Effect of density on kinetic performance in supercritical fluid chromatography with methanol modified carbon dioxide

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Journal of Chromatography A, xxx (2018) xxx–xxx

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Effect of density on kinetic performance in supercritical fluid chromatography with methanol modified carbon dioxide Terry A. Berger SFC Solutions, Inc, 9435 Downing St., Englewood, FL, 34224, USA

a r t i c l e

i n f o

Article history: Received 9 April 2018 Received in revised form 7 June 2018 Accepted 8 June 2018 Available online xxx Keywords: Supercritical fluid chromatography (SFC) Density of CO2 /MeOH Density vs. pressure Retention Efficiency Isopycnic operation

a b s t r a c t In a relatively recent reevaluation of the van Deemter Equation, Guiochon concluded that the mass transfer resistance in the mobile phase is independent of the retention factor. In the process he showed reduced plate heights ≈ 2 for a nearly unretained peak (k = 0.4) in high performance liquid chromatography (HPLC). In the present work, using supercritical fluid chromatography (SFC), efficiency was measured at various pressures, densities, and modifier concentrations. The highest efficiency, with a reduce plate height of hr = 1.63, was recorded with the lowest retention factor (k < 0.8). This is an extremely low hr for totally porous particles, at very low k, and appears to support Guiochon’s analysis. The density of methanol/carbon dioxide mixtures were calculated using the REFPROP program from the National Institute of Standards and Technology (NIST) over a wide range of pressures and % methanol. The density of higher methanol concentrations (>20%), commonly used in SFC, was found to be lower than the density of lower concentrations (<20%). At low methanol concentrations, density varies widely with pressure. However, at high methanol concentrations there is very little change in density, and very little change in retention with pressure. With increasing modifier concentration, density decreases, while viscosity increases (P increases). The pump and back pressure regulator (BPR) pressures are not necessarily good indicators of pressures or densities in the column. At high flow rates the extra-column pressure drop (P) can be much larger than the column P and can be unevenly distributed in front of and behind the column. In one extreme the P after the column was 3 times higher (105 bar) than the actual column P (32 bar). © 2018 Elsevier B.V. All rights reserved.

1. Introduction The relationship between density and retention in supercritical fluid chromatography (SFC) appears to be well established for pure CO2 , and to a lesser extent, with isocratic performance at low (such as 5%) modifier concentrations. Early workers used pressure programming of pure CO2 , to increase solvent strength. In 1982, Giddings [1] was the first to demonstrate linear density programming with pure CO2 , showing that retention was related to the density, not the pressure of the mobile phase. He was working with packed columns, and pure CO2 . This relation between retention and density had been anticipated by others but had not been demonstrated. He was the first to use relatively modern particles, but with home-made hardware. Capillary SFC, introduced in 1981, first employed pressure programming [2], but after the publications

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by Giddings, they adopted the use of CO2 , linear, and asymptotic density programming [3]. In 1982, Gere [4] demonstrated that the log of the partition ratios (retention factor) (log k) of several polycyclic aromatics, were non-linear functions of pressure, but linear functions of density, with pure CO2 on C18 packed columns using much more realistic, modern, commercial (3 ␮m) columns, and equipment, compared to previous reports. Gere also changed the outlet pressure on columns with different spherical particle sizes (10 ␮m to 3 ␮m) to maintain a constant average pressure in the columns, independent of particle size, in order to minimize differences in k due to differences in pressure drops (P). The intent was to try to keep k similar on the different particle sizes, to try to show similar resolutions at much shorted run times on the smaller particles, as is now typical with sub-2 ␮m particles. Gere’s minimum reduced plate heights (hr,min ) were 3 or below. Efficiencies were much higher than previously observed. Most of this work was done with pure CO2 . Pure CO2 is almost never used today, since it has a solvent strength similar to pentane (P’ ≈ 0), even at high pres-

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sures/densities. For polar solutes, methanol (P’ = 5.1)(water = 10.1) is usually the modifier of choice. SFC typically uses modifier concentrations between 5 and 65%. Such applications, with small drug-like solutes, and particularly chiral separations, have been the s¨ weet ¨ SFC. spotof The van Deemter Equation [5] is still the most widely used expression relating plate height to linear velocity, in both HPLC and, presumably, SFC. The original document did not actually present a clear version of the Equation, but explained the various aspects of diffusion as understood in the chemical engineering literature. It is generally written with 3 or 4 terms (A, B, C, D), each describing a separate form of diffusion of the solute. In its simplest form, each term contains a constant. The C¨ t¨ erm is meant to describe the effects of r¨ esistance to mass transfer¨in the particles, but is sometimes written with the term (k/(1+k))2 (where k is the retention factor) expanded out of the c¨ onstant¨, which according to Giddings was what van Deemter originally intended [6]. This term changes most radically at low values of k. This apparent relationship between retention (k) and efficiency (plate height) has more serious implications in SFC than in HPLC. In HPLC, k is independent of pressure, whereas in SFC differences in pressure, and the resulting differences in density, cause changes in k. According to the van Deemter equation, changes in k, due to changes in pressure/density ought to produce very different slopes of the C term for different values of k, and probably would dictate different optimum flow rates and potentially different values for hr,min. There are several other potential reasons why low k values should generally be avoided. Commercially available SFC’s have extra-column dispersions of 80-90 ␮L2 [7]. This is quite high, more than 20X higher than the dispersion of modern ultra high performance liquid chromatographs (UHPLC’s). The peak widths at low k must be such that the variance of the chromatograph must be much smaller (≈ <1/5th) than the variance of the peak, or extracolumn band broadening will cause loss of efficiency. It is also true that the use of a strong sample solvent can distort peaks. A typical mobile phase in SFC consists mostly of very non-polar CO2 , with a modest concentration of a polar modifier. If the sample is dissolved in a more polar solvent, larger injections can locally overwhelm the mobile phase solvent strength, distorting the peaks and causing a loss in efficiency. Neither of these effects involve questions about the van Deemter Equation. Since, in general, it appears that the use of low values of k potentially cause loss of efficiency, HPLC chromatographers have traditionally used values of k from 3 to 7 when generating van Deemter-like plots to minimize such potential effects. The van Deemter Equation doesn’t quite seem to fit empirical measurements quantitatively, but is quite effective in describing the general shape of efficiency vs. flow curves qualitatively. Since it doesn’t quite fit quantitatively there have been a number of modifications proposed. Horvath-Lin [8] developed a widely used variant that still contained the (k/(1+k))2 relationship in the C term, as per van Deemter. Knox-Scott [9] continued to relate plate height to k through density and through the term (k/(1+k)2 appearing in the C term. These equations were all developed for HPLC. Martire and Poe [10] continued to assume this dependency of h on k, and density and covered GC, HPLC and SFC. Somewhat surprisingly, this relationship between plate height, retention, and density was not seriously evaluated in HPLC, until much later. Guiochon [11], in 2013, suggested that, under most conditions, resistance to mass transfer (the C¨ t¨ erm) is independent of retention, and that, in this regard, the van Deemter Equation, the Horvath-Lin Equation, the Knox-Scott Equation, and Martire’s analysis are all slightly inaccurate, at least in liquid chromatography. These deviations are minor and actually have little impact on over-all curve fitting of the equations to empirical data. Guiochon presented both empirical data, with hr < 2, at k ≈ 0.4 and theoretical analysis (numerical solu-

