Protonation behavior of 6-deoxy-6-(2-aminoethyl)amino cellulose: a potentiometric titration study

Abstract

The protonation behavior of 6-deoxy-6-(2-aminoethyl)amino cellulose as a novel soluble aminated derivate of cellulose was studied by means of the potentiometric titration technique. The resulting proton binding isotherms exhibit two equivalent steps, which can be described by the standard macroscopic two-pK model, in which the degree of protonation is averaged over all the amine groups. In addition, a microscopic proton binding model was applied, in which the protonation sites are distinguished and the protonation free energy is expanded into an intrinsic term and an electrostatic repulsion between the primary and secondary amine groups. The protonation behavior of 6-deoxy-6-(2-aminoethyl)amino cellulose was compared with a model compound (N-methylethylenediamine).

Appendix: Protonation equilibria for diprotic molecules

The macroscopic proton binding isotherms

For a diprotic molecule, the two deprotonation steps are:

$$ {\text{H}}_{2} {\text{A}}^{n} \rightleftarrows {\text{H}}^{ + } + {\text{HA}}^{n - 1} ;\quad K_{2} = {\frac{{\left[ {{\text{HA}}^{n - 1} } \right]a_{{{\text{H}}^{ + } }} }}{{\left[ {{\text{H}}_{2} {\text{A}}^{n} } \right]}}} $$

(4)

$$ {\text{HA}}^{n - 1} \rightleftarrows {\text{H}}^{ + } + {\text{A}}^{n - 2} ;\quad K_{1} = {\frac{{\left[ {{\text{A}}^{n - 2} } \right]a_{{{\text{H}}^{ + } }} }}{{\left[ {{\text{HA}}^{n - 1} } \right]}}} $$

(5)

where the K _i denote the step-wise mixed dissociation constants. Please note that K _i are indexed according to the protons bound to the protonated species, thus the protonation (not deprotonation) step. For a base n = +2, while for an acid n = 0. The cumulative deprotonation reactions and the respective cumulative constants are defined as

$$ {\text{H}}_{2} {\text{A}}^{n} \rightleftarrows 2{\text{H}}^{ + } + {\text{A}}^{n - 2} ;\quad \bar{K}_{2} = {\frac{{\left[ {{\text{A}}^{n - 2} } \right]a_{{{\text{H}}^{ + } }}^{2} }}{{\left[ {{\text{H}}_{2} {\text{A}}^{n} } \right]}}} $$

(6)

$$ {\text{HA}}^{n - 1} \rightleftarrows {\text{H}}^{ + } + {\text{A}}^{n - 2} ;\quad \bar{K}_{1} = {\frac{{\left[ {{\text{A}}^{n - 2} } \right]a_{{{\text{H}}^{ + } }} }}{{\left[ {{\text{HA}}^{n - 1} } \right]}}} $$

(7)

The macroscopic protonation state is determined by the number of protons bound to the molecule. The total concentration of the molecule, [H₂A]₀ is the sum of all the protonation species

$$ \left[ {{\text{H}}_{2} {\text{A}}} \right]_{0} = \left[ {{\text{A}}^{n - 2} } \right] + \left[ {{\text{HA}}^{n - 1} } \right] + \left[ {{\text{H}}_{2} {\text{A}}^{n} } \right] $$

(8)

At a given pH = −loga ⁺_H , and provided [H₂A]₀ is known, Eqs. 6–8 or alternatively 4, 5, and 8 form closed sets, which permits calculating the probabilities of the three protonation species P ₀ = [Aⁿ⁻²]/[H₂A]₀, P ₁ = [HAⁿ⁻¹]/[H₂A]₀ and P ₂ = [H₂Aⁿ]/[H₂A]₀ from the dissociation constants. The overall degree of protonation is obtained as

$$ \theta = \frac{1}{2}\left( {P_{1} + 2P_{2} } \right) = {\frac{{\left[ {{\text{HA}}^{n - 1} } \right] + 2\left[ {{\text{H}}_{2} {\text{A}}^{n} } \right]}}{{2\left[ {{\text{H}}_{2} {\text{A}}} \right]_{0} }}} $$

