expected in an optimal system. Most importantly, the rotational velocity and the field strength must be increased to gain any significant advances. The degree to which performance might be enhanced is shown below. Under conditions of high retention, Equation 13 can be substituted into Equation 14 to yield the potential plate height
H = 24h'~~(~)/D
This equation shows that the efficacy of SFFF can be improved by manipulating a number of variables, including viscosity, temperature, and density. The method will, in theory, work much better with large particles than small, as reflected in the fifth power dependence on particle diameter d. However there are practical limits to this gain which will become apparent as d approaches either layer thickness 1 or surface-roughness dimensions in magnitude. Equation 37 shows that C for a given particle is inversely proportional to the square of the sedimentation field strength, G. The dependence on rotational velocity is therefore inverse fourth power. In the present study the maximum G was about 500g. If ultracentrifuges, with field strengths up to 300 times greater than this, were adapted to SFFF, C values could in theory be reduced by (300)2 = 90,000. While such gains would not be totally applicable to particles in the size range used here because of the previously mentioned restrictions on size, the above factor would be applicable to much smaller particles and to macromolecules, thus making their separation convenient also. Particle size analysis is significant in many fields of environmental control and industrial operation. The present method is promising in such analyses by virtue of its predictable dependence on simple mass and density parameters and its potential for further improvements in fractionating power.
(33)
With the aid of Equation 10 this expression becomes
H = 4h2w2v/D (34) where the zone velocity in the column, R ( u ) has been written simply as u. If now Equation 8 is used to eliminate X2, we have 144 ( k T ) 2 v H = (35) i72d6G2 (4p)2D If diffusion coefficient D is replaced by the Stokes-Einstein value, D = kT/3agd, where 7 is the coefficient of viscosity, we get 432 kTyv H = T dsGz(Ap)z If H is written as Cv,then the nonequilibrium coefficient C not only reflects the level of H achievable, but, more importantly, C can be shown to equal the minimum time in which a theoretical plate can be generated. We have
c = -432 T
kTy d5G2(4~)~
(37)
RECEIVEDfor review March 21, 1974. Accepted July 12, 1974. This investigation was supported by Public Health Service Research Grant GM 10861-17 from the National Institutes of Health.
Programmed Sedimentation Field-Flow Fractionation Frank J. F. Yang,' Marcus N. Myers, and J. Calvin Giddings Deparfment of Chemistry, University of Utah, Salt Lake City, Utah 84 7 72
This paper describes the development of two programming systems for sedimentation field-flow fractionation (SFFF): programmed field strength SFFF and programmed solvent density SFFF. The necessity for developing programming systems in SFFF is discussed, and both general and specific theories of programming are developed. A centrifugal SFFF system was adapted to programming by the controlled variation of rotation speed and solvent density. Polystyrene latex beads with diameters from 1756 to 3117A were fractionated by this device. Agreement between theoretical and experimental retention times was within about 5%, showing that the essential features of the technique are well characterized.
Sedimentation field-flow fractionation (SFFF) is a method designed for the separation of particles and large macromolecules. The essential features of the method, and a review of related work, appear in the preceding publication (1). Present address, Department of Chemistry, Oregon State University, Corvallis, Ore. 97331. (1) J. C. Giddings, F. J. F. Yang, and M. N. Myers, Anal. Chem., 46, 1917 (1974).
1924
ANALYTICAL CHEMISTRY, VOL. 46, NO.
All field-flow fractionation (FFF) methods, including SFFF, generate or are expected to generate elution patterns for macromolecules resembling the patterns generated by gas and liquid chromatography for smaller molecular species (2-4). When attempts are made to fractionate wideranging mixtures by chromatography, problems are encountered due to the incomplete resolution of early peaks and the excessive retention time and peak width of late peaks. The same difficulty can be expected to occur with most forms of FFF as a result of the analogous elution patterns. This problem in the field of chromatography has been referred to as the "general elution problem" ( 5 ) ;its scope appears, from the present analysis, to encompass certain nonchromatographic techniques as well. The solution to this problem is quite effectively realized in various programming techniques. Programmed temperature gas chromatography (6) and gradient elution liquid (2) G. H. Thompson, M. N. Myers, and J. C. Giddings, Anal. Chem., 41,
1219 (1969). (3) E. Grushka, K. D. Caldwell. M. N. Myers, and J. C. Giddings. Separ. Purificafion Mefh., 2, 127 (1973). (4) M. N. Myers, K. D. Caldwell, and J. C. Giddings, Separ. Sci., 9, 47 (1974). (5) L. R. Snyder, "Principles of Adsorption Chromatography,'' Dekker, New York. N.Y., 1968. (6) W. E. Harris and H. W. Habgood, "Programmed Temperature Gas Chromatography," Wiley, New York. N.Y., 1966.
