including organophosphorus pesticides. He obtained vastly different spectra for parathion and methyl parathion which, he felt, reduced the value of plasma chromatography as an analytical tool. We also studied these compounds, and fenitrothion, and found very reasonable relationships between the mobilities and masses of ions of this series (Table 111). When the data necessary to calculate the KOvalues for the two parathions are extracted from Moye's paper, it is found that his paper and this one agree precisely that the reduced mobility of the major peak of parathion (which we identify as (PAR)H+)is KO = 1.27 cm2 V-' s For methyl parathion, however, there is disagreement. We find K O= 1.36 cm2 V 1 s for the main peak (MPAR)H+,and an insignificantly small peak a t KO= 1.18 cm2 V-' s-'; Moye reported the latter as the mobility of the major peak. A likely explanation lies in the method of sample introduction. Throughout this work, and during Moye's work on parathion, the solvent was allowed to evaporate from the wire before insertion. However his methyl parathion spectrum was obtained using the insecticide in hexane solution; his mobility of 1.18cmLV s l corresponds (Equation 3) to an ionic mass of about 338 which could easily be (MPAR)(C6H1,)H+(mass 350). Thus the variations which Moye found and ascribed to unknown effects were probably due to interactions with solvent. Diffusion Coefficients. As mentioned earlier, the plasma chromatograph operates in the low-field domain. The mobility and diffusion coefficients are therefore related by (6)
cussed above, although electrostatic repulsion could also contribute. These factors were not rigorously considered by Spangler.
CONCLUSION The low-field mobility coefficients of 47 different ions formed by atmospheric-pressure ionic reactions of phosphorus esters have been measured. These ions are parent ions, hydrates, partial or full dimers, or decomposition products of the phosphorus esters. The excellent linear correlation between mobility and mass of these ions means that the mobility of similar ions can, with confidence, be determined from the regression fit of the data presented here. One must note, however, that only two of these ions were definitely identified, by use of a PC/MS instrument. While mobility spectra can be identified in the manner used here, and thus give useful data, exact identity of the ions is not assured. The Einstein equation was used t o calculate the diffusion coefficients from the experimental low-field mobility coefficients.
ACKNOWLEDGMENT We acknowledge comments on hydration reactions by Martin J . Cohen.
LITERATURE CITED R . D. O'Brien. "Toxic Phosphorus Esters", Academic Press, New York and London, 1960, pp 83-89 "The ProSlem of Chemical and Biological Warfare", Vol 2, Almqvist and Wiksell. Stockholm. 1973. DD 55-58. J A. Buckley, J. B. French,'and N. M. Reid, Can. Aeronaut. Space J . , 20. 231-233 (1974). F. W. Karasek, Anal Chem.. 46, 710A-717A (1974). C. S. Harden and T. C. Imenson. "Detection and Identification of Trace Quantities of Orqanic Vapors in the AtmosDhere bv Ion Cluster Mass Spectrometry a i d the Ionization Detector System".Edgewood Arsenal Technical Report (1973). E W. McDaniel and E.A. Mason, "The Mobility and Diffusion of Ions in Gases". John Wilev and Sons. New York. N.Y.. 1973. D. I. Carroll, I. Dzidic, R. N. Stillwell. and E. C. Hornina, Anal. Chem., 47, 1956-1959 (1975) H E Revercomb and E A Mason, Anal Chem 47, 970-983 (1975) G E SDanaler and C I Collins Anal Chem 47 403-407 11975) F W Karaskk. 0 S Tatone and D W Dennev J Chromatoar - 07 137-145 (1973) R. L. Holmstead and J. E. Casida. J , Assoc. Off. Anal. Chem., 47, 1050-1055 (1976). F. W. Karasek, S. H. Kim, and H. H. Hill Jr., Anal. Chem., 48, 1133-1137 (19761. H A. Moye, J Chromatogr S o , 13. 285-290 (1975)
kT D=-K e which is the Einstein equation. Here D is the diffusion coefficient, e the electronic charge, T the temperature, and k Boltzmann's constant. This relationship is exact, and thus diffusion coefficients can be calculated to the same accuracy as the mobility coefficients were measured. We have not listed these values, but note that k T / e = 23.54 mV for T = 0 "C. I t is of interest to compare the resulting diffusion coefficients with those calculated by measuring peak widths. Spangler (9) derived an equation between the peak width at half height, the diffusion coefficient, and the duration of the gate pulse, but it was found that the diffusion coefficients calculated in this manner were uniformly about five times too high, i.e., the experimental peaks were five times wider than predicted. This additional broadening was probably due primarily t o reactions continuing in the drift space, as dis-
RECEIVED for review April 26, 1977. Accepted July 19, 1977.
