From nanotubes to nanoholes: scaling of selectivity in uniformly charged nanopores through the Dukhin number for 1:1 electrolytes

Figure 2 illustrates the effect of the two relevant parameters ($\sigma$ and $\lambda_{\mathrm{D}}/R$) for fixed values of $R$, $H$, and $U$. The surface charge produces the separation of counterions and coions (DL formation), while the $\lambda_{\mathrm{D}}/R$ parameter accounts for the degree of overlap of the DL. The figure shows radial concentration profiles to depict the interplay of these two effects.

The columns correspond to different $\lambda_{\mathrm{D}}/R$ values (a larger $\lambda_{\mathrm{D}}/R$ value means stronger DL overlap), while the rows correspond to different surface charges (a larger $|\sigma|$ value results in a stronger separation of cation and anion profiles).

As we go from left to right (decreasing $\lambda_{\mathrm{D}}/R$), the gap between the cation and anion profiles around the centerline $r\approx 0$ decreases (note the log scale). As we go from top to bottom (increasing $|\sigma|$), the gap between the cation and anion profiles near the wall ($r\approx R$) increases.

This implies that the two parameters have two relatively distinct effects. The $\lambda_{\mathrm{D}}/R$ parameter rather determines the behavior in the middle of the pore, while $\sigma$ rather determines the behavior in the DL. It is common to express this distinction in terms of volume (or bulk) and surface conductions. The DL region (if exists) is responsible for the surface conduction, while the bulk region in the middle (if exists) is responsible for the volume conduction. With $\lambda_{\mathrm{D}}/R$ and $\sigma$, therefore, we have two parameters with which we can tune the weights of the volume and surface conductions in the total conduction.

These two effects, however, separate only in the nanotube limit ($H\rightarrow\infty$) that physically can be corresponded to a finite but long nanopore, where the influence of the DLs at the entrances of the nanopore is negligible in the middle of the nanopore. Voltage can also modify the DLs at the membrane’s two sides: it creates DLs of opposite signs on the two sides that, in turn, also modify the ionic distributions inside the pore. If the pore is short, therefore, these effects must be taken into account.

Considering the axial dimension, our main question is how the length of the pore influences the concentration profiles in the pore, and, consequently, selectivity. Fig. 3A shows the profiles for different values of the pore length. The cation concentration is larger in the $H\rightarrow\infty$ limiting case than in the $H=18$ nm case (the anion concentration, at the same time, is smaller). This is because charge neutrality is enforced in the GCMC simulations for $H\rightarrow\infty$, while the finite pore does not have to be charge neutral.

The pore charge of a short pore is only partly neutralized by the excess counterions inside the pore; it is partly neutralized by the excess cations accumulated at the entrances of the pore in the DLs near the membrane. This accumulation is larger if $\sigma$ is larger. Because the ions are charged hard spheres, volume exclusion works against ions entering the pore and ion accumulation outside (where there is more space) is advantageous and minimizes free energy.

This counterion accumulation is shown in Fig. 3B by the curves for $U=0$ mV. When voltage is increased, an excess cation accumulation can be observed on the right hand side, while an excess anion accumulation on the left hand side of the pore. This pair of charge accumulations of opposite signs creates a “dipole field” and arises because the electric field of the ions produces a counterfield against the applied field. More details about the origin of this “dipole field” can be found in the Appendix (Fig. 11), where it is explained that a larger charge accumulation is obtained both for shorter pores and for larger voltages. This is clearly seen from the concentration profiles of Fig. 3 as well.

As the pore length is decreased further from $H=18$ nm, the concentration profiles of both cations and anions elevate in the pore, because the DLs contain more charge in the case of short pores and the stronger counterion accumulation has a larger effect inside the pore. This modifies the imbalance of cations and anions in the pore for different $H$ values results in varying selectivity. Similarly, increasing voltage increases the degree of charge separation between the two sides of the membrane (Fig. 3B). By the same mechanism, the changed DL structure on the two sides of the membrane changes the electric field inside the pore, and, consequently, ionic distributions and selectivity.

In both panels of Fig. 3, the axial profiles have been obtained by normalizing with the cross section of the cell. Thus, the concentration profiles characterize the density of ions at a given $z$, not the quantity of ions. The quantity of ionic charge in the DLs outside the pore, therefore, is much larger than implied by this figure. Keeping this in mind, however, the figure shows the trends and describes the $H$ dependence (panel A) and the $U$ dependence (panel B) of the ionic profiles.