tion of the Navier–Stokes equation, with simulations) supporting his claim that hr is not a function of k. The relationship between k and efficiency, based on these equations [7–10] is likely a motivation for isopycnic (constant density) operation in SFC, where the outlet pressure is varied to maintain a constant average pressure and presumably constant average density in the column, in order to keep k constant. However, for large values of k, this term changes negligibly. Understanding the role of density in supercritical fluid chromatography (SFC) with modified fluids had been hindered by the lack of an accurate equation of state for binary mixtures until approximately 5 years ago, with the publication of the REFPROP, Version 9 program by NIST [12]. Still, the only binary system that can be accurately modeled is MeOH/CO2 . The REFPROP program is expensive to obtain, and maintain with annual fees. Despite its availability, the results have not been widely disseminated, so most SFC users are largely unaware of such results. Previously, there were only a limited range of empirical density measurements for MeOH/CO2 mixtures [for instance Ref. 13], at low pressures (<200 bar) and low methanol concentrations (max. 11.5 mol %). Under those conditions, increased methanol concentrations always caused the density to increase. It was unclear what would happen at higher modifier concentrations, and/or pressures, due to lack of empirical data, or appropriate theory. Most of the recent published work using the REFPROP program has concentrated on low modifier concentrations, such as 5%, and low densities. However, much of the literature uses higher modifier concentrations and pressures, in areas that have not been well characterized with respect to density effects. In 2005, Poe [14] characterized losses in the efficiency of packed columns caused by very large column pressure drops under varying conditions of pressure and temperature using pure CO2 near the critical point. In the process, he varied the outlet pressure so as to keep the average pressure and density in the column constant (isopycnic). Later [15], with the introduction of REFPROP, ver. 9, he used CO2 with low concentrations of methanol and used the same technique to maintain a constant average pressure. Traditionally, the optimum flow rate was found by varying flow at constant modifier concentration, and, in particular, at a constant outlet pressure, to find the flow rate producing the minimum reduced plate height, hr,min (maximum efficiency). Those opposed [16–19] to this approach have argue that increasing flow generates higher pump pressure, higher average system pressure, and higher average density in the column. Since k is proportional to density, k changes with higher flow rates and higher density. The proposed solution is to change the outlet pressure in such a way to keep k constant by keeping the average pressure and the average density constant (isopycnic). However, Guillarme [7] compared the isopycnic and traditional techniques and found them equivalent, while the traditional method was much easier to use. Others have reported an experimental apparatus that automatically delivered Mole%, and nearly constant mass flow (g/min) through changes in firmware [20]. With this apparatus, the volumetric flows of the CO2 and modifier pumps were automatically varied to generate any desired Mole%, at a nearly fixed mass flow rate (g/min) regardless of the pump pressure. The changes in k were nearly eliminated. Such a system has been a goal in SFC for decades. Unfortunately, this system is not commercially available, and appears to have been dropped. No apparent improvement in efficiency or optimum flow was reported. A relatively minor issue with the isopycnic approach is that the highest flow rate used, produces the largest P. This large P dictates a high average pressure. At lower flow rates the outlet pressure must be raised to maintain the same high average system pressure. With large particles and moderate flow rates this may not be a significant problem, since the P’s are probably small.

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However, with smaller particles, the P’s will be large, forcing high average pressures and high densities at all flows. The optimum velocity is inversely proportional to density. Forcing high average density results in slower chromatography. Decreasing the P by decreasing the maximum flow rate used, lowers the average pressure but limits the range of flows covered. Much of the SFC literature dealing with finding optimum linear velocity (at v/v% flow rate) has produced relatively poor efficiency with smaller particles, probably due to the large extra-column dispersion of the standard commercial designs [7,16,21]. Users appear reluctant to modify their standard configuration to improve extra-column dispersion. This has complicated interpretation of the causes of efficiency losses since it is likely that extra-column effects were significant, but not well characterized. The only pressure readings available to the typical user are the pump pressure and the pressure at the back pressure regulator (PBPR ). The average density between these 2 pressures is not necessarily the average density in the column. The extra-column pressure drop can be larger than the actual pressure drop across the column, particularly at higher flows, due to turbulent flow in connecting tubing [22,23]. Further, the magnitude of the pressure drops before and after the column are often quite different from each other. Thus, system pressure drops are likely to be poor indicators of actual column pressure drops or average column density.

