Anal. Chem. 2005, 77, 6292-6299
Linear Velocity Surge Caused by Mobile-Phase Compression as a Source of Band Broadening in Isocratic Ultrahigh-Pressure Liquid Chromatography Anton D. Jerkovich, J. Scott Mellors, J. Will Thompson, and James W. Jorgenson*
Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3290
A linear velocity surge caused by mobile-phase compression was investigated as a source of band broadening for ultrahigh-pressure liquid chromatography (UHPLC). To measure the effect of compression on mobile-phase velocity, ionic sample fronts were monitored using a contactless conductivity detector, as they migrated through long packed capillaries. Mobile-phase compression was found to cause an abnormally high linear velocity surge as the pressure was applied to the inlet of the column. An empirical equation was developed to describe the mobilephase flow velocity during and after compression. Data fits to this equation were then used to determine the amount of additional variance caused by mobile-phase compression. For a 10/90 v/v acetonitrile/water mobile phase, the velocity surge occurred over roughly 10-15% of the column length. In addition, the velocity surge caused by mobile-phase compression was found to be capable of causing a 50% increase in the measured van Deemter C-terms for reversed-phase UHPLC.
coefficient for van Deemter curves fit to the UHPLC data.1,2,7 The van Deemter equation, which relates column performance to mobile-phase linear velocity, expressed in reduced parameters, is as follows:
h ) A + B/v + Cv
where h is the reduced plate height, v is reduced linear velocity, and A, B, and C are empirical constants that describe components of the experimental curve corresponding to eddy diffusion, longitudinal diffusion, and resistance to mass transfer, respectively.8 Commonly reported values are roughly 1 for A and 2 for B. Common C-term values for “well-packed” columns vary from 0.1 to ∼0.2.9 Because of the possibility of coupling between the A- and C-terms in the van Deemter equation, the Knox equation has become a popular alternative fit equation for chromatographic data.
h ) Avn + B/v + Cv Ultrahigh-pressure liquid chromatography (UHPLC) is one method to generate very high efficiency reversed-phase separations (>500 000 plates/m) on a time scale shorter than conventional HPLC. UHPLC requires liquid mobile phase to be forced through capillary columns packed with very small particles (e1.5 µm) using extremely high pressures (up to 6800 bar).1-6 The effects of these pressures on chromatographic performance, although they have been investigated, have not been fully explored. A concern with the observed performance of UHPLC has been the deterioration of chromatographic efficiency at high linear velocities. This manifests itself as a higher than expected C-term * To whom correspondence should be addressed. E-mail:
[email protected]. (1) MacNair, J. E.; Lewis, K. C.; Jorgenson, J. W. Anal. Chem. 1997, 69, 983989. (2) MacNair, J. E.; Patel, K. D.; Jorgenson, J. W. Anal. Chem. 1999, 71, 700708. (3) Wu, N.; Lippert, J. A.; Lee, M. L. J. Chromatogr., A 2001, 911, 1-12. (4) Xiang, Y.; Maynes, D. R.; Lee, M. L. J. Chromatogr., A 2003, 991, 189196. (5) Mellors, J. S.; Jorgenson, J. W. Anal. Chem. 2004, 76, 5441. (6) Patel, K. D.; Jerkovich, A. D.; Link, J. C.; Jorgenson, J. W. Anal. Chem. 2004, 76, 5777-5786.
