### Numerical integration and comparison with some existing partial measurements

We show in Figure2 an example of the results obtained by numerically integrating Equations 5 to 7 using some representative values for the parameters involved (and always in the case of constant *P* and *C*_{imp}, and starting from a clean initial state *n*(*x* *t* = 0)=0). In particular, we have chosen parameter values that reproduce the case of channels coated with Y_{2}O_{3} nanopowders as measured in[5] (they are essentially valid also for the quite similar case of channels with ZrO_{2} nanocoating reported by the same group in[6]). In these filters, the channels have a typical value of the nominal radius *r*_{0} = 500 nm and length *L* = 7*.* 25 mm. They were shown[5] to efficaciously retain MS2 viruses (of radius *ρ*_{0} = 13 nm) carried by water with NaCl as background electrolyte and a conductivity of 400*μ* S/cm (corresponding then to *λ*_{D}≃5*.* 1 nm) feed at a pressure *P* = 3 bar. The incoming impurity number concentration was${C}_{\text{imp}}(x=0,t)=1{0}^{10}/{\text{m}}^{3}$. For the saturation areal density *n*^{sat}, we will estimate, based on figure nine of[5], a quite conservative value *n*^{sat} = 1*.* 5 × 10^{15}/m^{2}, corresponding to$0.25/{\rho}_{0}^{2}$. For the parameter *r*_{1}, we will use the value$4{\rho}_{0}^{3}$, also consequent in the order of magnitude with figure nine of[5]. These numbers imply that at saturation (*n* = *n*^{sat}), the effective radius of the channel is${r}_{e}^{\text{sat}}=487$ nm. Note that this value is rather close to the clean-state value of 500 nm, and then it would correspond to an increase of the hydrodynamic resistance of only about 10% (unfortunately, the nanocoatings in[5, 6] seem to be washed out before they can be fully saturated; however, other nanocoated filters[4, 7, 8] have been shown to have hydrodynamic resistance only moderately increased at saturation, what is indeed an advantage of paramount importance for applications). We will also assume a null${\mathrm{\Omega}}_{\text{trap}}^{\prime}$ value at the saturated state, i.e., Ω_{0} = 0 (so that we neglect conventional filtration mechanisms and focus on the effects of nanostructuring alone). In order to proceed with the numerical calculation of Equations 5 to 7, only two parameters remain to be given estimated values: Ω_{1}*z*_{0}(Ω_{1} and *z*_{0} do not appear separately in Equations 5 to 7) and *ρ*_{1}(or equivalently, via Equation 3, the effective impurity radius in the clean state of the channel,${\rho}_{\text{e}}^{\text{clean}}$). We have found that the values Ω_{1}*z*_{0} = 1*.* 2 × 10^{5}/m and *ρ*_{1} = 0*.* 11 produce results in reasonable agreement with the available experimental information, as we discuss below. The value *ρ*_{1} = 0*.* 11 corresponds to${\rho}_{\text{e}}^{\text{clean}}=33$ nm, or *ρ*_{0} + 4*λ*_{D}.

Figure2a presents the results corresponding to integrating Equations 5 to 7 using these parameter values, for the areal density of trapped impurities at the entrance of the channel *n*(*x* = 0,*t*) and at its exit point *n*(*x* = *L* *t*) and also for the global average areal density of trapped impurities$\stackrel{\u0304}{n}\left(t\right)={L}^{-1}{\int}_{0}^{L}n(x,t)dx$. Figure2b presents the corresponding logarithmic removal value (LRV), calculated as$-{log}_{10}\left[{C}_{\text{imp}}\right(x=L,t)/{C}_{\text{imp}}(x=0,t\left)\right]$. Note that in Figure2a,b, the time axis is logarithmic and that for convenience, it was normalized by the time *t*_{1/2} defined by the condition$\stackrel{\u0304}{n}\left({t}_{1/2}\right)={n}^{\text{sat}}/2$ (half-saturation time). The agreement of these numerical results with the measured filtration performance reported in[5, 6] is fairly good. In particular, we obtain an initial LRV of 6.5 log, equal to the LRV measured in[5, 6] when the actual filters (composed by a macroscopic array of microchannels) were challenged with only about 1 L of water (the authors of[5, 6] estimate that such volume carries a total amount of impurities that is orders of magnitude smaller than the total available binding centers in their filter, so the measurement is expected to correspond to almost clean channels, as in fact seems to be confirmed by microscopy images[5]). The calculated LRV is of 4 log at *t*/*t*_{1/2}≃0*.* 7, which is also in fair agreement with the observation of a 4 log filtration in[5, 6] after passing through the macroscopic filter approximately from 200 to 1,000 L, depending on the measurement. However, obviously, a more stringent determination of the parameter values, and in general of the degree of validity of our equations, would need more precise and detailed data. Unfortunately, to our knowledge, no measurements exist for the time evolution of the filtering efficiency of channels with nanostructured walls with a *t*-density and precision sufficient for a fully unambiguous quantitative comparison with the corresponding results of our equations; in fact, one of the main motivations of the present Nano Idea Letter is to propose (see our conclusions) that such measurements should be made, in order to further clarify the mechanism behind the enhanced impurity trapping capability of the channels with nanostructured inner walls.