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due to changes in the mobile phase composition. These conclusions have not been well characterized or understood in the past. In this report, the calculated density of methanol modified mixtures is evaluated in terms of its impact on retention, and efficiency, particularly at higher (>10%) methanol. Efficiency (reduced plate height, hr ) and retention (k) were measured over a wide range of methanol concentrations and outlet pressures. The flow was varied, while holding the back pressure regulator (BPR) pressure constant. The pump pressures were also recorded. In some cases an additional pressure transducer was used to measure the actual column inlet and outlet pressures (vs. the pump and BPR pressures). This allowed the measurement of system P’s before and after the column, and determine the actual inlet and outlet column pressures and densities and average column pressures and densities. The intent was to better elucidate relationships between k, hr , and density, especially at higher modifier concentrations. In addition a few isopycnic measurements were attempted to try to see differences in k and hr compared to the traditional approach. Most of the results are presented in terms of the instrument set points, in particular v/v% MeOH (resulting in variable Mole%), pump pressure, column inlet temperature, and PBPR , since the average user can only control these set points.

2. Experimental 1.1. Density vs. Viscosity 2.1. Instrumentation In SFC, modifier concentrations from 5% to 40%, or higher are commonly used. With the addition of modifiers, under isocratic conditions, increasing the outlet pressure results in decreasing retention. Peaks of similar solutes move closer together, but, peak reversals, due to changes in pressure, are quite rare. However, there are aspects of the relationship between density and retention, particularly with changes in modifier concentrations, that are still poorly understood by most users. Pure modifiers, like methanol, are much less dense than CO2 at high pressures, particularly above 200 bar. It has been pointed out [24,25] that, at high pressures, and modest temperatures, the density of mixtures high in methanol concentration (> 20%) can actually be less dense than mixtures with lower methanol concentrations (< 20%), at the same pressures and temperatures. During a composition program, the density at the column inlet initially increases but then decreases as the modifier concentration exceeds roughly 20%, particularly above ≈ 150 bar. This is counter-intuitive based on most earlier work, where density measurements always showed increasing density with increasing modifier concentration up to 11.5 mol% [13], but with no understanding of what would happen at much higher modifier concentrations. Most of the SFC literature seems to assume that any increase in column pressure drop, P, indicates increasing mobile phase density, and that density and viscosity are always closely related, regardless of the modifier concentration. This is true for isocratic operation with increasing flow rates, but often not true when the modifier concentration is increased. Tarafder reported that, with pure CO2 , increasing density always resulted in increased viscosity [26], and that viscosity increased non-linearly with density. Similarly, Guillarme [7] reported on the viscosity of several mixtures of methanol in CO2 at up to 40% as a function of pressure, showing dramatic increases in viscosity with increasing modifier concentration. This is consistent with wide experience, where increasing modifier concentration always results in increasing system P. However, while increasing modifier concentration produces increasing pump pressures, density is actually often decreasing. Such results indicate that column pressure drop due to increased modifier concentration is often not caused by higher mobile phase density, and that density and viscosity are no longer closely linked

A Model 4301 A SFC (Infinity II SFC)(this is a different system from those used in earlier reports from this lab) was controlled by a Model C.01.08 (210) OpenLabs, ChemStation, all from Agilent Technologies, Waldbronn, Germany. The system consisted of a SFC conversion module, a binary pump with a built-in 2 channel degasser, an automatic liquid multisampler (ALS), a column oven (MCC), a 160 Hz diode array detector (DAD) with a 2 ␮L, 3 mm tapered flow cell. The standard plumbing used the shortest possible ¨ lengths of 170 ␮m ID tubing, including 2 standardheat exchangers in the MCC (oven) employing 175 ␮m tubes. The column was connected to the plumbing with 2–5 cm lengths of 125 ␮m tubing with quick connect fittings and was isolated from the oven walls with plastic clips. The detector flow cell uses a 31 cm long 125 ␮m inlet tube, which tends to dominate system pressure drop at high flow rates. The multisampler can inject from 0.1 to 90 ␮L, in 0.1 ␮L increments, using a high pressure syringe. For this work, sample size was 2.0 ␮L, with a feed volume of 4 ␮L of isopropyl alcohol (IPA). The instrument was capable of 600 bar at up to 5 m L-min−1 . The Agilent SFC uses a unique approach to pumping. The SFC conversion module compresses the CO2 to ≈ 8 bar below the column head pressure, and the binary pump accurately meters the already compressed fluid. The binary pump head is not chilled. This largely eliminates Joule-Thompson heating since there is no compression in the metering pump, which minimizes CO2 pumping noise and makes it easier to calculate the Mole% of methanol in the CO2 . However, this results in certain issues outside normal operating conditions. At simultaneously low outlet pressures, low % modifier, and low flows the binary pump pressure can be as low as a few bar above the column outlet pressure. Under such conditions flow and composition can become erratic, with increased UV noise. Under such conditions, the data was not used. At the other extreme of high modifier concentration, high column outlet pressure, and high flow rates, the binary pump pressure sometimes approached the 400 bar limit of the column. Under those conditions the flow was limited to as little as 3 m L-min−1 to avoid damaging the column. The actual column inlet and outlet pressures were monitored using an additional pressure transducer. A 7 cm length of 175 ␮m