6292 Analytical Chemistry, Vol. 77, No. 19, October 1, 2005
(1)
(2)
In this equation, n typically has a value of 0.33.10 Over the practical range of reduced linear velocities for UHPLC (v e 30), however, there is virtually no difference in the goodness of fit between the van Deemter and Knox equation. Therefore, we favor the van Deemter equation for its simplicity and because all the terms are more easily related to physical phenomena. In a 2002 publication on band broadening, Knox recorded the Knox equation parameters for a variety of reversed-phase columns.7 To compare our van Deemter C-terms to the Knox C-terms from this paper, the data from the Knox paper were refit using the van Deemter equation for values of v less than 30. The resultant van Deemter C-terms ranged from 0.02 to 0.26. Experimental van Deemter C-terms for UHPLC performed in our laboratory are typically in the range of 0.25-0.35 for 10-30µm-i.d. columns.1,2,6 Originally, the larger-than-expected C-terms were suspected to be a result of poor packing in the capillary (7) Knox, J. H. J. Chromatogr., A 2002, 960, 7-18. (8) Giddings, J. C. Unified Separation Science; Wiley: New York, 1991. (9) Neue, U. D. HPLC Columns: Theory, Technology, and Practice; Wiley-VCH: New York, 1997. (10) Bristow, P.; Knox, J. H. Chromatographia 1977, 10, 279. 10.1021/ac0504924 CCC: $30.25
© 2005 American Chemical Society Published on Web 08/23/2005
columns, and this possibility was re-emphasized more recently by Knox.2,7 However, the overall high efficiency of the columns (hmin e 2), gives pause to the idea that the columns are poorly packed.1,2,5,11 Recently, a study examined the effect of column diameter on van Deemter parameters in UHPLC.6 Columns with 150-µm i.d., packed with 1-µm nonporous silica gave C-terms of 0.5, while 10-µm-i.d. columns gave C-terms of 0.25. Retention factor data and SEM images indicated that the larger columns were more densely packed in the wall region, an effect that was reduced in columns below 30 µm.6 Aside from decreasing column diameter, attempts in varying packing parameters have failed to significantly reduce C-terms. Therefore, causes of abnormally large C-terms other than poor packing are currently under investigation. Mobile-phase compressibility has historically been taken into account in gas chromatography (GC), because gases are readily compressible under low pressures. Measured flow rates at the outlet of a GC column must be corrected to give the average flow rate in a GC column because the gas expansion toward the column outlet causes an increase in observed linear velocity. Similarly to HPLC, injections in GC are performed by introducing the analyte into a compressed stream of mobile phase, so there is no mobilephase compression event that occurs at the beginning of a run, only expansion near the outlet. Thinking in a similar vein to GC, Martin et al. examined the theoretical effects of mobile-phase expansion in liquid chromatography at very high pressures (up to 1500 bar). Only small increases in velocity were predicted near the column outlet for aqueous mobile phases, because of the much lower compressibility compared to gases.12 This is especially true for LC under typical conditions of less than 350 bar, so HPLC theory was developed with the premise of a linear pressure drop and constant mobile-phase linear velocity (u) along the length of a chromatographic column.8 With the advent of UHPLC and liquid pressures approaching 7000 bar, this expansion may be more significant, although it will be gradual and continuous, as described by Martin et al.12 However, the focus of this paper is on a more significant compression event that occurs at the start of a run in UHPLC. The pressures used in UHPLC (up to 6800 bar) are high enough to significantly compress aqueous and organic mobile phases.1,2 In addition, the system examined in this paper is somewhat different from a typical HPLC or GC instrument in that the injection is performed near atmospheric pressure, followed by a quick pressurization (and thus compression) of the mobile phase to start the run. Therefore, the UHPLC systems described previously1-6 experience a dynamic compression event at the start of a run that decays within a few seconds, and that does not occur in typical HPLC. It is during this pressurization that the solvent front (and the injected band) will very quickly travel a distance into the column that is proportional to the compressibility of the entire mobile phase contained in the column. Therefore, the mobile phase near the column inlet would be flowing significantly faster than the terminal linear velocity after compression. We have chosen to call this event a velocity “surge” due to mobile-phase compression. It should be stated that a systematic study of mobile-phase compressibility, or change in partial volume with pressure, is not (11) Jerkovich, A. D. University of North Carolina at Chapel Hill, Chapel Hill, NC, 2003. (12) Martin, M.; Blu, G.; Guichon, G. J. Chromatogr. Sci. 1973, 11, 641-654.
