Band Streadins in liauid ChromatograDhy: Eli Grushka Department of Chemistry State University of New York at Buffalo Buffalo, N . Y . 14214
Chromatography is a routine separation method now used in all branches of science. The goal in any chromatographic separation is to completely resolve the components of the mixture in the shortest possible time; much theoretical, practical, and instrumental effort has been spent to attain this goal. Initially, major efforts, a t least in instrumentation and theoretical developments, were concentrated on gas chromatography; then some seven or eight years ago, attention was given to liquid chromatography. One important factor responsible for this development of liquid chromatography (LC) was the realization that band spreading theories in LC and gas chromatography (GC) are nearly identical; consequently, some of the technology employed in GC can be transferred to, and utilized in, liquid systems. As a result, to attain better column efficiencies, modern high-speed LC (HSLC) [sometimes referred to as high pressure (HPLC)] usually employs narrower columns and smaller support particles than those used a decade ago. Of course, a price has to be paid for increased performance: because of smaller packing size, high-pressure pumps must be used to overcome the low permeability (high resistance to flow of mobile phase) of such columns. This report concentrates only on chromatographic zone broadening in liquid chromatography. We describe, in a qualitative manner, the factors affecting the solute zone width and their relation to experimental parameters. The theoretical terminology used to ascertain the processes which take place in the column and cause the solute zone to broaden is explained and discussed, The impor5lOA
tance of zone width in relation to separations and resolution is discussed. Finally, a general set of rules is given which should help the practitioner to improve the efficiency of his chromatographic system. Plate Height and Plate Number
A mixture of solutes is injected into a chromatographic column. The solutes distribute themselves between the stationary phase and the mobile phase according to their partition coefficient, K (which depends on intermolecular forces). The partition coefficient, which is defined as the ratio of the solute concentration in the stationary phase to that in the mobile phase, determines the average velocity of each solute zone (more specifically. it determines the velocity of the zone center). Solutes having different partition coefficients will move down the column at different velocities, and the differential migration necessary to achieve separation is obtained. The partition coefficient, which is a thermodynamic quantity, depends on the nature of the mobile and stationary phases and on the temperature. As solute zones pass through the column, they broaden. The broadening is important as it can ultimately affect the resolution. It is not enough to just “pull apart”, by differential migration, the various solutes. The solute zones should be kept as narrow as possible by proper design of the experimental system. The quantity which measures the efficiency and is related to the peak width is called the plate height. H,or the height equivalent to theoretical plate, H E T P The plate height can be
ANALYTICAL CHEMISTRY, VOL. 46, NO. 6 , M A Y 1974
obtained directly from the chromatogram:
L is the column length, W is the peak (or zone) width at the base line, W1 2 is the width at half the peak height, and t R is the retention time. H has the dimension of distance. Since it is easier to measure the width at half the peak height rather than at its base, Equation 1includes the relation between H and W1 2 . The constant 5.54 has a theoretical basis and is related to the assumption that the peak is Gaussian in shape. Some workers prefer to measure the peak width at
Report
umn by calculating a dimensionless quantity called the plate number
(N):
Figure 1. Relation between peak width and resolution A Poor efficiency B More efficient system
From Equation 2, the larger the plate number, the more efficient the system is. The plate height rather than plate number is, however, a more meaningful measure of the efficiency since it is independent to a first approximation of the column length, and more importantly, at least from a theoretical point of view, it can be directly related to the experimental conditions and parameters. It is recommended that H be measured routinely when evaluating chromatographic columns. It is difficult to give an absolute value which Hshould be, but typical values may range from 0.05 to 5 mm. The comparison in Figure 1shows in a schematic manner the improvement in resolution which can be obtained, by keeping the peaks narrow, for a pair of components eluted under the same conditions. Before describing the connection between H a n d some of the column parameters, the relation between the plate height and the resolution must be discussed. Resolution Equation
about 0.6 of the peak height [for a Gaussian peak, this width is twice the standard deviation (2 u)]. If this width is used for the computation of H , then the numerical constant in Equation 1is 4. The smaller the value of H, the better the system, everything else being constant. Frequently, workers in the field measure the efficiency of their col-
In any chromatographic technique, the resolution between two adjacent peaks is defined as
where t~ and “indicate, respectively, retention times and peak base line widths, with subscript 1 identifying
