Effects of pressure drop, particle size and thermal conditions on retention and efficiency in supercritical fluid chromatography

Journal of Chromatography A, 1216 (2009) 7915–7926

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Effects of pressure drop, particle size and thermal conditions on retention and efficiency in supercritical fluid chromatography Donald P. Poe ∗ , Jonathan J. Schroden 1 Department of Chemistry and Biochemistry, University of Minnesota Duluth, 10 University Drive, Duluth, MN 55812, USA

a r t i c l e

i n f o

Article history: Received 18 June 2009 Received in revised form 27 August 2009 Accepted 31 August 2009 Available online 6 September 2009 Keywords: Supercritical fluid chromatography Pressure drop Efficiency Retention Temperature gradients Thermal conditions Modeling

a b s t r a c t The effects of particle size and thermal insulation on retention and efficiency in packed-column supercritical fluid chromatography with large pressure drops are described for the separation of a series of model n-alkane solutes. The columns were 2.0 mm i.d. × 150 mm long and were packed with 3, 5, or 10␮m porous octylsilica particles. Separations were performed with pure carbon dioxide at 50 ◦ C at average mobile phase densities of 0.47 g/mL (107 bar) and 0.70 g/mL (151 bar). The three principal causes of band broadening were the normal dispersion processes described by the van Deemter equation, changes in the retention factor due to the axial density gradient, and radial temperature gradients associated with expansion of the mobile phase. At the lower density the use of thermal insulation resulted in significant improvements in efficiency and decreased retention times at large pressure drops. The effects are attributed to the elimination of radial temperature gradients and the concurrent enhancement of the axial temperature gradient. Thermal insulation had no significant effect on chromatographic performance at the higher density. A simple expression to predict the onset of excess efficiency loss due to the radial temperature gradient is proposed. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Numerous studies have shown that excessive efficiency losses may occur when large pressure drops are used in packed-column supercritical fluid chromatography (pSFC). Among these are a few reports in which experimental results are compared to predictions based on theoretical models [1–4]. Good agreement was generally obtained as long as the outlet pressure was about 130 bar or higher. At lower outlet pressures poorly shaped peaks and poor agreement with theory were observed. In an early investigation [5] the use of larger particle sizes was suggested as one alternative to avoid excessive losses of efficiency. The effects of density gradients in solvating gas chromatography have also been the subject of recent modeling efforts [6,7]. The loss of efficiency under these conditions places important practical limits on operating conditions for packed-column SFC. This is unfortunate because at lower pressures increased diffusivity and decreased viscosity both favor increased speed and efficiency. Furthermore, there is currently no good theoretical framework that reliably predicts the impact of common operating parameters such as particle size, column length and column diameter on efficiency

∗ Corresponding author. Tel.: +1 218 726 7217; fax: +1 218 726 7394. E-mail address: [email protected] (D.P. Poe). 1 Present address: CNA, 4825 Mark Center Drive, Alexandria, VA 22311, USA. 0021-9673/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2009.08.089

under these conditions. A better understanding of the processes leading to excess efficiency loss might lead to better guidelines for operating conditions, and perhaps also to significant improvements in chromatographic performance. Excess efficiency loss in pSFC has generally been attributed to effects of density changes on the retention factor. Well-established theory for efficiency of nonuniform columns [8], where elution conditions change along the axis of the column, predict that the observed efficiency for such columns must be greater than the simple average. Recent studies in our laboratory on elution of an unretained solute using insulated, air- and water-thermostatted columns showed evidence of significant temperature drops in packed columns when low column outlet pressures (<100 bar) were used [9]. Efficiency losses were much greater than predicted by theory, and were attributed to the presence of radial temperature gradients. Thermal insulation on the column was used to effectively eliminate the radial temperature gradients and to minimize efficiency losses. The use of thermal insulation to improve efficiency for elution of retained solutes in pSFC has also been reported [10,11]. In this report we employ a mixture of medium-weight n-alkanes as test solutes and pure CO2 mobile phase to examine the effects of pressure drop, particle size and thermal insulation on retention and efficiency in packed columns. The data are from the same set of experiments reported earlier for studies on methane [9]. Isopycnic plate (constant density) height curves were generated at 50 ◦ C at average mobile phase densities of 0.47 and 0.71 g/mL.

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At the lower density the mobile phase is near its critical point and is highly compressible. The density is equal to the critical density c = 0.468 g/mL (reduced density r = /c = 1.00), and the temperature is slightly above the critical temperature of 304.2 K (reduced temperature Tr = 1.06). Under these conditions large pressure drops result in significant density gradients, and a column operated in this fashion is called a nonuniform column. At the higher density (r = 1.50) the fluid is not so compressible, and is much more liquidlike. This condition corresponds to a near-uniform column, much like the situation in typical liquid chromatography. By conducting experiments under these two conditions, and comparing the experimental results to theoretical predictions, we hope to provide further insight into the factors controlling efficiency and retention in pSFC. We also conduct similar experiments in which the column is covered with thermal insulation to evaluate the impact of radial temperature gradients, and to explore the potential benefits of using this approach to improve speed and resolution. One of the principle tools used in this study is the isopycnic plate height curve. Plate height curves in SFC are traditionally generated by increasing the flow rate while keeping the outlet pressure constant. The mobile phase density thus increases with increasing flow rate, resulting in large decreases in solute retention factors and complicating interpretation of the results. To generate an isopycnic plate height curve we keep the average mobile phase density constant while the flow rate is increased. This is accomplished by simultaneously increasing the inlet pressure and decreasing the outlet pressure to achieve the desired flow rate. In terms of mobile phase properties, solution retention in SFC is best described as a process that depends on the temporal average density [12]. The isopycnic approach is similar to the constant average pressure method used by Mourier et al. [13], but has the added benefit that retention factors and related mobile phase remain nearly constant with flow rate. Although computations in this paper predict that under isothermal, isopycnic conditions solute retention factors increase somewhat as the pressure drop increases, the changes are relatively small, simplifying the interpretation of the resulting plate height curves.

