High Performance Liquid Chromatography (HPLC) is a dominating separation technique routinely used in the industry due to its essential valuable characteristics, comprised of well-understood separation mechanisms, ease of use, sensitivity, selectivity and robustness. This technique has many applications including separation, identification, purification, and quantification of numerous compounds. Nevertheless, despite of all its worthy attributes, HPLC lacks high efficiency compared to GC and CE [1]. In addition, the demand of high-throughput analyses for fast screening of numerous pharmaceutical samples necessitates high-speed LC.

Different approaches have been explored in order to improve HPLC efficiency and speed of analysis. Successful results were achieved with the use of high temperature LC, monolithic columns and ultrahigh pressure. Due to lower viscosity and high diffusivity of mobile phase, high column temperatures allow operation at high flow rates on longer columns or smaller particles, resulting in higher efficiency and faster analysis. The limitations of this technique are uncertainties in column stability and possible on-column degradation of analytes. Monolithic columns offer high efficiency and high speed due to high porosity and small skeleton size which allow operation at high flow rates on longer columns [2, 3]. However, monolithic columns exhibit limited stability at higher pressures, temperatures and pH. In addition, the limited variety of monolithic columns does not meet the demands for a wide range of industrial applications [4].

Recent developments in the commercial manufacture of instruments that are able to produce and withstand pressures as high as 18,000 psi, along with the introduction of a new generation of columns packed with sub-2μm porous particles, have made it possible to achieve faster separations than in HPLC, while maintaining or increasing efficiency. The factors influencing chromatographic efficiency in Ultra High Performance Liquid Chromatography (UHPLC) that uses sub-2μm porous particles and the means to improve efficiency are reviewed herein.

2. Chemistry of small particles

Optimal separations in liquid chromatography mean symmetrical and sharp peaks. The following processes that happen inside the column are responsible for the band broadening [3]:
1. Eddy diffusion: multiple paths of different length and velocity;
2. Resistance to mass transfer: equilibration rate of the analyte between the stationary and mobile phase. In the case of high velocity of the mobile phase and strong affinity of the analyte for the stationary phase, the analyte in the mobile phase moves faster than the analyte in the stationary phase.
3. Longitudinal diffusion: diffusion of the analyte in the mobile phase.

In order to achieve optimal separations, band broadening must be minimized. Therefore, using small particles packings would reduce factors 1 and 2 and offer narrower bands and faster equilibration, and hence improve efficiency. This is also supported by the Van Deemter equation (1., 2., 3.), an empirical formula that describes the relationship between linear velocity, u (flow rate), and plate height, H (HETP or column efficiency).

1.[5]
or simplified 2.[1]
or simply,
3. H = A + B/u + Cu
where A represents the term for multiple paths, B - longitudinal diffusion in the mobile and stationary phases, and C - resistance to mass transfer in the mobile and stationary phases. The plate height is dependent upon the particle diameter so that the most significant impact on the efficiency would be introduced from the C-term since it is dependent upon the square of the particle diameter. The effect of particle diameter on plate height is demonstrated with the aid of Van Deemter plots (Figure 1.). Figure 1.[1]It is evident that when the particle size is reduced, the minimum plate height is decreased and the optimum velocity (point on the curve at the minimum plate height) is increased [1]. Then optimum flow rate is inversely proportional to particle size: Fopt ∞ 1/dp [6]. The flat part of the curve at higher linear velocities signifies that operating at the velocities above the optimal value results in an insignificant decrease in efficiency. Hence, columns with smaller particles can be used at higher linear velocity without sacrificing efficiency for analysis time. This suggests that the particle diameter should be decreased as much as possible to obtain the highest resolution [1], since N is inversely proportional to particle size, N ∞ 1/dp [6], and the resolution is proportional to the square root of N (4.). 4.[7]
Since N is inversely proportional to the square of the peak width, N ∞ 1/w^2, and peak height is inversely proportional to the peak width, Hp ∞ 1/w [6], then with the particle size decrease, and subsequent plate count N and resolution Rs increase, higher sensitivity is obtained as narrower peaks are taller. Narrower peaks also mean higher peak capacity in gradient separations.
It is also worth mentioning that efficiency is proportional to column length and inversely proportional to the particle diameter, N ∞ L/dp [6]. In this case, for a given efficiency, the column length can be reduced by the same factor as the particle size to achieve faster separations without sacrificing resolution.

