3.1. Variation of the thermo-osmotically induced electric field

We characterize the TOS flow by studying the thermo-osmotically induced dimensionless electric field $\bar {E}$ and the dimensionless flow velocity $\bar {u}$. We compare the results for three different cases of brush-grafted nanochannel and three corresponding cases of the brushless nanochannel. In the following discussions, the brushless cases are identified as ‘No-Brush’ case. More importantly, we assume that the walls of these brushless nanochannels have the same charge density ($\sigma _{c,eq}$) as that of the corresponding brush-grafted cases (where the walls are uncharged and the charges are distributed on the brushes). Of course, $\sigma _{c,eq}$ is expressed as $\sigma _{c,eq}=-e\int _{-h}^{-h+H}\phi n_{A^-} {\textrm {d} y}$ (see the theory section for definitions of $\phi$ and $n_{A^-}$).

Thermo-osmotically induced electric field $\bar {E} (=\bar {E}_{ion}+\bar {E}_{osm}+\bar {E}_{t})$ has contributions from ionic component ($\bar {E}_{ion}$) associated with the conduction of various ions, osmotic component ($\bar {E}_{osm}$) associated with the downward migration of the electrolyte ions, and thermal component ($\bar {E}_{t}$) due to the thermal gradient induced migration of the ions. The variation of $\bar {E}$ (shown in figure 2) is explained in detail by analysing each component contributing to the overall electric field. First, we study the variation of ionic component ($\bar {E}_{ion}$) (see figure 3(a)) of the induced electric field with the salt concentration ($c_\infty$). The ionic component of the electric field is due to the conduction of the EDL mobile ions. This component depends on the diffusivities of the mobile ions and the concentration gradient that is caused by the applied temperature gradient ($\textrm {d}T/{\textrm {d} x}$) due to the Soret effect. This induced concentration gradient acts on the EDL mobile ions (co-ions and counterions) of different diffusivities resulting in this conduction-based component. Figure 3(a) shows the variation of $\bar {E}_{ion}$ for all six cases: $\bar {E}_{ion}$ decreases in magnitude with an increase in the salt concentration ($c_\infty$). This is partly due to a decrease in the magnitude of the electrostatic potential $\bar {\psi }$ (see figure 4 which shows the typical variation of $|\bar {\psi }|$ with salt concentration) of the PE-brush-supported EDL with an increasing salt concentration. Such a decrease is caused by an augmented screening of the PE charges by a large concentration of the EDL ions. Equivalently, this decrease can be justified by noting that for a given average charge density $\sigma _{c,eq}$ (see figure 5), the magnitude of $\bar {\psi }$ is $|\bar {\psi }|\propto \sigma _{c,eq}\lambda _{EDL}\propto \sigma _{c,eq}/\sqrt {c_\infty }$ ($\lambda _{EDL}$ is the average EDL thickness). Of course, for very small electrolyte concentration (small enough to make $\lambda _{EDL}$ large enough to ensure an overlap of the EDL), the increase in $c_\infty$ causes a relatively weak decrease in $|\bar {\psi }|$. This is clearly seen for both the brush-grafted nanochannels and the corresponding cases in brushless nanochannels. The significant difference in the EDL potential distribution across the nanochannel between the cases of PE brush-grafted channel and the brushless channel can be attributed to the distribution of the charges in each of these cases. For instance, for the brushless nanochannels, the EDL potential has a finitely large magnitude (see figure 4(b)) only at locations near the walls of the nanochannel stemming from the fact that the charges (that induce the EDL) are localized at the walls. On the other hand, for the brush-grafted nanochannels, the PE brush charges are distributed along the brushes and hence the finitely large magnitude of the EDL potential is found to be distributed across the height of the channel (see figure 4(a)). It is also important to note that because of the localization of the charge on the walls for brush-free nanochannels, for the same charge density $\sigma _{c,eq}$, the EDL potential decreases sharply away from the wall (stronger near the wall and very weak away from the wall). However, for the brush-grafted nanochannels, the EDL potential decays much more slowly away from the wall owing to the PE charge distribution. In figure 5, we observe a monotonic increase in the magnitude of the charge density with increasing salt concentration. This charge density depends on the number of negative ions generated on the backbone of the PE brushes due to the ionization of the brushes. As the salt concentration increases, the EDL potential decreases due to an enhanced screening of the PE brush charge. This leads to a decrease in the local $\textrm {H}^+$ ion concentration ($n_H = n_{\textrm {H}^+,\infty }\exp (-\bar {\psi })$), which in turn leads to increased ionization of the brush. This increased ionization will lead to an increase in surface charge density. Similarly, with an increase in the grafting density (or decrease in $\ell$), we witness an increase in the number of PE chains grafted per unit area. This means an increase in the total charge per unit area, and hence we witness an increased surface charge density. Also, with increasing pH of the solution, we witness an increase in the magnitude of surface charge density. This is due to a decrease in $\textrm {H}^+$ ion concentration in the medium which increases the ionization of the PE brush.

