µV to flux conversion

by Joe Kunkel
University of Massachusetts Vibrating Probe Facility

Ion probe technologies should be designed to provide you with the information with which one can calculate the ionic flux, J, at a point {x,y,z} in space:
  Jxyz = D dC/dr ,
       where D = diffusion coef.; dC = concentration differential; and dr = spatial differential.
Because of the unique protocols of the measuring process used in each technology, the precise steps to calculating flux must consider the specific measuring system used.
We describe here the approach we use with the direct coupled amplifier system produced by Applicable Electronics and with measurements taken with either the 3DVIS software written by JG Kunkel or the ASET software from SCIENCEWARES.
We provide a downloadable zipped MsExcel spreadsheet which will aid in this process.   We also provide a downloadable zipped MathCAD worksheet which will perform the same process.
Each measurement of flux at a point in space requires two measurements made at that point in space (local probe voltage in mV and µV difference in the direction of the flux) plus a set of constants that are a function of the ion being measured, the LIX being used, the sampling protocol used by the software and the measured efficiency of that particular protocol and probe combination.  We will detail how these numbers are measured and how they are used to calculate flux.
The diffusion coefficient is tabulated for each ion but one must be conscious of the fact that ions travel in pairs and the diffusion coefficient in reality varies with concentration of the ion as well as the type and concentrations of counter-ions [Diffusion coefficients in aqueous solutions at 25°. Handbook of Chemistry and Physics, Chemical Rubber Co.]. This means that all flux measurements must be accepted with reservations.
The spatial differential, dr, is represented in the calculation by the distance between the points of the two discrete concentration measurements and may consist of a single distance if a single dimension of flux is being measured, or it may be made up of independent axes (X, Y and/or Z) if vectors of flux in space are being calculated.   In general practice, individual axes of flux are calculated and then a resultant flux is calculated using the hypotenuse square rule.   This is practical in all situations where the sources of ions are large and the individual spatial differentials are small compared to the dimensions of the source/sink being examined.
The concentration differential, dC, is a value that varies during an experiment and requires the most attention. Each point in space where one decides to measure flux, can be redefined temporarily as an origin of measurement with {x,y,z} = {0,0,0}. At that point one needs to determine the concentration of the ion of interest by using the mV potential measured at that point in combination with the equation for concentration that has been previously determined for that LIX and ion combination over a concentration span that spans the current conditions:
    C[0,0,0] = 10(mV[0,0,0] - A)/B , where A,B = Nernst intercept, slope.
Next the small measured differential voltage, µV, that results from moving the probe in the chosen direction (say dx of dx, dy, or dz) is measured by subtraction of the voltage measured at {0,0,0} and {dx,0,0}. It is this voltage differential, measured over a short time interval, that is subject to the efficiency correction that one has established for this LIX, ion and sampling conditions. The desire to make measurements in as short a time span as possible is in conflict with the time constant of the LIX. The LIX has a irreducible time that it takes to reach 95% of its expected voltage in a particular ion concentration. The Nernst callibration curve of mV vs ion concentration tells one the expected mV for a particular ion concentration one can measure when the probe is given abundant time to reach its theoretically correct voltage. The efficiency callibration tells one the percentage of the expected one can achieve during the dynamic measuring process. In the enclosed spreadsheet, knowing the efficiency corrected µV difference allows one to calculate the concentration of the ion of interest at the small distance from the origin {0,0,0}:
            C[dx,0,0] = 10(mV[0,0,0]-µVdiff/1000*eff) - A)/B) ,
      with A and B as above and eff = efficiency of measuring the µV difference.
Then dC can be calculated as:
    dCx = C[dx,0,0] - C[0,0,0].
After the above calculations are performed on each of the dimensions involved, one can proceed to finally calculate the flux:
  Jx = D dCx/dr , and
perhaps plot Jx, Jy, and Jz as a vector in space at the measurement point, or combine them according to the pythagorean theorem into a joint/total flux:
          J = [Jx2 + Jy2 + Jz2]0.5 .

last modified 9/22/98 by JG Kunkel