the intention of this paper. It is known that liquid mobile phases exhibit significant compression at the pressures used in ultrahighpressure liquid chromatography; the purpose of the studies described herein was to examine the effects that the velocity surge caused by mobile-phase compression may have on velocitydependent (C-term) band broadening in UHPLC. As stated previously, the hypothesis was generated that if mobile-phase compression were occurring at the column inlet, it would cause a temporary mobile-phase velocity surge at the head of the column. During this short period of time, the analyte zone would experience mobile-phase velocities that are far above the optimum of the van Deemter curve. Thus, additional C-term, or mass transfer, variance could be accumulated during the surge. The higher the run pressure used, the more mobile-phase compression would occur and the more intense the linear velocity surge would be, leading to an observed C-term with a higher slope. A loop-style injector would circumvent this problem by holding the column at the desired run pressure during the injection, which would be analogous to a typical LC or GC injector. However, a suitable injector of this type has not been developed for use at ultrahigh pressures. To test the hypothesis, fronts of a dead-time marker were injected and their velocity was monitored as they migrated down the column. Water compresses roughly 20% of its volume at 6800 bar,13 so we assumed that our 10/90 ACN/water mobile phase would have similar compressibility. Furthermore, we reasoned the average pressure in the column would be half the applied run pressure. Therefore, we assumed that the sample front would reach its terminal velocity after migrating ∼10% of the column length. Therefore, long columns were packed so that compression could be observed over an appreciable distance. A 117.3-cm-long capillary packed with 1.0-µm particles and a 206-cm-long capillary packed with 1.5-µm particles were prepared for this purpose. We believe these are the longest packed capillary columns ever reported for each corresponding particle size. EXPERIMENTAL SECTION Column Preparation. The 30-µm-inner diameter fused-silica capillary columns (Polymicro Technologies, Inc., Phoenix, AZ) were slurry packed with spherical 1.0- or 1.5-µm C18-modified nonporous silica (Eichrom Inc., Darien, IL) using the procedure described previously.1,2 Briefly, slurries were made with 40 mg of particles in 4 mL of 33% acetone/67% hexane (v/v). Slurries were then placed in an ultrasonic bath for 10-30 min to break up particle aggregates and subsequently passed through a 5-µm polycarbonate track-etched filter (Millipore Corp., Bedford, MA) to remove contaminants and remaining particle aggregates. Capillaries (Polymicro Technologies) were prepared with outlet frits by sintering 3.5-µm spherical silica beads in place using an electric arc device.14 Columns were packed using a previously described custom-built packing bomb, and pressure was slowly ramped to 2800 bar during the packing procedure.1 After the columns were packed to the desired length, they were removed from the packing bomb and flushed with acetone, followed by 10% acetonitrile/90% water (v/v) plus 0.1% trifluoroacetic acid. The (13) CRC Handbook of Chemistry and Physics, 77 ed.; CRC Press: New York, 1996-1997. (14) Hoyt, A. M., Jr.; Beale, S. C.; Larmann, J. P., Jr.; Jorgenson, J. W. J. Microcolumn Sep. 1993, 5, 325-330.
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inlet of the bed was then pressurized to greater than 4800 bar to ensure that the bed was thoroughly compressed. Pressure was then allowed to decrease very slowly to ∼700 bar, at which point the inlet frit was made by heating the end of the packed bed with a resistive heating wire stripper (Teledyne Interconnect Devices, San Diego, CA).11 Isocratic Elution Apparatus. The experimental apparatus and procedures used to perform isocratic UHPLC have previously been described in detail.1,2 In brief, a pair of pneumatic amplifier pumps (a single-stage pump used to pressurize the inlet of a triplestage pump; Haskel, Inc., Burbank, CA) were used to generate the necessary liquid pressures. The triple-stage pump was able to generate pressures up to 6800 bar. The outlet of this pump was connected to an in-house-machined static split-flow injector.1 Columns were inserted into the central channel of the injector and held in place with a capillary compression fitting.6 Contactless Conductivity Detection. To observe mobilephase compression in UHPLC, a contactless conductivity detector was used to detect ionic sample fronts; this detector was based on published examples and built in our laboratory.15 This conductivity detector has the advantage of being able to detect analytes through the fused-silica capillary wall and polyimide coating, allowing for detection at virtually any point along the length of a capillary without disturbing the capillary in any way. Furthermore, the detector can be moved during the course of a run, so a migrating peak or front can be detected at multiple points along the column. The disadvantage of this detector is that it can only detect ionic analytes. The conductivity detector built in our laboratory consisted of two 5-mm-long cylindrical electrodes machined from 23-gauge 304 stainless steel (Small Parts, Miami Lakes, FL) placed in line with a 1-mm gap between them. The inner diameters of the electrodes were 390 µm, just large enough to allow a 360-µm-outer diameter capillary to slide through the electrodes. The contactless conductivity detector functions as follows: An ac excitation signal is applied to one electrode and is capacitively coupled through the capillary wall to the mobile phase within the capillary. The signal conducts through the mobile phase and then back through the capillary wall to the second electrode. The signal received at the second electrode is proportional to the ionic conductivity of the solution. This signal is then amplified by a current to a voltage preamplifier and a lock-in amplifier (model 391A, Ithaco, Ithaca, NY) set to the excitation frequency. The signal is then received by a BNC-2090 digital acquisition board (National Instruments Corp., Austin, TX) and monitored using a program written in LABVIEW 6 (National Instruments Corp., Austin, TX), on a Dell Dimension XPST700r personal computer (Dell Inc., Austin, TX). Mobile-Phase Compression Test Parameters. The mobile phase used to investigate the compression phenomenon was 10% acetonitrile/90% water (v/v), with 0.1% trifluoroacetic acid, spiked with 8 mM NaNO3 to provide background conductivity. The “sample” used to generate the fronts was identical, but contained 10 mM NaNO3 to provide a conductivity increase over the baseline. Fronts were introduced with the column full of mobile phase by filling the central channel of the injector with sample and then fully pressurizing the column. The pressurization from (15) Zemann, A. J.; Schnell, E.; Volgger, D.; Bonn, G. K. Anal. Chem. 1998, 70, 563-567.