3. Effects of packing materials (porous versus nonporous) and efficiency of particle packing on column Efficiency

Modern column technology offers columns packed with porous and nonporous sub-2μm particles that are able to withstand high temperatures and high pressures, and are stable under acidic (pH < 1) and basic (pH > 11) conditions, which is essential in UPLC. An advantage of nonporous particles over porous particles is faster mass transfer due to the absence of sample diffusion in the stagnant mobile phase within the pores which results in reduction of band-broadening. Therefore, nonporous particles offer higher efficiency at high linear velocity, but the efficiency difference diminishes significantly when the particle size is reduced (to 1.5μm) [8]. An advantage of porous particles over nonporous is loading capacity which is 15 times higher for porous particles. Also, the average retention factors are higher for porous particles than nonporous, and pressure drops are smaller for porous particles than for nonporous [9]. In addition, nonporous particles are highly sensitive to extra-column contributions due to the low total surface area of the packing material in the column [10]. Figure 2. [10]
Waters Acquity BEH C18 column (50mm x 2.1mm, 1.7μm, pore diameter 135A)
Higher reduced plate heights, h, are commonly observed in experimental determinations, compared to theoretically predicted with Knox equation (5.), on columns of i.d.s below the standard 4.6mm, or packed with particles below 5μm due to low packing efficiency. 5.[1]
This is attributed to complications associated with producing and packing of 1.7μm particles in 2.1mm i.d. columns [1]. Increased bed heterogeneity results in increase in A-term and C-term constants which in its turn increase reduced plate heights [12]. Packed capillary columns with uniformly sized nonporous particles have a very tight and ordered packing bed structure, but columns with porous packing material with a wide size distribution have extensive nonuniformity as shown by scanning electron micrographs [10].

4. High pressure and its effect on Efficiency

Since the pressure drop, required to achieve optimal linear velocity, is inversely proportional to the square of the particle diameter (6.), high pressure drop across a column is expected for smaller particles. So that high pressure is required to operate an instrument with small particle size columns. 6.[3]
Aside from the engineering challenges associated with the use of high pressures in liquid chromatography, at high flow rates the friction of the mobile phase against the stationary phase generates heat [12]. The rate of heat generation, (power dissipation) is the result of pressure drop and flow rate: 7.[3]
The heat, convectionally transported by the mobile phase along the column, conducts in the axial and radial directions, and evacuates into the air through the column walls and end-fittings [14]. After asymptotically reaching a steady-state thermal equilibrium, the column temperature remains constant everywhere in the column and doesn't depend on time any more. At this point, the generated heat evacuates the column, producing radial and axial temperature gradients [13]. In the axial direction, the temperature increases from the column inlet to the outlet, and in the radial direction, the temperature decreases from the center to the column wall [14]. Figure 3.[15]
The amplitude of these gradients depends on the degree of column's thermal insulation. The longitudinal temperature gradient is the greatest when the column temperature is radially uniform (the smaller the radial heat loss, the smaller the radial temperature gradient, and the larger the axial temperature gradient). This happens when there is no radial heat loss through the wall (the column is kept adiabatic), and the generated heat is evacuated through the column ends. The radial temperature gradient is the greatest when the column wall temperature is kept constant [13, 16]. This can be achieved by placing the column in a liquid bath, so that the generated heat is evacuated through the column wall. Normally columns are operated under intermediate conditions of thermal insulation where radial and axial temperature gradients co-exist [13].

Temperature gradients have negative impact on the column efficiency due to heterogeneous distribution of the mobile phase linear velocity, viscosity and density throughout the column, which affect the equilibrium constant of a compound distribution between mobile and stationary phases (decreases with increasing temperature), and retention factors of the analytes. The difference between the temperatures at the inlet and the outlet of the column can be as much as 20K, and the difference between the temperatures of the center and the column wall can reach up to 6K, if the column is operated under regular convection conditions. The radial temperature gradient increases slowly along the column [17].

When the heat effects are significant and the radial thermal gradient is not flat, the classical theory of band broadening derived by Van Deemter (1., 2., 3. and Figure 1) is no longer accurate. More appropriate models are required to account for band spreading due to the combination of the radial dispersion (radial gradient of the retention parameters - since adsorption equilibrium constants decrease with increasing temperature, then retention factors are smaller in the central region of the column than near its wall) and radial gradient of the band velocity (due to radial viscosity distribution - since viscosity is higher in the colder region against the wall than in the warmer central region) that results from the radial temperature gradient. [The axial temperature gradient has only minor consequences since the longitudinal distribution of the mobile phase viscosity along the column affects the local pressure gradient that decreases along the column. This affects the efficiency only insignificantly [13]].

