Anal. Chem. 1989, 6 1 , 1937-1941 0.40 I
I
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of sample density enables accurate particle size distribution characterization of broad distribution latexes of this type.
CONCLUSION
0
18
54 TIME (MIN.)
36
72
90
F w e 8. Turbdi-time fractograms for latex D using standard mobile phase of 0.1 % Aerosol-OT (solid line; density = 0.9973 g/cm3) and methanoCmodifkd mobile phase (broken line; density
= 0.9882 g/cm3).
with a 95% confidence limit of the x intercept from 1.014 to 1.025 g/cm3. The linear R square value was 0.995. An assumed particle density value of 1.080 g/cm3 was used for latex C. For latex D, the density was measured by using the calculated particle size at the maximum of each of the two peaks in the fractogram shown in Figure 8. An assumed particle density value of 1.080 g/cm3 was used. Fractograms are shown for the standard mobile phase and the methanol-modified mobile phase of 0.9882 g/cm3. As expected, the fractograms are unchanged in shape and character for the two mobile phases. The sample in the methanol-modified mobile phase elutes approximately 7 min later than that of the one in the standard mobile phase. The densities of peaks 1and 2 for latex D in the fractogram are statistically equal, with measured values of 1.012 and 1.009 g/cm3, respectively. The 95% confidence limits of the x intercept for peak 1 are 1.003-1.032 g/cm3 and for peak 2, 1.009-1.019 g/cm3. The linear R square values are 0.980 and 0.960, respectively. This result demonstrates that for this particular vinyl acetate-acrylic latex, the copolymer composition is constant over the entire particle size distribution, although a distinct bimodal character is indicated. The lack of particle heterogeneity coupled with in situ determination
The use of methanol-modified mobile phases with SFFF expands the utility of this technique for characterizing densities of copolymer latexes with values near 1.00 g/cm3. Mobile phase densities from approximately 0.96-1.00 g/cm3 have been successfully used for characterization of these types of polymers. For particle size calculations of copolymers that exhibit suspected bimodal or broad distributions, it is important to establish that separation is by size alone and not due to polymer density heterogeneity.
ACKNOWLEDGMENT The author wishes to express his appreciation to S. C. Voth for assistance in these experiments and W. F. Tiedge and J. V. Martinez for helpful discussions. Registry No. Polystyrene, 9003-53-6; (styrene)(butadiene) (copolymer),9003-55-8.
LITERATURE CITED (1) Yang, F. J.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1974, 4 6 , 1924. (2) Giddings, J. C.; Caidwell. K. D.; Fisher, S. R.; Myers, M. N. Methods Biochem. Anal. 1980, 26, 79. (3) Kirkland, J. J.; Yau, W. W. Anal. Chem. 1983, 55, 2165. (4) Levy, G. B. Amer. Lab. 1987, 19, 84. (5) Kirkland, J. J.; Yau, W. W. Science 1982, 218, 121. (6) Yonker, C. R.; Jones, H. K.; Robertson, D. M. Anal. Chem. 1987, 59, 2574. (7) Giddings, J. C.; Karaiskakis, G.; Caldwell, K. D. S e p . Sci. Techno/. 1981, 16, 607. (8) Yau, W. W.; Kirkland, J. J. S e p . Sci. Techno/. 1981, 16, 577. (9) Po&mer Handbook; Brandrup, J., Irnmergut, E. H., Eds.; John Wiley 8 Sons: New York, 1967.
RECEIVED for review March 13,1989. Accepted May 23,1989. This work was presented in part at the First International Symposium on Field-Flow Fractionation, June 15, 1989, Park City, UT.
A Different Perspective on the Theoretical Plate in Equilibrium Chromatography Paul J. Karol Department of Chemistry, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
The differential rate model for equlllbrlum chromatography Is used to derlve both the continuous (Martin and Synge) plate model and the stepwlse (Cralg) plate model. The latter have previously been regarded as severely deflclent because of the ad hoc manner In whlch the plate height Is related to the chemistry and physlcs of the chromatographic process. However, It Is demonstrated that the “height equivalent to a theoretical plate” arises naturally from a finite different approach to solvlng the rate equatlon. I t Is also argued that the much mallgned step model Is a phenomenologically valld approximation to the continuous plate model and also to the rate model.
INTRODUCTION
of view, we will try to demonstrate that the plate concept is a phenomenologically and mathematically sound approximation to the chromatographic process and that its use is partially exonerated. After separately reviewing the differential rate model and the plate model, the link between them will be discussed. In so doing, it will be argued that the literature on this topic (1-7) routinely handicaps the plate model by prematurely invoking a unidirectional transport restraint in deriving chromatogram equations. It is that act which is the source of discord with the current procedure.