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Figure 1. Conductivity signal recorded for the 4800 bar run performed on the 206-cm column packed with 1.5-µm particles. The rises in the signal occurred when the 10 mM NaNO3 front reached the detector. The drops in the signal occurred when the detector was placed ahead of the front. The inset shows the first 200 s of the run in greater detail. The first 13 detection points were located 2 cm apart, from 6 to 30 cm from the inlet. All of the detection points after that were every 10 cm from 30 to 200 cm from the inlet. Note that the values on the y-axis correspond to a voltage output of the conductivity detector and is not an absolute measure of solution conductivity.
the pneumatic amplifier occurs very quickly, on the order of 1-2 s, after which the pressure is constant for the duration of the run. At the beginning of the run, the conductivity detector was placed as close to the inlet of the capillary as possible. Because a portion of the capillary must be held inside the fitting and injector, the conductivity detector could not be positioned closer than 6 cm from the column inlet. Detection points were measured and marked on the polyimide using a permanent ink marker prior to insertion into the injector. After the front was detected at 6 cm from the inlet, the detector was manually moved ahead of the migrating front to the next premarked detection point, farther from the inlet. The front was then detected at this position, and the detector was moved again. The procedure was repeated at multiple positions down the entire length of the column, thus tracking the analyte front during its migration. The arrival times of the front at each position were found by differentiating the sigmoidal fronts and performing Gaussian analysis on the resulting peaks using Igor Pro (Wavemetrics, Inc., Oswego, OR). Either a Savitzy-Golay smooth or digital frequency filter was performed to remove noise from the signal caused by differentiation. The peak maximum was used to represent the location of the front. From these data it was possible to plot the migration of a mobile-phase front through the capillary in the form of position (distance from the inlet) versus time. RESULTS AND DISCUSSION Column Position versus Migration Time Data. Utilizing the conductivity detector in the manner described above yields a series of consecutively rising and falling fronts on the same signal trace, as seen in Figure 1. Each rise in signal is in response to the increased conductivity of the solution due to the sample front reaching the detector, increasing the solution concentration of
Figure 2. Distance versus time plots for the 6300 (b), 5600 (0), 4800 (2), 4100 (O), 3400 (9), and 2500 bar (]) runs on the 117cm-long column packed with 1.0-µm particles.
NaNO3 from 8 to 10 mM. The signal recorded is a voltage output of the conductivity detector only and not an absolute measure of conductivity. Each drop in signal is caused by moving the detector down the capillary ahead of the migrating front, where 8 mM NaNO3 mobile phase exists. The progress of migration of a sample through the column can be monitored by measuring the migration time of each rising front. Figure 2 shows the distance versus time plot for the 117.3cm-long column packed with 1.0-µm particles. Detection points for this column were every 2 cm from 6 to 20 cm and then every 10 cm from 20 to 110 cm, with the final point near the outlet at 116 cm. Six runs were performed at pressures from 2500 to 6300 bar. The expected linear rise of the curves during the latter part of the runs is present, representing a constant flow velocity. At the beginning of each run is a sharp, nonlinear portion originating from the origin. The velocity at the beginning of the rise is greatest and decays exponentially until it is equal to the constant linear velocity observed in the latter portion of the column. As expected, the initial surge in velocity was most pronounced at 6300 bar and was less severe at lower pressures. Results from the 206-cm column packed with 1.5-µm particles are shown in Figure 3. For this column, detection was performed every 2 cm from 6 to 30 cm and every 10 cm from 30 to 200 cm. A greater amount of compression was observed on the 206-cm column, because the column volume is larger and a greater percentage of the column was within “reach” of the conductivity detector. The run pressures ranged from 2600 to 6300 bar, with the greatest compression observed for the highest pressure run. A significant increase in velocity at the outlet of the column, as is the case for GC, was not observed. In GC, the gaseous mobile phase expands as it enters a region of lower pressure. The expansion produces an increase in the velocity of the mobile phase toward the outlet of the column. A slight increase in velocity along the column, as predicted by Martin et al., cannot be ruled out in Figures 2 and 3, although it was not significant enough to warrant inclusion in our data analysis. It is our conclusion that an increase in velocity due to mobile-phase expansion is not readily observed in our case because such an effect is small and occurs gradually as the pressure drops linearly along the length of the column. The effect would likely be more noticeable if a mobile phase with higher compressibility, such as acetonitrile, were used.