Figure 2. Thermo-osmotically induced electric field due to the applied temperature gradient of $\boldsymbol {\nabla } T=\textrm {d}T/{\textrm {d} x}= 20~\textrm {Km}^{-1}$. The parameters used in obtaining this electric field are: $k_B = 1.38 \times 10^{-23}$ $\textrm {JK}^{-1}, T = 298\ \textrm {K}, e = 1.6\times 10^{-19}\ \textrm {C}$ (elementary charge), $\epsilon _0 = 8.8 \times 10^{-12}\ \textrm {Fm}^{-1}$ (permittivity of free space), $\epsilon _r = 79.8$ (relative permittivity of water), $N = 400, h = 100\ \textrm {nm}, L = 0.05\ \textrm {m}, a = 1$ nm (Kuhn length), $\gamma a^3=1, pK_a = 3.5, \nu = 0.5, \omega = 0.1$. Other parameters are $Q_+ = 3460\ \textrm {J}\ \textrm {mol}^{-1}, Q_- = 530 \ \textrm {J}\ \textrm {mol}^{-1}, Q_{\textrm {H}^+} = 13.3 \times 10^{3}\ \textrm {J}\ \textrm {mol}^{-1},$$Q_{\textrm {OH}^-} = 17.2\times 10^{3}\ \textrm {J}\ \textrm {mol}^{-1}$ (Agar, Mou & Lin Reference Agar, Mou and Lin1989), $D_+ = 1.330 \times 10^{-9}\ \textrm {m}^2\ \textrm {s}^{-1}, D_- = 2.030 \times 10^{-9}\ \textrm {m}^2\ \textrm {s}^{-1},$$D_{\textrm {H}^+} = 9.310 \times 10^{-9}\ \textrm {m}^2\ \textrm {s}^{-1}, D_{\textrm {OH}^-} = 5.270\times 10^{-9}\ \textrm {m}^2\ \textrm {s}^{-1}$ (Haynes Reference Haynes2014). Here $\bar {E}={E}/{E_0}$, where $E_0 = {k_BT}/{eL} = 0.514\ \textrm {V}\ \textrm {m}^{-1}$ (based on the parameters given above).

Figure 3. Variation of (a) $\bar {E}_{ion}$, (b) $\bar {E}_{ion,N}$ and (c) $\bar {E}_{diff}$ (see (3.1) for the definitions of $\bar {E}_{ion,N}$ and $\bar {E}_{diff}$) with salt concentration ($c_\infty$) in the presence of applied temperature gradient. In the insets of panels (b) and (c), we show the logarithmic plot of the variation $\bar {E}_{ion,N}$ and $\bar {E}_{diff}$, respectively, with salt concentration. All parameters are the same as mentioned in figure 2.

Figure 4. Variation of EDL electrostatic potential with salt concentration ($c_\infty$) at $\textrm {pH}_\infty = 4, \ell = 60$ nm for (a) PE-brush-grafted nanochannel, (b) brush-free nanochannel. All parameters are the same as mentioned in figure 2.

Figure 5. Variation of surface charge density ($\bar{\sigma}_{c,eq}$) with salt concentration ($c_\infty$) for PE brushes. All parameters are same as mentioned in figure 2.

To further understand the nature of variation of $\bar {E}_{ion}$, we first note that $\bar {E}_{ion}$ has contributions from all the ions ($\pm ,\textrm {H}^+,\textrm {OH}^-$) due to the induced gradients in the concentrations (caused by the Soret effect in presence of an applied temperature gradient ($\textrm {d}T/{\textrm {d} x}$)) of each of these different types of ions. Given that an identical surface charge is used for the brush-free nanochannels when compared to the brush-grafted nanochannel, the net charge $\int _{-h}^{0} \sum _i {z_i n_i} {\textrm {d} y}$ is identical for both the brush-free and brush-grafted systems. However, $\bar {E}_{ion}\propto \int _{-h}^{0} \sum _i {R_i z_i \boldsymbol {\nabla } n_{i,\infty }\exp (-({z_i}/{|z_i|})\bar {\psi })} {\textrm {d} y} \propto \int _{-h}^{0} \sum _i {R_i z_i n_i} {\textrm {d} y}$ (as ${\boldsymbol {\nabla } n_{i,\infty }} = - P_i ({\boldsymbol {\nabla } T}/{T})n_{i,\infty }$) is not identical for brush-grafted and brushless nanochannels. This is because each mobile ion has different diffusivity ($R_+ \neq R_- \neq R_{\textrm {H}^+} \neq R_{\textrm {OH}^-}$), and the fact that $(n_i)_{(Brush)} \neq (n_i)_{(No~ Brush)}$ due to the difference in the EDL potential distribution across the nanochannel. The variation of $\bar {E}_{ion}$ can be further explained by studying the conductive part ($\bar {E}_{ion,N}$) (see figure 3b), its components ($\bar {E}_{ion,N,i}$) (please see the online supplementary material) and the diffusive part ($\bar {E}_{diff}$), as expressed below (see (3.1) for the significance of each quantity present in these terms):