If we want to assess how the pore entrances influence the interior of the pore, we can use the relation of the screening length to the pore length, $\lambda_{\mathrm{D}}/H$, to quantify this effect. The charge accumulation at the entrance creates a double layer in the axial dimension that “penetrates” the pore. The degree of penetration and whether it reaches the middle of the pore can be characterized, at least, as a first approximation, with the ratio $\lambda_{\mathrm{D}}/H$. If $\lambda_{\mathrm{D}}/H>0.5$, the DLs extending axially from the two sides overlap in the middle, a phenomenon we call “bridging”.

From these analyses, we can see that the picture gets complicated as we go from the infinite pore to shorter pores and larger voltages. Every parameter has an effect that also influences the effects of other parameters. In the following, we try to cut a path through this jungle and provide a systematic analysis of the effects of all parameters. Because the nanotube limit is the simplest one, it is reasonable to start our analysis with that case.

Because there is a monotonic relationship between ionic selectivity and the ratio of surface and volume conductances that are present in the nanopore at the same time, the Dukhin number bazant_pre_2004 ; chu_pre_2006 ; bocquet_chemsocrev_2010 defined as

$$\mathrm{Du}\equiv\frac{|\sigma|}{eRc}$$

(7)

offers itself to be the scaling parameter we are after. This definition of Du contains exactly those variables that we want to see in our scaling parameter, $\sigma$, $R$, and $c$, although $H$ is missing.

Du was originally introduced by Bikerman bikerman_1940 to characterize the ratio of the surface and volume conductances focusing on electrokinetic phenomena. Later, Dukhin adopted the idea (see Ref. dukhin_advcollsci_1993, and references therein) to study electrophoretic phenomena. Although the name Bikerman number also occurs in the literature, Lyklema introduced the name Dukhin number to salute Dukhin. lyklema_book_1995 The Dukhin number (and a characteristic length called Dukhin length, $l_{\mathrm{Du}}=\mathrm{Du}\,R$) was used in several modeling studies describing nanopores, more specifically, when volume and surface transport processes competed inside the pore. bazant_pre_2004 ; chu_pre_2006 ; khair_jfm_2008 ; das_langmuir_2010 ; bocquet_chemsocrev_2010 ; zangle_csr_2010 ; lee_nanolett_2012 ; yeh_ijc_2014 ; ma_acssens_2017 ; xiong_scc_2019 ; poggioli_jpcb_2019 ; dalcengio_jcp_2019 ; kavokine_annualrev_2020 ; noh_acsnano_2020

The definition in Eq. 7 is in agreement with the traditional definition of Bikerman because it can be computed from the ratio of the surface excess of the cations, $|\sigma|2\pi RH$ assuming perfect exclusion of the anions, and the number of charge carriers assuming a bulk electrolyte in the pore, $2cR^{2}\pi H$ (the factor $2$ is needed because both cations and anions carry current).

For a 1:1 electrolyte, the Debye length can be written as $\lambda_{\mathrm{D}}^{2}=1/(8\pi l_{\mathrm{B}}c)=1/(l_{\mathrm{B}}^{*}c)$, where $l_{\mathrm{B}}^{*}=8\pi l_{\mathrm{B}}$ and $l_{\mathrm{B}}=e^{2}/4\pi\epsilon_{0}\epsilon kT$ is the Bjerrum length. The Dukhin number then can be rewritten in the form

$$\mathrm{Du}=\frac{|\sigma|l_{\mathrm{B}}^{*}\lambda_{\mathrm{D}}^{2}}{eR}$$

(8)

that makes it possible to relate quantities with the dimensions of distance to each other. We may also use other variations for the screening length that may depend on electrolyte properties ($z_{i}$ and $R_{i}$) such as the MSA screening length fertig_jpcc_2019 or on confinement levy_jcis_2020 .