the first eluted component and subscript 2 identifying the second component. The retention times are functions of the nature of the solute, the mobile phase, the stationary phase, and of the column length, the temperature, and the mobile phase velocity. By a judicious choice of the stationary and mobile phases, at a given temperature one can increase the difference in retention time between t(&) and t(R1) (i.e., increase the relative retention a as defined in Figure 2 ) and so improve the resolution. Guidelines as to the selection of phases are given elsewhere by other authors (1-3).The resolution equation can be approximated in a more useful form
lp- (L) (G)
Rs=q
1+k2/
(4)
The derivation of Equation 4 can be found in any chromatography text (for example, Appendix 2 of ref. 3 ) . The capacity ratio k’, the relative retention a , and N can be calculated directly from the chromatogram as indicated in Figure 2 . In this figure, t , is the retention time of an inert solute which is not retained by the stationary phase. Examination of Equation 4 shows that the resolution expression can be divided into three parts: an efficiency term depending on L and H, a “rate of migration” term depending on k‘, and a selectivity term depending on a . The selectivity term a depends solely on the molecular forces between the solute and the two phases. Once the mobile phase, nature of the stationary phase, and temperature are chosen, the relative retention is fixed and cannot be changed. As CY approaches unity, the resolution deteriorates rapidly. The rate of migration
ANALYTICAL CHEMIST13Y, VOL. 46, NO. 6, M A Y 1974
511 A
Figure 2. Calculation of some important parameters from chromatogram
term depends on the intermolecular interactions in the system, as well as on the amount of the stationary phase (or adsorbent surface area). Ask' increases for a given column system, the retention time also increases so that a trade-off between the resolution and analysis time must be made. Note that increasing k' beyond a certain value (say 5) does not greatly affect the second term of Equation 4. If, however, k' is small, say 0.5-1, then increasing k' (to, say, the value of 5) will increase markedly the resolu. tion. Lower k'values, although resulting in shorter analysis time, are more demanding in terms of-tbe efficiency needed for the resolution. As previously mentioned, the resolution can be improved by increasing the column length. However, the retention time and the pressure drop across the column are direct functions of the length, whereas the resolution increases only with the square root of the length (Equation 4). Finally, improving the efficiency (Le., decreasing H) also improves the resolution. Improving the resolution by decreasing H i s attractive since frequently it is easier to manipulate the factors influencing the plate height than other factors such as column length or nature of stationary phase. Now that the importance of the plate height or column efficiency has been demonstrated and the relation between H and the resolution established, we can proceed to discuss the various column processes that hroaden the solute zone.
Chromatographic Efficiency and Column Processes
The d a t e height is a function of thermddynamicand kinetic processes which take d a c e in the column. viz.. molecular diffusion in the mobile phase; mass-transfer phenomena in both thestationaryand the mobile phases; and flow irregularities. Theories such as the plate model, the random walk model, the nonequilihrium model. and the mass balance model discuss these facton quantitativelv. The first three are discussed by Giddings (4) in his monograph, and the last is described adequately by Kucera (5)and Grubner (6)for adsorption chromatography and by Grushka (7) for a partitioning system. This report does not attempt to review these theories. Rather. we describe rhe importance oithe various contributors 10rhe HETPand show bow they can he modified. Molecular Diffusion. I:pon injcction oia mixture 01 cihtesinto rhe chromau)graphir column. concentration " eradients wirh resoect to the solutes exist simply because a t the moment of injection the downstream parts of the column do not contain any of the solutes. According to Fick's laws of diffusion, the solute zones will begin to broaden by diffusion in the axial direction. In liquid chromatography, because of the low diffusion coefficient value in the liquid mobile phase (about 10W5cmal sec), this broadening mechanism is
5 1 2 A * ANALYTICAL CHEMISTRY. VOL. 46, NO. 6, MAY 1974
usually not of practical importance (unlike GC at low carrier velocities). Mass Transfer. Mass transfer is a phenomenon which takes into account the transfer of solute molecules from one region in the chromatographic column to another, and in chromatography the important parameter is the rate of mass transfer. Mass-transfer rate is important in the stationary phase, in stagnant pockets ofthe mobile phase (these pockets are found in the pores of the support particles), and in the mobile phase. Stationary Phase. Slow mass transfer in the stationary phase means longer time spent in the phase. For example, in liquid chromatography while some solute molecules are diffusing into and then out ofthe stationary phase, others are moving with the mobile phase. The result is a broadened zone. The rate of mass transfer in this instance can be improved by reducing the film thickness of the stationary phase and so reducing the distance that a molecule solute must diffuse. Solutes will have high mass-transfer rates when