(3.1)\begin{equation} \left.\begin{array}{c@{}} \displaystyle \bar{E}_{ion} = \dfrac{2\bar{E}_{ion,N}}{2 \bar{E}_{diff}}=\dfrac{\displaystyle \sum_i{\bar{E}_{ion,N,i}}}{\bar{E}_{diff}},\\ \displaystyle \bar{E}_{ion,N,i}={-}L P_+ \dfrac{\boldsymbol{\nabla} T }{T}\int_{{-}1}^{0} \left[\dfrac{z_i}{|z_i|} R_{i}\boldsymbol{\nabla} \bar{n}_{i,\infty}\exp\left(-\dfrac{z_i}{|z_i|}\bar{\psi}\right)\right]\, \textrm{d}\bar{y},\\ \begin{aligned} \bar{E}_{diff} & =\int_{{-}1}^0 [R_+\bar{n}_{+,\infty}\exp(-\bar{\psi})+R_-\bar{n}_{-,\infty}\exp(\bar{\psi})+ R_{\textrm{H}^+,\infty} \bar{n}_{\textrm{H}^+,\infty}\exp(-\bar{\psi})\\ & \quad + R_{\textrm{OH}^-,\infty} \bar{n}_{\textrm{OH}^-,\infty}\exp(\bar{\psi})]\, \textrm{d}\bar{y}. \end{aligned} \end{array}\right\} \end{equation}

In figure 3(c), we plot the variation of the diffusion component of dimensionless electric field $\bar {E}_{diff}$ with salt concentration. It is to be noted that $\bar {E}_{diff} \propto \sum _i {R_i \bar {n}_i}$. It mainly depends on the reduced number density of $\textrm {H}^+$ ions ($\bar {n}_{\textrm {H}^+}={n_{\textrm {H}^+}}/{n_\infty }$), given that the $\textrm {H}^+$ ions have the largest diffusion coefficient (or the largest $R_i$) among all the ions present in the system. We witness a large decrease in the $\bar {E}_{diff}$ with increasing salt concentration at very low concentrations ($c_\infty \ll 10^{-\textrm {pH}_\infty }$). This is because as the salt concentration (and hence $n_{\infty }$) increases, there is a decrease in reduced bulk number density of $\textrm {H}^+$ ions ($\bar {n}_{\textrm {H}^+,\infty }={n_{\textrm {H}^+,\infty }}/{n_\infty }$) in this region which decreases $\bar {E}_{diff}$. Also, the magnitude of EDL potential ($\psi$) decreases monotonically with increasing salt concentration which further reduces the magnitude of $n_{\textrm {H}^+}$ resulting in a steeper decrease of $\bar {E}_{diff}$. As a result of this we see a monotonically decreasing $\bar {E}_{diff}$ at low salt concentrations ($c_\infty \ll 10^{-\textrm {pH}_\infty }$). However, at much higher salt concentrations ($c_\infty \gg 10^{-\textrm {pH}_\infty }$), the variation in the EDL potential $\psi$ is negligible and also $|\bar {\psi }|\ll 1$ due to increased screening of EDL due to large concentration of mobile ions (see above for more discussions). At such salt concentrations, we witness a very small decline in $\bar {E}_{diff} (\approx \int _{-1}^{0} R_+ \exp (-\bar {\psi })+ R_- (1+\bar {n}_{\textrm {H}^+,\infty }) \exp (-\bar {\psi })+R_{\textrm {H}^+}\bar {n}_{\textrm {H}^+,\infty }\exp (\bar {\psi }))$, which eventually approaches a constant value ($\bar {E}_{diff}\approx R_++R_-$) as $\bar {n}_{\textrm {H}^+,\infty } \ll 1$, and $\exp (\bar {\psi }) \approx 1+\bar {\psi } \approx 1$. This can be clearly seen in the inset of figure 3(c): at these large salt concentrations, especially at the concentration range of $10^{-2}-10^{-1}$ M, we observe a nearly constant $\bar {E}_{diff}$. For any given concentration and grafting density, as pH of the fluid increases, there is a decrease in $\bar {E}_{diff}$ owing to a decrease in number density of $\textrm {H}^+$ ions. We can also witness an increase in $\bar {E}_{diff}$ by increasing the grafting density keeping other parameters constant. This is due to the increase in the EDL potential owing to an increase in overall charge density of the PE brush layer.