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tubing was inserted between the column inlet and a tee. The plumbing normally connected to the column was connected to another port of the tee. The additional transducer was connected to the side arm of the tee with a meter long piece of 250 ␮m ID tubing (with no flow). The transducer was alternately installed at the column outlet, to measure the pressure drop in the system after the column. During these pressure measurements no efficiency measurements were made. 2.2. Column and materials The column was 4.6 x 150 mm, packed with 5 ␮m RX-Sil totally porous, spherical type B silica from Agilent Technologies, Wilmington DE, USA, rated to 400 bar. The particles had 80 Å pores with180m2 /g surface area, according to the manufacturer. Food grade carbon dioxide in 50 pound steel cylinders, without a DIP tube, was purchased from Terry Supply Co., Bradenton, FL. HPLC grade methanol and IPA were purchased from SECO, Aston, PA. Theobromine (pKa = 9.9, LogP = -0.78, MW 180), was > 98% pure, purchased from Sigma-Aldrich, St. Louis, MO and used as received. A single solute was used throughout, in order to avoid any uncertainty due to potential differences in retention mechanism. This inherently means there will be a wide range of k over the ranges of % MeOH and PBPR used.

Fig. 1. Density vs. pressure plots for pure methanol and CO2 and methanol/CO2 mixtures (mole%) at 25 ◦ C.

Table 1 Top: Actual mole% at the nominal v/v% values as a function of pressure at the pump. Note that Mole% decreases with increasing pump pressure. Bottom: V/V% needed to achieve constant Mole %. Pump temperature 25◦ .

5%

v/v% Set 10%

20%

40%

P, Bar 100 150 200 250 300 350 400

6.644676 6.268177 6.067305 5.934589 5.838811 5.764261 5.707988

Mole% 13.0632 12.37121 11.99979 11.75355 11.57542 11.43652 11.33153

25.26644 24.10727 23.47791 23.05784 22.75256 22.51369 22.33265

47.41174 45.86004 44.99952 44.4179 43.9915 43.65566 43.39984

100 150 200 250 300 350 400

V/V% 3.76241 3.988401 4.120446 4.212592 4.281693 4.33707 4.379827

7.655094 8.083285 8.333476 8.508068 8.638998 8.743921 8.824935

15.83128 16.59251 17.03729 17.34768 17.58044 17.76697 17.911

33.74692 34.88876 35.55594 36.02152 36.37066 36.65046 36.86649

3. Results and discussion 3.1. Mole% vs. v/v% All SFC’s (and HPLC’s) use v/v% as the instrumental set point, as opposed to Mole%. In generating van Deemter-like plots, the flow is varied, which causes the pressure at the pump to change. The density of pure CO2 in one pump changes much more rapidly with changes in pressure compared to the density of MeOH in the other pump at the same pressures. However, the MeOH density does change significantly and must be accounted for. Increasing pump pressure results in decreasing Mole%, when v/v% is held constant. Constant mole ratio ought to yield the most consistent interactions between the solute and the mobile phase and between the mobile phase and the stationary phase. Accurate Mole% can only be generated at the pumps, but there are multiple problems implementing automated control of this possible capability. In SFC, a binary type pump is required since the CO2 arrives at an elevated pressure, typically at ≈ 65 bar, whereas the modifier reservoir is essentially at ambient pressure. Both pumps deliver at the same high pressure. However, the temperature of each pump head is as important as this shared pressure, in order to determine the density of each fluid at delivery. These densities in turn determine the required volume flow rate of each component, to obtain the desired Mole%. In some SFC’s, the pump head is chilled to between -20◦ and + 5 ◦ C. This does not mean that the fluid in the pump actually reaches these sub-ambient temperatures. In addition, compression of the fluid can raise the temperature of the fluid as much as 45 ◦ C. Some of this increased temperature will be dissipated into the metal parts of the pump head, but the mean temperature of the fluid, and subsequently the density, will be poorly understood. SFC instruments have, as a setpoint, v/v%. The typical user has no easy means of knowing or controlling Mole%, or translating from v/v% to Mole%. Accurate knowledge of the density of each pure fluid at the pump pressure at the temperatures of each pump is required. Much of this required information has not been easily available. As noted previously, the Agilent CO2 pump does not significantly compress the fluid and the data in Fig. 1 (25 ◦ C) closely represents the density of both the CO2 and MeOH in the pump heads. The

theoretical change in Mole% resulting from the use of a fixed v/v% at various pump pressures, assuming 25 ◦ C pump head temperatures is presented in the top of Table 1. These numbers indicate the change in Mole% with pressure, at fixed v/v%. The changes are fairly significant. The changes in v/v% required to produce constant Mole% at the mixing point are presented in the bottom of Table 1. Changing v/v% to maintain constant Mole% is likely to require changes in other parameters, such as BPR pressure, to maintain constant pump pressure, although such changes in BPR pressure will be small. Never-the-less, changing multiple control parameters simultaneously introduces more uncertainty. 3.2. Density calculations The densities of CO2 /methanol mixtures (Mole%), were calculated, as a function of pressure, using Version 9.1 of the REFPROP program from NIST [12], with appropriate physical chemical references [27,28] from the literature. This program is now widely used and appears to be reliable and reasonably accurate for mixtures of CO2 and methanol. The Agilent CO2 pump operates near 25 ◦ C. The density vs. pressure data at 25 ◦ C, between 100 and 400 bar, for each of the pure fluids, is presented in Fig. 1, and indicate the density of the pure flu-

