2 vc Lw v2m

(19)

It shows that the condition becomes more severe with increasing solute size. 2.4. Condition of high retention limit over the full length of the cell We would like to keep the high retention limit valid all over the cell length, whatever the variation of the velocities. For such a purpose, the aim is to determine the relation between entrance flow rate, membrane characteristics, channel geometry and ratio of flow rates which satisfies (Eq. (17)). As the membrane velocity is a

 e−sL/2

(24)

(25)

With the notations: tD∗ =

w3/2 d1/2 D

v∗c (0) =

vc 1w  vm 3 L

sinh U U

e−sL

DL

D D  vm  w d

L L2  2D D/vm



w3/2 d1/2 [F(L∗ , sL, rq )G(L∗ , sL, rq )]1/2

f (L∗ , sL, rq ) =

Moreover, in the derivation of the solute elution time, diffusion along the z direction was neglected which can be translated as convection time tc (at distance D/vm ) being much lower than the diffusion time td over the length of the cell.

sL L∗2 rq + ∗ 2 F(L , sL, rq )

A valuable estimation of the optimal entrance velocity is the geometric mean of the two boundaries:

Besides the condition D/vm  w, the characteristic length of the exponential distribution has to be also much larger than the particle size d to avoid drag induced interactions with interface, and/or particles aggregation, and/or significant adsorption phenomena. (17)

(23)

(26) e−sL [F(L∗ , sL, rq )G(L∗ , sL, rq )]1/2

(27)

The optimal velocity becomes: L f (L∗ , sL, rq ) tD∗

(28)

When the channel breadth is constant: F(L∗ , 0, rq ) =

L∗ (1 − cosh L∗ rq ) sinh L∗

(29)

G(L∗ , 0, rq ) =

L∗ (cosh L∗ − rq ) sinh L∗

(30)

f (L∗ , 0, rq ) =

L∗ [(1 − cosh L∗ rq )(cosh L∗ − rq )]1/2 sinh L∗

(31)

In the approximation of constant membrane velocity, i.e.

vm × channel area [b(0) (1 − e−sL )/s] = qc,in − qc,out = qc,in (1 − rq ), the optimal velocity is independent of the membrane permeability thus of L* and does not present any limit for rq :

v∗c (0)vm =constant =

D w3/2 d1/2

1 − e−sL s(1 − rq )

(32)

In addition, the corresponding optimal length lopt , according to the geometrical mean of the extreme values d and w, satisfies lopt /w = d/lopt = (d/w)1/2 . Such an approximation is valid for low values of rq and L* (Fig. 3), where the relative variation of membrane velocity is small over the length of the channel as approached in the work of Ahn et al. [22] (Fig. 4). Anyway the relative variation of the length l, and therefore of l/w is provided by the ratio vm (1)/vm (0) (Eq. (C11)).

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3. Illustration of the model on examples of the literature We address first the determination of the length m through the characterization of the membrane resistance or permeability. Strictly, the resistance of the membrane support has to be included but appears generally to be negligible with respect to the membrane resistance. Available data from membrane providers includes often the flow rate per unit area per unit pressure, under units such as LMH/psi, meaning litre per square metre per hour per psi (1 psi = 6895 Pa). For instance, the PLCGC or Ultracel 20 kD membrane used by Ahn et al. [22] is announced with an average water flux Qw = 9 LMH/psi at 25 ◦ C (viscosity 0.9 × 10−3 Pa s) (http://www.millipore.com/publications.nsf/a73664f9f981af8c85 2569b9005b4eee/ce8d86a76a8048f08525738e005ac086/$FILE/ pf1172en00.pdf). m is linked to Qw by the relation: m = 12  Qw

(33)

where  is the viscosity. For the membrane RC70PP used by Litzen [23], water flux at 25 ◦ C was measured (Fig. 2 of [24]) as 36 LMH/bar (1 bar = 105 Pa). The data available for the PH79 nitrocellulose or NC10 Whatman [21] are the permeability km = 9.72 × 10−12 cm2 of the material constituting the membrane assumed homogeneous and its thickness wm = 105 ␮m. Then the relation to m is:

Fig. 3. Optimal entrance velocity vc *(0) for particle with diameter d = 20 nm, as a function of ratio rq of flow rates for sL = 1.61 and w = 113 ␮m (data from Ahn et al. [22]). Full circles correspond to experimental velocity and rq . (full lines) From top to bottom: L* = 1.5; 1.0; 0.75; 0.50; 0.38 (experimental value according to m = 3.9 × 10−10 cm and L = 23.2 cm). (dashed line) approximation of constant membrane velocity.