For prediction of apparent retention factors in SFC we utilize the approach based on spatial and temporal average quantities developed by Martire and co-workers [14,15]. The dependence of the local retention factor on mobile phase density at constant temperature is described by the relation:

  t

B + A1/3 + C 

(3)

where  is the reduced velocity. Our attempts to fit experimental data obtained under near-uniform conditions resulted in poor fits to the Knox equation. We chose instead to utilize the simpler van Deemter equation, treating A as a constant and retaining the detail for the B and C terms in the Knox equation. Our modified Knox equation for local plate height is then: h=

B + A + C 

(4)

The equivalent form of the C term used by Knox and Scott is Ce e , where e is the reduced velocity of the mobile zone or excluded mobile phase. The two reduced velocities are related by the expression:  e = (5) 1 − K and Ce = C(1 − K ). K is the fraction of total eluent that is stagnant, given by K =

εt − εe εt

(6)

where εt and εe represent the total porosity and interparticle porosity of the column packing, respectively. Knox and Scott provide detailed expressions for the B and C terms. The expression for B is



s Ds Dm

B = 2 m + k



∼ = 2m

1 30

 k 2 1 −  K 1 + k

(8)

sm K

where k is the zone capacity factor and  sm is the obstruction factor to diffusion in the stagnant mobile phase. The two capacity factors are related by the expression: k = k (1 − K ) − K

(9)

Rewriting Eq. (8) in terms of

(2)

This approach has also been applied to Giddings’ theory of nonuniform columns [8] for prediction of apparent plate height [18]. Detailed expressions for the apparent plate height of unretained solutes were reported in a previous article [19]. Similar expressions for retained solutes, in which linear velocity and retention are expressed in terms of mass flow rate and mobile phase density, are developed here. Our equation for apparent plate height requires an appropriate expression for the local plate height. The Knox equation [20] was developed for liquid chromatography with bonded phases on

(7)

where Ds , Dm and  s ,  m are the diffusion coefficients and obstruction factors in the stationary and mobile phases, respectively. If the effective diffusion coefficient in the stationary phase is very small relative to the mobile phase, which is a reasonable assumption for SFC, then B can be treated as a constant, as indicated in Eq. (7). Likewise, for Ds /Dm = 0, the C term is given by Knox and Scott as

(1)

where k0 is the retention factor at zero density, and a and b are constants [16,17]. We use k to represent the local retention factor to distinguish it from the observed or apparent retention factor k for a solute eluted from a nonuniform column, which is the temporal average of the local value [15], k = k

h=

Ce =

2. Theory

ln k = ln k0 − aR + bR2

spherical porous silica packings, and has the general form:

Ce =

1 − K 30sm K

  + k  2 K

k

and K we obtain, (10)

1 + k

Combining Eqs. (5) and (10), we obtain for the C term in Eq. (4): C = Ck

  + k  2 k

(11)

1 + k

where Ck =

1 30sm K

(12)

The applicable form of the modified Knox equation is then: h=



B K + k + A + Ck  1 + k

2 

(13)

Information obtained under near-uniform column conditions, where the retention factor and velocity do not vary significantly along the column, can provide information about the constants B, A and Ck .

D.P. Poe, J.J. Schroden / J. Chromatogr. A 1216 (2009) 7915–7926

The equation for the apparent plate height of a column operated under nonuniform conditions (e.g., a large density drop) is [18]:



hˆ =

2

h(1 + k ) 



1 + k



2   t



t

(14)

z

Application of this expression to SFC is facilitated by writing the local plate height expressions in terms of mass flow rate, density and temperature. The reduced velocity can be written as =

˙ 0 dp um dp m = Dm εt Dm

(15)

where um is the linear velocity of the mobile phase, dp is the particle ˙ 0 is the mass diameter, εt is the total porosity of the column, and m flow rate per unit area, defined as ˙ m ˙0= m A0

(16)

˙ is the mass flow rate and A0 is the cross-sectional area where m of the empty column. At constant temperature and mass flow, the resulting expression is