As a further test, we have repeated the same numerical integration as in Figure2a,b but considering a radial impurity concentration profile${C}_{\text{imp}}\left(r\right)\propto 1+{z}_{e}{C}^{\prime}{e}^{-(r-{r}_{e})/{\lambda}_{D}}$, instead of a constant one as in Equation 4. We have obtained very similar results, provided that the parameter Ω_{1}*z*_{0} is conveniently varied: In particular, we observed that the filtration dynamics results obtained using Equation 4 and any given value *γ* for Ω_{1}*z*_{0} can be reproduced using the above Debye-like profile if employing for Ω_{1}*z*_{0} a new value (specifically, the new value can be estimated, by comparing the initial filtration performance, as$\gamma {\mathcal{I}}_{{r}_{0}-{\rho}_{e}^{\text{clean}}}^{1}{\mathcal{I}}_{0}^{{C}_{\text{imp}}(r)}/{\mathcal{I}}_{0}^{1}{\mathcal{I}}_{{r}_{0}-{\rho}_{e}^{\text{clean}}}^{{C}_{\text{imp}}(r)}$, where${\mathcal{I}}_{x}^{y\left(x\right)}={\int}_{x}^{{r}_{e}}y\left(x\right)u\left(x\right)dx$ ; for instance, taking${z}_{0}{C}^{\prime}=200$, which probably is a fair first approximation for the measurements in[5–8], the parameter values used in Figure2 correspond to 3*.* 2 × 10^{4}/m as equivalent Ω_{1}*z*_{0} value when using the Debye approach). These results indicate then that, as it could be expected, the wall charge effect on the radial gradient of the concentration may be safely summarized, for our present purposes, as one of the factors influencing the value of the trapping probability coefficient Ω_{1}*z*_{0} to be used when applying Equation 4.

### Linear, logarithmic, and saturated approximations

In Figure2a, it is possible to identify in our results for the areal density of trapped impurities some *t*-ranges in which the *t*-dependence is relatively simple: (1) The initial time behavior is an approximately linear *n*(*t*) growth; (2) in the intermediate regime, the growth of *n*(*t*) becomes approximately logarithmic; and (3) at sufficiently large *t* values, the saturation limit is reached, in which *n* approaches a value *n*^{sat} at a slow pace. These regimes are easily seen in Figure2a for *n*(*x* = 0,*t*), *n*(*x* = *L*,*t*), and$\stackrel{\u0304}{n}\left(t\right)$, albeit in each case they are located at different *t*/*t*_{1/2} ranges. The figure also evidences that it is possible for the linear and logarithmic *t*-ranges to overlap each other (the case of$\stackrel{\u0304}{n}\left(t\right)$ with the parameter values used in Figure2).

In the case of a very short cylindrical channel (so that all

*x*-derivatives may be neglected), it is possible to find analytical expressions for the

*n*(

*t*) evolution in the linear and logarithmic regions: For the linear regime, by just introducing in Equation

5 the condition

*t* ≃ 0, we find:

$n\left(t\right)\simeq {A}_{\text{lin}}t$

(8)

with

$\begin{array}{ll}\phantom{\rule{5pt}{0ex}}{A}_{\text{lin}}=& \frac{({\mathrm{\Omega}}_{0}+{\mathrm{\Omega}}_{1}{z}_{0}){C}_{\text{imp}}P}{16\eta L{r}_{0}}\\ \times {\left[{\left({\rho}_{0}-{r}_{0}+{\lambda}_{\text{D}}\phantom{\rule{2.56865pt}{0ex}}W\left(\frac{{\rho}_{0}}{{\lambda}_{\text{D}}{\rho}_{1}}\right)\right)}^{2}-{r}_{0}^{2}\right]}^{2}.\end{array}$

(9)

The logarithmic regime can be found by using the condition

*n* ≃

*n*^{sat}/2:

$n\left(t\right)\simeq \frac{{n}^{\text{sat}}}{2}+{A}_{\text{log}}ln\frac{t}{{t}_{1/2}}$

(10)

with

$\begin{array}{ll}\phantom{\rule{5pt}{0ex}}{A}_{\text{log}}=& \frac{(2{\mathrm{\Omega}}_{0}+{\mathrm{\Omega}}_{1}{z}_{0})\phantom{\rule{2.36043pt}{0ex}}{t}_{1/2}{C}_{\text{imp}}P}{512\eta L\phantom{\rule{0.3em}{0ex}}{r}_{0}}\\ \times {\left[{\left(2{\rho}_{0}-2{r}_{0}+{\lambda}_{\text{D}}\phantom{\rule{2.36043pt}{0ex}}W\left(\frac{{\rho}_{0}}{{\lambda}_{\text{D}}{\rho}_{1}}\right)+{r}_{1}{n}^{\text{sat}}\right)}^{2}-{\left(2{r}_{0}-{r}_{1}{n}^{\text{sat}}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\right]}^{2}\phantom{\rule{0.3em}{0ex}}.\end{array}$

(11)

In obtaining the above Equations 8 to 11, we have assumed that *n*(0) = 0 and that *ρ*_{
e
}< *r*_{
e
} at *t* = 0 or *t*_{1/2}.