Ultrahigh Pressure Diffusion Coefficient Determination for

The diffusion coefficient for hydroquinone in 50/50 acetonitrile/water with 0.1% TFA was determined from roughly 100 bar to 2000 bar using the CTOF instrument. Data was collected and evaluated as described in Section 4.4.1. Three data sets for hydroquinone are shown in Figure 4-10. Either two or three diffusion/viscosity determinations were performed at each desired pressure in each of the three data sets. All told, approximately 60 independent diffusion/viscosity determinations are plotted in Figure 4-10 for hydroquionone. The three data sets overlap reasonably well; there is very little scatter at low pressure and the scatter in the calculated diffusion coefficient values increase slightly as pressure increases. The relative standard deviation of the data set was determined at three different pressures; RSD was determined to be 1%, 3% and 5% at 200, 1000, and 1900 bar, respectively. This is likely due to the fact that as pressure increases it is more difficult to maintain the pressure exactly constant at the inlet and outlet of the capillary during the migration.

The average capillary diameter was calibrated using known viscosity data, as discussed in Section 4.4.2. In that case the capillary diameter of interest was the overall average capillary diameter, which along with the pressure determines the flow rate through the capillary. A fine point arises here, which is that the diameter of interest for the diffusion coefficient determination is not the overall average diameter, but only the average diameter

between the two detectors. This is because the variance accumulated between the detectors is dependent only on the local capillary diameter in that region, whereas the flow rate between the two detectors is a function of the average diameter over the entire capillary length. It is not uncommon for fused silica capillaries to vary in diameter by a micron or two along their length. It was therefore desirable to independently calibrate the 1.5-meter section of capillary between the two detectors for diffusion coefficient measurements.

The diffusion coefficient for hydroquinone in 50/50 acetonitrile/water at atmospheric

pressure was previously determined by a stopped-flow method to be 9.1x10-6 cm2 sec-1.4 As

a first approximation, a linear regression was performed to the three combined data sets shown in Figure 4-10 (linear regression not shown). The intercept of this linear regression

was then set to equal 9.1x10-6 by adjusting the capillary diameter to 50.0 µm. This capillary

diameter was then used as the “migration region” diameter for all further diffusion coefficient measurements.

As discussed in Section 4.2, elevated pressure is expected to shift the molecular diffusion coefficient based on the change in solution viscosity. The Stokes-Einstein

relationship (Equation 4-13) predicts an inverse relationship between D and η. To compare

the diffusion coefficient data obtained for hydroquinone from the CTOF instrument to the expected change in diffusion coefficient simply due to an increase in viscosity, the viscosity- predicted diffusion trend from Table 4-2 and Figure 4-2 was replotted as a dashed line on Figure 4-10. It was immediately obvious that the change in viscosity seemed to provide a very close estimation of the change in diffusion coefficient with pressure for hydroquinone.

The diffusion coefficient measurements from all three data sets were then collectively fit to the following equation, which allowed for nonlinearity due to a change in solution viscosity: P D α η δ + = 0 (4-18)

where δ is an adjustable fit parameter, P is the pressure in bar, and η0 and α are the viscosity

parameters for 50/50 acetonitrile/water from Table 3-2 (0.0081 Poise and 1.95x10-6 Poise

a prediction based on solution viscosity is a slight offset. It is possible that this offset is

simply due to a slight miscalibration (~0.5 µm) of the capillary diameter between the

detectors.

From the diffusion coefficient data obtained for hydroquinone in 50/50 acetonitrile/water using the CTOF instrument, it appears that the change in viscosity with pressure provides a very good estimation of the change in diffusion coefficient from atmospheric pressure to 2000 bar. Measurements from the CTOF instrument suggest a roughly 30% decrease in diffusion coefficient from 1 to 2000 bar. This is approximately

double the change that Mellors observed utilizing his stopped-flow technique.4 _{Figure 4-11}

shows the best-fit line from the CTOF diffusion data and the viscosity prediction from Figure 4-10, along with a comparison to Mellors data for hydroquinone at 1 bar, 1000 bar, and 2000 bar. Although Mellors’ data had a smaller standard deviation between measurements than our method, the data from the CTOF instument seems to agree much more strongly with the prediction based on viscosity. Mellors suggested that although solution viscosity increases with pressure, the diffusion coefficient of hydroquinone may decrease less than expected

because of the disruption of hydrogen bonds due to the applied pressure.5 However there is

no direct evidence to support such an effect in 50/50 acetonitrile/water. The discrepancy between diffusion coefficient data reported from the stopped-flow experiments and that observed using our CTOF instrument prompted further critical evaluation of the two methods to find any experimental flaws which might lead to the difference.