their diffusion coefficients in the stationary phase are high, but for a given LC system a t a given temperature, this is a variable beyond the control of the user. Obviously, the mass transfer in the stationary phase is a function of the partition coefficient (or the capacity ratio) since it indicates the affinity ofthe solute to that phase. Mass Transfer in Stagnant Pockets ofMobile Phase. This factor can contribute significantly to H in LC (because of the slow diffusion of the solute in it). In many respects, this phenomenon is similar to mass transfer in the stationary phase. The only mechanism by which the solute molecules can leave a stagnant pocket is by diffusion through the mobile phase. Hence, for fast mass transfer, the diffusion coefficient should he high and the depths of the pores small. This explains the success of pellicular packings, in which a porous layer is bonded to a hard core, and of porous microspheres. In these cases, the pores are shallow dr small, resulting in good mass-transfer properties in the stagnant pockets and correspondingly high efficiency. However, recent work by Done et al. (8)indicates that not all pellicular supports are equivalent in their behavior with respect to stagnant pocket mass transfer. They also indicate that the dependence of H o n k' is a function of whether the important contribution to the mass transfer is from that in the stationary phase or the stagnant mobile phase. In the first case (and only if it is the major contributor to the efficiency), the HETP will have a maximum a t h' = 1.In the second case, H will keep in-
creasing with the capacity ratio. Experimentally then, one can check the value of H a t a given mobile phase velocity as a function of k’ and get some insight as to the major contrihution to the HETP. The disadvantage of pellicular packings will be mentioned later. Mass Transfer in Moving Mobile Phase. This is a complicated phenomenon. In a packed column, the velocity of the mobile phase is differeut from point to point owing to the perturbation caused by the support particles. Streamlines near the particle boundaries moye slowly, and streamlines near the center between particles move relatively more rapidly. In addition, some regions in the column are packed more tightly than others, again resulting in flow inequalities ( 4 ) .Some solute molecules can be in slow streamlines, whereas others move with faster velocities, thus broadening the zone. The only mechanism that can overcome this phenomenon is the diffusion of the solute molecules from one streamline to another. In this manner, if the diffusion is fast enough, the solute molecules in the mobile phase \?ill ‘‘S(,L”’ all rhe \urious flow raws and will move wirh the samp a\’erape d o r i r ? . Thereiorc, tht: diffusion in the inoliile phase is an important parameter in determining the efficiency. Since the maldistrihution in the velocity of the carrier is due to the packing particles, it makes sense that the solute molecules should diffuse a distance which is proportional to the particle diameter in order to move from one streamline to another, The smaller the particle, the faster the exchange between the various velocities in the column and the more efficient the system. This is another reason for using smaller size support particles highapeed LC. Recently, for example, Kirkland (9,IO) and Majors (22, 221, among others, reported the use of extremely small particles (less than 10 wm) which gave efficient columns. Majors shows the improvement in the efficiency and resolution of several azo dyes that can be obtained in going from 13.2 fim average particlesizeto6.1pm (11). Mass transfer in the mobile phase can also he dependent on the partition coefficient. The nature of the dependence, however, is not known ex. actly; in fact, in many cases it may not be significant. Flow Irregularities. In addition to the velocity effects described above, there is also a distance effect. During the time that some solute molecules travel around particles, other go straight through and move further down the column. This obviously also broadens the zone. Actually, the picture is more complicated hecause so514A
lute molecules are diffusing in the mobile phase continuously. Thus, at one moment a molecule can he in a mobile phase streamline whose path is around a support particle. The next moment, the solute molecule can diffuse laterally to a different streamline whose velocity is different and whose path is relatively free (at least for a short distance) of obstruction by the packing. Hence, the tortuous path of solute molecules is due both to diffusing from streamline to streamline and to having to circumvent support particles. This simultaneous dual broadening mechanism is known as the “coupling” effect (molecular diffusion is coupled with uneven pathlines) first discussed by Giddings (4).The coupling effect manifests itself in giving the H vs. mobile phase velocity plot its usual convex shape in LC. It stands to reason that this “going around the particle” broadening mechanism is support size dependent and decreasing it should improve the efficiency. Effect of Mobile Phase Velocity. The mobile phase velocity affects the plate height since it determines the relatiw importanct. of the rcsi;tance ro mass-triinirer rerms. The solure molerulcs would like 10 distribute I hemselves h t a e e n the mobile and stationary phase and reach a state of thermodynamic equilibrium. The forward movement of the solute zone owing to the flow of the mobile phase does not allow this equilibration to take place except at one point, namely, the center of the zone (assuming linear isotherm). Oneitherside of the zone center, however, the constantly changing solute concentration (owing to mobile phase flow) keeps the system removed from equilibrium. Two points should he clear. The faster the flow rate is, the farther the system is removed from equilibrium (again with the exception ofthe zone
