Example 3 Optimisation proton binding parameters LFA master-curve
In this example the proton parameters are derived in two steps. In the first step the parameters for the Donnan model are derived. In the second step the parameters of the NICA model are derived, with the Donnan parameters fixed at the values obtained in step 1.
Step 1 Master Curve approach:
With the master curve procedure only the electrostatic part of the model, the Donnan model is used. The Donnan volume, which is assumed constant at a certain ionic strength, is optimized for each ionic strength such that the proton titration curves at different ionic strength merge into a single master curve. This can be done by relating the humic charge of the HA (Q) to the negative logarithm of the proton concentration in the Donnan volume (“pHdon”).
With the Donnan model the concentrations of protons and other cat- and anions in the Donnan volume are calculated such that the negative charge of the humic acid is counterbalanced by the surplus of positive charge in the Donnan volume. These calculations are done for similar series of humic charge at each ionic strength. For each point the input for the calculation consists of: the humic charge, the pH associated to that charge and the concentrations of the cat- and anions of the background electrolyte.
In order to have the input for the three different ionic strengths for similar values of humic charge we derived relations between Q and pH for each ionic strength. These relations where then used to calculate the pH associated to a certain Q at each ionic strength. For this purpose we used a 6th order polynomial relation with pH as a function of Q. For each ionic strength we constructed an input file with 100 points of humic charge (equidistant). To stay within the boundaries of the experimental values the lowest charge to be used is the maximum of the lowest charge of the 3 titration curves and the highest charge to be used is the minimum of the highest charges of the 3 titration curves. The construction of the Q-pH input data can be found in the Excel worksheet FH-23-MC.xls.
The Donnan model is implemented in the template for the chemistry file (chemistry1.tpl). For each ionic strength (in this case 3 datasets) the proton concentrations in the Donnan volume are calculated for a similar series of humic charge. For each ionic charge there is a separate chemistry-, input- and output file (i.e., “chemistryX.inp”, “inputX.dat” (Paragraph 3.2) and “outputX.dat”, with X=1,2,3). The proton concentrations in the Donnan volume are calculated for each point of humic charge for the three ionic strengths and their values (pHdonX with X=1,2,3 for the 3 different ionic strengths) are written to the three output files (“outputX.dat”). Thereafter ORCHESTRA reads the values of pHdonX from the 3 output files and calculates dif1 and dif2, being equal to pHdon1 - pHdon2 +1 and pHdon1 - pHdon3 +1, respectively. The instructions for this calculation are in the file “postprocess.inp”. The values of dif1 and dif2 are written to the file outputpost.dat which is the outputfile to be read by PEST according to the format of the outputfile described in “outputP.ins”. PEST optimizes the model parameters by minimizing the difference of dif1 and dif2 with the value of the observations (which is set to 1 for all cases). The value of 1 (and the addition of 1 in the calculation of dif1 and dif2) was chosen instead a value of 0 to enable the calculation of r2 by PEST.
The instructions for ORCHESTRA are defined in the file composer.xml.
Download all ORCHESTRA and PEST files for this example: Link to zip file MC-VdI.7z
Step 2 optimization of the NICA parameters
In this step the parameters K1, K2, Q1, Q2, n1 and n2 are optimized with the donnan parameters α and β fixed at their values as obtained in the first step
Download all ORCHESTRA and PEST files for this example: Link to zip file MC-step2.7z