Electrostatics of soft charged interfaces with pH-dependent charge density: effect of consideration of appropriate hydrogen ion concentration distribution

Abstract

A soft charged interface, represented by a nanoscopically thick charged polyelectrolyte layer (PEL) sandwiched between a rigid solid and an electrolyte solution, forms the central element in the theoretical description of electrodynamics of soft charged biological moieties (e.g., cells, bacteria and viruses), electrohydrodynamics in soft nanochannels, and analysis of properties of non-biological gels and films. In this paper we provide a free energy based theoretical model to quantify the electrostatics of such a soft charged interface with pH-dependent charge density. Our theory considers for the first time the explicit variation of the hydrogen ion concentration inside and outside the PEL and establishes that there cannot be a uniform (within the PEL) distribution of chargeable sites within the PEL. Such uniformity causes unphysical hydrogen ion concentration jump across the PEL–electrolyte interface. In fact we demonstrate that this distribution of chargeable sites on a given polymer must obey at least a cubic variation with distance (within the PEL) in order to address this hydrogen ion concentration jump and at the same time ensure zero hydrogen ion flux at the PEL–rigid-solid interface. We anticipate that these new results will serve as a major stepping-stone towards appropriate modeling of electrodynamics and electrohydrodynamics of soft charged reactive interfaces that are paramount in the description of various micro/nanoscale biological and non-biological problems.

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Introduction

A soft charged interface model and the related electrostatics and electrokinetics have often been used to model a number of biological and non-biological phenomena.^1–7 Some well-known examples include quantifying electrophoretic properties of cells, bacteria and viruses,^8–16 studying bacterial adhesion to surfaces,^17–19 interpreting agglutination and shape transition of red blood cells,^20,21 specifying properties of non-biological soft charged gels and films,^22–25 enhancing electrochemomechanical energy conversion efficiency in polymer-grafted nanochannels,²⁶ designing soft-charged-interface-based double layer capacitors,²⁷ and many more.

A soft charged interface is typically described by certain structural and chemical specifications. Firstly, the interface is assumed to be constituted by a layer of charged polyelectrolyte sandwiched between a rigid wall and an electrolyte solution.^1–7 This polyelectrolyte layer (or PEL) harbors a particular kind of ion, known as the PEL ions, which remain excluded from the regions outside the PEL; however the electrolyte ions can remain both inside and outside the PEL. For this reason there develops a Donnan potential deep within the PEL, with the PEL–electrolyte interface acting as a semi-permeable membrane.^4,28 The second important issue concerns the mechanism of charging of the PEL (or the generation of the PEL ions). Typically this charging occurs due to an acid or a base like reaction, where the chargeable sites on a polyelectrolyte (henceforth denoted as the Polyelectrolyte Chargeable Sites or PCS) dissociates to produce the PEL ions. Of course this dissociation reaction depends on the local value of the hydrogen ion concentration, and therefore the charge density of the PEL (or equivalently the number density of PEL ions) becomes a function of the pH. All the existing studies considering such pH-dependence of the PEL charge density has invariably described the hydrogen ion concentration within the PEL (which in turn dictates the charge density of the PEL) by the Boltzmann distribution.^{3,17,22–25,29–32} This is inappropriate, since the hydrogen ion concentration distribution within the PEL gets additionally dictated by the chemical reaction that produces the PEL ions. In this paper, we provide a free energy based model that indeed describes the variation of hydrogen ion concentration (within the PEL) taking into consideration the hydrogen-ion-concentration-dependent chemical reaction that produces the PEL ions. In fact, to the best of our knowledge, the present paper is the first study that describes the electrostatics of a charged soft interface, represented as a grafted PEL with a pH-dependent charge density, by explicitly accounting for the hydrogen ion concentration. Of course, there have been several studies on soft matter systems, other than such PEL-grafted systems, that have employed analysis based on the explicit consideration of the pH and pK_a of the system.^33–36