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Fig. 2. Density vs. pressure plots for pure methanol and CO2 mixtures of methanol in CO2 (v/v%) at 40 ◦ C.

ids in each of the pumps. The bottom curve is the pressure/density curve for pure methanol. At all pressures shown, this curve is well below any densities with pure CO2 or mixtures of methanol in CO2 . The methanol curve is fairly linear, with a low slope. At the other extreme is the curve for pure CO2 , which starts with the second lowest density at 100 bar. This curve is extremely nonlinear, particularly at lower pressures. Above ≈ 230 bar, pure CO2 has higher densities than any mixture of methanol in CO2 (40 ◦ C). A curve for 2% (Mole%) methanol is just above pure CO2 at 100 bar. At 100 bar, 2% methanol has slightly higher density than pure CO2 , but above ≈ 260 bar, has slightly lower density. Above ≈ 20%, all the density vs. pressure curves are nearly parallel, with quite modest slopes, similar to pure methanol. Increasing pressure, at any isocratic composition always results in increasing density, although the change is much smaller at higher methanol concentrations. In Fig. 2, some v/v% results at 40 ◦ C are presented, covering primarily the methanol concentrations used in this report (5 to 40 v/v% methanol, between 100 and 400 bar BPR pressure). This data represents the densities in the column. The Mole% values from the REFPROP program in Fig. 1, were converted into v/v% at the pumps, in order to correspond to the instrumental set point. This means that in this report Mole% does change with pump pressure, and % methanol means v/v% at the pumps. The curves in Fig. 2 (40 ◦ C) appear similar to those at 25 ◦ C except they are shifted to lower densities at the same pressures. Note that the data for 30% and 40% methanol do not extend down to 100 bar, since below ≈ 113, and 115 bar, respectively, 2 phases apparently exist. At low methanol concentrations, the curves are quite non-linear, and steep, particularly at low pressures. For 40% methanol, the density vs. pressure curve is very nearly linear, and the slope is quite shallow, and similar to pure methanol. The density of 5% methanol increases approximately 35% from 100 to 400 bar at 40 ◦ C. With 40% methanol, the density increases much less, at 5.1% between 115 and 400 bar. In comparison, the density of pure methanol increases 3.1% over the same change in pressure. This reaffirms that the mixture at 40% is much less compressible than at 5%, and that large changes in pressure only have a small effect on density, similar to a normal liquid. The compressibility of 40% MeOH appears to be only 1/3rd higher than pure hexane. The density of both 20% and 30% methanol is always higher than 40% at any pressure between 100 and 400 bar (40 ◦ C). Even 10% methanol is more dense than 40% above ≈ 180 bar, and 5% above 205 bar. This should be quite surprising to most users.

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Fig. 3. The pressure drops across the system with the 4.6 x 150 mm, 5 ␮m RX-Sil column installed, as a function of flow rate, at 5% and 40% methanol, with outlet pressures of 100 bar, 150 bar, and 300 bar, 40 ◦ C.

Fig. 4. The density at the pressures in Fig. 3, and 150 bar outlet pressure. Note that the density at 5% is often significantly higher than the density at 40% methanol.

3.3. Pressure drops and resulting densities 3.3.1. System pressure The system pressure drops were recorded over the extremes of v/v% MeOH and outlet pressure as a function of flow rate. The standard plumbing configuration was used with the column installed. The results for 5% methanol at 100, 150, and 300 bar at the BPR and 40% MeOH at 100, and 300 bar at the BPR are compared in Fig. 3. The results are unrevealing other than the use of 40% results in slightly higher pressure drops compared to the use of 5% MeOH, at the same BPR pressure. However, when the pressures are translated to densities, the comparison becomes more interesting as shown in Fig. 4. The densities shown are the average density in the column with the instrument set points (v/v% and PBPR ) which are indicated next to each curve. The actual Mole% at the pumps was used to determine the actual density in the column at column set temperature. As one would expect from Fig. 2, 5% methanol is sometimes significantly more dense than 40% methanol at the same temperature and pressure.

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Fig. 5. The actual pressure drops across the column, in the tubing and fittings before the column, and after the column, all at 5% methanol, 100 bar BPR pressure. The curvature of the pre- and post-column tubing pressure drops is probably due to turbulent flow in the tubing.

3.3.2. Column pressure The pressure was measured directly at the column inlet and outlet using an independent pressure transducer. This allowed the measurement of the true column inlet and outlet pressures as well as the pressure drops in the system before and after the column as a function of flow rate. The results are shown in Fig. 5 for 5% MeOH with 100 bar BPR pressure. The actual column pressure drop was small, and never exceeded 32 bar, even at 5 m Lmin−1 , whereas the total system pressure drop in an extreme case exceeded 184 bar. The pre-column P was larger than the column P above 3 m L-min−1 , while the post-column P was as much as 3 times the column P. This is probably largely due to the 31 cm length of 125 ␮m tubing connecting the detector cell to one of the heat exchangers. Much of the extra-column pressure drops can be attributed to turbulent flow of the mobile phase in the tubing and fittings [21,22]. Under normal conditions, the actual column inlet and outlet pressures are unknown, since there are usually no pressure transducers directly at the column inlet and outlet, but only at the pump

and at the BPR. Fortunately, near optimum flow rates (in this case ≈ 1–2.8 mL-min−1 ), the extra-column P’s are relatively small so that pump pressure and the pressure at the BPR give reasonable approximations to the column pressures. However, due to extracolumn P at higher flow rates, the pump and BPR pressures yield a poor indication of the true column inlet and outlet pressures, and subsequently, the local densities in the column.