The discrepancy between ultrahigh-pressure D data obtained by the stopped-flow

method and the CTOF instrument might be explained by a simple thought experiment, which explains how the ultrahigh pressure diffusion coefficients from the stopped-flow method may

be artificially elevated over the actual D value. The argument for the error in the stopped- flow method centers on compression of the solvent at elevated pressures.

In the stopped-flow method of measuring D at elevated pressures, recall that although

the variance was allowed to accumulate at elevated pressure, the actual variance measurement was at atmospheric pressure. For example, to measure the diffusion coefficient of hydroquinone at 1000 bar, an analyte front was migrated into the center of the column using gravity. Both ends of the capillary were then pressurized to 1000 bar, and diffusion was allowed to occur for the predetermined time (typically 24 hours). The pressure was then released from the capillary, and the variance was measured by migration of the diffusion- broadened front through the detector.

This experiment would perform without error for an incompressible solvent. However, for a solvent that compresses under pressure an unexpected (and heretofore unrealized) error is possible. For a solvent that compresses 10% under the aforementioned 1000 bar of pressure, the same 10% expansion will occur upon release of the pressure. It would therefore be expected for this case that any band (or front) in the capillary during the pressure release will experience the same 10% expansion. This means that whatever variance accumulated in the pause period will then be expanded by an amount directly proportional to the compressibility of the solvent, prior to measurement. It was postulated that the measured diffusion coefficient using the stopped-flow method would therefore be elevated by an amount directly proportional to the solvent compressibility at the pressure of the experiment.

To test this theory in a hypothetical sense, a calculation was carried out that incorporated the additional variance that would be generated due to expansion of the band

during decompression of the solution in the capillary. Assume an actual diffusion coefficient

of 8x10-6 cm2 sec-1 and a solution compressibility of 10% at 1000 bar. Assume an infinitely

narrow initial front was compressed 10% and then allowed to diffuse for 24 hours. After this

pause the spatial variance generated from diffusion would be 1.38 cm2 (calculated using

equation (4-3)). The 4σwidth while still under pressure was therefore 4.703 cm. If the

pressure was then released from the capillary containing the front, an expansion of 10%

would occur resulting in a 4σwidth of 5.226 sec. The measured variance at the detector

would therefore be 1.707 cm2 and D would be calculated to be 9.87x10-6 cm2 sec-1. This

evaluation therefore predicts that a solution compressibility of 10% would lead to 23% error

in the measured D value from Mellors’ stopped-flow ultrahigh pressure diffusion experiment.

A specific evaluation of the difference between the stopped-flow D measurements

and the CTOF measurements was carried out using the above hypothesis. For hydroquinone

in 50/50 ACN/water at 1000 bar, the CTOF measurement gave D = 7.50x10-6 cm2 sec-1 while

the stopped-flow method gave D = 8.4x10-6 cm2 sec-1 (see Figure 4-11). Performing an

evaluation as described above, solvent compression of 5.5% could result in this 12% error.

At 2000 bar, the reported D values were 6.28 x10-6 cm2 sec-1 and 7.5 x10-6 cm2 sec-1 for

CTOF and stopped-flow methods, respectively. Solvent compression of 8.5% could lead to this 19% error.

From density data previously reported for mixtures of acetonitrile/water at 30°C and up to 2000 bar, the percent compression of this mobile phase as a function of pressure was

calculated.15 Figure 4-12 is a plot of percent reduction in volume versus pressure and

composition. This plot reveals that for 50/50 (v/v) acetonitrile/water, the mobile phase is compressed approximately 5% and 8% at 1000 bar and 2000 bar, respectively. These

quantities agree very closely with the compression values calculated in the thought experiment outlined above (5.5% and 8.5%). Noting this agreement, it is highly likely that the error in the stopped-flow diffusion measurement method is due to compression and expansion of the mobile phase.

Citing the newly-recognized error in the stopped-flow D measurement at ultrahigh

pressure and the consistency of the CTOF data with that predicted by viscosity, it is now believed that the CTOF instrument provides a more reliable method of measuring diffusion at ultrahigh pressures. The stopped-flow method still provides the most accurate way to determine diffusion coefficients at atmospheric pressure. In addition, it appears that a

reasonably accurate estimation of D at ultrahigh pressures (<5% error) can be made by

adjusting the atmospheric pressure diffusion coefficient based on viscosity data at ultrahigh pressures with the Stokes-Einstein relationship (Equation (4-13)).