Figure 4 shows the selectivity (Eq. 1) as a function of $\mathrm{Du}$ (log scale for Du). The figure includes results from PB calculations (lines) and GCMC simulations (symbols). In the case of PB, the Debye length and pore radius are coupled, so the curves for different $\lambda_{\mathrm{D}}/R$ values collapse onto one single curve. The deviations for the small $\lambda_{\mathrm{D}}/R$ value are due to to the imprecision of the numerical methods applied). This problem only appeared at higher concentration values as the slope of the different functions ($\Phi(r),c_{i}(r)$) became larger. The figure also shows the Du values of the inflection points of the curves denoted as $\mathrm{Du}_{\infty}$. This value is estimated as $\mathrm{Du}_{\infty}\approx 0.546$ for PB.

The inflection point belongs to selectivity $S_{+}\approx 0.75$ and have a special role in our analysis. Fig. 4 and later figures show that the selectivity curves can be fitted with sigmoid curves

$$S_{+}(x)=0.5+\dfrac{0.5}{1+e^{-k(x-x_{0})}},$$

(9)

where $x=\lg(\mathrm{Du})$ and $x_{0}$ defines the inflection point ($\lg$ denotes logarithm of base $10$). The shape of the curves is very similar (the parameter $k$ is in the same ballpark between $2.5$ and $4$), so the major difference is that the selectivity curves are shifted along with the $\lg(\mathrm{Du})$ axis as $H$, $U$ or any other parameter is changed.

This way, we reduced the problem to the examination of the shift of the inflection point denoted as $\mathrm{Du}_{0}$. For the nanotube, therefore, $\mathrm{Du}_{\infty}=\lim_{H\rightarrow\infty}\mathrm{Du}_{0}(H)$. Because we have a good scaling for $H\rightarrow\infty$ , the $\mathrm{Du}_{\infty}$ values have a special role: we can relate the cases for finite pores to this limit. If we divide $\mathrm{Du}$ with $\mathrm{Du}_{0}$, we shift the selectivity curve into $\mathrm{Du}^{*}=\mathrm{Du}/\mathrm{Du}_{0}=1$, where

$$\mathrm{Du}^{*}=\frac{\mathrm{Du}}{\mathrm{Du}_{0}}$$

is a rescaled Dukhin number.

The $\mathrm{Du}_{0}$ inflection point, of course, depends on every parameter and the model (NP+LEMC or PNP). Our purpose is to map this dependence in a way that is useful and consumable for the reader (the number of variables is large).

For GCMC, for example, $\mathrm{Du}_{\infty}\approx 0.37$ as shown by Fig. 4. The deviation between PB and GCMC is twofold. First, the ions are point-like in PB, while they have finite size in the GCMC simulation (Eq. 3). Second, the GCMC (and LEMC) simulations include all the electrostatic correlations that are beyond the mean-field (BMF) treatment of PB (and PNP). For the 1:1 electrolyte studied here, the electrostatic BMF correlations are relatively weak, so the main source of the deviation between PB and GCMC is the finite size of particles.

This statement is supported by Fig. 4 that shows GCMC results for various $\lambda_{\mathrm{D}}/R$ ratios obtained from simulations for $R=1$ and $2$ nm using various concentrations. The points obtained for $R=2$ nm (open symbols) are closer to the PB curves because the finite size of ions has less importance in the wide pore.

The difference between PB and GCMC can also be seen in Fig. 5 that shows concentration profiles (top row) and selectivity profiles (bottom row) defined as

$$s_{+}(r)=\dfrac{j_{+}(r)}{j_{+}(r)+j_{-}(r)}\approx\dfrac{c_{+}(r)}{c_{+}(r)+c_{-}(r)},$$

(10)

where $j_{+}(r)$ and $j_{-}(r)$ are the $z$-components of the flux profiles for the cation and the anion, respectively, averaged over the pore in the $z$ dimension (from $-H/2$ to $H/2$). It is a quantity that depends on $r$ and characterizes to what degree a region of the pore at a distance $r$ from the $z$ axis contributes to the “global” selectivity, $S_{+}$. Note that the average of $s_{+}(r)$ is not equal to $S_{+}$, but we can draw conclusions from the shapes of the curves nevertheless.

The columns of Fig. 5 refer to different selectivities corresponding to specific scaled Dukhin numbers, $\mathrm{Du}^{*}$. These different $\mathrm{Du}^{*}$ values correspond to different combinations of $\sigma$ and $\lambda_{\mathrm{D}}/R$. If $\lambda_{\mathrm{D}}/R$ is small (red curves), for example, larger $\sigma$ values are needed to achieve the same selectivity. Red curves for $s_{+}(r)$ rise to a higher value at the wall due to the increased $\sigma$, but they decrease (the $c_{+}(r)$ and $c_{-}(r)$ profiles get closer to each other) in the centerline of the pore resulting in the same global selectivity, $S_{+}$, as, for example, the black curves. In the case of the black curves, $\lambda_{\mathrm{D}}/R$ is large, so the DL overlap is strong. As a consequence, a smaller surface charge is sufficient to get the same selectivity.