A few approaches were taken to calculate the influence of the combination of an axial and a radial gradient of temperature on the column efficiency [14, 16, 18, 19], but they were approximate. The correct prediction required the simultaneous solution of the heat and mass balance equations which was attempted and solved with the aid of 3 models: 1) a model of heat transfer, 2) a model of mass transfer, and 3) a model of the distribution of the mobile phase velocities [14, 17].
Since the detector measures the average concentration in the effluent from the column, and the radial concentration profiles of the analytes form an arc in the column, due to efficiency is lost due to radial temperature distribution that causes radial band velocity profile, the loss of efficiency is observed. When the temperature of the external column wall is kept constant, the efficiency loss is the largest [13] with the maximum amplitude of the radial temperature gradient (ΔT) between the column wall and its center can be expressed as: 8.[3]
This equation (8.) suggests that the frictional heating effect can be reduced by decreasing the column diameter. It was shown that the efficiency losses can be kept to a minimum for particle diameters of 1.5–2.0 μm if column i.d.s of < 2mm are used [10].

Columns should be operated under still-air conditions (inside the oven compartment of the instrument) since it offers better efficiency compared to under thermostated within a liquid bath conditions. At high flow rates, the shorter column should be used for better performance (under still-air conditions), and at optimal flow rates, longer columns will yield very high efficiency [16].

5. Extra-column band broadening and its effect on Efficiency

As mentioned above (chemistry of small particles), when particle size decreases, resolution increases as a result of narrower peak widths. From the other side, the narrower peaks are more susceptible to extra-column dispersion. The smaller the column diameter and the shorter the column, the smaller is the peak volume. Therefore, extra-column band broadening must be reduced in order to obtain efficient separations [12]. There are two main types of extra-column contributions: 1) volumetric - results from the volume of connecting tubing from the injector to the detector, the injection volume and the detector volume; and 2) time-related events - sampling rate and detector time constant [20].

Extra-column contributions are additive to the length variance of the peak in the column, σc. The sum of the volumetric and time-related extra-column contributions is expressed in equation 9. 9.[20]
The volumetric extra-column contribution can be decreased with increasing column length, column radius and retention factor, and the time-based extra-column contribution can be decreased with the column length and with increasing linear velocity [20]. In addition, optimization of connecting tubing, injection volume, flow cell volume and geometry leads to maximum performance of a column. For example, it was determined that for a typical UPLC porous particles column, the maximum injection volume is 8μL, but if other extra-column volume is taken into account (> 3μL from connection tubing and flow cell), the injection volume should be less than 5μL [3]. As shown in equation 10., reductions in the injection volume Vinj, the flow cell volume, Vcell, and column to detector connecting tubing internal radius, rc, would lead to improvements in efficiency.10.[8]
The volumetric contributions are also affected by temperature changes due to the change in the analytes's diffusion coefficient. The band spreading decreases with the temperature increase [20].

6. Temperature and its effect on Efficiency

(Note: the discussion is limited to elevated temperature (up to 90C), and not high temperatures).
Temperature reduces mobile phase viscosity and thus allows higher flow rates at the same pressure. Higher temperature also increases analyte diffusivity which decreases the mass transfer term (C-term), but increases the molecular diffusion term (B-term) of the Van Deemter equation (3.). A drawback of the temperature increase is that it increases linear velocity. That requires increased flow rates to avoid a loss in performance. Increasing flow rates, in its turn, will result in the pressure increase. This suggests that increasing temperature will not reduce backpressure. In any case, the advantage of the elevated temperature is the reduction in the ability to significantly shorten the analysis time with minimal loss in column efficiency [20].

It was shown that the elevation of column temperature while operating at constant flow rate below 0.4mL/min (below Van Deemter optimum) will result in the loss of efficiency. Higher flow rates should be used in order to maintain the same efficiency at the elevated temperatures [20]. In addition, experimental data demonstrated that at higher temperatures, certain efficiency can be achieved with bigger particle size (2.5μm), but not with smaller particle size (1.9μm). The best separations were achieved with larger particles and smaller operating pressure at higher temperatures. In addition to the use of larger particles, further improvement in efficiency was observed with small internal diameter (1mm) columns [21].