(9)

The discrete site binding model

Distinguishing the protonation sites leads to the microscopic protonation scheme for a diprotic molecule (Borkovec and Koper 1994; Borkovec et al. 2001; Szakacs et al. 2004) (Fig. 5)

Fig. 5

Microscopic protonation scheme of a diprotic molecule

In this picture, the particular sites are indexed and a microscopic protonation state (microstate) is defined by a set of the state variables {s _i }, where s _i  = 0 for deprotonated and s _i  = 1 for protonated. For a diprotic molecule, the index i can attain values of 1 or 2.

A microscopic dissociation constant (microconstant) $\hat{K}_{k} \left\{ {s_{i} |k} \right\}$ can be attributed to each site k and microstate {s_i≠k}. For modelling purposes, particularly useful are the microconstants for the first protonation step denoted as $\hat{K}_{k}$

$$ \left\{ {s_{i} = 0;\;s_{k} = 1} \right\} \rightleftarrows {\text{H}}^{ + } + \, \left\{ {s_{i} = 0;\;s_{k} = 0} \right\}\quad \hat{K}_{k} = {\frac{{\left\{ {s_{i \ne k} = 0;\;s_{k} = 0} \right\}a_{{{\text{H}}^{ + } }} }}{{\left\{ {s_{i \ne k} = 0;\;s_{k} = 1} \right\}}}} $$

(10)

In this step, a proton is bound at the site k of the fully deprotonated molecule. The site binding model equivalent to the Ising model, is defined by expanding the protonation free energy of a microstate F({s _i }) as (Borkovec and Koper 1994)

$$ {\frac{{F\left( {\left\{ {s_{i} } \right\}} \right)}}{kT\ln 10}} = \sum\limits_{i} {s_{i} \log \mathop {K_{i} }\limits^{ \wedge } } + \sum\limits_{i < j} {\varepsilon_{i,j} s_{i} s_{j} } + \cdots $$

(11)

where ε _i,j is the electrostatic interaction parameter between the protonated sites i and j, and neglecting the higher-order terms. NB this notation is valid only if the protonated site has the charge number +1, thus in the case of bases. ε _i,j is related to the free energy of the electrostatic repulsion as

$$ \varepsilon_{i,j} = {\frac{{E_{i,j} }}{kT\ln 10}} $$

(12)

For a diprotic molecule, Eq. 11 simplifies to

$$ {\frac{{F\left( {s_{1} ,s_{2} } \right)}}{kT\ln 10}} = s_{1} \log \hat{K}_{1} + s_{2} \log \hat{K}_{2} + \varepsilon s_{1} s_{2} $$

(13)

In this case, the two macroscopic cumulative constants defined in Eqs. 6 and 7 can be written as

$$ \bar{K}_{1}^{ - 1} = \hat{K}_{1}^{ - 1} + \hat{K}_{2}^{ - 1} $$

(14)

$$ \bar{K}_{2}^{ - 1} = \hat{K}_{1}^{ - 1} \hat{K}_{2}^{ - 1} 10^{ - \varepsilon } $$

(15)

For the purpose of the present work, the EDA functionality and the N-methylethylenediamine were modelled with the same $\hat{K}_{i}$ value for both protonation sites, in which case the above two equations simplify to

$$ \bar{K}_{1}^{ - 1} = 2\hat{K}^{ - 1} $$

(16)

$$ \bar{K}_{2}^{ - 1} = \hat{K}^{ - 2} 10^{ - \varepsilon } $$

(17)

Thus, for a diprotic molecule, one can calculate the macroscopic degree of protonation from the microscopic protonation constant $\hat{K}$ and the electrostatic interaction parameter ε by combining Eqs. 6–8 and 16, 17. A full derivation of the above relations and a comprehensive description of the above model for the present, as well as much more complex cases of polyelectrolytes and surfaces can be found in the afore mentioned papers.