The difference between the PB and GCMC curves is apparent. Because the centers of the ions are confined into a cylinder with the radius $R-R_{i}$ ($R_{i}=0.15$ nm), the ionic profiles do not extend to the wall. If we replot the figure by normalizing $r$ with $R-R_{i}$ instead of $R$, we obtain a much better agreement (see Fig. 2 of the SI). The $R-R_{i}$ radius can be considered an “effective” radius of the pore that the centers of the hard-sphere ions can reach.

If we start decreasing the pore length (voltage is $U=200$ mV), the DLs at the entrances of the pore “penetrate” the pore and modify the ionic distributions inside the pore as shown in Fig. 3A (“bridging”). This modifies the mutual effects of the parameters on selectivity, so also the shift of the $S_{+}(\mathrm{Du})$ function with respect to $\mathrm{Du}_{\infty}$. This is shown in Fig. 6, where the pore length decreases from left to right and different colors in a panel belong to different concentrations.

The curves are shifted to the right, so shorter pores require larger surface charges to produce the same selectivity. The curves for lower concentrations (at a given $H$) are also shifted to the right so electrolytes with more extended diffuse layers enhance the effect of the small length of the pore and also require larger surface charges to produce the same selectivity.

The main conclusion of this figure is that Du is not an appropriate scaling parameter for finite pores.

Fig. 7 shows the inflection point, $\mathrm{Du}_{0}$ normalized by the $H\rightarrow\infty$ value, $\mathrm{Du}_{\infty}$ as a function of $\lambda_{\mathrm{D}}/R$ (Fig. 7A) and $\lambda_{\mathrm{D}}/H$ (Fig. 7B). So, the limiting value $\mathrm{Du}_{0}/\mathrm{Du}_{\infty}=1$ corresponds to the nanotube limit. Fig. 7A shows the data as functions of $\lambda_{\mathrm{D}}/R$ for different aspect ratios $H/R$. With increasing $\lambda_{\mathrm{D}}/R$, we have wider DLs for a fixed $R$, so these wider DLs extend into the pore changing nanotube behavior and shifting the inflection point. With increasing $H/R$ (indicated by the arrow in the figure), we approach the nanotube limit, so the values of $\mathrm{Du}_{0}/\mathrm{Du}_{\infty}$ decrease for a fixed $\lambda_{\mathrm{D}}/R$ approaching the $\mathrm{Du}_{0}/\mathrm{Du}_{\infty}=1$ limit. The lines are fitted power functions. Unfortunately, we were not able to find any pattern in the exponent.

If we plot the results as functions of $\lambda_{\mathrm{D}}/H$, on the other hand, we can characterize the two limiting cases quantitatively (Fig. 7B). For a fixed pore radius $R=1$ nm, the curves of a given color correspond to a given concentration. As $\lambda_{\mathrm{D}}/H$ decreases, we approach the nanotube limit. Either the pore becomes long enough for a given electrolyte so that the pore entrances do not disturb the ions’ behavior in the middle of the pore, or the electrolyte becomes more concentrated so that the DLs extend less into the pore. The system approaches this limiting case with different trends depending on concentration. At large concentration ($c=1$ M, purple curve) the curve approaches the $\mathrm{Du}_{0}/\mathrm{Du}_{\infty}=1$ limit faster as it was already seen in Fig. 7A.