Shown in figure 3(b) , $\bar {E}_{ion,N}$ is the conduction component ($\bar {E}_{ion,N}$) of the ionic component of electric field. It can be seen that $\bar {E}_{ion,N}$ monotonically decreases in magnitude with increase in salt concentration. It is to be noted that $\bar {E}_{ion,N}$ mainly depends on the variation of reduced number density of $\bar {n}_{\textrm {H}^+}(=\bar {n}_{\textrm {H}^+,\infty }\exp (-\bar {\psi }))$ because $\textrm {H}^+$ ions have much greater diffusivity than other ions in the electrolyte (please see the online supplementary material for detailed explanation for contribution of each ion to the conduction component). The dimensionless bulk number density of $\textrm {H}^+$ ions ($\bar {n}_{\textrm {H}^+,\infty }={n_{\textrm {H}^+,\infty }}/{n_{+,\infty }}$) and the magnitude of the (negative) EDL potential decreases sharply with $c_\infty$ at low concentration ($c_\infty \leq \textrm {pH}_\infty$) leading to a steep decrease in the magnitude of $\bar {E}_{ion,N}$. At higher salt concentration ($c_\infty \gg \textrm {pH}_\infty$), as the magnitude of EDL potential $|\bar {\psi }|\ll 1, \bar {E}_{ion,N}$ decreases in magnitude primarily due to a decrease in $\bar {n}_{\textrm {H}^+}$ (due to increase in $\bar {n}_{\infty }$) with increasing salt concentration (see the inset of figure 3(b)). This is seen clearly for all three brush-grafted cases and the corresponding brushless cases. The ratio of $\bar {E}_{ion,N}$ and $\bar {E}_{diff}$ results in the variation of $\bar {E}_{ion}$ (see figure 3(a)). At very low concentrations ($c_\infty \leq 10^{-\textrm {pH}_\infty }$), the decrease in both $\bar {E}_{ion,N}$ and $\bar {E}_{diff}$ is steep resulting in a very gradual decrease in $\bar {E}_{ion}$. At very high concentrations ($c_\infty \gg 10^{-\textrm {pH}_\infty }$), we witness that $\bar {E}_{diff}$ remains nearly constant and $\bar {E}_{ion,N}$ decreases very gradually, which leads to a very gradual decrease in $\bar {E}_{ion}$. At intermediate concentrations, the decrease in the magnitude of $\bar {E}_{ion,N}$ with $c_\infty$ is much steeper for than that of $\bar {E}_{diff}$, resulting in a sharp decline in the magnitude of $\bar {E}_{ion}$. Secondly, we analyse the thermal component of the dimensionless electric field $\bar {E}_t$ which is shown in figure 6(a). The thermal component of the induced electric field is induced by the transport of the ions due to the applied temperature gradient. The magnitude the $\bar {E}_{t}$ depends on the EDL potential $\psi$,the bulk concentration ($n_{i,\infty }$), diffusivities, heat of transport of each species $i$ and temperature gradient ($\textrm {d}T/{\textrm {d} x}$) (see (2.28) ). Unlike ionic or osmotic component, the thermal component always remains positive. This is because the net thermal transport induced electric field primarily depends on the contribution from the $\textrm {H}^+$ ions (which is positive) as it has the largest heat of transport and diffusivity among the reported ions. The variation of $\bar {E}_{t}$ can be better understood by studying the variation of thermal conduction field $\bar {E}_{t,N}$ (see figure 6(b)) and its components $\bar {E}_{t,N,i}$ (please see the online supplementary material) which are expressed as

(3.2)\begin{equation} \left.\begin{array}{c@{}} \displaystyle\bar{E}_{t} = \dfrac{2\bar{E}_{t,N}}{2 \bar{E}_{diff}} = \dfrac{\displaystyle \sum_i \bar{E}_{t,N,i}}{ \bar{E}_{diff}},\\ \displaystyle\bar{E}_{t,N,i}=\dfrac{L \boldsymbol{\nabla} T}{T} \int_{{-}1}^{0}\left[\dfrac{z_i}{|z_i|} R_i \bar{n}_i\left(\bar{Q}_i +\dfrac{z_i}{|z_i|}\bar{\psi} \right)\right]\, \textrm{d}\bar{y}. \end{array}\right\}\end{equation}