Figure 3. Typical HETP vs. mobile phase velocity curve in HSLC
* ANALYTICAL CHEMISTRY, VOL. 46, NO. 6, MAY 1974
center or maximum). The faster the rate of mass transfer, the closer the system would he to equilibrium. These points indicate that a t high velocity and slow mass-transfer rates, the column efficiency would he lower than a t low velocity, all other conditions being equal. To express it in extremely simplified fashion, at high velocity the rate of mass transfer is not fast enough to catch up with the constantly changing solute concentration and the zone is broadened. At extremely low velocity of the mobile phase, the mass-transfer terms are less important, hut broadening owing to molecular diffusion might adversely affect H. In normal use, however, the velocity is such (between 0.1 to 10 cm/sec) that molecular diffusion is negligible, and an H vs. Uplot typically looks like the one shown in Figure 3. Some workers in the field prefer to use reduced plate height h (defined as H divided by the size of the support particle) and reduced velocity Y (defined as the velocity multiplied by the support size and divided by the diffusion coefficient of the solute in the mobile ohase). These are extremrlg inipwtanr piram( ters since rhey allou,th(.cornparisonotcolumn~ \r,ith dirierent packing sizvs and 1110hile phases. Hrnrcv( r. owing to the nature of this report, hand; will not he discussed here and the interested reader should consult ref. 8 (or 29 and references therein). Additional Factors Affecting Efficiency and Resolution A solute zone may, owing to injection of too large a mass or to injection of an overly large sample volume, broaden symmetrically and yield an elution peak of greater H value than the intrinsic capability of the column. A frequent question in this connection is whether a small hut concentrated or a larger but dilute sample volume should he injected. A recent study by DeStefano and Beachell (13) shows that, at least for large-diameter columns (10.8mm i.d.) and porous particles, it is more advantageous to introduce a large diluted volume over small but concentrated ones. This procedure, according to them, overcomes the problem of locally overloading the column a t the inlet with the solutes. Kirkland, on the other hand, shows (14) that in the case of porous silica microspheres, increasing the sample volume can increase unduly the peak width of early eluting solutes (Le., solutes having small partition coefficients). This increase in the peak width was attributed to extra column effects such as connecting tubes and detector cell volume. Thus, one must be careful not to inject such a large volume as to in-
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crease the width of early eluting solutes over and above the width that results from the usual processes in the column. Snyder (25)defined a term called the linear capacity of the system which is described as the weight of the sample per gram of packing which causes a 10%reduction in the specific retention volume relative to the constant retention volume ohserved for smaller samples. If constant injection volume is used, it is advisable to operate a t sample weights which are lower than the linear capacity, since this will ensure better efficiencies. The linear capacity of a chromatographic system can be obtained by experimentation ( 2 4 ) . Ingeneral, however, as the surface area of the support increases, larger sample injections can be made. It is also possible that partition coefficients can he solute concentration dependent, giving rise to asymi metrically shaped elution peaks. When the peaks tail (Figure 4, A), the position of the peak maxima decreases with increased solute concentration; when the peaks front (Figure 4, B), the peak maxima increase with increased solute concentration. In both bases the asymmetry results in a peak broadening which reduces the resolution between peaks. It is thus desired at times to run the experiment with several different solute concentrations to see whether or not the peak width, peak asymmetry, and retention time are constant.
Summary The efficiency H, then, is a function of the mobile phase velocity, the solute’s partition coefficient (or the capacity ratio), the diffusion coefficient of the solute in the stationary and mobile phases, the support particle size and nature, the amount of stationary phase on the support (film thickness), and the surface area of the support in adsorption chromatography. To lower H and increase the resolution, several courses of action are available. The following is only a rough guideline to some of the approaches the investigator can take. [A much more comprehensive and systematic discussion of improving the resolution in high-speed LC was recently given by Snyder (26, 2711. Decrease the carrier velocity. This, however, means longer analysis time. Also, one can increase the mobile phase velocity provided that the column length is increased proportionately. 0 Decrease amount of stationary phase in liquid-liquid system (but not to the point where adsorption on the naked support begins to adversely affect H ) .
ANALYTICAL CHEMISTRY, VOL. 46.