The central result of our analysis is that there cannot be a uniform distribution of PCS across the depth of the PEL. In case we consider that such a chargeable site is equivalent to a monomer in a polyelectrolyte molecule, this indicates to a non-uniform distribution (across the depth of the PEL) of monomers within the PEL. We show that such non-uniformity is necessary to ensure continuities in the value and in the gradient of the hydrogen ion concentration at the PEL–electrolyte interface. Of course, this extremely important issue has hitherto remained completely unidentified stemming from the sheer disregard of the explicit hydrogen ion concentration distribution in the modeling of electrostatics of soft charged interfaces with pH-dependent charge density. The second important issue on which our study sheds light is the nature of this non-uniformity in the PCS distribution. We establish that this PCS distribution must obey at least a cubic profile, arising from the fact that four different conditions must be satisfied by the PCS distribution. These conditions are the continuities in the value and in the gradient of hydrogen ion concentration distribution at the PEL–electrolyte interface, zero net flux of hydrogen ion concentration at the PEL–rigid-solid interface and constancy in the total number of PCS. We further identify that a profile other than this cubic one, which may satisfy these four conditions, may also represent a viable PCS distribution. For uncharged polymer brushes, where the equilibrium configuration is dictated by the balance of the elastic and volume exclusion effects, the monomer distribution has been identified as a unique quadratic function.^37–40 On the contrary, as we demonstrate here, for polyelectrolytes (which of course will also be in form of brushes), where we only consider pH-dependent electrostatic effects (and disregard elastic and volume exclusion effects) this PCS distribution (or equivalently the monomer distribution) is a non-unique function that must at least be cubic. Therefore this paper also provides possibly the first study in quantifying the equilibrium shape of the polyelectrolyte brushes constituting charged PEL – we also discuss certain experimental examples of pH-dependent shape changes of polyelectrolyte brushes,^41,42 where our proposed theory can provide useful insights and interpretations.

Results

Electrostatic potential and hydrogen ion concentration profiles for uniform distribution of polyelectrolyte chargeable sites (PCS) within the PEL

We consider a soft charged interface that constitutes the bottom plate of a nanochannel (see Fig. 1). The soft charged interface is represented by a layer of charged polyelectrolyte sandwiched between an electrolyte solution and a rigid surface. Thickness of this PEL is d, whereas the height of the nanochannel is 2h. Here we consider a uniform distribution of the PCS within the PEL. Under these circumstances, we are interested to obtain the equilibrium electrostatic potential and hydrogen ion concentration profiles. The free energy functional dictating the problem is.

F = ∫f[ψ,n_±,n_H⁺,n_OH⁻]d³r, (1)

where the free energy density f is a function of ψ, n_±, n_H⁺ and n_OH⁻ and can be expressed as (for the bottom half of the nanochannel):

Fig. 1 Schematic of a soft charged nanochannel, with soft charged plates. The PEL ions are shown in green.

In eqn (2), ε₀ is the permittivity of free space, ε_r is the relative permittivity of the medium (assumed identical for the media both inside and outside the PEL⁴), e is the electronic charge, k_BT is the thermal energy, ψ is the electrostatic potential, n_i and n_i,∞ are the number density and the bulk number density of ion i (i = ±, H⁺, OH⁻) and n_A⁻ is the number density of the PEL ions. Please note that both the PEL ions and the electrolyte ions (with number densities n_±) are assumed to be monovalent.

The PEL ions are assumed to be formed by the dissociation of an acid HA (producing A⁻ ions, which are the PEL ions), and therefore following Das,²⁷ we get

where γ is the maximum density of the PCS and K^′_a = 10³N_AK_a [N_A is the Avogadro number, K_a (having units of moles per liter) is the ionization constant of the acid HA dissociating to produce A⁻ ions (or PEL ions)]. Eqn (3) is used in eqn (2) to express the free energy density entirely in terms of the functions ψ, n_±, n_H⁺ and n_OH⁻.