ˆ  h=

⎧  ⎪ ⎨ Dm 2 (1+k )2 t

1

1+k

2   t



z

⎫

  B 2 ⎬ + (1 + k )  A ⎪ t ˙ 0 /εt )dp (m

⎪ ⎩ +( + k )2 /Dm  C (m˙ /εt )dp K 0 t k

⎪ ⎭

(17) This equation provides an estimate of the apparent plate height for a packed column using compressible mobile phases with substantial pressure drops under isothermal conditions. For pSFC this condition is seldom if ever achieved in practice, due primarily to the cooling effect and generation of temperature gradients that result from expansion of the mobile phase. These effects are treated in a later section. The principle use of Eq. (17) in this paper is to provide an estimate of the contribution of the axial density gradient to the overall efficiency of a column operated under such conditions. 3. Experimental work 3.1. Apparatus The SFC system was constructed in our laboratory and has been described elsewhere [11]. It consisted of an ISCO model 260D syringe pump, a helium-actuated Valco CI4W injector with a 60-nL internal sample loop, and a Varian model 2740 gas chromatograph with flame ionization detector. The injector was placed in the column oven and a thermal conditioning coil was placed within the oven upstream from the injector to preheat the mobile phase to the column oven temperature. Pressure transducers were connected to tees placed immediately before the injector and at the column outlet, and the column outlet pressure was adjusted with nitrogen. A single 50-␮m i.d. fused silica integral restrictor, approximately 30 cm in length, connected the column outlet tee to the FID, was used over the entire range of flow rates to generate the plate height curve. A secondary restrictor, a heated length of small-bore stainless steel tubing, was provided to accommodate mobile phase flows exceeding the capacity of the FID restrictor. Further details of the flow and pressure control are provided in earlier papers [9,11]. Data were acquired using a high-speed chromatography data acquisition system (VG Data Systems Chromatography Server, 22-bit A/D conversion rate at 960 Hz) and Thermo LabSystems XChrom software. The sampling frequency was varied from 30 to 240 Hz depending on the flow rate to provide at least 30 data points over the half-width of the narrowest peak. All peaks were analyzed using the manual integration utility included with the software.

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Three columns packed with Spherisorb C8 (Waters Corporation, Milford, MA 01757, USA) were used. Particle diameters were 3, 5, and 10 ␮m. The column dimensions were 2.0 mm × 150 mm with stainless steel walls and fittings. For studies on the effects of thermal conditions, the columns were configured in one of two ways. For the thermostatted case, the steel walls of the column were exposed to the oven air in typical fashion. For the insulated case, the column was covered with fiberglass and foam pipe insulation and suspended in the oven as described earlier [10]. The column inlet and outlet temperatures were monitored with small button-style surface-probe thermocouples. 3.2. Chemicals Carbon dioxide was SFC grade with no helium. Methane was 99 mol% pure. Both were obtained from Scott Specialty Gases, Troy, MI, USA. A neat alkane mixture containing equal masses of n-dodecane, n-tetradecane, n-hexadecane and n-octadecane was obtained from Alltech Associates, Deerfield, IL, USA. Solutions of the alkanes in CO2 were prepared by introducing up to 500 ␮L of the neat alkane mixture into an open 150-cm3 stainless steel vessel, followed by gaseous methane to a gauge pressure of 1–2 bar. The vessel was then sealed with a pressure relief valve on one end, and liquid CO2 was introduced through a valve on the opposite end to a pressure of 120 bar at ambient temperature. 3.3. Chromatography A connection was made between the valve of the pressurized sample container and the sample port of the injector with a length of 1/16 in. o.d. stainless steel tubing [11]. Loading sample was accomplished by temporarily opening a valve connected to the waste port of the injector and allowing the pressurized sample to vent through a restrictor fabricated from a short length of 1/16 in. o.d. stainless steel tubing which was crimped on the outlet end. Isopycnic (constant density) plate height curves were generated by setting the desired flow rate at the pump and adjusting the inlet and outlet pressures independently to achieve the desired temporal average mobile phase density. Pressures were computed as described elsewhere [19]. The mobile phase in the syringe pump was maintained at −2.0 ◦ C, and the detector temperature was 250 ◦ C. Injections for each set of conditions were done in triplicate, and an equilibration time of 10–15 min was allowed after changes in the flow rate. Stable pressure readings and a constant temperature at the column outlet were taken to indicate steady-state flow and thermal conditions [9]. 4. Results 4.1. Computations and modeling For modeling purposes, mobile phase velocity at any given mass flow rate was computed assuming Darcy’s law applies. Accordingly, the velocity of the mobile phase can be written as [21]: um =

dp2 p L

(18)

where dp is the particle diameter, p is the pressure drop,  is the column resistance parameter,  is the viscosity of the mobile phase, and L is the column length. We computed  based on the measured velocity L/tm and the temporal average viscosity. The value for each column was obtained by averaging data obtained for all flows for each plate height study at the two densities. The resulting values were 552 ± 30, 640 ± 22 and 1317 ± 33 for the 3-, 5- and 10-␮m columns, respectively. We did not determine the reason for the relatively large value for the 10-␮m column.

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The local retention factor at each of the two densities was determined by extrapolating the observed retention factor to zero pressure drop. We assumed a linear dependence between these two densities, setting b = 0 in Eq. (1). Solute diffusion coefficients were computed using the relation [22]: Dm =

(8.6 × 10

−15

0.6 Veb

1/2 )TMs

(19)

Table 1 Solute data for the 5-␮m column. Carbon number

Retention datab 0

12 14 16 18 a

where T is the temperature in kelvins, Ms is the molar mass of the solvent (44.0 g/mol),  is the viscosity of the solvent in Pa s, Veb is the molar volume of the solute at its boiling point in cm3 /mol, and Dm has units of m2 /s. Viscosity calculations utilized the data of Stephan and Lucas as described elsewhere [14,23]. Molar volumes for the nalkanes were based on the data of Vargaftik [24]. A summary of the solute data, including the Knox coefficients discussed below, appears in Table 1.