Figure 3. Distance versus time plots for the 6300 (b), 5600 (0), 4800 (2), 4100 (O), 3400 (9), and 2600 bar (]) runs performed on the 206-cm-long column packed with 1.5-µm particles. The inset shows the first 30 cm in greater detail.
Fitting of Compression Data. Because of the unique trend of the distance versus time data, a function had to be developed that would accurately represent the data. All points for each run in Figures 2 and 3 were fit to the following equation:
x(t) ) A(1-e-t/τ) + ut
(3)
where x(t) is the position of the front within the column in centimeters as a function of time (t) in seconds. The equation contains two parts, an exponential portion to describe the beginning of the run when the velocity surge occurs and a linear portion to describe the remainder of the run when the linear velocity is constant. τ is the time constant in seconds for the exponentially decaying velocity during compression. The distance coefficient A has units of centimeters and is proportional to the migration distance over which the velocity “surge” occurs due to mobilephase compression. It is equal to the y-intercept of the linear portion of the curve when extrapolated back to t ) 0 s. u represents the slope of the linear part of the curve or the terminal linear velocity (cm/s) after compression is complete. Equation 3 is empirically derived and is not rigorously correct. The surge of velocity may not necessarily exhibit a true exponential decay, given the complexity of the compression phenomenon. Furthermore, the column is not a closed system; while mobile phase is being compressed at the inlet, there is flow exiting the column outlet. Also, flow resistance of the packed bed affects the rate of compression. Finally, a pressure drop exists down the length of the column. This means the flow rate, viscosity, and compressibility of the mobile phase are different at every point along the column. Despite this complexity, the equation is accurate for the limits of very long time (t . τ, x(t) ) A + ut) and for t ) 0 (x(t) ) 0). In fact, the lines through the data points in Figures 2 and 3 are fits to eq 3. The fits are very good for all pressures and both columns, and since the coefficients of the fit each have a straightforward physical significance, they can be used to gain a clearer understanding of mobile-phase compression and how it affects band broadening. Results of Curve Fitting. The results of the curve fits for each run on the two columns analyzed are presented in Table 1. Values Analytical Chemistry, Vol. 77, No. 19, October 1, 2005
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Table 1. Results of Curve Fitting of Position versus Time Data for Both Columnsa length (cm)
pressure (bar)
tM (s)
A (cm)
τ (s)
u (cm/s)
A/τ + u (cm/s)
5τ (s)
5τ/tM (%)
n AA (cm)
nAA/L (%)
117.3
2500 3400 4100 4800 5600 6300 2600 3400 4100 4800 5600 6300
2299.8 1698.2 1436.1 1257.3 1086.4 980.5 3941.3 2914.9 2496.7 2166.6 1925.8 1703.4
6.2 ( 0.1 7.4 ( 0.1 8.3 ( 0.1 8.8 ( 0.2 9.5 ( 0.2 10.0 ( 0.2 7.8 ( 0.1 12.0 ( 0.2 13.4 ( 0.3 14.7 ( 0.3 15.7 ( 0.4 16.7 ( 0.4
32 ( 2 25 ( 1 22 ( 1 20 ( 1 18 ( 1 16 ( 1 32 ( 3 31 ( 2 29 ( 2 27 ( 2 25 ( 2 24 ( 2
0.048 ( 7 × 10-5 0.065 ( 9 × 10-5 0.075 ( 0.0001 0.086 ( 0.0002 0.099 ( 0.0003 0.109 ( 0.0004 0.050 ( 7 × 10-5 0.067 ( 0.0001 0.077 ( 0.0002 0.088 ( 0.0003 0.099 ( 0.0003 0.111 ( 0.0004
0.239 0.365 0.455 0.535 0.634 0.722 0.295 0.457 0.534 0.631 0.728 0.821
162.3 122.8 109.6 98.4 89.2 81.7 158.5 153.6 146.1 135.4 124.7 117.9
7.1 7.2 7.6 7.8 8.2 8.3 4.0 5.3 5.9 6.3 6.5 6.9
14.0 15.3 16.6 17.3 18.3 18.9 15.7 22.2 24.5 26.6 27.9 29.7
11.9 13.0 14.1 14.7 15.6 16.1 7.6 10.8 11.9 12.9 13.6 14.4
206
a
The curve-fitting parameters are listed ( one standard deviation. See text for detailed explanation of terms.