The equilibrium conditions can be obtained by minimizing F with respect to ψ, n_±, n_H⁺ and n_OH⁻ (see the ESI† for the detailed procedure), resulting in the following two sets of equations in dimensionless variables:

here the dimensionless variables are

Therefore there are 6 dimensionless parameters dictating the problem: n̄_H⁺,∞, n̄_OH⁻,∞, [small lambda, Greek, macron], d̄, K̄^′_a, [small gamma, Greek, macron]. Eqn (4) and (5) represent the coupled set of equations that are solved in presence of the following boundary conditions:

In order to solve this coupled equation, we first eliminate [small psi, Greek, macron] and obtain the governing equation and the boundary conditions entirely in terms of n̄_H⁺ (see the ESI†), which is then solved numerically to first obtain the hydrogen ion concentration profiles and then the electrostatic potential profiles.

Fig. 2(a–e) show the variation of the dimensionless hydrogen ion number density (n̄_H⁺) and the dimensionless electrostatic potential ([small psi, Greek, macron]) profiles for different ranges of the governing parameters. The most important result, evident in all the plots, is the distinct discontinuity in the n̄_H⁺ profile at the location ȳ = −1 + d̄ (i.e., at the PEL–electrolyte interface). Identification of this discontinuity is the most significant discovery of this study. Till date there have been many studies that have considered a pH-dependent charge density of a soft charged interface;^{3,17,22–25,29–32} however in none of the studies this discontinuity has been identified, since none of the studies cared to consider the explicit hydrogen ion concentration distribution. In other words, these studies simply consider a Boltzmann distribution for the hydrogen ion concentration [i.e., the form expressed in the 2^nd part of eqn (4)] both inside and outside the PEL. However, as we demonstrate through an appropriate free energy picture, the dependence of the PEL reaction on the hydrogen ion concentration ensures that the hydrogen ion concentration no longer obeys a Boltzmann distribution inside the PEL. Therefore, we encounter two different n̄_H⁺ − [small psi, Greek, macron] relationships – one inside the PEL and one outside the PEL [see eqn (4)]. It is precisely for this reason that we encounter two different n̄_H⁺ − [small psi, Greek, macron] relationships at the PEL–electrolyte interface (located at ȳ = −1 + d̄), depending on whether we approach the interface from the PEL side or the electrolyte side. Such difference ensures that if there are continuities (at ȳ = −1 + d̄) in the value and in the gradient of [small psi, Greek, macron], there cannot be continuities (at ȳ = −1 + d̄) in the value and in the gradient of n̄_H⁺.

Fig. 2 Variation of the dimensionless hydrogen ion number density (top panel) and the dimensionless electrostatic potential profile (bottom panel) for the case of uniform distribution of PCS. For all the figures we maintain the four parameters (i.e., n̄_H⁺,∞, [small lambda, Greek, macron], K̄^′_a, [small gamma, Greek, macron]) as equal to unity and d̄ = 0.3 except for (a) n̄_H⁺,∞ = 0.1, 1, 10, (b) [small gamma, Greek, macron] = 0.1, 1, 10, (c) [small lambda, Greek, macron] = 0.2, 1, 10, (d) d̄ = 0.1, 0.3, 0.5 and (e) K̄^′_a = 0.1, 1, 10. In all the figures, the value of the parameter (which is being varied) is indicated. Also for all the plots, we use n̄_H⁺,∞ = n̄_OH⁻,∞.

