Donnan theory (III, IV)

Effect of valence on the loading and release efficacy (III)

Let us consider the equilibrium of ion i between the aqueous external solution (w) and fiber (f) phases. The general equilibrium condition is the equality of its electrochemical potential ~ in both phases (Guggenheim, 1967):_i

) number of the ion (valence); F is the Faraday constant (96,487 C mol^–1), R the gas constant (8.314 J K^–1 mol^–1), T the absolute temperature, and the galvani potential.

Equation (1) can be rewritten in the following form:

) coefficients were to deviate significantly from one, their ratio would be close to one as all

Compound Acetonitrile (%) 0.1 % TFA pH 2.1 (%) Wavelength (nm) Retention time (min)

5-Aminosalicylic acid 5 95 235 3.6

Acetylsalicylic acid 40 60 235 3.2

5-Hydroxysalicylic acid 30 70 240 3.0

Salicylic acid 50 50 240 3.2

5-Fluorosalicylic acid 50 50 235 3.3

5-Methylsalicylic acid 60 40 240 2.9

5-Chlorosalicylic acid 60 40 240 3.1

5-Bromosalicylic acid 65 35 235 2.9

3-Isopropylsalicylic acid 75 25 245 3.0

5-Hydroxyisophthalic acid 20 80 235 3.0

the ion-exchange groups are exposed to aqueous environment, and we can neglect them as the first approximation (accurate ratio of the activity coefficients is impossible to measure in most cases). The first exponential on the right hand side of Eq. (2) is the chemical part of the ionic partition coefficient; we denote it by K^ch_p,i. (f) – (w) in the latter exponent is the Donnan potential, D. Thus, Eq. (2) can be expressed as

D

where = F /RT. Eq. (3) relates the concentration in the fiber phase with that in the external aqueous solution. As the purpose of our study was to evaluate the effect of the valence of the counter-ions and co-ions on the loading and release efficacy, the standard chemical potentials are set to unity, K^ch_p,i 1, in order to allow calculation of Donnan potential. Let us denote the fixed charge density of the fiber byX , in concentration units, and the charge number of the ion-exchange group byzf.

Electroneutrality condition in the fiber phase gives thus

i

The Donnan potential must be solved numerically from Eqs. (5-7). Yet, the limiting behaviour at large argument values can be expressed in closed form as _D ln X/c ,

c X /2

D ln and _D 1/2 ln X/2c for the 1-1, 2-1 and 1-2 electrolytes, respectively. Having the Donnan potential solved, the concentrations of the counter- and

6B with K^ch_p,i 1. For a 1-3 electrolyte (e.g. trisodium citrate) the limiting behaviour would be _D 1/3 ln X/3c .

Figure 6 tells the obvious result that the higher the fiber charge or the lower the concentration of the external solution, the stronger is the Donnan potential and co-ion exclusion. The Donnan potential is the highest for a 1-1 electrolyte, but the repulsion of a divalent cation is the strongest due to its charge number. The attraction of monovalent and divalent counter-ions is practically the same (K^ch_p,i 1), although the Donnan potential for the 1-1 case is more than 2-fold of that of the 1-2 case. It must be emphasised thatc in the x-axes ofFigure 6Aand B denotes the external aqueous salt concentration, but the y-axes inFigure 6B are ratios of the ionic concentrations in the fiber and aqueous phases.

Figure 6. (A) Donnan potential for a 1-1 electrolyte (NaCl; solid), 2-1 electrolyte (CaCl2; dashed), and 1-2 electrolyte (Na2SO4; dotted). (B) Concentration ratios of the co-ions (descending, left axis) and counter-ions (ascending, right axis) in the fiber and aqueous external phases for a 1-1 electrolyte (solid), 2-1-1 electrolyte (dashed), and 1-1-2 electrolyte (dotted). X/2c in the x-axis denotes the ratio of fixed charge density in the fiber and the external aqueous salt concentration.

Ion-exchange can also be analysed based on the Donnan theory. To calculate the effect of valence of the counter- and co-ions only, the chemical parts of the ionic partition coefficients are set again to one in the following derivations. Let us consider the release with a 1-1 electrolyte (NaCl) as an example. The fiber is initially loaded completely, i.e.

the initial concentration of a monovalent drug (SA) in the fiber (c_SA^f,0) is equal toX. The electroneutrality condition in the fiber after the ion-exchange equilibrium has been reached is

because c_Na^w c_Cl^w c_SA^w . Mass balance for Cl is

where c_Cl^w,0 is the initial concentration of Cl in the extracting aqueous solution, and r is the ratioV^f/V^w. Now, c_Cl^w can be solved from Eq. (9) as

Accordingly, c_SA^w can be derived to be

D

Inserting Eqs. (10) and (11) into Eq. (8), the Donnan potential can be solved numerically from Eq. (12):

Having the Donnan potential calculated, the fraction of drug released is obtained as

1 D

Similar calculations are carried out accordingly for 2-1 (CaCl₂) and 1-2 (Na₂SO₄) extracting electrolytes, as shown in Figure 7 with arbitrarily chosen values r = 0.1 and r = 1, with the order of releasing efficacy being NaCl CaCl₂ < Na₂SO₄.