Veb a

b c

287.18 340.75 396.14 453.34

Knox coefficientsc

ln k

a

B

A

Ck

4.928 5.694 6.440 7.197

3.816 4.164 4.509 4.867

1.342 1.398 1.346 1.441

1.241 1.289 1.387 1.343

0.4021 0.3747 0.3051 0.2797

Molar volume in cm3 at the boiling point. Eq. (1). Eq. (13).

Predicted plate height curves at the two reduced densities were generated using Eq. (17) in the theory section. Values for the coefficients B, A, and C in Eq. (3) were obtained by fitting data obtained at r = 1.5 to Eq. (13). Values of Ck in Eq. (17) were computed as

Fig. 1. Separation of alkane mixture using 10-␮m particles at 50 ◦ C at two densities with and without thermal insulation. Conditions for each chromatogram (A, B, C): reduced density (1.0, 1.0, 1.5); thermal condition (air bath, insulated, air bath); flow rate, mL/min (1.200, 1.200, 0.500); Pin , bar (113.0, 113.0, 153.0); Pout , bar (100.8, 100.8, 147.3). Other conditions given in Section 3.

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Fig. 2. Separation of alkane mixture using 5-␮m particles at 50 ◦ C at two densities with and without thermal insulation. Conditions for each chromatogram (A, B, C): reduced density (1.0, 1.0, 1.5); thermal condition (air bath, insulated, air bath); flow rate, mL/min (1.265, 1.265, 0.492); Pin , bar (118.4, 118.5, 155.9); Pout , bar (93.5, 93.5, 144.9). Other conditions given in Section 3.

described in the theory section. The value for K was obtained from Eq. (6). The total porosity of 0.70 for the 5-␮m column was measured by pycnometry. We assumed an interparticle porosity of 0.4, yielding K = 0.43. This value was used for all three columns. 4.2. Retention, efficiency and temperature gradients At low flow rates and pressure drops, separations on all three columns yielded good chromatograms with well-shaped peaks for both mobile phase densities. At the higher density peak shapes were generally good over the entire range of flow rates, up to about 2 mL/min. At the lower density peak shapes and efficiency became progressively worse as the flow rate was increased. Representative chromatograms for elution of the alkane mixture at high flow rates are shown in Figs. 1–3 for columns filled with 10,

5 and 3-␮m particles. In each figure, the top two chromatograms were obtained at reduced density 1.0 and at the same flow rate, with and without thermal insulation. For separations on the thermostatted columns, the peaks become broad and distorted as the particle size decreases. For the 3-␮m column, the peaks are severely distorted and overlapping. In this case the last eluting peak, labeled C18 for n-octadecane, reaches a maximum at about 56 s and extends well beyond the displayed window to approximately 140 s. For all columns at reduced density 1.0 the addition of thermal insulation results in a decrease in the retention time of each solute, and peaks are relatively narrow and well-formed, especially for the 10- and 5-␮m columns. For the 3-␮m column the peak shapes are greatly improved, but some significant tailing is observed. Chromatograms for the same separation performed at reduced density 1.5 are also shown in Figs. 1–3 for comparison. At this

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Fig. 3. Separation of alkane mixture using 3-␮m particles at 50 ◦ C at two densities with and without thermal insulation. Conditions for each chromatogram (A, B, C): reduced density (1.0, 1.0, 1.5); thermal condition (air bath, insulated, air bath); flow rate, mL/min (1.245, 1.245, 0.514); Pin , bar (129.1, 129.0, 163.2); Pout , bar (73.8, 73.9, 138.6). Other conditions given in Section 3.

pressure the addition of thermal insulation had no obvious effect, and the chromatograms shown were obtained in the thermostatted mode with no insulation on the column. In each case a flow rate was selected so that the overall separation time is approximately the same as for the insulated case at the lower density. The apparent retention factor k for each retained solute was computed assuming that methane is unretained [9]. The effect of flow rate on the retention factor of n-octadecane is shown in Fig. 4. At the lower density especially, significant decreases in the retention factor occur with increasing flow rate. The corresponding plate height curves for elution of noctadecane appear in Figs. 5–7. We plot the apparent reduced plate height against the average reduced velocity, where: ˆ H hˆ = dp

(20)

is the apparent reduced plate height, and ¯ =

Ldp



tr Dm



(21) t

ˆ is the apparent plate height, dp is the average reduced velocity, H is the particle size, L is the column length, and Dm is the diffusion coefficient of the solute in carbon dioxide at the average mobile phase density [22]. The apparent plate height was calculated as ˆ =L H

wh2 5.54tr2

(22)

where tr is retention time and wh is the peak width at half height. In many cases this equation yields only a rough estimate of the plate height because it assumes Gaussian peaks, which many of these clearly are not.