plus or minus one standard deviation are presented for the distance coefficient (A), time constant (τ), and linear velocity (u), fit to the data using eq 3. As pressure is increased and compression becomes greater, A is observed to increase, meaning that compression occurs over a larger portion of the column. τ is observed to decrease, indicating that the mobile-phase velocity decreases toward the equilibrium velocity at a faster rate. For practical purposes, the compression and velocity surge were considered to be complete after a time equal to five time constants (5τ). After this time, the exponential decay of the velocity surge is more than 99% complete (1 - exp(-5τ/τ) ) 0.993), and the mobile-phase velocity levels off at a value equal to u. The percentage of the dead time (tm) over which compression occurs (5τ/tm) is listed in Table 1. For a given column, 5τ/tm increases with increasing pressure, indicating that despite the decrease in time constant with pressure, the compression takes up a larger portion of the overall migration time. To calculate the percentage of total column length over which compression occurs, the distance along the column (x(t)) that experiences a velocity surge can be calculated by substituting 5τ for t in eq 3:
x(t) ) A(1 - e-5τ/τ) + u(5τ)
(4)
Simplifying the exponential term, we obtain
x(t) ) 0.993A + 5τu
(5)
Using the coefficients from each fit yields values from 1.78 to 2.26, with an average nA value of 1.93 (see Table 1). This agrees well with Figures 2 and 3, in which the compression appears to be finished after a distance equal to about twice the distance coefficient (2A). The percentage of the total column length (L) over which compression occurs can therefore be calculated as (nAA/L), and values are listed in Table 1. It was observed that the velocity surge occurs over a larger percentage of the column length at higher pressures, as expected. While the absolute compression distance is greater for the longer column, the relative values are in about the same range for both columns. This is expected since the mobile phase will compress a certain percentage of its volume at any given pressure, and assuming a column is packed uniformly along its length, the volume of mobile phase within the column should correlate with the column length. It is important to note that these percentages are not equivalent to the compressibility of the solvent mixture at any given applied pressure. This is, again, because the solvent’s compressibility is actually different at every point along the column’s length due to the pressure drop. Effect of Compression on the C-Term. The velocity surge due to compression at the beginning of a run contributes considerably to the spatial variance of a peak. With the distance versus time data acquired using the conductivity detector, this broadening contribution can be quantified. The instantaneous mobile-phase velocity (u′(t)) at any time during the run can be calculated by differentiating eq 3:
u′(t) ) (dx/dt) ) (A/τ)e-t/τ + u Compression therefore pushes an unretained sample zone into the column a distance of 0.993A farther than the distance it would have migrated due to normal flow during the time equal to 5τ. The total distance traveled by an unretained sample zone during this time can be described as a multiple of distance coefficient (A) values, nAA, corresponding to a compression time of 5τ: Substituting nAA for x(t) in eq 5 yields
nAA ) 0.993A + 5τu
(6)
nA ) 0.993 + (5uτ/A)
(7)
Solving for nA:
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(8)
Solving this equation for t ) 0 yields the linear velocity at the start of the run when the surge of velocity is greatest.
u′(0) ) (A/τ) + u
(9)
The mobile-phase linear velocity at the initial application of pressure is about 5-7 times higher than the terminal velocity (u) for any given run, as can be seen in Figures 4 and 5. Initial velocity values for each run are listed in Table 1 as (A/τ) + u. Since the C-term is proportional to mobile-phase velocity, and we are quantifying only the velocity contribution to the C-term (all other things constant), the velocity can be used as a relative
cumbersome expression that is very difficult to integrate. Therefore, an approximation approach was taken that gives a more practical, albeit approximate, solution. Instead of integrating over an infinite number of infinitesimally narrow slices of column length, a summation of a finite number of slices of column length was taken to approximate spatial variance (σL2): x)L
σL2 ∝
dx
∑ dt ∆x
(14)
0
Figure 4. Velocity (A) and integration of velocity (B) versus distance from column inlet for the 2500 (‚‚‚‚), 4100 (- - -), and 6300 bar (s) runs performed on the 117-cm-long column packed with 1.0-µm particles.