7. Hypothesis of additional contributor to reduced Efficiency - mass transport from the stationary phase

In order to compare the data for different particle sizes, it is easy to use reduced parameters: h = H/dp and ν = (udp/Dm) in Van Deemter equation (2.) which gives the Knox equation (5.). If equation (2.) is the right model, then Knox plots using equation (5.) should superimpose for all particle sizes, but the don't [22]. Experimental data is in agreement that Hmin is smaller (Figure 1.), but hmin is higher for sub-2μm particles (Figure 4.)[1]. Figure 4.[1]
Thus demonstrated discrepancy minimizes the value of using smaller particles to achieve high efficiency and speed. This disagreement triggered extensive research for possible causes. Inhomogeneous packing density, frictional heating, extra-column volume are reasonable contributors to this phenomena, but it is not justified to neglect mass transport from the stationary phase in the C term contributions. Thus far only the contribution from intraparticle diffusion, Cm was considered in C term of equations (2.) and (5.) , but the contribution from the desorption of the analyte from the stationary phase, Cs, was neglected, although it is well accepted that C = Cm + Cs. When Cs contributions are included, C-term becomes larger than previously used. It is hypothesized that the value of Cm for the smallest particle size, has a substantial contribution from Cs on the basis of the desorption time that thought to be significant compared to the time it takes to diffuse in and out of the particle. There is no proof to this consideration, but it is thought that slow desorption is a possible contributor to the reduced plate height for sub-2μm particles [22].

8. Conclusions

Efficiency is a very important measure in liquid chromatography that is used to assess capacity of a column to restrain peak dispersion and with that to provide high resolution. With the increased complexity of samples analyzed, the ability to resolve multiple peaks is essential for quantitation and qualification. The theory of advantages of small particles was developed over 40 years ago, but only now the technology is able to take advantage of it. The necessity to separate highly complex mixtures of protein digests increased interest in smaller particles in the late 1990s and became the driving force in further changes of chromatographic instrumentation. The development of ultrahigh pressure pump systems and sub-2μm packing materials, able to withstand very high pressures, permitted significant improvement in separation speed and efficiency of modern liquid chromatography. This review demonstrates that understanding of the "chemistry" of small particles and factors influencing chromatographic efficiency at high pressures, allows further improvement of efficiency.

9. References

1. DOI Andre de Villiers, Francois Lestremau, Roman Szucs, Sylvie Gelebrat, Frank David, Pat Sandra (2006), Evaluation of ultra performance liquid chromatography. Part I. Possibilities and limitations, Journal of Chromatography A, 1127, p. 60-69
2. DOI Nobuo Tanaka, Hiroshi Kobayashi (2003), Monolithic columns for liquid chromatography, Analytical Bioanalytical Chemistry, 376, p. 298–301
3. DOI Naijun Wu, Andrew M.Clausen (2007), Fundamental and practical aspects of ultrahigh pressure liquid chromatography for fast separations, Journal of Separation Sciences, 30, p. 1167-1182
4. DOI Davy Guillarme, Elia Grata, Gaetan Glauser, Jean-Luc Wolfender, Jean-Luc Veuthey, Serge Rudaz (2009), Some solutions to obtain very efficient separations in isocratic and gradient modes using small particles size and ultra-high pressure, Journal of Chromatography A, 1216, p. 3232-3243
5. DOI J.J. Van Deemter, F.J. Zuiderweg and A. Klinkenberg (1956), Longitudinal diffusion and resistance to mass transfer as causes of nonideality in chromatography
6. Link Michael Swartz (2005), Ultra Performance Liquid Chromatography (UPLC): An Introduction, Supplement to LC-GC, 23 (5), p. 8-14
7. DOIJoe P. Foley (1991), Resolution Equations for Column Chromatography, Analyst, 116, p.1275-1279
8. DOI Szabolcs Fekete, Jeno˝ Fekete, Katalin Ganzler, (2009), Shell and small particles; Evaluation of new column technology, Journal of Pharmaceutical and Biomedical Analysis, 49 p. 64-71
9. DOINaijun Wu,Yansheng Liu,Milton L. Le, (2006), Sub-2mm porous and nonporous particles for fast separation in reversed-phase high performance liquid chromatography. Journal of Chromatography, A, 1131(1-2), p.142-150
10. DOI Jeffrey R. Mazzeo, Uwe D.Neue, Marianna Kele and Robert S. Plumb, (2005) Advancing LC Performance with Smaller Particles and Higher Pressure, Analytical Chemistry 77(23), p. 460 A–467 A
11. DOI Deirdre Cabooter, Jeroen Billen, Herman Terryn, Frederic Lynen, Pat Sandra, Gert Desmet (2008), Detailed characterization of the flow resistance of commercial sub-2-μm reversed-phase column, Journal of Chromatography A, 1178, p. 108-117
12. DOI Hamed Eghbali, Vincent Verdoold, Lieselot Vankeerberghen, Han Gardeniers, and Gert Desmet (2009), Experimental investigation of the band broadening arising from short-range interchannel heterogeneities in chromatographic beds under the condition of identical external porosity, Analytical Chemistry, 81, p. 705-715
13. DOI Fabrice Gritti, Michel Martin, and Georges Guiochon (2009), Influence of viscous friction heating on the efficiency of columns operated under very high pressures, Analytical Chemistry, 81, p 3365-3384.
14. DOI Krzystof Kaczmarski, Joanna Kostka, Wojciech Zapala, Georges Guiochon (2009), Modeling of thermal processes in high pressure liquid chromatography I. Low pressure onset of thermal heterogeneity, Journal of Chromatography A, 1216, 6560-6574
15. DOI Fabrice Gritti and Georges Guiochon (2009), Optimization of the thermal environment of columns packed with very fine particles, Journal of Chromatography A, 1216, p. 1353-1362
16. DOI Fabrice Gritti and Georges Guiochon (2008), Complete temperature profiles in ultra-high pressure liquid chromatography columns, Analytical Chemistry, 80, p. 5009-5020
17. DOI Krzystof Kaczmarski, Fabrice Gritti, Joanna Kostka, Georges Guiochon (2009), Modeling of the thermal processes in high pressure liquid chromatography II. Thermal heterogeneity at very high pressures, Journal of Chromatography A, 1216, p. 6575-6586
18. DOI Krzystof Kaczmarski, Fabrice Gritti, Georges Guiochon (2008), Prediction of the influence of the heat generated by viscous friction on the efficiency of chromatography columns, Journal of Chromatography A, 1177, p. 92–104
19. DOI Gert Desmet (2006), Theoretical calculation of the retention enthalphy effect on the viscous heat dissipation band broadening in high performance liquid chromatography columns with a fixed wall temperature, Journal of Chromatography A, 1116, p. 89-96
20. DOI Kenneth Fountain, Uwe Neue, Eric Grumbach, Diane Diehl (2009), Effects of extra-column band spreading, liquid chromatography system operating pressure, and column temperature on the performance of sub-2-μm porous particles, Journal of Chromatography A, 1216, 5979-5988
21. DOI Sabine Heinisch, Gert Desmet, David Clicq, Jean-Louis Rocca (2008), Kinetic plot equations for evaluating the real performance of the combined used of high temperature and ultra-high pressure in liquid chromatography. Application to commercial instruments and 2.1 and 1mm I.D. columns, Journal of Chromatography A, 1203, p. 124-136
22. DOI Mary J. Wirth (2007), Mass transport in sub-2-μm high-performance liquid chromatography, Journal of Chromatography A, 1148, p. 128-13