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Fig. 5. Efficiency for elution of n-octadecane on the 10-␮m column. Conditions same as for Fig. 1. Legend lists reduced density (1.0 or 1.5) and thermal condition (Air = air bath; Ins = insulated; Pred = predicted). Error bars are shown where standard deviation is greater than 0.1.

Fig. 4. Change in retention factor with reduced velocity for each of the three columns. Conditions given in Figs. 1–3. The horizontal reference lines mark the extrapolated values of the retention factor at zero flow rate for each density.

For all three columns the experimental plate height curves at the higher density are similar to what one might expect for a wellpacked column in HPLC. The curves at this pressure were nearly identical whether or not thermal insulation was used, and only the curve for the thermostatted case is shown. For the 3-␮m column thermal insulation showed small improvements in efficiency at high velocities (less than 5%, not shown). At the lower density catastrophic efficiency losses are observed at moderate to high velocities for the thermostatted case. The observed plate heights are much greater than predicted. The addition of thermal insulation yields significant improvements, especially for the 10-␮m column. For the 5-␮m column thermal insulation yielded peaks that were well-formed with asymmetry factors less than 1.1 except at the highest velocities where some distortion was observed. For the 3-␮m column with thermal insulation the plate heights are still quite large and the plate height curve has an irregular shape. Even though the peak shapes are greatly improved, the overall performance of the insulated 3-␮m column at the lower density is poor. The decrease in the apparent plate height

Fig. 6. Efficiency for elution of n-octadecane on the 5-␮m column. Conditions same as for Fig. 2. Legend items and error bars defined in Fig. 5.

above reduced velocity 3 is an artifact of the measurement method, which is based on the width at half height. Peaks showed significant distortion over the entire range of flow rates. At the highest flow rate, n-octadecane produced a broad peak with a flat plateau. The temperature drop along the column, as measured by sensors placed on the inlet and outlet connectors, is shown in Fig. 8. The values are outlet minus inlet temperature, and must be considered as only approximations of the actual temperature drop because the probes were not in direct contact with the mobile phase. Except for flow rates under 1 mL/min on the 5- and 10-␮m columns at the

Table 2 Properties of carbon dioxide at 50 ◦ C at some representative pressures used in this study. Reduced density

Compressibility as d/dp (g cm−3 bar−1 )

Viscosity (Pa s) (×105 )

Pressure drop (bar)

Average pressure (bar)

Inlet density (g/cm3 )

Outlet density (g/cm3 )

1.0

1.15 × 10−2

3.48

0 15 30

106.9 106.6 105.5

0.468 0.541 0.589

0.375 0.289

0 15 30

150.1 150.7 151.2

0.702 0.720 0.736

0.684 0.662

1.5

2.43 × 10−3

5.62

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Fig. 7. Efficiency for elution of n-octadecane on the 3-␮m column. Conditions same as for Fig. 3. Legend items and error bars defined in Fig. 5.

higher density, significant temperature drops are observed under almost all conditions. For a given column, the largest temperature drops occur at the lower density, and are enhanced by the presence of thermal insulation on the column. At a given flow rate, the magnitude of the temperature drop increases with decreasing particle size. 5. Discussion 5.1. Compressibility, pressure drop and flow rate The two mobile phase densities used in this study were <r >t = 1.0 and 1.5, where r = /c is the reduced density and c = 0.468 g/cm3 is the critical density of carbon dioxide. At the lower density the average column pressure decreases from 106.9 bar at zero pressure drop to 105.5 bar and a density drop of 0.300 g/cm3 for a 30-bar pressure drop. The higher reduced density of 1.5 corresponds to a pressure of 150.1 bar, and the average column pressure at a 30-bar pressure drop is 151.2 bar with a density drop of only 0.074 g/cm3 . Table 2 provides a brief listing of representative pressures and densities for this study.

Fig. 8. Temperature drop vs. flow rate for each of the three columns. Temperatures measured at surfaces of end fittings. Flow rates are for liquid mobile phase at the pump. The particle size in microns is given in the top panel. Conditions given in Figs. 1–3. Legend items defined in Fig. 5. Corrections for bias in one of the thermometers were applied to data for two datasets at r = 1.5 (3-␮m air-thermostatted and 5-␮m insulated).

Fig. 9. Pressure drop vs. flow rate for the 5-␮m column at reduced densities 1.0 and 1.5. Legend items defined in Fig. 5.