variance. Substituting the velocity expression of eq 8 for u in the van Deemter C-term (HC ) Cu) yields the following expression for plate height:
HC ) Cu′(t) ) Cu + C(A/τ)e-t/τ
(10)
It can be seen from this equation that, in addition to the normal C-term (Cu), there is a contribution to broadening solely due to the velocity surge. In the compression region, the velocity is changing with time, so eq 10 describes the instantaneous plate height contribution at time t. Since variances are additive, the total C-term plate height would be the accumulation of all the instantaneous variance contributions throughout the column. The total variance is therefore proportional to the integral of the velocity expression. As plate heights are defined in terms of spatial variance, the integration is performed over the length of the column, L:
σL2 ) LCu′(t) σL2 ) C
∫
x)L
u′(t) dx
0
σL2 ) LCu + C
∫
x)L
0
A -t/τ dx e τ
(11) (12) (13)
The second term on the right-hand side of eq 13, however, is expressed not in terms of space, but in time. Obtaining this expression in spatial coordinates is tedious and leads to a
With the use of a spreadsheet, position (x(t), from eq 3) and velocity (dx/dt) values were calculated for time values ranging from 0 s through the end of each run, at intervals of 0.1 s. While the slices of time are uniform in size, the slices of column length are not; however, the intervals in length are small enough (∼1 mm) to accurately approximate the integral. The velocity values were then plotted versus their corresponding position in the column, x(t), and this line was used to determine the velocity at exact 1-mm position intervals using an interpolation. Plots of dx/ dt versus x(t) are shown in Figures 4A and 5A. The velocity surge is most evident in these figures. At the beginning of each run, the velocity starts at its initial value, listed in Table 1, and decays exponentially until it levels off at the terminal velocity u. If the velocity were constant throughout the entire run, the velocity curves in Figures 4A and 5A would simply be lines of zero slope, equal to a value of u, and flow-induced broadening (proportional to the integration) would accumulate linearly as the sample zone migrated down the column. Figures 4A and 5A approximate the velocity versus column length, which is proportional to the local variance contribution. Equation 14 was then used to sum the velocity values over a desired length interval, therefore generating curves that are proportional to the total accumulated spatial variance at any position along the column. Examples of this are shown in Figures 4B and 5B. Looking at these figures, it becomes obvious that a disproportionate amount of variance is acquired toward the inlet of the column due to the velocity surge. The information from these plots can be used to more clearly measure the effect compression has on band broadening. Quantification of Compression Effect on the C-Term. The van Deemter C-term coefficient is equal to the slope of the highvelocity portion of a van Deemter curve. In this experiment, we did not generate van Deemter curves by measuring plate heights and plotting those values versus linear velocity. Instead we attempted to determine what effect the linear velocity surge, caused by mobile-phase compression, would have on chromatographic performance. Therefore, we plotted the area under the velocity versus distance curves (from Figure 5A) versus the u value (the terminal linear velocity obtained after compression) for each run (Figure 6). The slope of the line fit to these points is proportional to the C-term that would be calculated for these runs. The compression effect was then removed by taking the area under a line of constant velocity, u (uL). These values were also plotted versus the u value for each run to illustrate what the C-term would be if no mobile-phase compression had occurred. The steeper slope of the compression C-term is evident in Figure 6. It is interesting to note that while the constant velocity Analytical Chemistry, Vol. 77, No. 19, October 1, 2005
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Figure 5. Velocity (A) and integration of velocity (B) versus distance from column inlet for the 2600 (‚‚‚‚), 4100 (- - -), and 6300 bar (s) runs performed on the 206-cm-long column packed with 1.5-µm particles.
