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Theory Answers
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GPC Theory: Deflection Refractometer |
The most common RI detector in use today is based on the deflection of a beam of light as it passes through a dual compartment flow cell as shown in Figure 1. One side of the compartment contains the reference solvent of refractive index n0, which is static during the measurement process. The other side contains the sample solution, i.e., the column eluent, having refractive index n.
The beam is refracted at the liquid-glass interfaces separating the two compartments and also at the liquid-glass interface on the exit wall, and again at the glass-air interface on the exit wall. A rigorous analysis must take each of the refractions into account. Fortunately, a simplified analysis that ignores the glass interfaces as shown in Figure 1(b) yields substantially the same result as the rigorous analysis, so that derivation will be reviewed instead.
Snells law of refraction is applied to the hypothetical liquid-liquid interface shown in figure 1(b).
For the very small angles of deflection encountered here we may approximate with no loss of accuracy as follows.
Combining equations 1 and 2 shows that the incremental refractive index is proportional to the deflection x.
The differential signal will likewise be proportional to x because the beam, which actually has a finite width, will deliver more light to one photo-detector and less to the other as the beam is moved across the beam splitter due to refraction. So we can combine instrument constants into one detector calibration factor. And we can replace n by n0 because they are very nearly the same value at low concentrations.
Batch Method
If the RI detector is to be useful as a concentration detector, then the refractive index of the solution must be proportional to sample concentration. With batch analysis, one can test this assumption by plotting the measured signal against sample concentration. This procedure is shown in Figure 2, where the RI signal is plotted against concentration for two different samples, polystyrene and polybutadiene, in the same solvent, THF. It can be seen that each sample yields a straight line slope. But the intercepts are not zero for either sample, and the slopes are different.
The reason the lines do not intercept at zero is because the reference side of the refractometer unintentionally had a slightly different solvent in it than that used to make up the sample solutions. Typically, for a solvent like THF atmospheric contaminants like water and air itself will tend to decrease the RI of the samples to a significant extent. It is impossible with the batch experiment to make sure that the reference side of the RI detector is purged with solvent having the exact composition of atmospheric contaminants as is in the sample solutions, because the sample solutions continually absorb moisture and air during the course of the measurements. So we must be content to find a linear relationship between RI signal and concentration.
The fact that the slopes of the polystyrene sample and polybutadiene sample are different means that the RI is not only proportional to concentration, but also to a sample-dependent parameter. This sample-dependent parameter is called the refractive index increment, or dn/dc. The general equation for the RI detector is therefore expressed as follows.
dn/dc is the slope of a plot of refractive index against concentration, but since the proportionality assumption requires the slope to be constant and the intercept equal to n0, the dn/dc is simply defined for a single solution of refractive index n and concentration C as follows.
Substitution of equation 6 into equation 5 yields equation 4 again.
In order to use the RI detector quantitatively, the calibration constant RI Cal must be determined. Figure 2 and equation 5 together provide a logical way to do this calibration. The slope of either of the plots in Figure 2 is a composite of three variables.
If any two of the three variables are known, the other can be calculated from the measured slope.
If dn/dc and the refractive index of the solvent are known, then the calibration constant is calculated from equation 7. This can be applied to the polystyrene data in Figure 2, with dn/dc taken to be 0.185 and refractive index taken to be 1.405. The value of RI.Cal calculates to be 3.63 x 106. Using this value for RI.Cal, the dn/dc of polybutadiene calculates to be 0.137, which is a bit higher than the literature values, which range from 0.125 to 0.130.
Chromatographic Method
The batch method is hardly convenient for calibrating or using a chromatographic detector to calculate dn/dc. Columns must be removed and solvent and solutions must be injected by hand, a laborious and tedious procedure. In addition, there is the atmospheric contamination problem, which can lead to a great deal of error if special care is not taken. It is much more convenient to inject the samples in the ordinary way with an auto-injector. Computer data acquisition and processing then permit the calibration and calculations with a high degree of precision. It is necessary to reformulate the equations previously developed for this purpose. Equation 5 can be restated as follows.
Here RIi is the RI signal and Ci is the concentration at data interval i. The data intervals will be evenly spaced in time, ∆t, and therefore in elution volume data interval, ∆V.
Here Q is the flow rate in ml/min, ∆t is in seconds, and the data interval is therefore in ml.
Suppose a sample of concentration Conc.S and volume V.inj is injected into the chromatograph. The RI response will follow equation 8 and is shown experimentally in Figure 3.
Note that the atmospheric contaminants peak is resolved from the polymer. The area of the RI signal peak is computed as follows.
Substitution from equation 8 shows that the measured area of the RI peak is proportional to the area of the (derived) concentration peak.
Equation 11 is derived with the assumption of conservation of mass, that is, the entire sample injected is recovered under the RI peak. Furthermore, the concentration peak area must be equal to the total mass injected.
Insertion of equation 12 into equation 11 yields the following general expression for RI detector peaks.
Now one can use chromatographic peak areas to calibrate and calculate dn/dc or sample concentration. The same polymer solutions used in the batch analysis of Figure 2 were injected onto a chromatographic column. The RI peak areas were measured and plotted against sample concentration in Figure 4.
The slope of the plots is related to the calibration constant, dn/dc, etc. according to the derivative of equation 13.
Computing the calibration constant from the polystyrene standard we obtain the value of RI.Cal to be 3.60 x 106, which is very close to that obtained by the batch method. The dn/dc of the polybutadiene sample calculates to be 0.129, which is closer to the expected range. The lines extrapolate very close to the origin because the atmospheric contaminants are removed from the peak. And the precision of data fit is evidently better because this variable is removed. The batch method of calibration is still championed by some. However, it is implicit in the present context that the sample of interest will be chromatographed whether the standard is or not, so it is impossible to avoid introducing the injection volume and flow rate variables. It is therefore better to introduce them in the calibration process, and in the dn/dc measurement process, so that any potential errors in these variables will tend to cancel out of the calculations. As far as the conservation of mass question is concerned, if one is concerned about absorption on the columns, the peak calibration method can be performed with columns removed.
The atmospheric contamination will still interfere, as in the batch analysis, so the precision will be poorer than with column chromatography. Nevertheless, it provides a way of testing whether the standard is being absorbed.
In summary, one should calibrate the RI detector by the batch method if it is to be used in the batch mode, but by the chromatographic method if it is to be used in the chromatographic mode.
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