For a given flow rate the pressure drop is greater at the higher reduced density, as shown in Fig. 9. This difference is clearly due to the increased viscosity at the higher density. The plot for the insulated column at reduced density 1.0 is slightly nonlinear, showing a small increase in slope as the flow rate increases, suggesting a slight increase in viscosity with flow rate. This small increase is relatively insignificant, but deserves some comment. In order to maintain a constant temporal average density the computational model requires a slight decrease in the simple average column pressure from 106 bar at low flow to 102 bar at the highest flow rate. This decrease in pressure is accompanied by a predicted increase of approximately 3% in the temporal average viscosity assuming isothermal conditions. Under experimental conditions cooling of the mobile phase at high flows, especially for the insulated column, would also lead to increased viscosity and larger pressure drops. 5.2. Axial temperature gradients and retention Theoretical treatments indicate that in pSFC there will be both frictional heating and cooling due to adiabatic expansion. Model calculations by Schoenmakers et al. [25] indicate that under typical conditions cooling outweighs the heating, and a temperature drop may exist from the column inlet to the outlet. For the systems they studied, the temperature drop was relatively small, typically only a few tenths of a degree Celsius. Our results for columns packed with 5- or 10-␮m particles at the higher density do show relatively small temperature drops, but at the lower density the temperature drops are much greater. The 3-␮m column produced large temperature drops at the higher flow rates under all conditions used in this study. The influence of the axial temperature gradient on the retention factor can be seen clearly in Fig. 4. At r = 1.0 our model predicts that under isothermal conditions the retention factor should increase as the flow rate increases while maintaining the temporal average density constant. This effect is due entirely to the pressure drop, and has been reported previously [12]; for small to moderate pressure drops the retention factor was approximately constant at a given temporal average density. In our study, we observe significant decreases in the retention factor as the flow rate increases. This behavior is consistent with the presence of a negative axial temperature gradient. The mobile phase density near the outlet, and the temporal average density, are therefore greater than the prescribed value, leading to decreased retention. The decrease in retention is enhanced by addition of thermal insulation. Although somewhat smaller in magnitude, a decrease in retention factor at higher flow rates is also evident at the higher density.

D.P. Poe, J.J. Schroden / J. Chromatogr. A 1216 (2009) 7915–7926

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We consider a typical packed SFC column operated at constant mass flow rate in which the column wall is thermostatted at the temperature of the incoming mobile phase Te . The heat balance equation for an infinitesimal volume element of the column, written as the change in temperature (T) with time (t), is [25]: ∂ ∂T = ∂t ∂z −

Fig. 10. Ratio of predicted apparent plate height (Eq. (17)) for n-octadecane at reduced density 1.0 to the predicted plate height of the same column under uniform conditions (Eq. (4)) at the same density for the three columns used in this study. Knox constants given in Table 1. A horizontal reference line marks the value of one.

The predicted effect of the density drop on efficiency for elution of n-octadecane under isothermal conditions on each of the three columns used in this study is shown in Fig. 10. Each curve shows the ratio of the apparent plate height computed from Eq. (17) to the plate height for the same column operated at the same mobile phase density but without a density gradient. Over the range of reduced velocities used in this study, the predicted apparent plate height due to the density gradient alone is never more than twice the plate height for the column operated under uniform conditions. The experimentally observed apparent plate heights, which are based on the width at half height and may thus underestimate the value for distorted peaks, are in some cases many times greater than the predicted values, as illustrated in Figs. 5–7. We refer to this difference as the excess efficiency loss. In our previous studies on the unretained solute methane, we attributed this excess efficiency loss to the presence of radial temperature gradients [9].

5.3. Radial temperature gradients and efficiency The large axial temperature drops observed for the 3- and 5-␮m columns (Fig. 8) will result in the generation of radial temperature gradients if the column wall is thermostatted. The magnitude of the radial temperature gradient should be the principle factor leading to excess efficiency loss. A recent study on the effects of viscous heat dissipation in HPLC [26] indicates that the impact of the radial temperature gradient on efficiency can be accounted for in the plate height equation by inclusion of an Aris dispersion term that depends on the magnitude of the radial profiles of the mobile phase velocity and radial diffusion coefficient, both of which depend on the radial temperature profile. The supporting equations are quite complicated and estimation of the apparent plate height requires numerical computation. A complete mathematical description of the temperature and density profile of a column in SFC can be provided by solution of the appropriate heat balance equation, and in principle the impact on efficiency evaluated in a similar fashion. However, the situation is even more complicated due to added cooling effects associated with expansion of the mobile phase. We therefore desire an approximate solution for the magnitude of the radial temperature gradient, in the form of a simple analytical expression, that would provide a reasonable estimate of this important parameter.



lon ∂T c ∂z



+

1 ∂ r ∂r



rad r ∂T c ∂r



−

u0 m cm ∂T c ∂z

u0 ∂p u0 p ∂m − c ∂z m c ∂z

(23)

where lon and rad represent the average effective heat conductivities of the column content in the longitudinal (z) and radial (r) direction, respectively; c is the volume average heat capacity of the heterogeneous system; u0 is the superficial velocity; m , cm , and m are the density, heat capacity, and molar volume of the mobile phase; and p is the pressure. The last two terms on the rhs represent viscous heat dissipation and adiabatic expansion. The viscous heat dissipation term takes on a positive sign with the negative pressure gradient, resulting in heating, whereas the adiabatic expansion term results in cooling. The net effect will depend on which term is dominant. Modeling of analogous HPLC systems [27–29] has shown that the radial temperature gradient is much larger than the longitudinal gradient, and that at large enough distance z the temperature is invariant along the column axis. Although the case for SFC is complicated somewhat by the axial variation of the density and related parameters, we neglect the ∂2 T/∂z2 and ∂T/∂z terms in Eq. (23), as Poppe does for HPLC [30], on the basis that they should be much smaller than the radial terms. We can eliminate the complicated dependence on z for the variables in the last two terms on the rhs by working with a virtual system in which these variables assume some average value and are invariant along the axis of the column. The equation can then be written in a single variable r, and after expanding the radial term and rearranging, the differential equation can be written in simple form as c

dT = rad dt

1 dT d2 T + r dr dr 2



− u0

u0 p dp − m dz

∂m ∂z



(24) T

where the last two terms are treated as constants. In particular, the ∂m /∂z term is evaluated at an appropriately selected temperature and density. Noting that (∂/) = −(∂/) and introducing the pressure variable, we obtain: dT c = rad dt