Here we discuss the effect of individual parameters in dictating the profiles of n̄_H⁺ and [small psi, Greek, macron], as well as the discontinuity in the n̄_H⁺ profile. Fig. 2(a) shows the effect of variation of the dimensionless bulk hydrogen ion concentration n̄_H⁺,∞. Insets of Fig. 2(a) clearly reveal that the extent of discontinuity in n̄_H⁺ values across the two sides of ȳ = −1 + d̄ increases with a decrease in the value of n̄_H⁺,∞. Such a behavior can be attributed to the corresponding variation of [small psi, Greek, macron]. Lowering of n̄_H⁺,∞ will enhance the reaction (HA ↔ H⁺ + A⁻) that produces the polyelectrolyte ions A⁻, thereby augmenting the number density of the A⁻ ions. This happens since lowering of hydrogen ion concentration signals a depletion of the hydrogen ions from the system, thereby favoring the forward reaction that produces H⁺ ions (of course, along with A⁻ ions). Enhanced number density of A⁻ ensures an enhanced magnitude of [small psi, Greek, macron] for smaller values of n̄_H⁺,∞, which in turn enhances the consequence of the disparity in the n̄_H⁺ − [small psi, Greek, macron] relationship [see eqn (4)] leading to a more enhanced discontinuity in the n̄_H⁺ profile. Fig. 2(b) demonstrates the effect of variation of the parameter [small gamma, Greek, macron]. Enhanced [small gamma, Greek, macron] implies enhanced value of the PCS (that dissociate to produce the polyelectrolyte ions A⁻), thereby enhancing the number density of A⁻ ions. This in turn augments the magnitude of [small psi, Greek, macron] and the extent of discontinuity in the value of n̄_H⁺ (for reasons discussed above) at ȳ = −1 + d̄. Fig. 2(c) represents the effect of variation of the parameter [small lambda, Greek, macron]. Enhanced [small lambda, Greek, macron] signifies larger extent of the EDL overlap in the nanochannel, leading to a much weaker gradient in the value of n̄_H⁺ and [small psi, Greek, macron] values – as a consequence it is the gradient of n̄_H⁺ that shows a large discontinuity for weaker [small lambda, Greek, macron] at ȳ = −1 + d̄. However, since there is relatively less deviation in the value of [small psi, Greek, macron] for different [small lambda, Greek, macron] at ȳ = −1 + d̄, the discontinuity in the value of n̄_H⁺ (at ȳ = −1 + d̄) is similar for different [small lambda, Greek, macron] values. Fig. 2(d) reveals the effect of variation of the parameter d̄. Enhanced d̄ increases the magnitude of [small psi, Greek, macron] (ref. 26) (since there is greater total number of A⁻ ions), which in turn increases the extent of jump in n̄_H⁺ the profile at ȳ = −1 + d̄. Variation of K̄^′_a [the corresponding effect on n̄_H⁺ and [small psi, Greek, macron] profiles is shown in Fig. 2(e)] produces the most non-trivial and non-monotonic variation of the n̄_H⁺ profile. Increase in K̄^′_a obviously enhances the reaction that produces A⁻ ions, thereby leading to an enhanced magnitude of [small psi, Greek, macron] and a greater extent of discontinuity in the n̄_H⁺ profile (for reasons discussed above). This description remains perfectly valid for the increase of K̄^′_a from 0.1 to 1. However, when K̄^′_a becomes very large (=10), it effectively indicates a nearly complete dissociation of the chargeable groups (that produce the polyelectrolyte ions A⁻). Hence for such a case the number density of the PEL ions become effectively pH independent, thereby favoring the fact that n̄_H⁺ profile will now be dictated by Boltzmann distribution both inside and outside the PEL. Accordingly, there is much weaker discontinuity in the n̄_H⁺ profile for K̄^′_a = 10, although the magnitude of [small psi, Greek, macron] keeps on increasing.

Electrostatic potential and hydrogen ion concentration profiles for non-uniform distribution of polyelectrolyte chargeable sites (PCS) within the PEL: appropriate choice of the distribution of chargeable sites