m = 12

km wm

(34)

However the experiments in that study implied two reservoirs in parallel to the sample channel whose one wall was without membrane. Therefore, the “effective m ” (2. 1 × 10−7 cm) differed significantly from the membrane alone value (1.1 × 10−8 cm). The values of the characteristic lengths of these membranes are reported in Table 1, with other details relative to the flow cells. Fig. 5 presents a comparison of the entrance velocity from the work of Ahn et al. [22], with the theoretical optimal velocity vc *(0) (Eq. (25)). Experimental data were obtained for an exponential channel (sL = 1.61) and a rectangular one (sL = 0). It appears that the experimental values 120 cm/min (exponential) and 240 cm/min (rectangle) are in the optimal range for the smaller particle of diameter 20 nm but away of it for the larger particles (Table 2), as expected from Eq. (25) (as D ∼ d−1 , vc *(0) ∼ d−3/2 ): vc *(0) ≈ 120; 30; 12; 6.8 cm/min for particle diameter 20; 50; 93;135 nm, respectively. However the width of the channel was deduced from the elution time of that smaller particle. It would be of interest to infer such value from a more direct analysis.

4. Determination of channel height from pressure differential

Fig. 4. Velocities as a function of relative distance Z for (top) exponential and (bottom) rectangular channel. Parameters from Ahn et al. [22]: sL = 1.61, w = 113 ␮m and L* = 0.38 according to m = 3.9 × 10−10 cm and L = 23.2 cm (see Table 1). Channel velocity (left axis; full lines; increasing amplitude in the series rq = 0.1; 0.2; 0.3) and membrane velocity (right axis; dashed lines; decreasing amplitude in the series rq = 0.1; 0.2; 0.3).

Besides the knowledge of m , the height w of the channel has to be known for the estimation of , which characterizes the channel/membrane assembly. Generally w is deduced by measuring the elution time of a solute of known diffusion coefficient in conditions of the high retention limit and applying relations like Eq. (15). The obtained height is often smaller than the nominal thickness of the spacer (Table 1). Indeed are occurring compression of the spacer in the cell mounting, and/or protrusion of the membrane in the channel [21,25]. Pressure measurements should provide in situ access to this parameter, typically by recording the pressure at different flow rates as performed by Litzen [23]. For an exponential channel, the pressure drop P(0) − P(1) between entrance and exit of the sample channel is related to the

P. Déjardin / J. Chromatogr. A 1305 (2013) 213–220

217

Table 1 Characteristic length m of membranes. Channel height w and length L. Characteristic length  of the channel/membrane assembly. L* = L/. Italic: data where w = spacer thickness. Membrane

Cut-off

RC70PP PLCGC (Ultracel 20 kDa) PH 79 nitrocellulose (NC10 Whatman) a

m (cm) −10

Ref.

w (␮m)

L (cm)

 (cm)

L*

120 113 155 150 260

28.5 23.2 23.2 41 41

126.5 61 98 4.0a 9.0a

0.23 0.38 0.24 10a 4.5a

10 kDa 20 kDa

1.1 × 10 3.9 × 10−10

[23,24] [22]

0.1 ␮m Pore size

1.1 × 10−8

[21]

With “effective m ” 2.1 × 10−7 cm.

Table 2 Comparison of the ratios d/l and l/w for different particles in a channel of height w = 113 ␮m: (i) when membrane velocity vm ≈ 0.1 cm/min (Fig. 4) as in Ahn et al. [22] and (ii) when membrane velocity is related to the optimal entrance velocity according to Eq. (28) (vm ∝ vc *(0) ∼ d−3 / 2 ). d (nm) 20 20 50 50 93 93 135 135

D (␮m2 s−1 ) 24 24 9.7 9.7 5.2 5.2 3.6 3.6

vc (0)

vm

(cm/min)

(cm/min)

120

0.10 120

120

0.10 0.10

30 120

0.025 0.10

12 120

0.010 0.10

6.8

0.0057

102 × d/l

l (␮m) 4.05 4.05 1.62 6.48 0.87 8.71 0.60 10.6

102 × l/w

0.49 0.49 3.1 0.77 11 1.1 22 1.3

102 × (d/w)1/2

3.6

– 1.3 – 2.1 – 2.9 – 3.5

3.6 1.4 5.7 0.77 7.7 0.53 9.4

entrance flow rate qc (0) through the relation (Appendix D): 12 P(0) − P(1) = qc (0) b(0)Lm