## Improving LC Efficiency with Smaller Particles

Nomenclature## 1. Introduction

High Performance Liquid Chromatography (HPLC) is a dominating separation technique routinely used in the industry due to its essential valuable characteristics, comprised of well-understood separation mechanisms, ease of use, sensitivity, selectivity and robustness. This technique has many applications including separation, identification, purification, and quantification of numerous compounds. Nevertheless, despite of all its worthy attributes, HPLC lacks high efficiency compared to GC and CE [1]. In addition, the demand of high-throughput analyses for fast screening of numerous pharmaceutical samples necessitates high-speed LC.Different approaches have been explored in order to improve HPLC efficiency and speed of analysis. Successful results were achieved with the use of high temperature LC, monolithic columns and ultrahigh pressure. Due to lower viscosity and high diffusivity of mobile phase, high column temperatures allow operation at high flow rates on longer columns or smaller particles, resulting in higher efficiency and faster analysis. The limitations of this technique are uncertainties in column stability and possible on-column degradation of analytes. Monolithic columns offer high efficiency and high speed due to high porosity and small skeleton size which allow operation at high flow rates on longer columns [2, 3]. However, monolithic columns exhibit limited stability at higher pressures, temperatures and pH. In addition, the limited variety of monolithic columns does not meet the demands for a wide range of industrial applications [4].

Recent developments in the commercial manufacture of instruments that are able to produce and withstand pressures as high as 18,000 psi, along with the introduction of a new generation of columns packed with sub-2μm porous particles, have made it possible to achieve faster separations than in HPLC, while maintaining or increasing efficiency. The factors influencing chromatographic efficiency in Ultra High Performance Liquid Chromatography (UHPLC) that uses sub-2μm porous particles and the means to improve efficiency are reviewed herein.

## 2. Chemistry of small particles

Optimal separations in liquid chromatography mean symmetrical and sharp peaks. The following processes that happen inside the column are responsible for the band broadening [3]:1.