Peak Broadening Factors in Thermal Field-Flow Fractionation LaRell K. Smith, Marcus N. Myers, and J. Calvin Giddings" Department of Chemistry, University of Utah, Salt Lake City. Utah 84 112
Previous observations of plate-height effects in field-flow fractionation (FFF) are briefly noted. The theory of plate height in FFF is reviewed and expanded. Contributing terms are divided into ideal and nonideal categories, the latter resulting mainly from relaxation, polydispersity, and multipath effects. A method for distinguishing these terms from one another is presented. When applied to data from a standard column, it shows behavior approaching ideality, dominated by the nonequilibrium plate height term. Cases in which the channel surface was roughened and distorted exhibited nonideal behavior, establishing the critical role of surface finish. 1750
A N A L Y T I C A L CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
The retention of peaks in field-flow fractionation has been found to occur much as predicted by theory (1-8). The dependence (or lack of it) on channel dimensions, particle size or molecular weight of solute, field strength, flow velocity, and other variables is well understood. Peak width, by contrast, has not generally yielded results in accord with theory ( I , 5, 8). Peaks have, in most cases, been broader than predicted, and this divergence has increased with the approach to the conditions of high resolution that we theoretically may expect of FFF. This work represents an attempt to study and isolate the
role of various experimental parameters in determining peak width and thus plate height in present-day FFF systems. The last comprehensive discussion of this subject was based on a single prototype thermal field-flow fractionation (thermal FFF) system (1) which is now recognized to have produced some erroneous results, even in retention effects (9). The present results derive from several thermal FFF columns, all yielding retention data of a generally reproducible nature. These columns have been described elsewhere ( 4 ) . Various modifications are detailed below. All subclasses of FFF are closely related, a fact making the conclusions derived from one subclass applicable to all others. Not only do the present results provide guidelines for other members of the F F F class, but the results of all other systems are relevant to the present studies. We briefly summarize the conclusions of the prior studies below after noting the reasons for expecting favorable plate heights in FFF based on the rapid equilibration within solute layers. In F F F , narrow solute layers are compressed against one channel wall. T h e thinness of the layer encourages rapid equilibration and thus rapid separation, a positive characteristic of FFF noted in the first paper on the FFF concept (10). Rapid equilibration is tantamount to low plate height values. Detailed theoretical expressions for plate height as it relates to equilibration (the so-called nonequilibrium plate height contributions have been worked out (11-13). (A more exact theory has been developed recently by Krishnamurthy and Subramanian (14). Contributions to plate height from longitudinal diffusion, relaxation effects, and solute polydispersity have also been developed (1). T h e associated experimental study showed t h a t plate heights for linear polystyrenes in toluene were generally several times larger than the values predicted based on equilibration considerations. T h e experimental values were usually lower than those expected from reported polymer polydispersities, however. A considerable uncertainty in the actual polymer polydispersity cast doubt on the latter conclusion. Flow velocity studies showed t h a t the plate height behavior was intermediate between t h a t demanded by nonequilibrium theory and polydispersity theory. Since publication of the study cited above, more limited observations of plate height have been reported for several FFF systems. Plate height vs. velocity plots for polystyrene latex beads in sedimentation F F F were found to intercept the H axis rather than the origin, as expected from theory ( 5 ) . Agreement with theory was good, however, a t high flow rates. Relaxation, polydispersity, and end effects were ruled out as causing this trend. A concentration effect resulting from repulsion between the latex beads that are forced to the lower wall provided an explanation consistent with the data, but by no means proven. A study in flow F F F showed that plate height decreased with retention volume, but a t a rate far less than predicted (8). The plate height vs. flow velocity plots resembled those from the sedimentation studies, except that the experimental curves were more or less parallel with the theoretical curves, thus failing t o show convergence a t high velocities. Results from an electrical F F F system using rigid membranes also yielded parallel H vs. velocity plots, but for lysozyme the plots were almost coincident with the theoretical lines, indicating a very good agreement between theory and experiment (15). Other proteins showed the usual divergence, possibly due to polydispersity. Two separate studies of sample-size effects-one with thermal FFF (4)and one with flow FFF (16)-have shown that the plate height increases with increasing quantities of sample.