THEORY The (simplified) rate model for continuous flow equilibrium chromatography ( I ) , ignoring velocity profiles, usually starts with a differential rate equation such as
For the past 25 years the theoretical plate model has been viewed as failing to describe the physical and molecular events occurring in chromatography ( 1 ) . In contrast to this point 0003-2700/89/0361-1937$01.50/0
0 1989 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 17, SEPTEMBER 1, 1989
which describes the simultaneous drift and dispersion of the eluting species. With convenient boundary conditions, eq 1 yields a normalized chromatogram as a function of time t for a column of length L (8) c(t) =
CO
(47rDet)'/'
On occasion, reference in the literature to the function of eq 2 incorrectly terms it a Gaussian function. Statistical moments of the equilibrium chromatogram have also been derived (9, 10) for a column of finite length L. Below are the mean, variance, and skewness taken from previous work (10) t, = L/u,
Implicit in eq 1 for equilibrium chromatography is the stipulation that the mobile phase velocity is low enough to justify ignoring interphase kinetic effects. Kinetic considerations would necessitate additional terms and additional equations. In the above equations, u, is the eluite velocity (the mobile phase velocity divided by 1 + k'where k'is the capacity factor). De is the longitudinal dispersion constant divided by 1 k'and c(z,t) is the concentration of solute at position z and time t on the column. What we will show is that the solution to the differential rate equation can lead cleanly to the so-called plate model and that there is no prerequisite for a large number of plates. Finite Difference Solution. Equation 1 is a second-order linear differential equation which, although it has an exact analytical solution, will be solved for our ultimate purposes by numerical approximation. For convenience, we will abbreviate eq 1 as ct = DecZz- U,C,
+
The numerical solution to the above equation will be accomplished by use of the finite difference method ( 1 1 , I 2 ) . Briefly, multidimensional variable space is replaced by a grid whose mesh size can be made smaller and smaller. As is usually done, the mesh sizes in the z and t directions are labeled h and k , respectively. Any point in z,t space can be represented by the number of mesh steps from the origin: 2, = mh and t , = nk. A Taylor series expansion replaces partial derivatives with corresponding difference quotients C(Z
C(Z
+ h,t) = c(z,t) + hc,(z,t) + h2 -c,,(z,~) + ... 2 - h,t) = c ( z , ~ -) hc,(z,t)
h2 + -c,,(z,~) 2
- ...
Subtraction and addition of the above truncated equations give the following approximations with errors of order 0 (h2) C,
c,,
-
1, respectively. Substitution of the above finite difference approximations into the partial differential equation 1then gives
-
L [ c ( z + h , t ) - c(z-h,t)] 2h
1
-[c(z+h,t) h2
- 2c(z,t)
+ c(z-h,t)]
Additionally c(z,t+k) = c(z,t)
+ kc,(z,t) + ...
yields
in which X = k/h2. In this approximation, the concentration at grid point m at time step n 1on the left is equal to the sum of three terms on the right, all determined at the previous time step n: a forward-feed term from the previous grid point m - 1, a retention at the given grid point m, and a backward-feed from the next grid point m + 1. By "feed", we imply a combination of a forward-only velocity effect and a forward-and-backward diffusion effect. Traditional finite difference developments of the rate model using the Taylor series ( I , 2 ) invariably exclude the last term corresponding to backward feeding. Our approach is thus a departure from the conventional one. Equation 4 can be used to generate the approximate oncolumn profile even in the diffusion-only limit, u, 0. This observation is relevant to the criticism ( I ) that plate treatments do not include effects of longitudinal diffusion. However, we will defer further discussion of the finite element solution until the plate models are addressed. Plate Models. Martin and Synge's plate model (3) envisions the chromatographic column as being a linear sequence of hypothetical cells or plates, each of which consists of identically repeating proportions of stationary phase and mobile phase. The loading plate, numbered zero, contains the sample solute which partitions between the two phases according to the equilibrium retention ratio, R. R, the fraction of solute in the mobile phase, is related to the capacity factor k' = (1- R ) / R . Mobile phase is then transferred downstream to the next plate, numbered 1; a portion of what was in 1 is similarly transferred to 2; etc. Simultaneously, pure mobile phase is added to the original cell containing stationary phase with retained solute. We will symbolize the time step for this transfer by 7. Whether these transfers are truly stepwise or infinitesimally stepwise (continuous) can be dealt with according to a very simple argument presented below. Picture each cell as being divided into u identical (hypothetical) "subcells". For