Molecular theory of glyphosate adsorption to pH-responsive polymer layers

Abstract

By means of a molecular-level theory we investigate glyphosate adsorption from aqueous solutions to surface-grafted poly(allylamine) layers. Our molecular model of glyphosate and the polymeric material includes description of size, shape, conformational freedom, and state of protonation of both components. The composition of the bulk solution (pH, salt concentration and glyphosate concentration) plays a critical role to determine adsorption. Adsorption is a non-monotonic function of the solution pH, which can be explained in terms of the pH-dependent protonation behavior of both adsorbate and adsorbent material. Lowering the solution salinity is an efficient way to enhance glyphosate adsorption. This is because glyphosate and salt anions compete for adsorption to the polymer layer. In this competition, glyphosate deprotonation, to increase its negative charge upon entering the polymer layer, plays an critical role to favor its adsorption under a variety of solution conditions. This deprotonation is the result of the higher pH that establishes inside the polymer. Our results show that such pH increase can be controlled, while achieving significant glyphosate adsorption, through varying the grafting density of the material. This result is important since glyphosate degradation by microbial activity is pH-dependent. These polymeric systems are excellent candidates for the development functional materials that combine glyphosate sequestration and in situ biodegradation.

Appendix

1.1 Theoretical approach

In this section, we give expressions for the different contributions to the free energy, Eq. (1). The first term in the right-hand side of that equation contains the conformational entropy of the polymer layer,

$$\begin{aligned} \begin{aligned} -\frac{S_{conf}}{k_{B}}=N\sum _{ { \alpha } }{ { P }(\alpha )\ln {(P(\alpha ))} } \end{aligned} \end{aligned}$$

(5)

where $P(\alpha )$ is the probability of finding a polymer chain in its conformation $\alpha$, $k_B$ is the Boltzmann constant, and N is the number of chains grafted to the surface of total area A.

In the second contribution to the free energy, the translational (mixing) entropy of mobile species (except glyphosate) is

$$\begin{aligned} \begin{aligned} -\frac{S_{mix}}{k_{B}}&=A\sum _{ \gamma =w,H^+,OH^-,+,-}\int _{0}^{\infty }dz\,\rho _{\gamma }(z)\\&\quad \times \left( \ln (\rho _{\gamma }(z)\nu _{w})-1+\beta \mu _{\gamma }^{0}\right) \end{aligned} \end{aligned}$$

(6)

where the subindex $\gamma$ runs over water molecules (w), hydronium ($H^+$) and hydroxyde ions ($OH^-$), salt cations ($+$) and anions (−); $\rho _{\gamma }(z)$ gives the local number density of the corresponding species; $\nu _w$ is the volume of a water molecule, and $\beta =\frac{1}{k_BT}$. This contribution also includes the standard chemical potential (self-energies) of each of these free species, $\mu ^{0}_{\gamma }$.

The translational and configurational (rotational) entropy of the glyphosate molecule is

$$\begin{aligned} \begin{aligned} -\frac{S_{gly}}{k_{B}}=&A\int _{0}^{\infty }dz\sum _{\alpha _{gly}}\rho _{gly}(\alpha _{gly},z) \\&\times \left( \ln (\rho _{gly}(\alpha _{gly},z)\nu _{w})-1+\beta \mu _{gly}^{0}\right) \end{aligned} \end{aligned}$$

(7)

where $\rho _{gly}(\alpha _{gly},z)$ is the local density of glyphosate in conformation $\alpha _{gly}$. This contribution also contains the self-energy of the molecule, given by its standard chemical potential $\mu _{gly}^{0}$. Then, the total density of glyphosate at z is

$$\begin{aligned} \begin{aligned} \left\langle { \rho }_{gly}(z) \right\rangle&=\sum _{\alpha _{gly}}{\rho _{gly}(\alpha _{gly},z)} \end{aligned} \end{aligned}$$

(8)

where 〈〉 indicate ensemble average over molecular conformations.

