-\major difference between ZC, in its simple form described above, and other cthroinatographic techniques, as described b>- Keulemans ( I ) is that no material mores through the colunmthere is no “percolation.” In its fundamental aspect, however, the separation of solutes into bands, ZC resembles chromatography. Moreover, simulated or actual percolation of the solid solvent can be realized in ZC. If each time a molten zone is at one or the other end of the column it is removed and replaced by pure solvent, the effect, will be to remove solutes just as in conventional chromatography, even though there is not a net flow of solvent through the column. Allso,by using techniques of matter transport described in connection with continuous zone refining, (8) bulk flow of mat’erial along the column can be realized simultaneously with the zone-travels. Anot 1 in the solvent, or more properly, in the multicomponent solution. example of the former is benzene as a solvent for polystyrene. .\ wide range of k’s < 1 was found for certain growth conditions in this system by Loconti and Cahill ( 2 ) . An example of the latter is xylene as a solvent for polyethylene. Bands of solutes having k’s < 1 move forward, as mentioned above, and generally move more rapidly and spread out more rapidly than solutes having k’s > 1. The latter move backward and tend to be more concentrated. I t is also possible to select a solvent that will separate a mixture of solutes into two groups of bands, one moving forward, the other backward. The only published experiments the writer is aware of that might fall in the class of ZC are those of Peaker and Robb (4) on the partial fractionation of polystyrene in naphthalene and those of Loconti and Cahill ( 2 ) on polystyrene in benzene. Because the distributions of k-values for their systems were continuous, and because short columns and few zone-passes were used, imited
separations were achieved. Application of ZC to groups of distinct compounds, in apparatus utilizing an orderof-magnitude-greater number of zonelengths per column and more zonepasses can be expected to result in the kind of separations associated with conventional chromatography. T h i l e ZC is much slower than GLC, although perhaps comparable with some of the slower liquid-liquid chromatographic separations, it can be used for solutes of low volatility. It exploits a phase transformation hitherto unused for chromatography, and hence it may find application for entirely new classes of materials. ACKNOWLEDGMENT
The writer is indebted to E. Helfand; H. L. Frisch, K. A. Jackson, and P. R. Story for helpful discussions. LITERATURE CITED
I. M.,“Gas Chromatography,” Reinhold, 2nd Ed., New York 1959. (2) Loconti, J. D., Cahill, J. W., J . Polymer Sci. A , 1, 3163 (1963). (3) Martin, A. J. P., Synge, R. L. M., Biochem. J . 35, 1358 (1941). ( 4 ) Peaker, F. W., Robb, J. C., Nature No. 4649, 1591, (December 6, 1958). (5) Pfann, W. G., J . A p p l . Phys. 35, 258 (1964). (6) Pfann, W. G., Trans. A I M E 194, 747 (1) Keulemans, A.
f1052). \ - - - - ,
(7) Zbid., 197, 1441 (1953). (8) Pfann, W. G., “Zone Melting,” Wiley, Sew York, 1958. (9) Reiss, H.. Helfand., E.,. J . ADDL Phys. 32, 228 (1961). L
.
RECEIVEDfor review April 23, 1964. Accepted August 25, 1964.
Gradient Loaded Columns in Gas Chromatography DAVID C. LOCKE’ and CLIFTON E. MELOAN Department of Chemistry, Kansas State University, Manhattan, Kan,
b The use of gradient loaded columns in gas chromatography is a new technique involving the systematic variation of the partition ratio, k, during the course of an analysis by varying the liquid loading from the beginning to the end of the column. The case of a continuous linear decrease in k down the column is considered theoretically and experimentally. Equations are derived describing solute retention behavior, column efficiency, and solute resolution on columns with a linear gradient in k. Good agreement is obtained between the theoretical predictions and the experimental results on a 16-stage step-wise approximation to a continuous linear gradient column. For linear gradient columns, the partition ratio i s reduced to 50% of that which it would be on a regular 2234
ANALYTICAL CHEMISTRY
column of liquid loading corresponding to the initial k on the gradient column. Column efficiency is improved for solutes of low or intermediate partition ratio on the gradient column. The combination of these effects results in resolution of solutes of low retention which is superior to that which could be obtained on a regular column.