3.3.3. Isopycnic operation Isopycnic operation as used here involves changing the outlet pressure when the flow rate is changed in order to maintain the same average system pressure. The average user has only the pump and BPR pressures to work with. An outlet pressure of 100 bar is near the lower limit to avoid 2 phase regions. Using 100 bar outlet pressure with the highest flow rate means that all lower flow rates will have a higher outlet pressure. The results with 5% (v/v%) MeOH, 4.5 m L-min−1 , and 100 bar outlet was used as a starting point. The pump pressure was 238 bar making the average pressure 169 bar. Thus, the average pressure at all the flow rates must be 169 bar for

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Fig. 6. The change in density with flow rate using 5% methanol at 40 ◦ C. The average column density was calculated using the densities at the actual column inlet and outlet pressures. The average system density was calculated using the densities at the pump pressure and the BPR pressure. The “isopycnic” density is based on an average pressure of 169 bar. See text for details.

isopycnic operation. The P at each of the other flow rates was estimated from the P’s at 100 bar BPR pressure with the various flow rates. These P values were divided by 2, and the result subtracted from 169 bar, to yield a new BPR pressure for each flow rate. From Fig. 2, the densities 1.) across the system with 100 bar BPR pressure, 2.) across the column with 100 bar BPR pressure, and through 3.) isopycnic operation with an average pressure of 169 bar, were estimated as a function of flow rate. The results are presented in Fig. 6. At the lowest flow rates, the average system density and the average column density are similar and much lower than the average density with isopycnic operation. However, at higher flows the true average column density is noticeably higher than the average system density. This is reasonable since most of the extra-column P is after the column, so the average pressure in the column ought to be higher than the average pressure in the system. The average density in the isopycnic approach is quite high simply because the average pressure must be high in order to include the higher flow rates. The average density using the isopycnic approach was not quite constant across flows because the density vs. pressure curves in Fig. 2 are non-linear, particularly with low methanol concentrations, like 5%. Using different P’s around the same average density results in somewhat distorted results due to this non-linearity. One could, by trial and error keep adjusting the BPR pressure at each flow rate until the average of the pump pressure and the BPR was 169 bar but this is rather tedious because one needs to wait until a steady state is reached after each change in BPR pressure. However, the results in Section 3.3.2. showed that, at high flows, the average system pressure was not the same as the average column pressure. The retention factor, k, of theobromine was plotted vs. density, ¨ using system P, column P, and ¨isopycnicdensities as shown in Fig. 7. The actual column density yielded a fairly flat curve at lower flow rates but the ¨isopycnic¨results indicate that near constant k was achieved although constant density was not. 3.4. k Vs. % methanol The retention factor, k, for theobromine at various v/v % methanol concentrations, was measured as a function of flow rate. The results using 100 bar BPR pressure is presented in Fig. 8a, and show a modest decrease in retention at the lower methanol concentrations with increasing flow rate. The slope is only significant when k is large and % methanol is small. With 5v/v% MeOH, k decreased

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Fig. 7. Retention factor vs. density for the 3 scenarios, as described in Fig. 6.

from ≈ 16 at 1.5 m L-min−1 to ≈13 at 4.75 m L-min−1 . The higher flow rates resulted in higher pump pressure, and higher average pressure and density, and lower retention. If there was an effect related to (k/(1+k))2 , it would involve a decline of ≈ 3%, since the values for k were always large. The slope of the curves becomes progressively flatter with increasing methanol concentration, until by 40% there is little obvious slope (very little change in k). With a BPR pressure of 150 bar, the results shown in Fig. 8b, indicate a modest shift to lower retention, with a further flattening of the curves at each % methanol compared to Fig. 8a. Additional data with a BPR pressure of 200 bar is presented in Fig. 8c. The data make it even more obvious that k changes very little except at low methanol concentrations, and there, the values of k are still quite large, making changes in (k/(1+k))2 small. Such minimal changes in k are unlikely to distort efficiency measurements, even at low modifier concentrations. At 300 bar PBPR , only the results at 5% and 40% are presented, in Fig. 8d. The retention at 5% MeOH dropped only about 10% between 0.75 and 3 m L-min−1 , but the average value of k was still large at 8.99, which is still larger than the typical HPLC user uses for finding the optimum flow rate. From this analysis. it appears that k only changes significantly with pressure/density when the solute is strongly retained (large k), and any effect on (k/(1+k))2 is minimal. 3.5. k vs. Density The retention factor was also determined as a function of the density at pump pressure (and 40 ◦ C) for the same extremes used in Fig. 4. The results are shown in Fig. 9. With 40% methanol at either 100 or 300 bar BPR pressure, density changed very little, as did k when the flow rate was varied. At the lower methanol concentrations, and 100 bar BPR pressure, the range of densities resulting from changes in flow, is much larger, as one would expect from the slopes of the curves in Fig. 2. The actual densities with 5% methanol bracketed (i.e., some were higher, some were lower) the densities at 40%, meaning almost all the change in retention was due to the modifier concentration and not a density effect. 3.6. Efficiency The hr vs. flow rate plots at 5% and 40% methanol at 3 constant BPR pressures, are presented in Fig. 10a, and b respectively. The arrows under the curves refer to the optimum flow rate at each

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Fig. 8. The retention factor as a function of flow rate with: a.) 100 bar, b.) 150 bar, c.) 200 bar. and d.) 300 bar BPR pressure and 40 ◦ C. At low modifier concentrations there is a noticeable change in retention caused by the increased pump pressure at higher flow rates. However, at higher modifier concentrations retention hardly changes with increased pressure at higher flow rates. Increasing the BPR setting causes a slight shift to lower retention with the slope of the curves decreasing. At high methanol concentrations the curves are virtually flat.