+e

sL



sL U(cosh U − esL/2 ) + 2 sinh U





U(cosh U − e−sL/2 ) sL rq − 2 sinh U

(35)

In the conditions of equal entrance and exit velocities (rq = e−sL ): P(0) − P(1) 12 = qc (0) b(0)Lm



2U(cosh U − cosh sL/2) sinh U

(36)

For a rectangular channel (sL = 0; U = L*) of resistance Rc the expression is shortened to: P(0) − P(1) Rc tanh(L∗ /2) (1 + rq ) = 2 qc (0) L∗ /2

(37)

Therefore, the height w, as L and b are known, can be deduced from measurements of pressure and flow rates when m has been determined. If the resistance of the connections between the pressure sensors providing Pin and Pout and the channel extremities

Fig. 5. Plots of the optimal entrance velocity (dashed line) as a function of the ratio rq of the sample channel exit over entrance flow rates. The upper and lower boundaries (Eq. (22)) are also plotted. (left) exponential decrease of the channel breadth; (right) channel of constant breadth. From top to bottom: particle diameter d = 20; 50; 93; 135 nm. Full circles correspond to experimental velocity and rq of Ahn et al. [22]. Fig. 6. Pressure drop over entrance flow rate as a function of the ratio of channel exit flow rate over entrance flow rate. Data from Table 1 in Litzen’s work [23]. Rectangular channel.

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P. Déjardin / J. Chromatogr. A 1305 (2013) 213–220

are negligible with respect to channel resistance and if L*  1, then: Rc Pin − Pout ≈ (1 + rq ) 2 qc (0)

(38)

Rc = 12L/(bw3 ) where  is the viscosity. The expected order of magnitude from the Litzen data (Table 1 in [23]) was obtained through fit with Eq. (38) (w = 105 ␮m for a 120 ␮m thick spacer) with a quite large dispersion of the data (Fig. 6). More precise pressure sensors would be needed to check really the model. It suggests however that a quantitative determination of the width of the channel could be reached in situ through the linear relation of the pressure differential over entrance flow rate as a function of rq for exponential channels. 5. Conclusion For exponential to rectangular channels, we determined the optimal entrance velocity to satisfy the conditions of high retention limit for a particle of diameter d as a function of the membrane resistance, the channel geometry and the ratio exit over entrance channel flow rates. The model does not assume the cross-flow (membrane) velocity constant and provides analytical solutions, with an optimal velocity dependent on the membrane resistance through the parameter L*. At high membrane resistance (L*  1) and small rq , the model provides an optimal velocity close to the value obtained under the assumption of constant membrane velocity. Using such an assumption leads to an optimal entrance velocity independent of the membrane resistance and presenting no limitation on the ratio of flow rates (Eq. (32)). Examples have been given to determine the characteristic lengths of membranes (m ), therefore of membrane/channel assemblies () leading to the parameter L*· Comparison with previous experimental data is quite satisfactory. The consequence of a too high membrane velocity for the largest particles is the accumulation of the sample material very close to the membrane favouring parasite phenomena such as aggregation and/or adsorption. In addition, we proposed a quantitative analysis of the differential pressure to infer in situ the thickness of the channel which could be an alternative to using solutes of known diffusion coefficient. High retention limit elution time – defined as the mean velocity of the stationary concentration profile, assumed to be instantaneously established – does not depend on the channel geometry. We did not analyze the peak broadening, which was the source of comments [18] leading to a reply [26] where the position of the recent model with respect to previous ones was pointed out. It could be extended anyway to the “symmetric” configuration as leading to a linear differential system relative to the three pressures in the three compartments. The author of the comments recognized in a recent overview [17] that there are still discrepancies on this matter. Ahn et al. [22] found experimentally some advantages to using the exponential breadth geometry; accumulation of such kinds of data certainly would be welcome for a better understanding of the separation power. Appendix A.

0

C0 e−x/l dx

 and l are depending on Z via the mean axial velocity vc and the membrane velocity vm , respectively: (Z) =6vc (Z)/w and l(Z) = D/vm (Z). Hence Vs (Z) =

6D vc (Z) w vm (Z)

(A1)

(A2)

Complete derivation, taking into account the height of the channel and allowing the transverse field to depend on the distance to the wall, is presented below, within the important assumption that the Poiseuille parabolic profile is maintained in the z direction. When considering the local law of mass conservation when breadth is non-constant (Fig. A1), we arrive at the differential equation with the axial velocity field Vc (x, z) and transverse velocity field Vm (x, z): ∂Vm (x, z) + ∂x