At low aqueous concentrations ( 3 meq of extracting electrolyte/meq of ion-exchanger), the order of decreasing affinity towards the anion-exchanger is usually citrate

> sulfate oxalate > chloride (Kunin and McGarvey, 1949), as can be easily shown with the Donnan theory. At high concentrations, however, the ion-exchange affinity of ions with similar valence may differ (e.g. oxalate > sulfate) and, even, oxalate (-2) > citrate (-3) (Kunin, 1949). Theoretically, multivalent ions can simultaneously bind to several ion-exchange groups in the fiber, divalent with two and trivalent with three ionic groups (Bhandari et al., 1993; Liu et al., 2001).

Figure 7. The released fraction of a monovalent drug (z = -1) with the volume ratios r = 0.1 and r = 1 in the case of a 1-1 electrolyte (NaCl; solid), 2-1 electrolyte (CaCl2; dashed) and 1-2 electrolyte (Na2SO4; dotted). X/[A(w)]⁰ in the x axis label is the ratio of initial concentrations of anionic drug in the fiber and the extracting anion (Cl , SO²₄ ) in the releasing aqueous solution.

Effect of loading solution concentration on fiber occupancy and loading efficiency (IV) Ion equilibrium between the fiber (f) and aqueous (w) phases is given by

, D partition coefficient, and _D = F _D/RT the dimensionless Donnan potential. As there is no a priori knowledge of the values of K^ch_p,i, they are all set to one as the first approximation in order to assess the effect of the charge of the fiber and the model ions.

In the SA solution, there are three mobile ions: H⁺, S^– (salicylate), and Cl^– from the fiber. Equation (14) is thus written for all of them:

Cl D

where n denote for the amount of ions (mol) and r = V(f )/V(w), the volume ratio of the phases. The amount of S^– transferred into the fiber is denoted by x and that of H⁺ by y.

Hence, at the equilibrium, n w n y n y n w nw x n f x

the initial amount of SA in the aqueous phase.

From the electroneutrality condition, n_Cl(w) n_H(w) n_S(w) x y. Correspondingly, in the fiber phase, n_Cl(f) n_H(f) n_S(f) X y x X , where X is the amount (mol) of the monovalent ion-exchange groups (binding sites). Dividing all the amounts by n₀^w, Eq. (15) can be brought into a dimensionless form:

w

In the case of the simultaneous loading of SA and divalent di-COOH, Eq. (16) becomes where only a divalent compound is in the solution, the second equation in Eq. (17) is left out. The solutions of Eqs. (16) and (17) are described in paperIV,Appendix A in detail.

Figure 8A shows the calculated Donnan potentials. As can be seen, the Donnan potential is the highest with the solution of monovalent compound and the lowest with the divalent compound; the equimolar mixture of them gives an intermediate value. The abscissa inFigure 8, ^–1, is defined in Eq. (16); hence the Donnan potential decreases with increasing the initial amount of SA in the aqueous loading solution and/or decreasing the ionic charge of the fiber. InFigure 8B a comparison of the fiber occupancy (left axis) and loading efficiency (right axis) is given between a monovalent and a divalent compound. It has to be realised that although the occupancy is higher with a monovalent compound, a divalent compound occupies two sites and, therefore, its occupancy can be considered higher. InFigure 8D a magnification to the low concentration range, ^–1 < 1 is given. As can be seen, at low concentrations the divalent compound is bound to the fiber to a higher extent than a monovalent compound. In Figure 8C, both the monovalent and divalent compounds are simultaneously present in the aqueous solution in an equal amount of

n₀w 2

1 . Both the fiber occupancy and loading efficiency are higher for the divalent compound when the chemical part of the partition constants are equal to one.

Donnan failure

An interesting feature in the fiber occupancy curves is that they can exceed 100% (Figure 8B). This phenomenon is generally known as the Donnan failure, which means that the solution concentration is so high compared with the fiber charge that coulombic repulsion fails to impede the entrance of co-ions (here H⁺) into the fiber. After all the ion-exchange sites are occupied, compound (counter-ion) transfer continues into the fiber, carrying the respective co-ion along with it in order to maintain electroneutrality, until the chemical potentials are balanced.

Figure 8. Simulations to the Donnan theory. (A): Donnan potentials for monovalent (solid) and divalent compounds (dashed), and their mixture (dotted). (B): Comparison of the fiber occupancy (ascending, left y axis) and loading efficiency (descending, right y axis) between monovalent (solid) and divalent compounds (dashed). (C): Fiber occupancy (ascending, left y axis) and loading efficiency (descending, right y axis) in the equimolar mixture of monovalent (solid) and divalent compound (dashed). (D): Magnification of (B) in the low concentration range. ^–1 in the x axis denotes the ratio of initial amount of salicylic acid in the external aqueous phase (mol) and the amount of ionic binding sites in the fiber phase (mol).