1 dT d2 T + r dr dr 2



dp − u0 dz

p ∂m 1− m ∂p



(25) T

This equation has the same general form as the one given by Poppe for HPLC, and the corresponding solution is

T (r, z = ∞) = Te + 

1−

r2 R2



(26)

where R is the radius of the column and  is the maximum temperature difference between the mobile phase at the column wall and the center of the column. The value of  is given by =

u0 R2 (dp/dz)(1 − (p/m )(∂m /∂p))T 4 rad

(27)

If there is net cooling due to expansion of the mobile phase the difference in parentheses becomes negative and Eqs. (26) and (27) predict a parabolic temperature profile with the temperature at the axis of the column lower than the column wall. Corresponding radial gradients in the mobile phase density and viscosity will result in a parabolic flow profile with fluid velocities highest near the column wall. The temperature and density profiles will also

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The results for all four solutes on the three different columns are listed in Table 3. The data listed are the middle value and range defining the window of u2m /dp2 values for the onset of excess efficiency loss for each solute, obtained by visual estimation as described above. For a given solute, the onset of excess efficiency loss occurs at approximately the same value of um /dp for all columns, but this value varies with the solute retention factor. Lower retention correlates with higher velocities for the onset of efficiency loss, and corresponding larger radial temperature gradients. A recent study on the effects of viscous heat dissipation in HPLC predicts some dependence of radial dispersion on the retention factor [26]. 5.4. Effect of thermal insulation

Fig. 11. Reduced plate height for n-octadecane vs. the square of the ratio of mobile phase velocity to particle diameter for the three columns. Conditions for experimental curves given in Figs. 1–3. Legend items defined in Fig. 5. The predicted curves include effects of density gradients under isothermal conditions. The vertical lines bracket the region of the onset of excess efficiency loss.

result in a radial variation in the solute retention factor which is not accounted for in this treatment. The approximate solution given by Eq. (27) may provide some insight into the relation between the radial temperature gradient and the onset of excess efficiency loss. According to Darcy’s law, the local pressure gradient is dp u0 = dz εt dp2

(28)

where  is the flow resistance parameter and  is the viscosity, which is a function of temperature and density [23]. Combining Eqs. (27) and (28), and replacing u0 with um /εt yields: =

 4 rad ε2t

u2m R2 dp2



p ∂m 1− m ∂p



(29) T

For columns of the same radius with CO2 mobile phase at a given average density and temperature, the magnitude of the radial temperature gradient is thus proportional to the square of the ratio um /dp . Fig. 11 shows plots of the observed and predicted apparent plate height for n-octadecane on all three columns plotted against u2m /dp2 . The onset of excess efficiency loss for each column was estimated by visual inspection to occur at the velocity where the slope of the experimental HETP curve begins to increase significantly relative to the predicted curve for the same particle size. The range for all three particle sizes is represented by the window defined by the two vertical lines in the figure. The increase in HETP occurs over a narrow range of u2m /dp2 values, and according to Eq. (29), at approximately the same value of  for all three columns. Although the dataset in the velocity range is limited, these results clearly suggest that the radial temperature gradient is directly correlated with the onset of excess efficiency loss.

Table 3 Values of u2m /dp2 at onset of excess efficiency loss. Solute carbon number

u2m /dp2 (×10−6 )

12 14 16 18

3.14 2.20 1.93 1.58

± ± ± ±

0.50 0.45 0.18 0.19

Average k (values for 3, 5, 10-␮m column) 2.24 (2.25, 2.45, 2.02) 3.55 (3.64, 3.94, 3.08) 5.22 (5.40, 5.85, 4.41) 7.68 (7.96, 8.58, 6.50)