Figure 6(b) shows the variation of $\bar {E}_{t,N}\propto \sum _i({n_{i,\infty }}/{n_\infty }) (\bar {Q}_{i}+z_i\bar {\psi })\exp (\overline {-z_i\psi })$, which is the summation of all the conduction components of the dimensionless thermal electric field ($\bar {E}_{t,N}=\sum _i \bar {E}_{t,N,i}$). It is worthwhile to note that $\bar {E}_{t,N,\textrm {H}^+}$ is dominant at most concentrations owing to its heat of transport being much greater than that of other ions (please see theonline supplementary material for detailed explanation for $\bar {E}_{t,N,i}$). We witness a steep decrease of $\bar {E}_{t,N}$ with $c_\infty$ in the concentration range ($10^{-6}-10^{-\textrm {pH}_\infty }$). This stems from the fact that both the reduced number density of $\textrm {H}^+$ ions $\bar {n}_{\textrm {H}^+,\infty }$ and $|\bar {\psi }|$ decreases with $c_\infty$ in this concentration range. However, for higher concentration values ($c_\infty \gg 10^{-\textrm {pH}_\infty }$), $\bar {E}_{t,N}$ decreases weakly and approaches a nearly constant value with increasing salt concentration as the EDL potential is weak ($\exp (-\bar {\psi })\approx 1-\bar {\psi } \approx 1$) and $\bar {n}_{\textrm {H}^+,\infty } \ll 1$. The gradual, steep and again gradual nature of decrease in the magnitude of $\bar {E}_{t}$ with respect to increasing salt concentration can be explained by the variations of $\bar {E}_{t,N}$ and $\bar {E}_{diff}$. For instance, for the case of $\textrm {pH}_\infty =3, \ell = 10$ nm, we observe a steep decrease in both $\bar {E}_{t,N}$ and $\bar {E}_{diff}$ in the concentration range $c_\infty \sim 10^{-6}-10^{-3}$ M, which is reflected by a slight decrease in $\bar {E}_{t}$. At intermediate concentration range $c_\infty \sim 10^{-3.5}-10^{-2}$ M, the decrease in $\bar {E}_{t,N}$ is much steeper compared with $\bar {E}_{diff}$, leading to a sharp decrease in the magnitude of $\bar {E}_{t}$. At higher concentration of $10^{-2}-10^{-1}\ \textrm {M}, \bar {E}_{diff}$ reaches a nearly constant value while there is a decrease in magnitude of $\bar {E}_{t,N}$. At these concentrations, therefore, $\bar {E}_t$ mirrors the variation seen in $\bar {E}_{t,N}$ and decreases very gradually with $c_\infty$. Similarly, for the case of $\textrm {pH}_\infty =4, \ell = 60$ nm, we see a slight decrease in magnitude of $\bar {E}_{t}$ in the salt concentration range $c_\infty \sim 10^{-6}-10^{-4}$ M owing to large decreases in both $\bar {E}_{t,N}$ and $\bar {E}_{diff}$. At intermediate concentrations $c_\infty \sim 10^{-4}-10^{-2}$ M, we observe a large decrease owing to a much larger decrease in $\bar {E}_{t,N}$ in comparison with that of $\bar {E}_{diff}$. At concentrations $c_\infty \sim 10^{-2}-10^{-1}$ M, $E_t$ follows a variation similar to that of $\bar {E}_{t,N}$. Likewise, for the case of $\textrm {pH}_\infty =3, \ell = 60$ nm, we see a gradual decrease, followed by a sharp decrease, which is followed by a gradual decrease in $|\bar {E}_{t}|$ in the concentration ranges of $10^{-6}-10^{-3.5}\ \textrm {M}, 10^{-3.5}-10^{-2}\ \textrm {M}, 10^{-2}-10^{-1}$ M, respectively. Finally, we study the osmotic component of the dimensionless electric field $\bar {E}_{osm}$ (see figure 7(a)). The osmotic contribution to the induced electric field is due to the migration of the mobile ions due to the background flow. It is clear from figure 7(a) that the magnitude of $\bar {E}_{osm}$ for the brush-grafted case is always greater than that of the corresponding brushless case across all $c_\infty , \textrm {pH}_\infty$ and $\ell$. This can be attributed to the significant enhancement of TOS velocity for the brush-grafted nanochannel compared with its brushless counterpart. Such an enhancement in the TOS velocity can be attributed to two factors. The first is the fact that the presence of brushes ensures that the location of the EDL charge density is localized away from the nanochannel walls where the velocity is maximum (away from flow-retarding walls) which results in an enhanced contribution of the migrated EDL ions to $\bar {E}_{osm}$. This is not the case in the bare nanochannels, where the EDL charge density is localized near the walls where the velocity is impeded by the presence of the wall. The dimensionless osmotic electric field $\bar{E}_{osm}$ has contributions from all the mobile ions ($\pm ,\textrm {H}^+,\textrm {OH}^-$). The second is the possible presence of the molecular slip that the liquid experiences along the brush surface. Such slip, which has been hypothesized for cases of other types of PE (e.g. DNA molecules in a background fluid flow (Galla et al. Reference Galla, Meyer, Spiering, Sischka, Mayer, Hall, Reimann and Anselmetti2014; Hirano et al. Reference Hirano, Iwaki, Ishido, Yoshikawa, Naruse and Yoshikawa2018)), is expected to be present on the PE brush surfaces and ensures that the brushes do not behave as rigid cylinders that stagnate the flow on their surfaces. It is to be noted that the contribution of $\textrm {H}^+$ ions is very significant at very low salt concentrations, whereas this contribution is very small at very high concentrations. The contributions of the ions towards the osmotic component follows a non-monotonic variation with respect to salt concentration, which can be better explained by studying the advection electric field $\bar {E}_{adv}$ (see figure 7(b)) and its components $\bar {E}_{adv,i}$ (please see the online supplementary material) which are expressed as