Eddy diffusion:multiple paths of different length and velocity;2.

Resistance to mass transfer:equilibration rate of the analyte between the stationary and mobile phase. In the case of high velocity of the mobile phase and strong affinity of the analyte for the stationary phase, the analyte in the mobile phase moves faster than the analyte in the stationary phase.3.

Longitudinal diffusion:diffusion of the analyte in the mobile phase.In order to achieve optimal separations, band broadening must be minimized. Therefore, using small particles packings would reduce factors 1 and 2 and offer narrower bands and faster equilibration, and hence improve efficiency. This is also supported by the Van Deemter equation (1., 2., 3.), an empirical formula that describes the relationship between linear velocity, u (flow rate), and plate height, H (HETP or column efficiency).

1.[5]

or simplified

2.[1]

or simply,

3.

H = A + B/u + Cuwhere A represents the term for multiple paths, B - longitudinal diffusion in the mobile and stationary phases, and C - resistance to mass transfer in the mobile and stationary phases. The plate height is dependent upon the particle diameter so that the most significant impact on the efficiency would be introduced from the C-term since it is dependent upon the square of the particle diameter. The effect of particle diameter on plate height is demonstrated with the aid of Van Deemter plots (Figure 1.).

Figure 1.[1]It is evident that when the particle size is reduced, the minimum plate height is decreased and the optimum velocity (point on the curve at the minimum plate height) is increased [1]. Then optimum flow rate is inversely proportional to particle size: Fopt ∞ 1/dp [6]. The flat part of the curve at higher linear velocities signifies that operating at the velocities above the optimal value results in an insignificant decrease in efficiency. Hence, columns with smaller particles can be used at higher linear velocity without sacrificing efficiency for analysis time. This suggests that the particle diameter should be decreased as much as possible to obtain the highest resolution [1], since N is inversely proportional to particle size, N ∞ 1/dp [6], and the resolution is proportional to the square root of N (4.).

4.[7]

Since N is inversely proportional to the square of the peak width, N ∞ 1/w^2, and peak height is inversely proportional to the peak width, Hp ∞ 1/w [6], then with the particle size decrease, and subsequent plate count N and resolution Rs increase, higher sensitivity is obtained as narrower peaks are taller. Narrower peaks also mean higher peak capacity in gradient separations.

It is also worth mentioning that efficiency is proportional to column length and inversely proportional to the particle diameter, N ∞ L/dp [6]. In this case, for a given efficiency, the column length can be reduced by the same factor as the particle size to achieve faster separations without sacrificing resolution.

## 3. Effects of packing materials (porous versus nonporous) and efficiency of particle packing on column Efficiency

Modern column technology offers columns packed with porous and nonporous sub-2μm particles that are able to withstand high temperatures and high pressures, and are stable under acidic (pH < 1) and basic (pH > 11) conditions, which is essential in UPLC. An advantage of nonporous particles over porous particles is faster mass transfer due to the absence of sample diffusion in the stagnant mobile phase within the pores which results in reduction of band-broadening. Therefore, nonporous particles offer higher efficiency at high linear velocity, but the efficiency difference diminishes significantly when the particle size is reduced (to 1.5μm) [8]. An advantage of porous particles over nonporous is loading capacity which is 15 times higher for porous particles. Also, the average retention factors are higher for porous particles than nonporous, and pressure drops are smaller for porous particles than for nonporous [9]. In addition, nonporous particles are highly sensitive to extra-column contributions due to the low total surface area of the packing material in the column [10].Figure 2. [10]

Waters Acquity BEH C18 column (50mm x 2.1mm, 1.7μm, pore diameter 135A)

Higher reduced plate heights, h, are commonly observed in experimental determinations, compared to theoretically predicted with Knox equation (5.), on columns of i.d.s below the standard 4.6mm, or packed with particles below 5μm due to low packing efficiency.

5.[1]

This is attributed to complications associated with producing and packing of 1.7μm particles in 2.1mm i.d. columns [1]. Increased bed heterogeneity results in increase in A-term and C-term constants which in its turn increase reduced plate heights [12]. Packed capillary columns with uniformly sized nonporous particles have a very tight and ordered packing bed structure, but columns with porous packing material with a wide size distribution have extensive nonuniformity as shown by scanning electron micrographs [10].