THEORY Plate height equations have been established for FFF
processes that we shall term ideal. These occur in ideal FFF columns that consist of the channel space between two infinite parallel surfaces. The surfaces for the purposes of theory are absolutely flat, smooth, and unyielding t o distortion. They do not absorb solute. The flow in the channel between them is laminar and the flow profile is parabolic. There are no end effects, The diffusion coefficient, D , and field-induced drift velocity, U , of solute between the two surfaces are assumed constant, Transients due to the lateral relaxation of the solute to its steady-state configuration are absent. Each peak consists of a monodisperse solute in which the molecules or particles are of negligible size compared to the channel width or mean height of the solute layer. Ideal FFF. For such an ideal FFF system, the plate height can be written as the sum of the two terms (12),the first for
longitudinal molecular diffusion
HD = 2 D / R ( v ) and t h e second for nonequilibrium
HN = x w (v)}D
(2)
where D is the solute-solvent diffusion coefficient, R is the retention ratio, ( u ) is the mean flow velocity, LL' is the width of the gap between the two surfaces, and x is a complicated function of the dimensionless parameter A. T h e equation expressing x as a function of A is presented elsewhere (13); calculated values for this parameter will be plotted shortly. Quantity X is the ratio of the characteristic altitude of the exponential solute cloud divided by column width, w. I t can be expressed in the form
X = D/Uw (3) where U is the average velocity induced by t h e field across the width of the channel. T h e retention ratio R (the ratio of void volume to retention volume) is to a good approximation a function of X only (17).
R = GX[coth('/z X )
2h]
(4) Nonideal FFF. In real columns, the ideal terms above are augmented by plate height contributions arising in FFF -
nonidealities. Expressions have been derived for some of these contributions (1). These are shown below with some modifications, and the effect of column irregularities is discussed. When a sample is introduced into ;an FFF channel, there is an inevitable time lag in which it relaxes into its characteristic quasi-equilibrium layer. During this time there will be a nonuniform displacement of solute down the channel leading to peak broadening. This is referred to as the relaxation contribution to plate height. This is described approximately by
where n is the effective number of relaxations which occur and L is the column length. This equation is based on the assumption that X is close to zero. Real polymer fractions invariably consist of a distribution of molecular weights. This contributes to peak broadening through a term described as t h e polydispersity contribution ( 1 ) .