The dramatic improvement in efficiency obtained by the use of thermal insulation has been reported previously [10], and was attributed to the elimination of radial temperature gradients. The improvement in efficiency far exceeds that observed in similar HPLC experiments [27,29]. The present study examines the effect of thermal insulation in SFC in greater detail. While the addition of thermal insulation causes a larger axial temperature gradient, it should also effectively eliminate the generation of a radial gradient. The very significant improvements in efficiency at the lower density are probably due in large part to the effective elimination of the radial temperature gradient. The recovery of the excess efficiency loss is evident for all three columns, but to different degrees. It is notable that, for the 10-␮m column, the efficiency for the insulated column is significantly better than the predicted efficiency for an isothermal column. It appears that in the absence of a radial temperature gradient, an axial temperature gradient may actually lead to improved performance. This is in fact consistent with the theory of nonuniform columns. Eq. (14) indicates that for isothermal conditions large variations in mobile phase density and retention factor are primary factors leading to increases in the apparent plate height. Relative to isothermal conditions, an axial temperature drop would result in smaller changes in the density. If the retention factor is controlled primarily by mobile phase density (LC-like), and not so much by temperature (GC-like), axial variation in the retention factor should be minimized by a negative temperature gradient. From an efficiency standpoint, an SFC column with negative axial temperature gradient is thus more uniform, and therefore more efficient. Other experiments where long columns were subjected to an externally imposed temperature gradient have also led to improved efficiency [31]. The observed efficiency for the 10-␮m column with thermal insulation does show significant improvements in efficiency over the predicted efficiency for an isothermal column (Fig. 5). For the 5-␮m column the two curves almost overlap (Fig. 6) up to a point at higher velocity where a large efficiency loss occurs. For the 3␮m column the efficiency for the insulated column worse than the predicted isothermal case at all flow rates used. These results suggest that the thermal insulation does not completely eliminate the radial temperature gradients. The relatively high thermal conductivity of stainless steel may result in a smaller temperature gradient along the wall than exists in the bulk packing, leading to a temperature difference between the wall and bulk packing [27]. This is one potential explanation for the large efficiency losses observed for the 3-␮m insulated column. Finally, we examine the effect of thermal insulation on the overall separation performance. The chromatograms in Figs. 1–3 were selected so that the overall separation time is approximately the same in all cases. The chromatograms for the 10-␮m column show adequate resolution at the higher density, but the resolution is much better at the lower density. The improvement is a result of greater retention and increased flow rate at the lower density. For

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The deleterious effects of radial temperature gradients can be partially or completely eliminated by the use of thermal insulation on the column. In some cases, with larger particles and smaller pressure drops, the observed efficiency is better than what is predicted for an isothermal column. The larger axial temperature gradient induced by thermal insulation can lead to improved efficiency because the density gradient is minimized. In such cases, especially for weakly retained solutes, significant improvements in column performance can be achieved by use of a thermally insulated column. Nomenclature

Fig. 12. Resolution vs. time for elution of n-hexadecane and n-octadecane on the 5-␮m column. Conditions same as in Fig. 2. Legend items defined in Fig. 5.

the thermostatted 5-␮m column the improvement in resolution at the lower density is negated by the excess efficiency loss caused by radial temperature gradients, whereas a significant improvement in the efficiency is gained by using thermal insulation. Fig. 12 shows the resolution vs. time for the separation of n-octadecane from nhexadecane on the 5-␮m column. Except possibly at very high flow rates, the best performance is obtained using thermal insulation at the lower density. For situations in SFC where temperature gradients are a problem, or for solutes that are weakly retained at high mobile phase density, the significant improvements in resolution per unit time can be provided by use of a thermally insulated column at lower mobile phase densities. 6. Conclusions Efficiency losses associated with large pressure drops in pSFC arise from primarily three different sources. These include: • The normal chromatographic band broadening processes. These are described by the van Deemter equation and related expressions for local plate height, such as Eq. (3). • The axial density gradient that results from the pressure drop. This source of band spreading is present when using any compressible mobile phase, including primarily GC and SFC. This effect has been treated by the general theory for nonuniform columns developed by Giddings [8], and the observed plate height for such systems is normally referred to as the apparent plate height. For SFC under isothermal conditions it is described by Eq. (14). For the system in our present study, the predicted apparent plate height reached twice the plate height for a uniform column. • The generation of radial temperature gradients. In SFC near the critical point, large pressure drops result in axial temperature gradients due to expansion of the mobile phase. If the column wall is thermostatted, a radial temperature gradient may develop resulting in excess efficiency loss. The increased plate height can be many times greater than the increase due to density gradients alone. For a given solute, preliminary data indicate that the onset of excess efficiency loss occurs at a fixed value of u/dp for a fixed column diameter. The square of this ratio is proportional to the magnitude of the radial temperature gradient based on an approximate solution for the heat balance equation. A simple expression is proposed to estimate the magnitude of the radial temperature gradient and predict the onset of excess efficiency loss.

a, b fitting constants for retention equation cross-sectional area of empty column A0 B, A, C, Ce van Deemter or Knox coefficients Ck Knox C term for SFC Dm , Ds diffusion coefficient in: mobile phase, stationary phase dp particle diameter ˆ H ˆ h, h, reduced plate height, apparent reduced plate height, apparent plate height apparent retention factor, local retention factor, zone capacity factor k0 retention factor at zero density L column length ˙ m ˙0 m, mass flow rate, mass flow rate per unit area of empty column Ms molar mass of solvent (g/mol) p, p pressure, pressure drop r, R radial position in column, radius of column T, Tr , Te temperature in kelvins, reduced temperature, temperature of incoming mobile phase and column wall time, elution time for mobile phase t, tm u0 superficial velocity Veb molar volume at normal boiling point z axial position in column

lon , rad average effective heat conductivity of the column content in the longitudinal and radial directions c volume average heat capacity of the heterogeneous column content  flow resistance parameter  viscosity  radial temperature difference at large z between the column wall and center of the column  t,z temporal (t) or spatial (z) average of quantity in brackets reduced velocity of: mobile phase, mobile zone , e , r , m density, reduced density, density of mobile phase K fraction of total eluent in column that is stagnant m molar volume of the mobile phase  m ,  s ,  sm diffusion obstruction factor in: mobile phase, stationary phase, stagnant mobile phase εt , εe total porosity, interparticle porosity k, k , k

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