## 4. High pressure and its effect on Efficiency

Since the pressure drop, required to achieve optimal linear velocity, is inversely proportional to the square of the particle diameter (6.), high pressure drop across a column is expected for smaller particles. So that high pressure is required to operate an instrument with small particle size columns.6.[3]

Aside from the engineering challenges associated with the use of high pressures in liquid chromatography, at high flow rates the friction of the mobile phase against the stationary phase generates heat [12]. The rate of heat generation, (power dissipation) is the result of pressure drop and flow rate:

7.[3]

The heat, convectionally transported by the mobile phase along the column, conducts in the axial and radial directions, and evacuates into the air through the column walls and end-fittings [14]. After asymptotically reaching a steady-state thermal equilibrium, the column temperature remains constant everywhere in the column and doesn't depend on time any more. At this point, the generated heat evacuates the column, producing radial and axial temperature gradients [13]. In the axial direction, the temperature increases from the column inlet to the outlet, and in the radial direction, the temperature decreases from the center to the column wall [14].

Figure 3.[15]

The amplitude of these gradients depends on the degree of column's thermal insulation. The longitudinal temperature gradient is the greatest when the column temperature is radially uniform (the smaller the radial heat loss, the smaller the radial temperature gradient, and the larger the axial temperature gradient). This happens when there is no radial heat loss through the wall (the column is kept adiabatic), and the generated heat is evacuated through the column ends. The radial temperature gradient is the greatest when the column wall temperature is kept constant [13, 16]. This can be achieved by placing the column in a liquid bath, so that the generated heat is evacuated through the column wall. Normally columns are operated under intermediate conditions of thermal insulation where radial and axial temperature gradients co-exist [13].

Temperature gradients have negative impact on the column efficiency due to heterogeneous distribution of the mobile phase linear velocity, viscosity and density throughout the column, which affect the equilibrium constant of a compound distribution between mobile and stationary phases (decreases with increasing temperature), and retention factors of the analytes. The difference between the temperatures at the inlet and the outlet of the column can be as much as 20K, and the difference between the temperatures of the center and the column wall can reach up to 6K, if the column is operated under regular convection conditions. The radial temperature gradient increases slowly along the column [17].

When the heat effects are significant and the radial thermal gradient is not flat, the classical theory of band broadening derived by Van Deemter (1., 2., 3. and Figure 1) is no longer accurate. More appropriate models are required to account for band spreading due to the combination of the radial dispersion (radial gradient of the retention parameters - since adsorption equilibrium constants decrease with increasing temperature, then retention factors are smaller in the central region of the column than near its wall) and radial gradient of the band velocity (due to radial viscosity distribution - since viscosity is higher in the colder region against the wall than in the warmer central region) that results from the radial temperature gradient. [The axial temperature gradient has only minor consequences since the longitudinal distribution of the mobile phase viscosity along the column affects the local pressure gradient that decreases along the column. This affects the efficiency only insignificantly [13]].

A few approaches were taken to calculate the influence of the combination of an axial and a radial gradient of temperature on the column efficiency [14, 16, 18, 19], but they were approximate. The correct prediction required the simultaneous solution of the heat and mass balance equations which was attempted and solved with the aid of 3 models: 1) a model of heat transfer, 2) a model of mass transfer, and 3) a model of the distribution of the mobile phase velocities [14, 17].

Since the detector measures the average concentration in the effluent from the column, and the radial concentration profiles of the analytes form an arc in the column, due to efficiency is lost due to radial temperature distribution that causes radial band velocity profile, the loss of efficiency is observed. When the temperature of the external column wall is kept constant, the efficiency loss is the largest [13] with the maximum amplitude of the radial temperature gradient (ΔT) between the column wall and its center can be expressed as:

8.[3]

This equation (8.) suggests that the frictional heating effect can be reduced by decreasing the column diameter. It was shown that the efficiency losses can be kept to a minimum for particle diameters of 1.5–2.0 μm if column i.d.s of < 2mm are used [10].

Columns should be operated under still-air conditions (inside the oven compartment of the instrument) since it offers better efficiency compared to under thermostated within a liquid bath conditions. At high flow rates, the shorter column should be used for better performance (under still-air conditions), and at optimal flow rates, longer columns will yield very high efficiency [16].

## 5. Extra-column band broadening and its effect on Efficiency

As mentioned above (chemistry of small particles), when particle size decreases, resolution increases as a result of narrower peak widths. From the other side, the narrower peaks are more susceptible to extra-column dispersion. The smaller the column diameter and the shorter the column, the smaller is the peak volume. Therefore, extra-column band broadening must be reduced in order to obtain efficient separations [12]. There are two main types of extra-column contributions: 1) volumetric - results from the volume of connecting tubing from the injector to the detector, the injection volume and the detector volume; and 2) time-related events - sampling rate and detector time constant [20].Extra-column contributions are additive to the length variance of the peak in the column, σc. The sum of the volumetric and time-related extra-column contributions is expressed in equation 9.