Here, X is the distance migrated by the species of molecular weight M , and cru is the variance of the weight distribution curve. This variance rather than the number variance is used because the detector senses polymer mass rather than polymer numbers ( 1 ) . A more useful form of this equation is obtained by writing it in differential logarithmic form A N A L Y T I C A L CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
1751
-
Table I. Values of the Exponent, r , as Defined by Equation 1 2 , f o r Different Plate Height Mechanisms. The Corresponding Equation Numbers Are Shown In that displacement X is proportional to retention ratio R which is inversely proportional to retention volume V, , we can replace d In X by either d In R or (4 In VJ. Furthermore, the X and M terms outside the parentheses can be replaced by the appropriate mean values under observation: column length L and weight-average molecular weight M,i,respectively. These substitutions give
From Equation 32 of Ref. 1, u , ~ M,,L can be approximated is number average by ( p - l),/p, where p = M k / M nin which molecular weight. Thus Equation 7 becomes
an
dlnVr2p-1 (d In M I t has been shown that d In VJd In M approaches -0.5 under ideal conditions of high retention (17). A more realistic
H,=L-
)
empirical value is -0.6 a t high retention, ranging down to zero a t the void peak. Another dispersion effect can be expected to stem from various imperfections in the channel geometry. Small departures from perfect smoothness and flatness will create inequalities from point to point in the solvent flow velocity and therefore in the solute velocity. A distribution of solute velocities will therefore exist over the breadth of the channel. In the limit of zero solute diffusivity, every flow streamline would carry its load of solute through the column in a time characteristic of that streamline. With finite diffusivity and in any given time t , however, streamlines within the approximate range (Dt)' will exchange solute. This includes the streamlines that lie above one another between the wall where the stream velocity is zero and the center of the channel where velocity is a t its maximum. The exchange (mass transfer) of solute between these streamlines leads to the nonequilibrium term, Equation 2 . A similar term will arise if systematic lateral velocity differences exist also over distance (Dt)' ', which is typically -1 mm. However, the term would acquire the same coupling form applicable to chromatography (18) if the velocity differences were random. For purposes of the present discussion, we consider a simplified model in which the flow exists as a series of filaments, each occupying (Dt)' of the column breadth. Solute exchange will occur rapidly within the filament, but will not occur between adjacent filaments. Two kinds of dispersion exist for the above model: that for the mean dispersion within individual filaments and that caused by the unequal displacement of solute in the different filaments. We can write for this f l o u effect
'
-
'
H f = Hint + H , where the two terms represent the internal and the multifilament or rnultipath contributions just noted. T h e term Hint includes the term H.\,Equation 2. For the present we assume that the other contributions to H,,, are generally negligible in a well-constructed channel. T h e multipath term originated in the inequalities in solute velocity, v , found in different filaments. These inequalities are reflected in the variance in v: ut,'. In the course of passage through the channel the velocity variations translate into variations in migration distance, z , measured by uz'. In the mean retention time t,, these variances are related by uZL= t?u,,'. T h e plate height contribution is simply
1752
A N A L Y T I C A L CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
Equation Plate height mechanism
No.
r
Longitudinal diffusion Nonequilibrium Re lax at ion Polydispersity Multioath
1 2 5 9 11
1 1 2 0 0
and is independent of mean solvent velocity ( c ) because t , and ut are affected oppositely by ( c ); t , and u, are proportional t o l / ( u ) and to ( u ) , respectively. The difficulty with Equation 11 compared to the previous plate height expressions is that it cannot be explicitly evaluated in terms of known parameters. Equations 1,2, 5,9, and 11 are contributions to plate height all of which can be cast in the general velocity-dependent form
H = const (u)'
(12)
where r is a value (generally an integer) that depends on the specific plate height term(s) under consideration. The r values associated with the different plate height mechanisms are summarized in Table I. Equation 1 2 can be written in logarithmic form
log H
=
log const.
+ r log ( u )
(13)
A plot of log H vs. log ( u ) should, over any range where a single term is dominant, yield a straight line whose slope equals exponent r for that dominant plate height mechanism. In this way, a log-log plot becomes a diagnostic tool for reducing the number of candidate mechanisms for the major plate height effect in FFF. This method can be applied to different parts of the velocity range, with the possible result that different principal mechanisms will emerge upon passing from one region to another. The diagnosis is further crystallized by the fact that the first four mechanisms of Table I are subject to explicit expressions. These are given in the equations whose numbers are indicated in the table. The basic diagnostic approach outlined above was developed during the initial study of plate height effects in thermal FFF (1). The above analysis is incomplete in the sense that other perturbations from ideality exist. These include such factors as dispersion in the dead volume of the apparatus and finite injection volume and time. I t is necessary to minimize these effects by using judicious experimental design.