13, NOVEMBER 1974
chromatography (7) are classic examples. The reason that programming techniques accomplish the separation-and thus the rationale of programmed FFF-can be understood as follows. For each peak (or close-lying pair of peaks), there is an ideal level of retention during migration involving a compromise between resolution, time-economy and peakheight factors (8, 9). The ideal range in chromatography is generally somewhere in the region described by R = 0.2 to 0.5, where R is the retention ratio. For gas chromatography, ideal retention occurs near the boiling point (9). Effective programming, as has been shown in an analysis of programmed temperature gas chromatography, continuously alters the retention parameter of each component so that the bulk of its migration can occur in the ideal range (8,9). The first requirement for an effective programming system in any method of separation is the existence of some parameter whose variation will alter the retention time of solute peaks. Temperature and solvent composition serve in this capacity in most programmed systems of chromatography. In programmed FFF, these same two parameters can be used in many instances, but a still more versatile parameter exists. This parameter is the strength of the imposed field. As retention is caused by this field, control ranges all the way to zero retention. Because the field is externally applied, it can be varied with precision according to any desired program (10). The present work describes results with two programmed parameters in SFFF: solvent composition (density) and the external centrifugal field. Development of the theory of these programmed systems will be followed by a description of preliminary experimental results in which polystyrene beads of different diameters are fractionated.
GENERAL THEORY OF PROGRAMMED RETENTION The theoretical prediction of retention in programmed
SFFF is complicated by the variable velocity of the zone in the column. One must. develop integral expressions that properly sum up each small increment of migration. Similar problems exist for programmed forms of chromatography (6,8,9). In this section, we will subdivide the general programming technique into two subclasses and derive the essential integrals describing each. In the next two sections, we will obtain specific solutions for a select group of programs in
SFFF. Basic Retention Parameters. The initial step in developing the theory of programmed FFF, in any of its forms, lies in stating the basic retention equations. For uniform fields and under ideal conditions, the retention ratio, R, is given by ( 1 1 )
R = 6A[coth(1/2h) - 2h] where X is the dimensionless retention parameter
(1)
h = l/U! (2) in which 1 is the exponential length constant (very roughly the mean layer thickness) describing the solute distribution and w is the width of the column. Equation 1 is a mathematical form too complex to work (7)P. R. Brown, "High Pressure Liquid Chromatography." Academic Press, New York, N.Y., 1973. (8) J. C. Giddings. "Gas Chromatography." Academic Press, New York,
N.Y.. 1962,Chap. 5. (9) J. C. Giddings, J. Chern. Educ.. 39,569 (1962). (10)J. C. Giddings. Separ. Sci.. I,123 (1966). (11) M. E. Hovingh. G. H. Thompson, and J. C. Giddings. Anal. Chern., 42, 195 (1970).
with conveniently under the variable conditions of programming systems. Simple approximations work under some circumstances, as for example
R = 6X which is a limiting form approached as X
R
-
0, and
= 6h - 1 2 h 2
(4)
an approximation which, by including a term of higher order, considerably extends the range of applicability of Equation 3. Equation 3 overestimates R by 20% at R = 0.25 whereas Equation 4 is correct to well within 10% up to R = 0.7 (12). In SFFF, parammeter X takes the general form ( I )
kT l
=
m
s
(5)
where K is Boltzmann's constant, T is temperature, m is particle mass, G is gravity, p s is particle density and Ap = ps- P; in which p is solvent density. For spherical particles, of the type used in the experimental work, we have
6kT 7id3CwAp where d is particle diameter. General Theory of Programming. Programmed elution methods can be divided into two basic categories which require different theoretical treatments. In the first class, which we term uniform programming, some parameters like temperature are changed simultaneously and equally in all parts of the column. In the second class, termed solvent programming, change is induced by the inflowing solvent and it thus affects the head of the column before inducing change further down. Gradients exist, therefore, in distance as well as in time. The programming of field strength falls in the category of uniform programming while programmed solvent density falls in the solvent programming category. In both categories there exists, in general, some parameter Y which controls retention and which can be systematically varied to yield the desired program in R. We must, first of all, determine by theory or experiment the functional dependence A =
R = R(Y) which relates the R- program to the Y-program. Instantaneous zone migration hinges on the instantaneous value of R in the same fashion as in chromatography (8, 13)
dZ
= R(l')dt
where d Z is the incremental distance moved by the zone center in the time dt. Quantity ( u ) is the solvent velocity expressed as a cross-sectional average. Equation 8 is the basic differential equation for programming theory. Uniform programming in FFF can include both field strength and temperature variations. These parameters are forced to follow some program
Y = Y(t) This program applied equally over the column length and thus R follows some corresponding program, R ( t ) ,in a uniform fashion throughout the column. (For temperature programming a slight lag exists for temperature equilibration, but the lag, too, is essentially constant throughout the column.) Therefore Equation 8 can be integrated over the total migration path from injection to elution (12)J. C. Giddings. J. Chern. Educ., 50,667 (1973). (13)J. C. Giddings, J. Chrornatog.,4, 1 1 (1960).