Figure 6. Relative variance versus linear velocity plots with compression (O) and without compression (9) for the 206-cm-column packed with 1.5-µm particles. These plots illustrate what the C-term dominated portion of the van Deemter curves would look like for each case.
curve intersects the origin, the compression curve has a negative y-intercept. This is because the mobile phase does not compress nearly as much at pressures corresponding to the slower velocities in these plots and thus contributes much less, if any, additional variance. If data were obtained at the lower velocities, it is expected that the points from the compression curve would simply join those of the constant velocity curve and intersect the origin as well. From these plots it was found that the surge of velocity due to mobile-phase compression caused the C-term to increase by 46% for the 117.3-cm, 1.0-µm particle column and by 51% for the 206-cm, 1.5-µm particle column. Figure 7 illustrates the effect that the increased C-terms would have on a van Deemter curve. This figure includes a reduced parameter van Deemter curve with coefficients typically reported for well-packed columns (A ) 1, B ) 2, C ) 0.15), as well as an additional curve that has the same A- and B-terms but has a C-term that is inflated by 50%. From this figure it can be seen that, at ultrahigh pressures, mobile-phase compression will have a significant effect on the chromatographic performance of capillary columns. For example, if a hypothetical column operated without mobile-phase compression exhibits a C-term of 0.2, compression at ultrahigh pressures would make this column appear to have a C-term of 0.3. Practical Considerations for Mobile-Phase Compression in UHPLC. It is important to note that experiments reported here were performed under isocratic conditions with an unretained analyte. For a retained analyte, the expected increase in variance 6298 Analytical Chemistry, Vol. 77, No. 19, October 1, 2005
Figure 7. Hypothetical reduced parameter van Deemter curves illustrating an ideal curve without compression (s), where the coefficients were set at A ) 1, B ) 2, and C ) 0.15; and a curve with the C-term coefficient inflated by 50% to 0.225 (- - -).
may be much less, as a function of its retention factor. An analyte with a k′ of 5 would spend only 1/5 of its time in the mobile phase and, therefore, would only migrate 1/5 the distance during the time under which the linear velocity surge is occurring. Thus. the encountered variance due to the velocity surge is lower for a highly retained peak. For gradient liquid chromatography, where analytes are strongly sorbed onto the column head under the initial conditions of a run, mobile-phase compression would not cause band broadening. While the mobile phase compresses, these analytes remain completely sorbed and therefore experience no elevated linear velocity. If one were interested in performing isocratic UHPLC and avoiding the compression effect, it may be possible to decrease the linear velocity “surge” by ramping the inlet pressure so that the linear velocity is constant while mobile-phase compression is occurring. Alternately, a conventional loop-type injection valve that allows the column to be maintained at run pressure during the injection could be developed for UHPLC, although this would require significant advances in injector design. CONCLUSIONS We have shown with the experiments reported herein that a surge in linear velocity caused by mobile-phase compression at the start of isocratic UHPLC runs can cause C-terms of experimentally determined van Deemter curves to be ∼50% above the values commonly reported for standard-bore columns. Therefore, mobile-phase compression in UHPLC could justify a C-term increase from 0.1 to 0.15 or from 0.2 to 0.3. Although mobile-
phase compression cannot completely account for the discrepancy in our typical C-term data from that of traditional “well-packed” columns, it can make up a sizable portion of the difference. It is our suspicion that a number of small factors each play a part in making the C-terms for capillary UHPLC larger than those of typical HPLC.11 Packing structure characteristics that lead to larger C-terms cannot be dismissed. For example, the nonporous packings used in our experiments are much more monodisperse in size than normal porous packings used in HPLC columns. For capillary-toparticle diameter ratios greater than 50, we have observed highly ordered packing structure along the capillary walls and much more random packing in the center.6 A column that had a densely packed structure along the walls would certainly exhibit a higher flow resistance along the wall than in the center region. It is not clear whether such a packing structure would produce increased variance due to A-term or C-term type phenomena; however, it is fairly certain that uneven flow resistance across a packed column would cause additional band broadening.
Capillaries packed with particles in the 1.0-1.5-µm size range, even with the C-terms observed, exhibit excellent performance in the form of reduced plate heights of less than 2 at their optimum linear velocity.1,2,5,6 Further investigation into processes unique to capillary UHPLC are ongoing, and these investigations will hopefully shed new light on chromatographic theory. ACKNOWLEDGMENT The authors acknowledge Waters Corporation for graciously funding this project. We thank former and current group members Dr. Keith Fadgen (Waters Corp.), Dr. Luke Tolley (Southern Illinois University), Steve Johnston, and Dawn Stickle for the development of the conductivity detector used in this work. Finally, we acknowledge Dr. Kevin Lan (GlaxoSmithKline) for his thoughts and conversations concerning this work. Received for review March 23, 2005. Accepted July 17, 2005. AC0504924
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