EXPERIMENTAL The two TFFF columns used for the plate height vs. velocity study were of the standard type used in this laboratory. Their construction and operation have been described ( 4 ) . The channel dimensions of each were 0.0254 X 2.54 X 37.5 cm; the volume was 2.42 mL. The columns are distinguished here as No. 1 and No. 2, respectively. All samples but one were ultra-narrow fraction polystyrenes from Waters Associates. For these the manufacturer specified ,VLt/M,,-1.009, The 200000 MW sample was from Pressure Chemical Co. The solvent was ethyl benzene. All precautions were taken to minimize dead volumes and injection effects. T w o different systems were used for the column surface study. One was a copper bar system of standard dimensions with the column surfaces polished to different degrees of smoothness: the other was a 13.2-cm long glass plate system of the same breadth and width in which the column walls were formed by pieces of plate glass held together with a Plexiglas clamping apparatus. The latter column allowed the samples t o be observed and photographed as they traveled down the column. No field was applied to the glass plate column and the sample zone was therefore
,,.lll
/
, 0.1
q,iIwi
. .:. 1
/, ,
, 1 1 1
I
1
( .
-
L
J
, ,#,, I
1;. 1 (s.)
i',) 1 '
Figure 1. Logarithmic plot of plate height H vs. mean solvent flow velocity ( v ) for 2000 MW polystyrene. The constituent (straight-line) plots are calculated using the theoretical equations and parameters of t h e text. The linear summation of these constituent plots yields the curve labeled I H . T h e B H curve is to be compared with t h e experimental points from columns No. 1 and No. 2, which were made to virtually identical specifications but nonetheless yield slightly different results. For both columns, temperature increment l T i s 50 'C; R =
I
.'I
1
~
Figure 3. Logarithmic plate height plot for 51 000 MW polystyrene. Conditions are those of Figure 1. Retention ratio R = 0 . 5 4
10.92
11.11;
-
,L
-1 8' . .~ A -1 . ~~
i1.01
11.1
i s )
(,)
( I '8
-.I)
Figure 2. Logarithmic plot analogous to that in Figure 1 for 20000 M W polystyrene. 1 T = 50 OC; R = 0.73 diffused evenly across the channel width LC. In addition to the normal glass plate system having smooth upper and lower surfaces, three different configurations were used in which width, ri , was varied slightly from point to point across the breadth of the column. In one case a piece of Mylar approximately one-third the breadth of the channel and 0.001 in. (0.0254 mm) thick was placed as a strip down the center of the bottom plate creating. in effect, three separate channel regions: two of width, rL' = 0.010 in. (0.254 mm) at the sides of the channel. and one of width, rc = 0.009 in. (0.229 mm) in the center of the channel. In a second case, the entire bottom plate was covered with 0.001 in. (0.0254 mm) thick Mylar which was cut to produce a pattern of small, randomly shaped and spaced cutouts, resulting in the column having in some places a width of 0.010 in. (0.254 mm) and in others 0.011 in. (0.279 mm). The third surface study made with the glass plate system used a Mylar bottom surface sanded in the direction of the long axis with 120 grit sandpaper. In each case, the spacer used to form the channel was 0.010 in. (0.254 mm) thick. All the studies in the glass plate system were made using water as the solvent and a solution of malachite green oxalate as the
( L '
1.0 - i . J
Figure 4. Logarithmic plate height plot for 97 000 MW polystyrene run at AT = 50 'C. The R value is 0.36. F o r this polymer, stop-flow experiments were done, eliminating the effect of relaxation. T h e Z H curve calculated on this basis is shown as t h e broken line sample. Detection in this system was accomplished by means of a Laboratory Data Control Model 1205 Ultra Violet Monitor. Values of plate height were determined by the width-athalf-height method. Polymer diffusion coefficients were calculated from a formula given in a previous paper (9). Each experimental value reported for plate height represents 3-8 measurements, except in the case of the rough-wall column where 1-4 measurements have been used.