ANALYTICAL CHEMISTRY, VOL. 46, NO. 13, NOVEMBER 1974
1925
Figure 2. Plots of @ as a function of Xo fcr I q y d - .
1 7
A”
C
Figure I.plots of R vs. XO for different T value5
G ( t ) = G , - at Equation 15 integrates to
These ciirv~swere obtarned by numerical methods from Equation 18
where t , is the elution time. The program R ( t ) must, of course, he evaluated by combining Equations 7 and 9. InteBration provides a solution for t , and thus allows the desired prediction of ehtion patterns. Programming in the second category (solvent programming) is more complicated. Here some essential solvent property, Y , is changed continuously a t the column inlet according to the program
Y’ = Y ’ ( t ) (11) A given iv1i.t value of this property is normally carried down the column a t mean solvent velocity, ( u ) . Its arrival a t a migrating zone a distance Z downstream lags behind its presence a t the origin by the time Z/( u ) Y = Y‘(t -
z/(?’))
(12)
Thus the value of Y , and the associated value of R, is both time and distance dependent. In this case, the basic differential expression of Equation 8 is of the form
dZ
dt = (v)R’(t
-
various 7 values
These plots were obtained by numerical methods from Equation 24 Crosshatched region IS only part excluded from validitv by Equation 27
z/(v))
A change of variables from t to y = t - Z/( u ) yields the dif-
ferential equation d Z = ( u ) R’(y) dy/(l - R’(y)). Integration over the total miqration path leads to the basic integral eauation of solvent programming
(0
5
f
5
G/ff)
A solution for t , can be obtained from this equation by numerical means. The calculated results are made more general by obtaining the value of G a t retention, G,, through the rearranged equation
l n ( l / @ ) - ( 2 h o / @ )= (7/6ho) - 2 h o
where t o is the elution time of a nonretained peak. Equation 14 is applied to solvent programming in the same way that Equation 10 is used to obtain solutions for uniform programming.
f-,
Go(1 - 9 ) / 0
(19) Since Equation 4 has been employed for R, this approach is only valid so long as X remains under 0 7 This condition is expressed by zz
@ 2 XO/O.? (20) A first simple approximation to retention time calculations can be obtained by using Equation 3 for R instead of Equation 4. T o do this, one simply retain!: the first term on the right of Equation 16 and l7b and the first terms on the left and right of Equation 18. This vields the retention time
t ? --- C %[1 Y
-
e x p ( - ~ / 6 ~ , )=]
%[1 - exp(- aL/6A(2)))]
x
= A/G (15) where A = kT/mw(Ao/p,). Ry combining this with Equations 4 and 10, we obtain
(21)
The theory developed above can be applied to any form of FFF in which linear field programming is applied. In the case of SFFF, a linear program in the angalar velocity is likely to be more convenient experimentally than a linear program in field strength
w
= w o --
pt
(0
5
t
5
w/a)
(22)
The value X is
x --
THEORY O F PROGRAMMED FIELD SFFF In this case the acceleration term, G,replaces Y of the general diwuwion. By virtue of Equation 5 . Equation 7 has the fwm
(18)
where 8 = G,/Go, is the initial retention parameter which from Equation 14 is seen to be A/Go,and T is the ratio of void-peak elution time. Golo. This equation descrihes B in 2nd 7. Tn this form. uniterms of only two parameters, versal plots can be generated for the 0 solutions (Figure 1) which then yield t , in the simple form
cy
(Solvent P r o q r a m m i n g )
(l7a)
R/W2
(23)
where B = kt/rnwroAp/pF, as can be seen hy comhining Equation 5 and G = d r o . A procedure to integration and rearrangement similar to the one above. using R = 6X 12X2 from Equation 4, gives the equation
where A0 and T ( T = @ L / q( u ) ) have the same meaning!: as before, and in which
J f the G-program i s linear in time 1926
ANALYTICAL CHEMISTRY, VOL. 46, NO. 13, NOVEMBER 1974
Quantity defines the point of elution and provides elution time t , through an expression analogous to Equation 19 (26) = ~o(1 - @)/P Numerical means must be employed to obtain @ for different 7 and XO values. The results of numerical calculations are shown graphically in Figure 2. The criterion that A, should not exceed 0.7 is given by tr
a2 2
7 (27) When X is small enough throughout the run to employ R = 6X, Equation 3, we obtain the simplified retention equation
8+!
t*\\
I
-Ape’ -1ME. t
__t
Figure 3. A schematic illustration of the density increment (Ap’) program described by Equation 38,with Ap, negative
the column a t elution time t,. The elution time itself is obtained from the above { definition and is the following function of