(3.3)\begin{equation} \left.\begin{array}{c@{}} \displaystyle\bar{E}_{osm} = \dfrac{2\bar{E}_{adv}}{2 \bar{E}_{diff}} = \dfrac{\displaystyle\sum_i \bar{E}_{adv,i}}{\bar{E}_{diff}},\\ \displaystyle\bar{E}_{adv,i}= Pe \int_{{-}1}^{0} \bar{u}\left[-\dfrac{z_i}{|z_i|}\bar{n}_{i,\infty}\exp\left(-\dfrac{z_i}{|z_i|}\bar{\psi} \right)\right]\, \textrm{d}\bar{y}. \end{array}\right\}\end{equation}

Figure 6. Variation of (a) $\bar {E}_{t}$ and (b) $\bar {E}_{t,N}$ (see (3.2) for the definition of $\bar {E}_{t,N}$) with salt concentration ($c_\infty$) in the presence of applied temperature gradient. In the inset of panel (b), we show the logarithmic plot of variation of $\bar {E}_{t,N}$ with salt concentration. All parameters are the same as mentioned in figure 2.

Figure 7. Variation of (a) $\bar {E}_{osm}$ and (b) $\bar {E}_{adv}$ (see (3.3) for the definitions of $\bar {E}_{adv}$) with salt concentration ($c_\infty$) in the presence of applied temperature gradient. In the inset of panel (b), we provide a magnified view of the variation of $\bar {E}_{adv}$ at higher salt concentration. All parameters are the same as mentioned in figure 2.