RESULTS AND DISCIJSSION Figures 1-4 show plate height results obtained from two different columns in the approximate solvent velocity range, 0.01 cm/s-0.1 cm/s. T h e limitations of the present solvent delivery system prevented operating a t flows much lower than 1 mL/h-equivalent t o about 0.01 cm/s. This lower limit, as demonstrated by the plots, is well above the plate height minimum in all cases. T h e failure of the longitudinal diffusion term to appear in any plot but Figure 1 is a result of its generally small magnitude, decreasing with molecular weigbt. T h e different scale ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
1753
used with Figures 2-4 excludes it entirely. All polydispersity contributions were calculated by means of Equation 9 using for p = M J M , the value 1.009 specified as the maximum by the manufacturer. The polydispersity contribution increases with MW (but, of course, not with velocity) because of the increasing value of d In V,/d In M with MW. For our calculations we have assumed t h a t d In VJd In M reaches a maximum value of 0.6 for high M values, in agreement with our empirical observations. We add here some general comments about polydispersity contributions in thermal FFF t h a t would also apply to techniques such as gel exclusion chromatography. For most available samples, the polydispersity is larger than t h a t applicable to the Waters' samples used here. It is large enough to seriously affect plate height measurements. Yet, errors make the effect difficult to calibrate. This may explain the results of our earlier study ( 1 ) where plate heights less than the calculated values were observed. As an example of the uncertainty, we refer t o 51 000 MW polystyrene reported by Pressure Chemical Co. to have p = (MU/&f,,)= 1.06. This polymer was used in the cited study ( I ) . If one takes the full range of the separate uncertainties in MtLand M , for this polymer, the calculated p range is 0.92-1.10. Quite obviously, with such uncertainties, the polydispersity contribution cannot be calculated very reliably. In this study the samples used were ultra-narrow fractions obtained by repeated gel-permeation chromatography. The reported M , / M , ratio was "less than 1.009". No error limits or individual measurements were given. Our experimental values substantiate, for the most part, a value near 1.009. However, the 2000 MW sample showed consistently high plate height values for several different columns with the discrepancy increasing a t lower velocities. This is exactly the result which would be expected if the polydispersity were higher than t h a t reported. This effect is far from certain, but in view of the uncertainties noted above, the quantitative contributions of polydispersity must be regarded as tentative. Fortunately, the slope of the experimental plots suggests that contribution is usually small in the present case. The most satisfactory feature of the plots of Figures 1-4 is the good agreement between theory and experiment. This agreement indicates t h a t the major plate height terms have been reasonably accounted for, and that the other terms-for this system at least-we small. The agreement is exceptional in comparison with most prior plate height studies as summarized earlier in the paper. Most prior studies, however, allowed neither for polydispersity nor relaxation. Furthermore, the present results are limited to relatively low retention volumes, no more than about three void volumes. Experimentally, one can eliminate the effect of relaxation on plate height by the stop-flow method: the solvent flow is halted immediately after injection for a period of time sufficient to allow the solute to "relax" to its quasi-equilibrium position above the cold wall ( 4 ) . Figure 4 shows two experimental values obtained with stop flow. The dashed line in this figure is the theoretical ZH curve without the relaxation contribution. The reduction in plate heights agrees well with the calculated reduction, supporting the applicability of Equation 5. T h e plots of Figures 1-4 show that the plate heights from column No. 1 are consistently lower-although not by a large margin-than those from column No. 2. The reasons for the difference in performance are not completely understood, but the observations are in good accord with previous experience with thermal FFF. This has shown that very consistent values of retention can be obtained from different columns, but that every column exhibits somewhat different performance characteristics as measured by plate height values. 1754
A N A L Y T I C A L CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
r
1
~ 1 -
-
1.11
I
tI!.lll
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