∂V (x, z) 1 db(z) + Vc (x, z) = 0 b(z) dz ∂z

(A3)

Under the assumption of a Poiseuille parabolic axial velocity field all over the length of the cell Vc (x, z) = (z) f(x) with f(x) = x (1 − x/w):



d 1 db(z) ∂Vm (x, z) + f (x) +  =0 dz b(z) dz ∂x

(A4)

We will determine first the transverse velocity field, then the solute stationary concentration profile under its influence combined with diffusion. A.1. Determination of the transverse velocity field In the high retention limit, it was assumed that vm was not dependent on x. As we are considering now such dependence, the condition of conservation of mass (Eq. (14)) can be written, taking into account the orientation of x axis and using the non-reduced variables, as: −Vm (0, z) = −w

Let us consider an exponential distribution of the solute C(x) = C0 e−x/l where x is the distance to the membrane wall in a domain where the axial velocity in the sample channel varies linearly with distance Vc (x) = x. The mean velocity Vs of the solute, within the approximation of integration over full half-space, is:

 ∞ −x/l C0 e xdx 0 Vs =  = l ∞

Fig. A1. Expression of the local conservation of mass with a non-constant breadth b(z) at any position between the two walls.

 dv

c

dz

+



1 db vc (z) b dz

(A5)

With the mean axial velocity vc (z) = (z) w/6: −Vm (0, z) = −

w2 6

 d dz

+

1 db  b dz



(A6)

Then, the differential equation for the transverse velocity is obtained from (A4) as: ∂Vm (x, z) 6 + f (x)Vm (0, z) 2 = 0 ∂x w

(A7)

P. Déjardin / J. Chromatogr. A 1305 (2013) 213–220

219

which in terms of the reduced variables X = x/w becomes: ∂Vm (X, Z) + 6X(1 − X)Vm (0, Z) = 0 ∂X The solution is

(A8)



X

X (1 − X )dX

Vm (X, Z) − Vm (0, Z) = −6Vm (0, Z)

(A9)

0

or Vm (X, Z) = Vm (0, Z)(1 − 3X 2 + 2X 3 )

(A10)

It can be verified that Vm (1, Z) is zero as it must be. Moreover the transverse velocity field is not dependent on the variation of the breadth with distance. A.2. Determination of the stationary solute concentration profile We consider the transverse velocity field Vm (x) at some position z and neglect any transport along the z direction. The stationary distribution C(x) of the solute satisfies the equation: D

Fig. A2. Migration distance as a function of time for particles of diameter 20 nm, according to the simple exponential distribution of solute and half infinite space of integration (Eq. (A2); full line) and taking into account the finite width and nonuniform transverse velocity (Eq. (A17); dashed line). From bottom to top (e−sL = 0.2): rq = 0.1; 0.2; 0.3.

Appendix B.

d2 C d(CVm ) =0 − dx dx2

(A11)

Putting the expression (A10) of the transverse velocity into Eq. (A11) provides the differential equation for C(X):

Let us start from the expression of the membrane velocity (Eq. (4)). The derivative of the membrane velocity with respect to the normalized distance to the entrance is, with notation UZ = U(1 − Z):

d2 C 1 dC 1 + 6 (−X + X 2 )C = 0 + (1 − 3X 2 + 2X 3 )  dX  dX 2

v m (Z) = [coeff > 0]

(A12)



with



D l= |Vm (0)|

l = w

X

C(X) = C(0) exp −



1 1 − X + X3 2 2



 2

sL sL + +ˇ 2 2

(A13)

The solution is, with the condition of non-adsorption on the walls, especially (dC/dX)X=1 = 0:

− U cosh UZ

w 0

C(x)(z)x(1 − (x/w))dx

w 0

C(x)dx



Vs (Z) = 6vc (Z)

0

C(X, (Z))X(1 − X)dX

1 0

t(Z) = L 0

Z

dZ Vs (Z )

2

− ˇL∗2 sinh UZ



esLZ/2



sL 2

 eUZ 2



− U+

sL 2

 e−UZ 2

−ˇL∗2 sinh UZ

esLZ/2

(B3)

(A15) As 0 < UZ < U, U > sL/2 and ˇL*2 > 0, the derivative is always negative. (A16)

(A17)

C(X, (Z))dX

which for   1 provides the limit (A1). Time of migration is:



sL

(B1)

(B2)

v m (Z) = [coeff > 0] − U −

or, with the reduced variables and exhibiting the dependence (Z):