S
EVERAL
CHROMATOGRAPHIC
TECH-
have been developed which involve the systematic variation of a critical parameter during an analysis. Programmed temperature gas chromatography has provided a simple method for the analysis of wide-boiling mixtures in a minimum of time while enhancing resolution. Gradient elution liquid chromatography ( I ) , which involves an NIQUES
increase in the eluting power of the mobile phase during the course of an analysis, has offered similar advantages to the column rhromatographer. Porath (14) has described a step-graded adsorption column constructed of six sections, each successive section down the column containing charcoal deactivated with stearic acid to a lesser extent than the preceeding section. Satisfactory results were claimed for the analysis of pea-root exudate. I t has been suggested, notably by Purnell ( 1 5 ) ,that a gas chromatographic column in which the partition ratio, k , was varied down the column, might exhibit capabilities similar to those Present address, Department of Chemistry, Stevens Institute of Technology, Hoboken, PI’. J.
I
L
Figure 1. Schematic diagram of gradient loaded gas chromatographic column
offered by programmed temperature gas chromatography. This technique, involving a gradient liquid loading (for which the anacronym GLGC is used), has been considered theoretically and experimentally in this work. In GLGC, the partition ratio is varied continuously (or discontinuously) from the beginning i o the end of the column by a continuous (or discontinuous) variation in the liquid loading on each (infinitesimal) increment of column packing. Columns possessing a linear variation in k from the beginning (where the initial k is k,) to the end of the column (where k is 0) have been treated theoretically from the points of view of solute retention, column efficiency, and resolution. These equations have been verified experimentally on a column which closely approximates a linear gradient-a 16stage step-wise gradieint. In linear GLGC, the partition ratio is reduced by 50% to a value characteristic of the mean liquid loading on the gradient column. For solutes of low or intermediate retention, column efficiency is improved to a level matching that of a column constructed of the mean quantity of stationary phase on the gradient column. Most important is the improvement in resolution, which is superior for solutes of low or intermediate partition ratio on the gradient column to that on a regular column containing the average liquid loading on the gradient column. The gain in resolution results from the combination of the reduced partition ratio and the improved efficiency.
where v, = u(x) is the velocity of an unsorbed substance -Le., the carrier gas velocity-and R, is the ratio of solute velocity to carrier gas velocity. u is given as a function of 2, u ( z ) , to emphasize that the carrier gas velocity will vary down the column because of the compressibility of the mobile phase. Using ordinary gas chromatographic columns, an expression for retention time can be derived from Equations 1 and 2 :
where ii is the time average carrier gas velocity and t, is the retention time of an unsorbed component, In GLGC, k is a function of x and hence changes as the solute moves d o m the column. Many relationship5 can be written to describe an x-dependence of k . The case to be considered in this paper is a linear decrease in k with x, as described in Figure 1. For a linear gradient, the partition ratio a t any point down the column is
k
=
k,
+
CY2
(4)
where k , is the initial partition ratio and (Y = dk/dx. Thus Equation 2 becomes V, = U(X)
Rf
= U(X)
(1
Figure 2. Dependence of partition ratio on a = p,/po on GLGC column
f ( a ) Is deflned
by Equation 16
where c is the permeability constant. Second, from Boyle’s lawv,pu = pouo = constant. From these equations, it is readily derived that
Combining these equations, we arrive a t a manageable form for the second integral in Equation 6:
X
+ k , + ax)
(5) whose solution is
Retention time is now
Thus retention time is given by To evaluate the first integral in this equation, the general derivation of the Martin and James compressibility factor is followed, as detailed in reference ( 3 ) . We find:
tr
=
tm
(1
+ ko) +
-
5 I n terms of the partition ratio,
THEORETICAL
Solute Retention Behavior in GLGC. Solute retlsntion time, t,, may be defined by t,
=
LL$
where v, is the solute belocity and x is its position down the column of length L. From the fundamental theorems of chromatography, solute velocity is related to the partition ratio, k , by
(””; where a is the ratio of inlet to outlet pressures, p , / p o , and j is the gas compressibility factor. To arrive at this result, we note first that the carrier gas velocity is related in a general way to the pressure drop across the gradient column by
u
=
-c(dp,’dx)
1
a5 - 1)
5
uL2
+-7-=k,+
u&
5
+ aL
(12)
From Equation 9, it is readily shown that
(8) VOL. 36, NO. 12, NOVEMBER 1964