The more interesting case is the nanohole limit ($H/R\rightarrow 0$ with constant $R$) where the curves seem to collapse onto a line, at least, for concentrations with Debye lengths smaller than $R$. It is important that the slope of the line is $1$, so it can be given by the equation

$$\lg\left(\dfrac{\mathrm{Du}_{0}}{\mathrm{Du}_{\infty}}\right)=b+\lg\left(\dfrac{\lambda_{\mathrm{D}}}{H}\right).$$

(11)

where $b$ is a fittable parameter. The resulting value is $b\approx 1.4$. For the NP+LEMC simulations ($\mathrm{Du}_{\infty}=0.37$), the inflection point in the $H/R\rightarrow 0$ limit for $R=1$ nm and $U=200$ mV can be expressed as

$$\mathrm{Du}_{0}=9.29\frac{\lambda_{\mathrm{D}}}{H}.$$

(12)

Now if we want to bring the selectivity curves onto each other, namely, we want the scaling to work, we need to divide $\mathrm{Du}$ by this $\mathrm{Du}_{0}$ to define a new scaling parameter that we call the modified Dukhin number:

$$\mathrm{mDu}=\mathrm{Du}\frac{1}{\beta}\frac{H}{\lambda_{\mathrm{D}}}=\frac{|\sigma|l_{\mathrm{B}}^{*}\lambda_{\mathrm{D}}H}{eR\beta},$$

(13)

where the $\beta$ parameter depends on $R$ and $U$. We have the most data for $U=200$ mV, so at this point, we can state the existence of the above scaling parameter only in the large voltage limit.

The selectivity curves as functions of this modified Dukhin number are shown in Fig. 8 for the same cases included in Fig. 6. It is seen that the new scaling parameter works quite well for short pores: it brings the curves for different concentrations together and also the curves for different pore lengths because the inflection points are close to $1$. As $H$ is increased ($H=18$ nm and $H\rightarrow\infty$), the curves become more separated and the inflection points are shifted.

The advantage of the mDu parameter is that it contains $H$ so it can account for the length of the pore. Unfortunately, it is not appropriate for the long pores ($H\rightarrow\infty$). The question arises whether we can construct a scaling parameter that is appropriate for both long and short pores. At this point of our research, we have only a heuristic formula for the scaling parameter (let us denote it with $\mathrm{Du}^{\prime}$) that is based on the two limits:

$$\mathrm{Du}=\lim_{H\rightarrow\infty}\mathrm{Du}^{\prime}\quad\quad\mathrm{and}\quad\quad\mathrm{mDu}=\lim_{H\rightarrow 0}\mathrm{Du}^{\prime}.$$

(14)

This implies that $\mathrm{Du}^{\prime}$ can be written as a mix of the two limiting cases. The parameter that tuned how strongly the DLs at the entrances influence the pore interior is $\lambda_{\mathrm{D}}/H$, so we propose the following equation for the “universal” scaling parameter:

$$\mathrm{Du}^{\prime}=\mathrm{Du}\dfrac{1}{1+a(c)\dfrac{\lambda_{\mathrm{D}}}{H}}.$$

(15)

If $H$ is large, the $a(c)\lambda_{\mathrm{D}}/H$ term can be neglected next to $1$, so we recover $\mathrm{Du}$. If $H$ is small, $1$ can be neglected next to $a(c)\lambda_{\mathrm{D}}/H$, so we recover $\mathrm{mDu}$. The adjustable $a(c)$ parameter depends on the concentration, although we found it independent of concentration in the case of PNP.

In any case, by plotting $S_{+}$ against $\mathrm{Du}^{\prime}$ we obtain a scaling that works for every value of $H$ investigated (see Fig. 3 of the SI). This statement is valid rigorously only for $U=200$ mV and $R=1$ nm because that is the case that we investigated in detail. We are confident that $\mathrm{Du}^{\prime}$ can be mixed for other values of $R$ as well, but we are not so sure about voltage.

Voltage dependence has been studied with the PNP theory. The results are shown in Fig. 9. The general trend is that $\mathrm{Du}_{0}/\mathrm{Du}_{\infty}$ depends on $U^{2}$: the lines in Fig. 9 are quadratic fits. The dependence on voltage is stronger at small concentrations, when the ionic diffuse layer is extended over a large space influencing the behavior inside the pore and voltage can distort this more diffuse DL more efficiently. Phrasing in a different way, voltage produces more extended positive/negative DLs (shown in in Fig. 3B) for dilute electrolytes.

Also, the effect of voltage is stronger if the pore is shorter. This is because the charge in the DLs is larger in the case of the shorter pore (see Fig. 3A and the Appendix for more detailed discussion) and the DLs can extend into the pore more deeply and can modify the ionic distributions.

Summarized, the interplay of the various parameters of the system ($U$, $R$, $H$, $c$, and $\sigma$) is so intricate that it is difficult to come up with a “universal” scaling parameter. However, we can draw conclusions for the behavior of the nanopore for other combinations of the parameters on the basis of knowledge about the behavior for a specific parameter set.