NO. 6 . MAY
1974
Figure 4. Peak asymmetry A Tailing B Fronfing
Decrease the size of the support particle. It must he emphasized here that when using small particles (Le., less than 20 pm), packing becomes a problem and special care must he exercised when filling the column. This point will he discussed shortly. Use a mobile phase having low viscosity so that the diffusion of solutes in it is rapid. Also, for a desired mobile phase velocity, lower viscosity means lower inlet pressures. Lower the viscosity of the mobile phase by increasing the temperature. However, the effect of temperature on the plate height is less clear cut. Schmit et al. (18)indicated the H improves with increasing temperature, whereas Knox and Vasvari (19) found no dependence of H on the temperature. An increase in the temperature results in a decrease of h’ (or the retention time) and most frequently in a. That decrease in LC is most often more pronounced than any improvement in H, and the resolution can deteriorate with inkeasing temperature (Equation 4): Use a support where the likelihood of large stagnant pockets of mobile phase are minimized (pellicular packing or microspheres with small pores). As mentioned previously, the HETP can he decreased by decreasing the size of the packing. H varies roughly as the particle diameter to the power of about 1.8.As the particlesize decreases, the permeability of the column decreases, and higher inlet pressures are needed to drive the mobile phase. The permeability is roughly a function of the square of the particle diameter. However, because of the higher efficiency of the
system, the column length can he decreased, thus decreasing the needed pressure drop. Particles smaller than 20 fim are more difficult to pack uniformly, and special techniques such as balanced density slurry (20) are needed. However, with care, particles down to 20 fim can he dry packed to give efficient columns (8).Recent work also shows that the shape of small particles need not be spherical for efficient packing. Small porous particles have one distinct advantage over pellicular packings. The latter packing usually gives short retention times and lowcapacity ratiio value. It is, therefore, easy to overload the pellicular packings. In addition, their price is rather high. Porous supports, on the other hand, are usually much less expensive and, owing to their large surface area, can handle larger charges of solute injections.
(3) N. Hadden et al., "Basic Liquid Chro-
matography,'' Varian Aerograph, 1971.
(4) J. C . Giddings, "Dynamics of Chromatography," Marcel Dekker, New York,
N.Y., 1965. (5) E. Kucera, J. Chromatogr., 19,237 119fiS) ~---~,.
(6) 0. Grubner, in "Advances in Chrama-
tagraphy," J. C. Giddings and R. A. Keller, Eds., Vol6, p 173, Marcel Dekker, New York, N.Y., 1966. (7!,Eirwhka, J. Phys. Chem., 76,2586 ,'"l
I A,.
(8) J. N. Done, G. J. Kennedy, and J. H. Knax, "Gas Chromatography-1972," S. G. Perry and E. R. Adlard, Eds., p 145, Applied Sciences, Essex, England, 1973. (9) J. J. Kirkland, J. Chromatogr. Sci., 10, t (10) J
(12) R. E. Majors and F. R. MacDonald, J. Chromatogr., 83,169 (1973). (13) J. J. DeStefana and H. C. Beachell, J. Chromatogr. S+, 10,654 (1972). (14) J. J. Kirkland in "Gas Chromatography-1972," S. G. Perry and E. R. Ad-
lard, Eds., p 39, Applied Science, Essex,
1471 -Fnrrlanrl _.-.. _~
(15) L. R. Snyder,Anal. Chem., 39,398
References
(1967). '(16) L. R , Snyder,J . Chromatogr. Sei.,
i n i n n i1-" i q 7.-,. m _",-I" (17) L. R. Snyder, ibid.,p 369. (18) J.A. Schmit, R. A. Henry, R. C. Wil-
liams, and J. F. Diechman, ibid.,9,645 11971) I_".-,.
(19) J. H. Knox and G. Vasvari, J. Chromotogr., 63,181 (1973). (20) R. E. Majors,Anol. Chem., 44, 1722 (1972).
Eli Grushka is assistant professor of chemistry a t the State University of New York at Buffalo. His BS was obtained in 1963from Long Island University, and his PhD from Cornell University in 1968. He did postdoctoral work with Professor Giddings at the University of Utah (1967-1QfiQ) T h (;nshk;l', research inrere;tsare in rhe arra of chromatograptiic theories, .--"- h:n-no enhancement of chromato,La,...L olution, the utilization (If bonded phases and of liquid cry:3tals in chromatography, and in me;rsuring physicochemical constants h: i chromatography. He has numeralJS scientific papers in the field of seFiaration. Dr. Grushka is a member of ACS, M A S , and the Chromatographic Discussion GrouD. His current address is Hoffman;-La Roche In(:., Nutley, N.J.
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