9.[20]

The volumetric extra-column contribution can be decreased with increasing column length, column radius and retention factor, and the time-based extra-column contribution can be decreased with the column length and with increasing linear velocity [20]. In addition, optimization of connecting tubing, injection volume, flow cell volume and geometry leads to maximum performance of a column. For example, it was determined that for a typical UPLC porous particles column, the maximum injection volume is 8μL, but if other extra-column volume is taken into account (> 3μL from connection tubing and flow cell), the injection volume should be less than 5μL [3]. As shown in equation 10., reductions in the injection volume Vinj, the flow cell volume, Vcell, and column to detector connecting tubing internal radius, rc, would lead to improvements in efficiency.10.[8]

The volumetric contributions are also affected by temperature changes due to the change in the analytes's diffusion coefficient. The band spreading decreases with the temperature increase [20].

## 6. Temperature and its effect on Efficiency

(Note: the discussion is limited to elevated temperature (up to 90C), and not high temperatures).Temperature reduces mobile phase viscosity and thus allows higher flow rates at the same pressure. Higher temperature also increases analyte diffusivity which decreases the mass transfer term (C-term), but increases the molecular diffusion term (B-term) of the Van Deemter equation (3.). A drawback of the temperature increase is that it increases linear velocity. That requires increased flow rates to avoid a loss in performance. Increasing flow rates, in its turn, will result in the pressure increase. This suggests that increasing temperature will not reduce backpressure. In any case, the advantage of the elevated temperature is the reduction in the ability to significantly shorten the analysis time with minimal loss in column efficiency [20].

It was shown that the elevation of column temperature while operating at constant flow rate below 0.4mL/min (below Van Deemter optimum) will result in the loss of efficiency. Higher flow rates should be used in order to maintain the same efficiency at the elevated temperatures [20]. In addition, experimental data demonstrated that at higher temperatures, certain efficiency can be achieved with bigger particle size (2.5μm), but not with smaller particle size (1.9μm). The best separations were achieved with larger particles and smaller operating pressure at higher temperatures. In addition to the use of larger particles, further improvement in efficiency was observed with small internal diameter (1mm) columns [21].

## 7. Hypothesis of additional contributor to reduced Efficiency - mass transport from the stationary phase

In order to compare the data for different particle sizes, it is easy to use reduced parameters: h = H/dp and ν = (udp/Dm) in Van Deemter equation (2.) which gives the Knox equation (5.). If equation (2.) is the right model, then Knox plots using equation (5.) should superimpose for all particle sizes, but the don't [22]. Experimental data is in agreement that Hmin is smaller (Figure 1.), but hmin is higher for sub-2μm particles (Figure 4.)[1].Figure 4.[1]

Thus demonstrated discrepancy minimizes the value of using smaller particles to achieve high efficiency and speed. This disagreement triggered extensive research for possible causes. Inhomogeneous packing density, frictional heating, extra-column volume are reasonable contributors to this phenomena, but it is not justified to neglect mass transport from the stationary phase in the C term contributions. Thus far only the contribution from intraparticle diffusion, Cm was considered in C term of equations (2.) and (5.) , but the contribution from the desorption of the analyte from the stationary phase, Cs, was neglected, although it is well accepted that C = Cm + Cs. When Cs contributions are included, C-term becomes larger than previously used. It is hypothesized that the value of Cm for the smallest particle size, has a substantial contribution from Cs on the basis of the desorption time that thought to be significant compared to the time it takes to diffuse in and out of the particle. There is no proof to this consideration, but it is thought that slow desorption is a possible contributor to the reduced plate height for sub-2μm particles [22].

## 8. Conclusions

Efficiency is a very important measure in liquid chromatography that is used to assess capacity of a column to restrain peak dispersion and with that to provide high resolution. With the increased complexity of samples analyzed, the ability to resolve multiple peaks is essential for quantitation and qualification. The theory of advantages of small particles was developed over 40 years ago, but only now the technology is able to take advantage of it. The necessity to separate highly complex mixtures of protein digests increased interest in smaller particles in the late 1990s and became the driving force in further changes of chromatographic instrumentation. The development of ultrahigh pressure pump systems and sub-2μm packing materials, able to withstand very high pressures, permitted significant improvement in separation speed and efficiency of modern liquid chromatography. This review demonstrates that understanding of the "chemistry" of small particles and factors influencing chromatographic efficiency at high pressures, allows further improvement of efficiency.## 9. References

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