2235
k
AIakiiig substitution for , I C 2 in Equation 12, and recognizing that for a linear gratliciit column, a = - k , 'I,>
=Bo
c
+
3a5 - 5a3 2 5(a* - 1) ia3 - I) ~~~
]
(14)
Xote that when a = O-i.e.! when no gradient exists, t, = t, (1 k , ) as usual. Equation 14 predicts that the partition ratio on the GLGC column is a function only of the initial partition ratio, k , , and the inlet, outlet pressure ratio, a (or p , , when p , = 1 atin.). Figure 2 presents a plot of the coefficient of 12, in Equation 14,
+
as a function of a. As a increases, solute retention on the more heavily loaded end of the column must be di+1)roportionately weighted. In their recent study of combination columns, Hildebrand and Reilley (9) o h ~ w w lsuch behavior with a column consisting of two stages in series, the simplest case (and worst approximation) of a gradient column. Equation 14 provides soiiie theoretical basis for their observations. .ilthough a situation \$-here a < I is physically impossible, the curve in Figure 2 is drawn to a = 0 to show the niathematical form of Equation 14. Of (souwe, if the column was used with the heavily loaded end at the outlet, that portion of the curve in Figure 2 for a < 1 would be relwent-i.e., the applicable to situations involving both a linear decrease and a linear increase in k . [-sing the GLGC column, a t least a 507, decrease in partition ratio is predicted hy Equation 14. In terms of retmtion time, this corresponds to a minimum decrease given by t, = t,(l
+ k") t,(l
+ k",I2)
=
0.5 t,ka
The limit of f(a) as a becomes very large is 0.4, indicating that the maximum decrease in partition ratio is 6070, corresponding to a retention time difference of 0.6 t , k,. For purposes of predicting retention values from a knowledge of k , and Equation 14, the value of f ( a ) at a = 1 is used. This represents an extrapolation to zero pressure droll across the column, and yields a static value for k . -1 quite simplified derivation of a retention equation in GLGC lends support to this view: From Equation 6, for a gradient column, tr
=
s,
1
2
This is the same result ab Equation 14 a i t h u = 1.
Column Efficiency in GLGC. T o arrive a t a suitable formulation for plate height in GLGC, \\e mu-t first recognize that the local plate height, H , will change with poqition down the gradient column becau.e of the k-dependence of the mas< transfer resiktance contributions Furthermore, as has been stressed in detail by Giddings ( 6 ) , experimentall) one does not meaiure the local plate height, but lather the apparent plate hqght, H These two quantities, H and H , cannot be equated hecaube of the existence of nonuniform condition. (in most cases, of pressure, and in this nork, of pressure and k ) down the column To evaluate the effec: of these column nonuniformities on I € , Giddings ( 5 ) recommend- use of the formula (his Equation 10)
However, in GLGC, H , R,, and u all vary with x , which leads to an overly complicated integral to evaluate. We have thus adopted an approximate procedure which is somen hat different than that of Giddings. Folloning a general derivation of the rlassical (noncoupling) plate height equation, such as has been outlined in part by Dal Sogare and .Juvet (S), for the gradient column me arrive at
H
B
+ + CZU+ C,U B dz2 k , + ax A + + u Di (1 + ko + =
A
-
5
~
U
c1-
us
H dx
=
(:I
+ B,lx)dx +
or
where the "or" emphasizes the alternative expressions for the C, terms. Integration of Equation 19 and algebraic simplification of the result leads to an expression for the apparent (observed) plate height :
fl =
:l
+ B/u + k,
+1
I+
1
+ k , + 1) ( k o + 1) 1 ( a L + ko + 1 1 2 - 1 -DT (2[- a L + k, + 1 + 2 ) - In ( a L + k , + k, + 1 ko + 1 -
(aL
')I)
kO(k0
or where A , B , cII and c2' are constants, D , and D i are solute diffusion coefficients in the gas and liquid phaqes, respectivel), and d , and di are the reqpective gay and liquid film thicknesses The C, term in this equation is due to Jones (IO) Mole recent nork b) Giddings (4) and Perrett and Purnell (13) has shonn the inadequac) of wch a k-dependence of the C, term They iuggest that the gak 1)ha.e mass transfer resistance contribution to H is rather given b\ an equation of the form
.
u(x)
ANALYTICAL CHEMISTRY
part,icle diameter and f (I'l) represents a liquid phase nialdistribution function. For the sake of completeness, in this work we will study the effect of gradient liquid loading on both the Jones and the Giddings-Purnell C, terms. With the gradient column, d i , d,, c2', f ( V i ) , and u all vary with x . To simplify Equation 17, we employ mean values for these parameters, di, do, c2 = F~', and f'(Vi)> some average value for the maldistribution function. Furthermore, we assume that velocity gradients in the column are negligible such that u z u ( x ) . Incorporating these approximations into Equation 17 and mult,iplying through by dx, we have
Czd,2U
+ k" + ax dx
Extracting the mean value of u ( x ) over the interval O