Fig. 9. Some of the data from Fig. 8 is replotted as k vs. density. This makes it clear that density changes very little at high modifier concentration, regardless of the outlet pressure.

outlet pressure. In Fig. 10a, hr,min < 2, at all the BPR settings. Higher pressures, at 5% MeOH, suppress the optimum flow rate significantly. The optimum flow rate depends on BPR pressure, but the

excellent maximum efficiency (minimum reduced plate height) was the same, even though increasing pressure resulted in lower k. With the BPR set to 100 bar, and 5% MeOH the value for k changed nearly 25% with flow and resulted in a fairly large change in density, as previously shown in Fig. 9. With the BPR set to 300 bar, with 5% methanol, k changed less than 8.5% (in Fig. 9) while the density was much higher. Both sets of conditions, yielded the same low hr,min despite the changes in k and density and the differences in densities within and between the 2 sets of data. The peaks change their speed of migration significantly when passing through the column, particularly at the higher flows, yet the curves in Fig. 10 are quite flat even at the highest flows. This suggests that even large changes in k, when k is large, have minimal effect on efficiency measurements, which peripherally supports Guiochons analysis. In Fig. 10b, with 40v/v% MeOH, hr,min is even lower, as low as ≈ 1.63, and nearly independent of PBPR . The higher modifier concentration suppressed the optimum flow rate substantially, so that increased BPR pressures appear to have little additional effect in suppressing the optimum flow rate. The same, extremely high efficiencies obtained at all PBPR pressures, and the corresponding very low values for k, suggests k has little to do with efficiency, baring extra-column effects. The peak generating the highest efficiency, is presented in Fig. 11. It was collected with 40% MeOH (v/v%), 1 m L-min−1 , 300 bar, at 40 ◦ C. The efficiency, as reported by the ChemStation at w1/2 (2.35␴) was >18,300 plates, with a reduced plate height of 1.63. Manual measurements using peak width at the inflection

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Fig. 10. a) Reduced plate height vs. flow rate for 5% methanol in CO2 at 40 ◦ C with the 4.6 x 150 mm, 5 ␮m RX-Sil column with 100, 150, and 300 bar outlet pressure. The arrows under the curves indicate the approximate optimum flow rates. The optimum flow rate decreases with increasing outlet pressure. Note that at all 3 outlet pressures the minimum reduced plate heights are well below 2. This indicates that extra-column band broadening is not significant. b) reduced plate height vs. flow rate again at 100, 150, and 300 bar outlet pressure, 40 ◦ C, but with 40% methanol in CO2 . At high modifier concentration the optimum flow is depressed and pressure no longer has a significant effect. Efficiency is even better than at 5%, even though retention is very low, with k < 1.

Fig. 11. Expanded scale of the theobromine peak exhibiting the highest efficiency on the 4.6 x 150 mm, 5 ␮m RX-Sil column, at 40% methanol in CO2 , at 300 bar BPR pressure, 40 ◦ C. The k was only 0.761 but hr,min = 1.63.

Fig. 12. The minimum reduced plate height is fairly independent of the BPR pressure, as shown for a.0 5%MeOH, and b.) 40% MeOH. Each set of data points uses multiple pump pressures caused by changes in flow rate. In all cases the minimum reduced plate height was below 1.70. Conditions: 4.6 x 150 mm column packed with 5 ␮m RX-Sil, at 40 ◦ C.

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Fig. 13. Reduced plate height vs. retention factor at a.) 5% MeOH and b.) 40% MeOH. The retention factor changed only moderately with flow (pressure drop) except at 100 bar outlet pressure. The minimum reduced plate height was less than 1.7 at all pressures even though retention was minimal (k mostly <1).

points (2␴), and at 4␴ (13.4% of peak height) yielded similar high efficiencies although the peak was apparently slightly asymmetric. The reduced plate height obtained using 5% modifier, at outlet pressures between 100 and 300 bar outlet pressure are presented as a function of pump pressure (caused by varying the flow) in Fig. 12a. Note that hr approaches or goes below1.9 at all outlet pressures at each appropriate optimum flow. Similar results were obtained with 40% MeOH, only the minima all approached hr = 1.67 as shown in Fig. 12b. These are outstanding efficiency measurements and rival some using superficially porous particles. The same reduced plate heights are presented as a function of k, at 5% and 40% methanol, in Fig. 13a & b. At 5% MeOH, increasing PBPR results in narrowing distributions of k, but in all cases the same hr < 2 was obtained at a characteristic optimum flow rate. At 40% MeOH, parts of the curves are nearly vertical, meaning k is nearly constant, but hr varies significantly. The data were collected with k < 1.1, but at optimum flow, hr,min is near or below 1.7, which is outstanding. Traditionally, solutes are chosen for van Deemter like plots that have retention factors of 3–7 to avoid the effects of extra-column band broadening, associated with low k. In this work, extra-column effects with the plumbing used are shown to be minimal to non-existent. However, the use of columns packed with smaller particles require additional efforts to significantly reduce system dispersion. This is possible since hr < 2 was recently demonstrated with 1.8 m particles with k ≈ 2 [24], but only after extensive modification of the flow path. The results reported here all support Guiochon’s [11] analysis that plate height is independent of k. 4. Conclusions It is often stated that isopycnic operation (constant average column density) is preferable to isobaric operation, because it more nearly produces constant retention. The justification for such usage is based on the classic van Deemter Equation, where plate height is thought to be a function of retention. This characterization was continued by Horvath-Lin, Knox-Scott, and Martire. Decreases in retention, particularly at very low values of k, were expected to result in decreased efficiency. However, Guiochon has shown that efficiency is NOT a function of retention (k) and the van Deemter Equation and the others are inaccurate, in this regard, at least with respect to reversed phase high performance liquid chromatography. In the present work it was shown that the highest efficiencies were obtained with the lowest retention, with reduced plate heights as low as 1.63 using totally porous particles, and k values < 0.8. These results very strongly challenge the idea that efficiency can only be achieved at high values for k, or that changes in k, result-