The Dukhin number was originally introduced as a parameter characterizing the ratio of the surface and volume conductions. This works well in the limiting cases when either volume conduction or surface conduction dominates, e.g., for $\mathrm{Du}\ll 1$ or $\mathrm{Du}\gg 1$, or, equivalently, for $S_{+}\approx 0.5$ or $S_{+}\approx 1$. In this work, however, we are interested in the domain in between, when the nanopore may contain both a region where rather the surface conduction dominates (near the wall) and a region where rather volume conduction dominates (in the middle of the pore around the centerline).

In this domain (around the inflection point), it is not obvious from the “global” selectivity value ($S_{+}$) or the Dukhin number alone to what degree surface and volume conductions are present in the two respective regions of the pore (close and far from the surface). For example, if we have a large selectivity value $S_{+}=0.85$, it does not necessarily mean that there is no volume conduction in the pore. Similarly, if we have a mildly selective value $S_{+}=0.65$, it does not necessarily mean that there is no surface conduction.

If we want more information, we need to look into the “black box” and investigate concentration and selectivity profiles as we already did in Fig. 5 for the nanotube limit. In Fig. 5, we already observed that the same selectivity can be achieved in different ways depending on the relation of $\lambda_{\mathrm{D}}/R$ and $\sigma$. Different combinations of these parameters can produce the same Du, and, consequently, the same $S_{+}$, but the corresponding profiles behave differently. Larger surface charge enhances separation of the cation and anion profiles near the wall, while large $\lambda_{\mathrm{D}}/R$ value enhances DL overlap in the centerline of the pore.

Figure 10 shows similar results for a finite pore ($H=6$ nm). The top row shows selectivity and concentration profiles for two simulations that, due to scaling, produce about the same “global” selectivity, $S_{+}=0.787$ (blue) and $0.757$ (red). The blue curve is for a smaller surface charge ($\sigma=-0.1$ $e$/nm²) and a smaller concentration ($c=0.025$ M), while the red curve is for a larger surface charge ($\sigma=-0.5$ $e$/nm²) and a larger concentration ($c=1$ M) for fixed $R$.

The $s_{+}(r)$ curves (left panel) show that approximately the same $S_{+}$ value can be achieved in two ways. In the case of the blue curve, the DL overlaps (small $c$, large $\lambda_{\mathrm{D}}$) so the coion (anion) is excluded over the whole cross section and selectivity is at a high level even if $\sigma$ is relatively small. In the case of the red curve, the DL overlaps less (large $c$, small $\lambda_{\mathrm{D}}$) so a less selective bulk region is formed in the middle, but the larger surface charge ($\sigma=-0.5$ $e$/nm²) produces a large counterion-coion separation near the wall and pulls up selectivity there.

The bottom panels show results for two simulations that produce approximately the same “global” selectivities, $S_{+}=0.77$ (blue) and $0.772$ (red) for the same concentration ($c=0.1$ M). The blue curve is for a smaller surface charge ($\sigma=-0.25$ $e$/nm²) and a smaller pore radius ($R=2$ nm), while the red curve is for a larger surface charge ($\sigma=-0.5$ $e$/nm²) and a larger pore radius ($R=4$ nm).

The $s_{+}(r)$ curves (left panel) show similar behavior to those in the top panel. In the case of the blue curve, the DL overlaps (small $R$) so there is less space for the formation of a bulk region in the middle. Therefore, the selectivity is uniformly large even if $\sigma$ is relatively small. In the case of the red curve, the degree of overlap of the DL is smaller (large $R$) so a less selective bulk region is formed in the middle, but the larger surface charge ($\sigma=-0.5$ $e$/nm²) produces a large counterion-coion separation near the wall and pulls up selectivity there.

Looking into the “black box”, however, is usually not possible in experiments. Therefore, for a rule of thumb to decide whether volume or surface conduction dominates in a state point, we propose using the inflection point.

The inflection point offers itself as a transition point separating “rather non-selective” and “rather selective” regions. The derivative of the $S_{+}$ curve as a function of $\lg(\mathrm{Du})$ (or any suitable scaling parameter) is maximal in the inflection point where small changes in Du lead to relatively large changes in selectivity.