1

esLZ/2

v m (Z) = [coeff > 0] −U cosh UZ +

Let us reintroduce the Z variable by vm (Z) which is the previous Vm (0) dependent on Z:

Vs (z) =

sinh UZ

(A14)

A.3. Expression of the solute average velocity

l(Z) (Z) = w

−ˇU



or, with the definition of U:

or

D , l(Z) = vm (Z)

 2

Appendix C. To derive the condition on channel velocity at the entrance from the condition D/w  vm (1), we need to link vm (1) to vc (0). We use the conservation of mass:

vm (Z) = −

w L

 dv

c

dZ

+



1 db vc (Z) b dZ

(C1)

Eq. (3) provides: rv = (A18)

An example of distance as a function of time according to Eqs. (A2) and (A17) is provided in Fig. A2.

vc (1) U esL/2 = vc (0) U cosh U + (ˇL∗2 − (sL/2)) sinh U

(C2)

and thus is transformed as, with the notation UZ = U(1 − Z):

vc (Z) rv = vc (0) U





U cosh UZ + ˇL∗2 −

sL 2





sinh UZ

e−sL(1−Z)/2

(C3)

220

P. Déjardin / J. Chromatogr. A 1305 (2013) 213–220

rv dvc = vc (0) U dZ

 +

ˇL



breadth b(0) and same length L and width w.

(−ˇL∗2 + sL)U cosh UZ

∗2 sL

2

−

 sL 2 2

 −U

2

P(0, L∗ , sL) − P(1, L∗ , sL) g(U, 0, ˇ) − esL/2 g(U, 1, ˇ) = Rc0 qc (0) f (U, 0, ˇ)

sinh UZ

e

−sL(1−Z)/2



(C4)

f (U, 0, ˇ) = U cosh U + ˇL∗2 −

Therefore,

 dv  c

dZ

1

rv = vc (0){(−ˇL∗2 + sL)U} = vc (0)(sL − ˇL∗2 )rv U

(C5)

Then, the membrane velocity at the position Z = 1 is deduced by means of Eq. (C1) as:

vm (1) =

w w rv vc (0)ˇL∗2 = vc (0) ˇL∗2 rq esL L L

(C6)

From Eq. (10a): ˇ rq L∗2 = F(L∗ , sL, rq ) =

Ue−sL/2 − (U cosh U − (sL/2) sinh U)rq (C7) sinh U

 sL

g(U, 0, ˇ) = ˇU cosh U + ˇ

w D  vc (0)F(L∗ , sL, rq )esL w L or

vc (0) 

(C8)

e−sL DL ∗ 2 F(L , sL, rq ) w

(C9)

The other condition vm (0)  D/d can be written as:

v (1) D vm (1)  m vm (0) d



(C10)



sL L∗2 rq + ∗ 2 F(L , sL, rq )



−1

sinh U

(C12)

The condition (C10) becomes, introducing vm (1) from (C6):

vc (0) 

DL e−sL wd G(L∗ , sL, rq )

+ 1 sinh U

(D3)



(D4)

P(0) − P(1) 1 = Rc0 ∗2 qc (0) L



+e

sL



sL U(cosh U − esL/2 ) + 2 sinh U





U(cosh U − e−sL/2 ) sL rq − 2 sinh U

(D5)

where Rc0 is the resistance of a channel of constant breadth b(0) and length L. As L*2 = Rc /Rm , Rc0 /L*2 is the membrane resistance of breadth b(0) and length L and does not depend on w. Rc0

[1] [2] [3] [4] [5] [6]

We define: F(L∗ , sL, rq ) vm (1)/vm (0)

(D2)

Introduction of Eqs. (D2)–(D4) in Eq. (D1) with the relation of ˇ to parameter rq (Eq. (10a)) leads to:

(C11)

G(L∗ , sL, rq ) =

sinh U

1 12 = Rm0 = b(0)Lm L∗2

(D6)

References

The ratio of membrane velocities is obtained from Eq. (4) and expression of ˇ (Eq. (C7)):

vm (1) U U cosh U + = vm (0) e−sL/2



g(U, 1, ˇ) = ˇU

Therefore, the condition D/w  vm (1) becomes the condition on

vc (0):

2

sL 2

(D1)

(C13)

The combination of the inequalities (C9) and (C13) provides the relation (22). Appendix D. The difference of pressure between entrance (Z = 0) and exit (Z = 1) as a function of the ratio of flow rates for an exponential channel is obtained from Eqs. (A1)–(A3) in Appendix of [20] where – P 00 = Rc0 qc (0). Rc0 is the resistance of a rectangular channel of

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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