We have identified that the appropriate consideration of hydrogen ion concentration distribution forbids simultaneous continuities in the gradient and in the value of both the electrostatic potential and the hydrogen ion concentration at the PEL–electrolyte interface. We have further identified that such discontinuities result from the contribution of the electrostatic effect of the PEL ions; this effect introduces a situation where the relationship between [small psi, Greek, macron] and n̄_H⁺ are different at the PEL–electrolyte interface [see eqn (4)], depending on whether the interface is approached from the PEL side or from the electrolyte side. This can be addressed only if it is ensured that the electrostatic contribution of the PEL ions is such that there are simultaneous continuities in the values and in the gradient of [small psi, Greek, macron] and n̄_H⁺ at the PEL–electrolyte interface. Additionally, the net flux of the hydrogen ions must be zero at the PEL–rigid-wall interface. Given the fact that at that interface , we must have zero diffusive flux for the hydrogen ions at that interface [i.e., ]. In a framework that does account for all the necessary contributions to the free energy [see eqn (1) and (2)], these issues can only be addressed, by considering a dimensionless concentration (or volume fraction) distribution φ(y) of the chargeable sites. This is exactly analogous to consideration of monomer distribution for uncharged polymer brushes.^33–36 Also, φ(y) must satisfy the constraint:where σ is the area corresponding to a single polyelectrolyte chain, a is the chain thickness and N_p is the total number of chargeable sites on a polyelectrolyte chain [for uniform distribution of PCS, i.e., when φ(y) = 1, we get ].

Therefore the revised free energy density f will be expressed as:

where α is the Lagrange multiplier and n_A⁻ is described by eqn (3).

Here the minimization is carried out in terms ψ, n_±, n_H⁺, n_OH⁻ and φ(y). The resulting equations (in dimensionless form) are (see the ESI† for detailed derivation):

Here too we first eliminate [small psi, Greek, macron] and write all the governing equation and the boundary conditions [see eqn (6)] entirely in terms of n̄_H⁺ (see the ESI†). Of course these equations and the boundary conditions will depend on the profile φ. Below we discuss the procedure to obtain φ, which will then allow the solution of the hydrogen ion and the electrostatic potential profiles by employing numerical simulations.

The first thing that should be noted about the quantification of the profile φ is that the minimization of F with respect to φ yields only a value for the Lagrange parameter α (see the ESI†), and no information on the functional form of φ. This happens since none of the free energy terms (except for the electrostatic contribution of the PEL ions) depend on φ [see eqn (8)] and this electrostatic contribution of the PEL ions depends only linearly on φ. Therefore, φ can be any function that satisfies all of the three following conditions, along with eqn (7) (see the ESI† for more details)

The simplest possible choice of φ that satisfies all the above conditions is a cubic equation, expressed as (see the ESI†):

where is another dimensionless parameter quantifying the grafting density of the polyelectrolytes. Therefore, now the problem is dictated by seven parameters n̄_H⁺,∞, n̄_OH⁻,∞, [small lambda, Greek, macron], d̄, K̄^′_a, [small gamma, Greek, macron], β.