The resulting summation of all the advection components is plotted in figure 7(b). The non-monotonic behaviour of the $\bar {E}_{osm}$ as witnessed in figure 7(a) can be easily explained from the variation of $\bar {E}_{adv}$ and $\bar {E}_{diff}$. Firstly, it has to be noted that the variation of $\bar {E}_{adv}$ (please see figure 7b) is determined by the contribution from $\textrm {H}^+$ ions at lower concentration region ($c_\infty < 10^{-\textrm {pH}_\infty }$) and the contribution of $+ve$ ions at concentrations $c_\infty > 10^{-\textrm {pH}_\infty }$ merely due to the extent of concentration of these ions in the solution (please see the online supplementary material for detailed explanation for contributions of each ions). In the lower salt concentrations ( $c_\infty < 10^{-\textrm {pH}_\infty }$), dominated by $\textrm {H}^+$ ion contributions, we witness a steep decrease in $|\bar {E}_{adv}|$ with increasing salt concentration due to the decrease in magnitude of reduced bulk number density of $\textrm {H}^+$ ions ($\bar {n}_{\textrm {H}^+,\infty }$), EDL magnitude ($|\psi |$) and the TOS velocity ($\bar {u}$) (see figure 8). However, at higher salt concentrations (where the contributions due to $+ve$ ions dominate), variation of $|\bar {E}_{adv}|$ mirrors that of the magnitude of the TOS velocity $u$. At these higher concentrations, the EDL magnitude is weak ($\exp (-\bar {\psi }) \approx 1-\bar {\psi }\approx 1$) and hence its effect on $|\bar {E}_{adv}|$ is negligible (also note that $\bar {n}_{+,\infty } = 1$ for all salt concentrations). In lower salt concentration ($10^{-6}-10^{-\textrm {pH}_\infty }$), the ratio of sharply decreasing $\bar {E}_{adv}$ and $\bar {E}_{diff}$ results in a slow increase in the magnitude of $\bar {E}_{osm}$ as witnessed in figure 7(a). However, at higher salt concentrations ($10^{-\textrm {pH}_\infty }-10^{-1}$), a nearly constant value of $\bar {E}_{diff}$ means that the variation of $\bar {E}_{osm}$ mirrors that of $\bar {E}_{adv}$ with salt concentration. For instance, for the cases of $\textrm {pH}_\infty =3, \ell = 10$ nm, and $\textrm {pH}_\infty =3, \ell = 60$ nm, we observe a slight incline of the magnitude of $\bar {E}_{osm}$, followed by a variation in its magnitude consistent with that of $\bar {E}_{adv}$ in this region as witnessed in figure 7(b). Likewise, for the case of $\textrm {pH}_\infty = 4, \ell =60$ nm, we see a gradual increase in $|\bar {E}_{osm}|$ from $c_\infty \sim 10^{-6}-10^{-4}$ M, followed by a monotonic decrease in $|\bar {E}_{osm}|$ similar to the decrease seen in $|\bar {E}_{adv}|$ at concentration range $10^{-4}-10^{-1}$ M. It is worthwhile to note that for the brushless case, in the lower concentration regime, the decline in the magnitude of $\bar {E}_{adv}$ with increasing salt concentration is not as steep as that of the brush-grafted channel owing to the small magnitude of TOS velocity. However, the variation in $\bar {E}_{diff}$ is still as steep as that of brush-grafted nanochannels. As a result of this, for brushless nanochannels, we observe a monotonic decrease in the magnitude of $\bar {E}_{adv}$ at lower salt concentrations ($c_\infty < 10^{-\textrm {pH}_\infty }$). However, for larger salt concentrations, similar to the brush-graftednanochannels, the variation of $\bar {E}_{osm}$ follows the variation of $\bar {E}_{adv}$. The combined contributions of these three components of dimensionless electric fields, namely $\bar {E}_{ion}, \bar {E}_t$ and $\bar {E}_{osm}$, result in the overall variation of the thermo-osmotically induced electric field ($\bar {E}$) as shown in figure 2.

Figure 8. Velocity profile in the presence of applied temperature gradient for various salt concentration, pH and grafting density. Solid line represents brush-grafted cases while dotted line represents brushless case for (a) $\textrm {pH}_\infty =3,~\ell = 60$ nm, (b) $\textrm {pH}_\infty =3,~\ell = 10$ nm and (c) $\textrm {pH}_\infty =4,~\ell = 60$ nm. All parameters are the same as mentioned in figure 2. Here $\bar {u}={u}/{U}$, where $U = {\epsilon _0 \epsilon _r (k_BT)^2}/{e^2 \eta L} = 1.046 \times 10^{-8}~$ m s$^{-1}$ (based on the parameters given in figure 2).

3.2. Variation of the TOS velocity field

We plot the transverse variation of dimensionless TOS velocity ($\bar {u}$) in figure 8 for different $c_\infty , \textrm {pH}_\infty$ and $\ell$. The results clearly demonstrate that for most of the parameter combinations, the presence of the brushes significantly enhances the overall nanofluidic TOS transport. From (2.21), we can see that there are three contributions to the TOS velocity. First is the COS component due to the induced concentration gradient. The second is the thermal component due to the thermal gradient. The third is the EOS component due to the induced electric field $E$. Similar to our previous work (see Sivasankar et al. Reference Sivasankar, Etha, Sachar and Das2020a), we compare the dimensionless COS component ($\bar {u}_{COS}$), thermal component ($\bar {u}_{T}$) and the total TOS velocity ($\bar {u}_{total}$) for better understanding the influence of each of these components. We obtain the COS component by excluding the effect of the induced electric field and the temperature gradient on the velocity. The thermal component of the velocity is obtained by switching off the effect of the electric field and the concentration gradient. The equations governing the COS and thermal velocities are given by

(3.4)\begin{gather} \left.\begin{array}{c@{}} \begin{aligned} & \dfrac{\textrm{d}^2\bar{u}_{COS}}{\textrm{d}\bar{y}^2}-\left(\dfrac{\textrm{d}^2\bar{u}_{COS}}{\textrm{d}\bar{y}^2}\right)_{(\bar{y}=0)}= {\bar{n}'}\bar{\kappa}^2{(\cosh(\bar{\psi})-\cosh(\overline{\psi_c}))}\\ & \quad +\dfrac{h^2}{\kappa_d}\bar{u}_{COS}\quad ({-}h\leq y\leq{-}h+H_0), \end{aligned}\\ \displaystyle\dfrac{\textrm{d}^2\bar{u}_{COS}}{\textrm{d}\bar{y}^2}-\left(\dfrac{\textrm{d}^2\bar{u}_{COS}}{\textrm{d}\bar{y}^2}\right)_{(\bar{y}=0)}= {\bar{n}'}\bar{\kappa}^2{(\cosh(\bar{\psi})-\cosh(\overline{\psi_c}))} \quad ({-}h+H_0\leq y\leq 0). \end{array}\right\}\end{gather}