ing from changes in pressure/density are relevant. These results further seriously challenge the widely promulgated idea that isopycnic operation is required, for measuring efficiency in SFC. In fact isopycnic operation involves a good deal of extra work for no apparent gain compared to the traditional approach using isobaric outlet pressures. On the other hand, for comparing columns with different packing particle sizes, it seems appropriate to adjust pressures to try to get similar selectivities on columns with wildly different pressure drops, due to large differences in packing particle diameter. Similarly, the extremely important field of linear solvation energy relationships (LSER) [29], used to characterize and compare different stationary phases, is based in part on an unequivocal value for k, to deconvolute the effects of dipole, proton donor, and proton acceptor interactions. For such work it appears to be important to minimize extra-column pressure drops, and use larger particles, with modest pressure drops at, or below, optimum flow, to achieve the smallest change in density across the column in order to obtain a pure value for k. It is often stated that the density of modified mobile phases is a major control variable affecting the retention of solutes. At low modifier concentrations and low outlet pressures, retention does change noticeably with pressure, but not dramatically, because the density vs. pressure curves are very steep under these conditions. However, the modifier concentration continues to dominate retention. At higher outlet pressures, curves of k vs. flow become much flatter and retention changes much less with pressure. At higher modifier concentrations pressure has a minimal effect on retention. Higher modifier concentrations are usually thought to result in higher density. Higher density is thought to significantly lower retention. However, the density of higher methanol concentrations is usually lower than the density of lower concentrations. The effect of higher modifier concentrations on reducing retention is not the result of increased density, but is the result of increased solvent strength. During gradient elution from 5% to 40%, at 40 ◦ C, the density of the mobile phase in the column should initially rise with increasing methanol concentration but near 20% the density should begin to decrease and continue decreasing to 40% and beyond. Increasing modifier concentration causes increased pressure drops but the density is decreasing. The role of density on retention has been distorted by overemphasis on low modifier concentrations, at low column outlet pressures, in efforts to use low densities to maximize speed. Few realistic solutes elute under such conditions. Most of the useful applications of SFC have involved higher modifier concentrations, and often higher pressures. Modern applications attempt to shift usage to smaller particles such as sub-2 ␮m, that have much

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higher average pressures. When higher concentrations and higher pressures are use, the role of density on retention diminishes significantly, as shown by the data presented. The pump and BPR pressures are not necessarily good indicators of pressure or density actually in the column. At high flow rates the extra-column P can be larger than the column P and can be unevenly distributed in front of and after the column. The actual densities at the inlet and outlet of the column depend on the actual pressures, which are typically not measured. The system plumbing does affect pressure drops, and retention, and cannot be ignored, but the relationships can be complex. Acknowledgements The author wishes to thank Dr. Sam O. Colgate, Professor Emeritus of Physical Chemistry, University of Florida for using the REFPROP program to calculate the data used in Figs. 1 and 2. References [1] Lyle M. Bowman Jr., Marcus N. Myers, J. Calvin Giddings, Supercritical fluid (dense gas) chromatography with linear density programming, Sep. Sci. Technol. 17 (1982) 271–287. [2] Milos Novotny, Stephen R. Springston, Paul A. Peaden, John C. Fjeldsted, Milton L. Lee, Capillary supercritical fluid chromatography, Anal. Chem. 53 (1981) 407A–414A. [3] W.P. Jackson, P.A. Peaden, M.L. Lee, Density programming in capillary supercritical fluid chromatography, J. Chromatogr. Sci. 21 (1983) 222–225. [4] Dennis R. Gere, Supercritical fluid chromatography with small particle diameter packed columns, Anal. Chem. 54 (1982) 736–740. [5] J.J. van Deemter, F.J. Zuiderweg, A. Klinkenberg, Longitudinal diffusion and resistance to mass transfer as a cause of non-ideality in chromatography, Chem. Eng. Sci. 5 (1956) 271–289. [6] J.C. Giddings, Dynamics of Chromatography, Marcel Dekker, New York, NY, 1965. [7] Alexandre Grand-Guillaume Perrenoud, Chris Hamman, Meenakshi Goel, Jean-Luc Veuthey, Davy Guillarme, Szabolcs Fekete, “Maximizing kinetic performance in supercritical fluid chromatography using state-of-the-art instruments, J. Chromatogr. A 1314 (2013) 288–297. [8] C.S. Horvath, H.-J. Lin, Band spreading in liquid chromatography: General plate height equation and a method for the evaluation of the individual plate height contributions, J. Chromatogr. 149 (1978) 43–70. [9] J.H. Knox, H.P. Scott, B and C terms in the Van Deemter equation for liquid chromatography, J. Chromatogr. 282 (1983) 297–313. [10] D.P. Poe, D.E. Martire, Plate height theory for compressible mobile phase fluids and its application to gas, liquid and supercritical fluidchromatography, J. Chromatogr. 517 (1990) 3–29. [11] Fabrice Gritti, Georges Guiochon, The van Deemter equation: assumptions, limits, and adjustment to modern high performance liquid chromatography, J. Chromatogr. A 1302 (2013) 1–13. [12] E.W. Lemmon, M.L. Huber, M.O. McLinder, NIST Reference Database 23: Reference Fluid Thermodynamic and Transport Properties- REFPROP, Version 9.1, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, 2013.

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