Fig. 3(a–f) show the variation of n̄_H⁺ and [small psi, Greek, macron] profiles for these parameter values. The most important observation that should be made from these figures is that for all the parameter values, there is no longer any discontinuity in the n̄_H⁺ profile. Therefore, using the prescribed remedy of considering a non-uniform distribution of the PCS within the PEL, we do ensure removal of the utterly non-physical result of discontinuous hydrogen ion profiles. Fig. 3(a) demonstrates the effect of considering different values of β. Enhanced β implies enhanced grafting density of the polyelectrolytes in the PEL. This signifies larger number density of the polyelectrolyte ions and hence larger magnitude of [small psi, Greek, macron]. There are two crucial issues associated with Fig. 3(a). Firstly, we find that β = 150 represent a value that reproduces the [small psi, Greek, macron] profile very similar to that of Fig. 2(a). Therefore, for the subsequent plots [Fig. 3(b–f)], we shall work with this value of β. More importantly, consideration of non-uniform distribution of chargeable sites in order to remove the non-physical behavior of the n̄_H⁺ profiles (see Fig. 2) leads to a much steeper n̄_H⁺ profile within the PEL. This behavior is true for the n̄_H⁺ profiles for all other plots as well [see Fig. 3(b–f)]. This steepness can be attributed to the fact that n̄_H⁺ now spans larger ranges of values over a given distance (which is the PEL thickness) in order to remove the corresponding discontinuities (see Fig. 2); hence larger the extent of discontinuity in Fig. 2, steeper will be the corresponding n̄_H⁺ profile within the PEL. Fig. 3(b) provides the effect of the variation of the parameter n̄_H⁺,∞. Qualitatively the variation is very similar to that depicted in Fig. 2(b), except for the fact that there is no discontinuity in n̄_H⁺ profile here. Fig. 3(c) represents the effect of the variation of [small gamma, Greek, macron]. For large [small gamma, Greek, macron] (=10), the discontinuity in n̄_H⁺ profile is very large [see Fig. 2(b)]; therefore the n̄_H⁺ profile is extremely steep within the PEL. More importantly, we can actually find that n̄_H⁺ value at the PEL–rigid-solid interface (i.e., at ȳ = −1) is slightly higher as compared to that in Fig. 2(b). This can be attributed to the corresponding cubic distribution of the PCS constituting the PEL – this cubic distribution ensures greater fraction of PCS at ȳ = −1, which in turn causes greater n̄_H⁺ concentration at ȳ = −1. Effect of this cubic profile in causing a more augmented value of n̄_H⁺ at the PEL–rigid-solid interface is also reflected in Fig. 3(d) that demonstrates the effect of the parameter [small lambda, Greek, macron]. From eqn (12) we get [φ(ȳ)/β]_ȳ=−1 = d̄³/2 – hence this enhancement of n̄_H⁺ at the PEL–rigid-solid interface is most significant for larger values of d̄ (=0.5) [see Fig. 3(e), which illustrates the effect of variation of parameter d̄]. Finally, the non-monotonic behavior of the discontinuity for varying K̄^′_a [see Fig. 2(e)], leads to a non-monotonic steepness variation of n̄_H⁺ (within the PEL) for the present case where a non-uniform distribution of chargeable sites within the PEL is considered [see Fig. 3(f) that demonstrates the effect of variation of the parameter K̄^′_a].

Fig. 3 Variation of the dimensionless hydrogen ion number density (top panel) and the dimensionless electrostatic potential profile (bottom panel) for the case where the chargeable sites in a polymer constituting the PEL obey a cubic distribution expressed in eqn (12). For all the figures we maintain the four parameters (i.e., n̄_H⁺,∞, [small lambda, Greek, macron], K̄^′_a,[small gamma, Greek, macron]) as equal to unity, d̄ = 0.3 and β = 150 except for (a) β = 100, 150, 400, (b) n̄_H⁺,∞ = 0.1, 1, 10, (c) [small gamma, Greek, macron] = 0.1, 1, 10, (d) [small lambda, Greek, macron] = 0.2, 1, 10 and (e) d̄ =0.1, 0.3, 0.5, (f) K̄^′_a = 0.1, 1, 10. In all the figures, the value of the parameter (which is being varied) is indicated at a given plot. It can be clearly seen that there is no more discontinuity in the hydrogen ion concentration at the PEL–electrolyte interface (this interface is located at ȳ = −1 + d̄). Also for all the plots, we use n̄_H⁺,∞ = n̄_OH⁻,∞.

Discussions

Here we shall first like to discuss the elements in which the present model compares to the classical equilibrium description of the polymer brushes.^37–40 First and foremost, as has been already identified, the present model disregards the contribution of the elastic and the volume exclusion effects, which are paramount in the description of the polymer brushes. The second issue, correlated with the above issue, is that for the present case we can use a fixed value of the thickness of the PEL d. This is an important difference with respect to the case of polymer brushes, where one needs to obtain an equilibrium value of the brush height d.³⁶ We can make such a simplification in the present case since the free energy functional does not depend explicitly on brush height, except for φ [see eqn (8)]. Thirdly, as has already been mentioned in this paper, for the case of polymer brushes, the monomer distribution obeys a quadratic profile^37–40 – the monomer concentration is higher at near-wall locations and decays at locations way from the wall. For the present case of the polyelectrolyte brushes, the PCS (which can be considered equivalent to monomers) will obey a non-unique and non-uniform distribution; however, just like the case of polymer brushes, this concentration of the chargeable sites will also decay away from the wall.