(3.5)\begin{gather} \left.\begin{array}{c@{}} \begin{aligned} & \dfrac{\textrm{d}^2\bar{u}_{T}}{\textrm{d}\bar{y}^2}-\left(\dfrac{\textrm{d}^2\bar{u}_{T}}{\textrm{d}\bar{y}^2}\right)_{(\bar{y}=0)}= L\dfrac{\boldsymbol{\nabla} T}{T}\bar{\kappa}^2(\cosh(\bar{\psi})-\cosh(\overline{\psi_c})- \bar{\psi} \sinh(\bar{\psi})\\ & \quad + \overline{\psi_c} \sinh(\overline{\psi_c}))+ \dfrac{h^2}{\kappa_d}\bar{u}_{T}\quad ({-}h\leq y\leq{-}h+H_0), \end{aligned}\\ \begin{aligned} & \dfrac{\textrm{d}^2\bar{u}_{T}}{\textrm{d}\bar{y}^2}-\left(\dfrac{\textrm{d}^2\bar{u}_{T}}{\textrm{d}\bar{y}^2}\right)_{(\bar{y}=0)}= L\dfrac{\boldsymbol{\nabla} T}{T}\bar{\kappa}^2(\cosh(\bar{\psi})-\cosh(\overline{\psi_c})- \bar{\psi} \sinh(\bar{\psi})\\ & \quad + \overline{\psi_c} \sinh(\overline{\psi_c}))\quad ({-}h+H_0\leq y\leq 0). \end{aligned} \end{array}\right\}\end{gather}

It is to be noted that the term $\overline {n^\prime }={L(\boldsymbol {\nabla } n_\infty +\boldsymbol {\nabla } n_{\textrm {H}^+,\infty })}/{(n_\infty +n_{\textrm {H}^+,\infty })}$ in (3.4) depends on the concentration gradient, which in turn is determined by the applied temperature gradient. It is important to note that the concentration gradients $(\boldsymbol {\nabla } n_\infty ,\boldsymbol {\nabla } n_{\textrm {H}^+,\infty })$ in this term are a direct consequence of the temperature gradient due to the Soret effect. It is also important to note that there is no additional gradient imposed beyond the Soret effect induced ionic concentration gradient. Also, the three contributions to the overall TOS velocity, namely the COS, TOS and EOS contributions cannot be solved separately. However, to understand the extent of contribution of each of the driving mechanisms, we solve for the velocity due to each of these mechanisms separately, i.e. separately obtain the COS velocity $({\bar {u}}_{COS})$ and the thermal velocity $({\bar {u}}_T)$ components. It is to be noted that these components give a qualitative measure of the contributions of each driving mechanism. However, one cannot add these components to obtain the overall TOS velocity. We compare the non-dimensional total TOS velocity ($\bar {u}$ or $\bar {u}_{total}$), COS velocity ($\bar {u}_{COS}$) and thermal velocity ($\bar {u}_{T}$) for both brush-grafted and brushless nanochannels for various $c_\infty , \textrm {pH}_\infty$ and grafting density (see figures 9–11). It could be seen that the COS and thermal component of the velocity are generally greater in magnitude for the brush-grafted case when compared with its corresponding brushless case. This can be ascribed to two factors. The first is the localization of the EDL charge density away from the flow-retarding wall: as a result, any driving force that is associated with the net EDL charge density is manifested to a much larger extent. The second is the possible molecular slip experienced by the liquid along the PE brush surface. In fact, a qualitative comparison of $\bar {u}_{COS}, \bar {u}_{T}$ and $\bar {u}_{total}$ also confirms that the same is true for the EOS components: the localization of the EDL away from the nanochannels wall, along with the possible presence of molecular slip along the PE brush surface, leads to a much larger manifestation of the EOS body force. Finally, the direction of both the $u_{COS}$ component and the thermal component is always from left to right. Therefore, the net thermal and the COS flow is either retarded or augmented by the EOS flow based on the induced electric field. A positive (negative) dimensionless induced electric field ($\bar {E}$) results in an EOS flow component from left to right (right to left) and this, in turn, dictates the overall direction of the TOS transport. In summary, therefore, as explained earlier, the overall TOS velocity (see figure 8) is much enhanced for the brush-grafted nanochannels when compared with brushless counterparts due to the combined influence of the localization of EDL density away from the walls by the brushes and the possible presence of molecular slip on the brush surfaces.