v = 1, we will have the stepwise m we will have the (Craig) plate model, whereas for v (Martin and Synge) continuous plate model describing the transfers at equilibrium. The amount being transferred will be calculated from the equilibrium concentration in a volume of one subcell and is identical with the concentration in all other subcells in the given plate. Such a picture is consistent with the plate definition as "the length of a cell whose mean concentration is in equilibrium with its own effluent" (I). It is similar to the rectangular step approximation used in numerically integrating smooth functions that actually vary over the step's width. In terms of fractions of the original sample amount, the zero-numbered plate's total solute amount after the first transfer is calculated from the sum of 1 - R, the fraction remaining in the stationary phase, and R ( l - (l/v)), the fraction remaining in the mobile phase. Here we introduce the abbreviation
+
-
-
p =R/Y
for the fraction of solute in a subcell, and
-
c(z,t+k) - c(z,t) Ct k The grid coordinates can be identified by just their indices. For example, z - h, z , and z h become m - 1, m, and m +
+
T = T/V for the steps to transfer a volume of mobile phase equal to that in a complete plate. Note that R T = p r . The letter "p" has frequently been used to represent R or p or sometimes
ANALYTICAL CHEMISTRY, VOL. 61, NO. 17, SEPTEMBER 1, 1989
both (4). To avoid this ambiguity, we have chosen not to use ,p". With the above definitions, the zeroth plate contains a solute fraction 1 - p after the first partial transfer. At equilibrium, repartition yields fractional compositions of R ( 1 - p ) and ( 1 - R ) ( 1 - p ) in the mobile and stationary phases, respectively. In the first plate down the sequence, the transferred total fraction is p and repartitioning yields fractions Rp and ( 1 - R ) p in the mobile and stationary phases, respectively. After a second partial transfer and equilibration, the original, first, and second plates contain total fractions ( 1 - P ) ~ 2, p ( l - p ) , and p2, respectively. In terms of amounts rather than fractions, the continued development as above the amount corresponds to a recursion relationship for c,,+~, of solute in plate m following the ( n + 1)st transfer. In terms of c,,-~,, and c, the amounts in the given and upstream plates prior to the transfer, we have for the continuous plate model cm,n+1
-- PCm-l,n + [ 1 - PIC,
(5)
scanned the column as a function of distance from the load point. It is incorrect, however, to conclude the derivation at this stage as is occasionally done. The binomial and Poisson distributions pertain only to the on-column development. That is, those distributions express the column spatial profile for fixed number of transfers and with plates j ranging between 0 and T . The chromatogram develops off the column and the elution (time) profile is a different function, one which properly involves a fixed plate index j which we will now call r. Then T for the chromatogram ranges from r through m. If the last plate is r, determined by the length of the column, then the elution chromatogram is determined by the fraction of solute in the rth plate. Equilibrium following transfer T = r determines that a fraction p of the amount in cell r is in the mobile phase and will move out of the column on the next transfer. Consequently, the effluent chromatogram is functionally given by terms of the form T!
and, parenthetically
= Rcm-l,n + [ 1 - Rlcm,n
cm,n+1
(6)
(1 - p)3
+ 3p(1 - p)' + 3p2(1 - p ) + p3 = 1
After a total of T transfers, plates 0 through T have total compositions corresponding to the T + 1 terms in the binomial expansion generalized as
[(I - P) Each of the written as
T
+ PIT = 1
+ 1 terms in the binomial expansion can be T!
pq1 - p)?j j!(7 - j ) !
(for r = constant)
- p)"
p-pr(l
for the step plate model. After a third transfer, four plates have total compositions given sequentially by the terms on the left in the sum below
1939
r!(r - r ) !
which are to elute beginning with the ( T + 1)sttransfer. The elution profile is neither a binomial nor Poisson distribution. Nor is the Gaussian approximation necessarily valid for large transfer numbers as is evident in eq 2 or in the nonzero third central moment of the chromatogram (9, IO).
RESULTS Stepwise Plate Model Result. Bearing in mind that the terms with 0 5 T C r are explicitly equal to zero, reflecting the displacement of the interstitial volume that precedes the original load in the Craig model, the normalized chromatogram, f ( j = r , T ) = c ( T + l ) ,for the stepwise case and the normalization constraint are correctly given by the infinite series m
with plate index j running from 0 to T . This is identical with what has been given in most developments of the plate model where attention is immediately drawn to the binomial distribution and its mathematical properties. In particular, the Craig stepwise model in which v = 1,p = R , and T = T gives
E c ( T + 1) = 1
(10)
T=r
where c ( T + 1) = 0
for O < T