Here we also attempt to highlight the relevance of the present study in qualitatively interpreting different experimental results on pH-responsive polyelectrolyte brushes. Yameen et al.⁴¹ demonstrated that poly (4-vinyl pyridine) brushes undergo a pH-dependent reaction leading a reversible switching between the swollen charged hydrophilic state (at low pH; here the polymer is pyridine) and collapsed, neutral hydrophobic state (at high pH; here the polymer is pyridinium). Here the charging occurs via a base-like reaction, when pyridinium, at a small pH, accepts a proton to form pyridine – this ensures that the charging of the PEL is pH-dependent. Since this charging occurs at a small pH, the pH-dependent PEL charge density will be typically witnessed at this small pH. Hence at such a pH, by the proposed analysis of this study, there needs to be a given non-uniform distribution of the PCS or monomers with the PCS concentration decaying away from the wall, and this ensures this swollen hydrophilic state. On the contrary, at large pH (pH > pK_a) when the pyridine loses proton to become uncharged pyridinium. Therefore, at such pH the PEL is uncharged, and therefore there is no constraint on the distribution of the monomers, enforcing them to remain in collapsed, neutral hydrophobic state.

Hou et al.⁴² employed the pH-dependent shape modulation of poly acrylic acid (PAA) to develop a pH-gating ionic transport in a conical nanochannel. PAA contains a carboxylic acid group, which is assumed to form an intramolecular hydrogen bonding at low pH, whereas at high pH ionizes to form COO⁻ and in the process is assumed to form hydrogen bonding with the surrounding water molecules. Hou et al.⁴² hypothesized that such a transition from intramolecular to intermolecular hydrogen bonding enforced a structural transition of the PAA molecules, enforcing the conical nanochannel to allow water imbibition at the configuration where there is intermolecular hydrogen bonding (i.e., at larger pH). We can provide an alternate explanation, based on our proposed theory. At large pH, where the PAA molecules have ionized to form COO⁻ (i.e., the PEL has pH-dependent charge density), the PCS (or monomers) will demonstrate a non-uniform distribution, with the PCS concentration decaying away from the wall. Therefore, the drag is also much larger near the wall; since the flow velocity (which is a surface tension driven imbibition) is much weaker closer to the channel walls, this PEL configuration (with greater PCS or monomer concentration at near-wall locations) ensures much weaker overall drag and hence a more enhanced flow rate. On the contrary, for smaller pH, the PEL is uncharged, and hence the PCS or monomers can remain uniformly distributed, thereby inducing a much larger drag force and a weaker flow rate.

Above two examples show that our theory can help to interpret the effects associated with the pH-response-dependent shape changes of polyelectrolyte brushes, and the associated variations in the transport characteristics. Please note that in the above two examples, we only provided qualitative interpretations of the existing experiments based on our new theoretical understanding. Similar such examples, where our theory can provide useful qualitative insights, include the host of experimental studies on transport in soft (polyelectrolyte-grafted) nanochannels/nanopores that are responsive to pH or concentration of a specific ion (Wen et al.⁴³ reviewed several such nanochannels/nanopores). Of course, a more rigorous quantitative comparison will be possible when we actually obtain the unique distribution of the PCS or monomers by considering the elastic and volume exclusion effects in addition to the electrostatic effects. This will be studied in a future publication.

Conclusions

In this paper we provide a rigorous free energy based approach to pinpoint a gross existing error in the modeling of electrostatics of soft charged interfaces, represented as a grafted-PEL system with pH-dependent charge density: we show that for such soft charged interfaces, it is impossible to have a uniform PCS (or monomer) distribution within the PEL. We arrive at this very important conclusion on the basis of the fact that we consider for the first time explicit number density distribution of the hydrogen ions inside and outside the PEL. We argue that the monomer distribution is non-uniform and non-unique; it must at least be a cubic distribution, which is very insightful given the fact that the monomer distribution of the polymer brushes (resulting from the balance of the elastic and volume exclusion effects) should be uniquely quadratic. We also demonstrate that our theory can provide qualitative interpretation of a number of experiments involving pH-response-dependent shape transition in polyelectrolyte brushes.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c4ra13946a

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