Next term in Eq. (1) is the chemical free energy that describes the acid-base equilibrium of PAH units. This contribution can be expressed as

$$\begin{aligned} \begin{aligned} \beta F_{chm,pol}&=N \int _{0}^{\infty }dz\,\left\langle \rho (z) \right\rangle f_{AH}(z) \\&\quad \times \left( \ln f_{AH}(z) + \beta \mu _{AHp}^{0}\right) \\&\quad +N \int ^{\infty }_{0}dz\,\left\langle \rho (z)\right\rangle (1-f_{AH}(z)) \\&\quad \times \left( \ln (1-f_{AH}(z)) + \beta \mu ^{0}_{AHd}\right) \end{aligned} \end{aligned}$$

(9)

where $f_{AH}(z)$ is the local degree of charge (or protonation) of PAH units, and $\left\langle \rho (z)\right\rangle$ is the ensemble average density of segments belonging to a single chain, such that

$$\begin{aligned} \begin{aligned} \left\langle \rho (z)\right\rangle&= \sum _{ { \alpha } }{ { P }(\alpha )\rho (\alpha , z)} \end{aligned} \end{aligned}$$

(10)

with $\rho (\alpha , z)$ being the local density of segments of a chain in conformation $\alpha$. In Eq. (9), $\mu ^{0}_{AHp}$ and $\mu ^{0}_{AHd}$ are the standard chemical potentials of the protonated and deprotonated unit, respectively. These quantities are related to the thermodynamic equilibrium constant, $K^0_{AH}$, through the following equation:

$$\begin{aligned} \begin{aligned} K^0_{AH}&= \exp \left( \beta \mu ^{0}_{AHp} - \beta \mu ^{0}_{AHd}-\beta \mu ^{0}_{H^+} \right) \end{aligned} \end{aligned}$$

(11)

The chemical free energy of glyphosate, which accounts for the acid-base equilibria of its titratable units is

$$\begin{aligned} \begin{aligned} \beta F_{chm,gly}&=\sum _{\tau } A \int _{0}^{\infty }dz\, \left\langle \rho _\tau (z) \right\rangle \\&\quad \times g_{\tau }(z)\left( \ln g_{\tau }(z) + \beta \mu _{\tau p}^{0}\right) \\&\quad +\sum _{\tau } A \int ^{\infty }_{0}dz\,\left\langle \rho _\tau (z)\right\rangle (1-g_{\tau }(z))\\&\quad \times \left( \ln (1-g_{\tau }(z)) + \beta \mu ^{0}_{\tau d}\right) \end{aligned} \end{aligned}$$

(12)

where $\tau$ runs over the different titratable units of glyphosate, which are the diacidic phosphonate group, the carboxylate group and the amine group ($\tau =php_1,php_2,cbx,amn$); the standard chemical potentials of these units are $\mu ^{0}_{\tau p}$ and $\mu ^{0}_{\tau d}$ for the protonated and deprotonated unit, respectively. The local degree of protonation of $\tau$ units is $g_\tau (z)$. In the case of the amine group, this quantity is equal to its local degree of charge $f_{amn}(z)$, while for the acidic units $f_\tau (z)=1-g_\tau (z)$ (with $\tau =php_1,php_2,cbx$). In addition, the local density of glyphosate units can be expressed as:

$$\begin{aligned} \begin{aligned} \left\langle \rho _\tau (z) \right\rangle&=\sum _{\alpha _{gly}}\int _{0}^{\infty }dz' \,{\rho _{gly}(\alpha _{gly},z')}\\&\quad \times n_\tau (\alpha _{gly},z',z) \end{aligned} \end{aligned}$$

(13)

where $n_\tau (\alpha _{gly},z',z)$ is the number (density) of units type $\tau$ that a glyphosate molecule with center of mass at $z'$ contributes to z.

The next term in Eq. (1) is $U_{ste}$, which describes the steric repulsions at the excluded volume level. This contribution is incorporated through requiring each element of volume to be fully occupied by some of the chemical species. This constraint to the the free energy can be expressed as:

$$\begin{aligned} \begin{aligned} 1=N\left\langle { \rho (z)} \right\rangle \nu _{AH}\; &+&\sum _{ \gamma =w,{ H }^{ + }{ OH }^{ - },+,-}\rho _{\gamma }(z)\nu _\gamma \\&+& \sum _{\tau = amn, cbx, php}\left\langle \rho _\tau (z)\right\rangle \nu _\tau \end{aligned} \end{aligned}$$

(14)

where $\nu _{AH}$ is the volume of a PAH segment, $\nu _\gamma$ is that of free species $\gamma$, and $\nu _\tau$ the volume of glyphosate unit $\tau$.

The last contribution to F is the electrostatic energy,

$$\begin{aligned} \begin{aligned} U_{elec}&=A \int ^{\infty }_{0}dz\, \Big [ \left\langle \rho _{q}(z)\right\rangle \beta \psi (z) \\&\quad -\frac{1}{2}\beta \epsilon (\nabla \psi (z))^{2}\Big ] \end{aligned} \end{aligned}$$

(15)

where $\epsilon$ is the medium permittivity, and $\psi (z)$ is the position-dependent electrostatic potential. In this equation, the local density of charge is

$$\begin{aligned} \begin{aligned} \left\langle \rho _{q}(z)\right\rangle\,=\,N f_{AH}(z)\left\langle { \rho (z)} \right\rangle q_{AH}\; &+\sum _{ \gamma ={ H }^{ + }{ OH }^{ - },+,-}\rho _{\gamma }(z)q_\gamma \\&+\sum _{\tau = php_1,php_2,cbx,amn}f_\tau (z)\left\langle \rho _\tau (z)\right\rangle q_\tau \end{aligned} \end{aligned}$$

(16)

$q_{AH}$, $q_\gamma$ and $q_\tau$ are the electric charges of the different species.

This system is in equilibrium with a bulk solution of controlled composition, which fixes the chemical potentials of all free species ($\mu _\gamma$ and $\mu _{gly}$). Under these conditions, the proper thermodynamic potential whose minimum yields equilibrium is the semi-grand potential, which can be written as

$$\begin{aligned} \begin{aligned} \Omega&= F - \sum _{\gamma =w,{ H }^{ + }{ OH }^{ - },+,-} \mu _\gamma N_\gamma - \mu _{gly} N_{gly}\\&=F - A \sum _{\gamma =w,{ H }^{ + }{ OH }^{ - },+,-} \int _0^\infty dz\, \mu _\gamma \rho _\gamma (z) \\&\quad - A \int _0^\infty dz\, \mu _{gly} \left\langle \rho _{gly}(z)\right\rangle \end{aligned} \end{aligned}$$

(17)

where $N_\gamma$ and $N_{gly}$ are the corresponding number of molecules.

With all these expressions for its different contributions, the grand potential can expressed in terms of integrals and configurational sums the following functions:

• (i) $P(\alpha );$

• (ii) $\rho _{w}(z), \,\rho _{H^{+}}(z),\, \rho _{OH^{-}}(z),\, \rho _{+}(z),\, \rho _{-}(z),\rho _{gly}(\alpha _{gly},z);$

• (iii) $f_{AH}(z),\,f_{php_1}(z),\, f_{php_2}(z),\, f_{amn}(z),\, f_{cbx}(z);$

• (iv) $\psi (z).$

Optimization of $\Omega$ with respect to functions (i) to (iii) leads to expressions for these functions that only depend on the two interaction potentials: $\psi (z)$ and $\pi (z)$, which is the Lagrange multiplier introduced to enforce satisfaction of the incompressibility constraint, Eq. (14). The extremum of $\Omega$ with respect to the electrostatic potential results in the Poisson equation:

$$\begin{aligned} \begin{aligned} \frac{\partial ^{2}\psi }{\partial z^{2}}&=-\frac{\left\langle { \rho }_{ q}(z) \right\rangle }{\varepsilon } \end{aligned} \end{aligned}$$

(18)

Finally, $\psi (z)$ and $\pi (z)$ are obtained though iteratively solving the Poisson equation (Eq. (18)) and the incompressibility constraint (Eq. (14)) at each position. Once these interaction potentials have been calculated, functions (i) to (iv) are known, and consequently the free energy of the system is determined.