Anal. Chem. 1987, 59, 28-37
28
to low molecular weight components (of the order of magnitude of lo3) than is liquid-based thermal FFF technology. In addition, theory predicts a higher selectivity S for thermal FFF systems using supercritical fluids (S 1) than those using liquids (S 0.6). Theory consequently implies that the selectivity may nearly double using supercritical fluids in place of liquids, thus approaching the selectivity level (unity) found for sedimentation FFF (17),but applicable over a much lower molecular weight range than that accessible to sedimentation FFF.
-
-
LITERATURE CITED (1) Hovingh, M. E.; Thompson, G. H.; Glddings, J. C. Anal. Chem. 1970, 42, 195. (2) Glddings, J. C.; Smith, L. K.; Myers, M. N. Anal. Chem. 1978, 4 8 , 1587. (3) Glddings, J. C.; Martin, M.; Myers, M. N. J . Chromatogr. 1978, 158, 419. (4) Giddings, J. C.; Myers, M. N.; Janca, J. J . Chromatogr. 1979, 186, 37. (5) DeGroot, S. R. Thermodynemics of Irreversible pmxsses ; Interscience: New York, 1951. (6) Giddings, J. C.; Smith, L. K.; Myers, M. N. Anal. Chem. 1975, 4 7 , 2389.
Giddhgs, J. C.; Myers, M. N.; McLaren, L.; Kelier, R. A. Science (Washlngton, D . C . ) 1908, 162, 67. Myers, M. N.; W i n g s , J. C. I n Progress in Separation and Purification; Perry, E. S.,Van Oss, C. J., Eds.; Wiiey: New York, 1970; Vol. 3, p 133. Giddings. J. C.; Myers, M. N.; King, J. W. J. Chromatogr. Sci. 1969, .7 ,. 276. Czubryt, J. J.; Myers, M. N.; W i n g s , J. C. J. Phys. Chem. 1970, 7 4 , 4260. Gunderson, J. J.; Caldwell, K. D.; Giddings, J. C. Sep. Sci. Techno/. 1984, 19. 667. Gunderson, J. J. Doctoral Thesis, University of Utah, Salt Lake City, UT, 1986. Stephan. K.; Lucas, K. Vlscoslty of Dense Fluids; Plenum: New York, 1979. Varbnik, N. 0. Tables on the Thsrmophysical Properfies of Liquids and Geses; Wlley: New York, 1975. Giddings, J. C.; Caldwell, K. D.; Myers, M. N. Macromolecules 1978, 9, 106. Scott, J. R.; Roff, W. J. Handbook of Common Polymers; Chemical Rubber Co.: Cleveland, OH, 1971. Myers, M. N.; Giddings, J. C. Anal. Chem. 1982, 5 4 , 2284.
RECEIVED for review May 19, 1986. Accepted September 3, This is based work supported by Grant CHE82-18503 from the National Science Foundation.
Fractionating Power in Programmed Field-Flow Fractionation: Exponential Sedimentation Field Decay J. Calvin Giddings* and P. Stephen Williams Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 Ronald Beckett
Water Studies Centre, Chisholm Institute of Technology, Caulfield East, Victoria, Australia
I n order tofill a maJorvoid In the opthrlzakmof programmed fleld-flow fractionation (FFF), we have devdoped general theoretlcal expressions for resdutbn, expremed as fractknating power F d , applkable to essentla#y ail fiekl-programmed FFF systems. The rewklng Integral expressions have then been worked out for a constant fldd/exponentlal Held decay programmlng sequence that Includes the Kbklad-Yau tbnedelay-exponentlal program as a specla1 case. The flnal Fd equation has bean applkd to FFF, yiekwns pkts of F d vs. partlcle dlameter that account for varlatlons In Fd wlth changes In void t h e (or flow veloctty), expmntlal programmlng rate, constant-fkld t h e , InHlal field strength, partkle density, and channel thlcknera Based on the resuits of these plots, we have developed an optlmlzatlon strategy for exponentlally programmed sedlmentatlon FFF.
The importance of programmed operation in field-flow fractionation (FFF) was recognized a t an early stage in the development of sedimentation FFF ( I ) . The philosophy of programming, both in chromatography and field-flow fractionation, has evolved from the recognition that no single set of conditions is suitable for the resolution of each of the components in a mixture of widely variable properties. Consequently, various retention-influencing parameters 0003-2700/87/0359-0028$01.50/0
(temperature, solvent properties, field strength, flow velocity, etc.) are varied in the course of a run in order to expose in an orderly sequence each of the components to effective separation conditions. Sedimentation FFF separates colloidal components on the basis of their effective mass (2). Under ideal conditions, without programming, the retention time or volume is roughly proportionalto particle mass. Thus, a 10-fold range in particle diameter corresponds approximately to a 1000-fold range in retention time. The widely spaced retention times yield a high selectivity leading to very high resolution levels for similar particles, but the process is often impractical because of the inordinate time required for elution of the most massive components. Consequently, programming is invoked. Most commonly, field programming is used. In this technique a high initial field strength is employed for the separation of the small colloidal particles, following which the field strength is programmed downward to provide suitable conditions for the larger particles ( I , 3 ) . Accordingly, the full particle range can be eluted in practical experimental times. The potential and limitations of such programming methods have been discussed in a recent paper (3). Field programming has also been used with other FFF techniques, notably thermal FFF and flow FFF ( 4 , 5 ) . However, selectivity is not as high in these cases as in sedimentation FFF and the need for programming is somewhat blunted by the smaller range in elution time and volume as 0 1986 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 59, NO. 1, JANUARY 1987
compared to that exhibited by sedimentation FFF. Two other programming approaches in FFF utilize changes in carrier properties such as density ( I ) and changes in flow velocity (6). Theoretical analysis has shown that flow programming is particularly promising, but this promise is offset some by experimental complications (6). The theory of retention in programmed FFF operation has been developed and applied to various program forms. The general theory ( I ) provides a rigorous equation for the elution spectrum (elution time vs. particle diameter) for each program. Thus, particle diameter can be obtained in terms of elution time much as in nonprogrammed operation. The advantages of field programming in sedimentation FFF have been particularly exploited by Kirkland, Yau, and coworkers (7-9). These authors have focused on a program in which the field strength, after a specified initial time lag, decays exponentially with time. For this program, the elution spectrum is represented approximatelyby a linear relationship between elution time and the logarithm of particle diameter (7). (The validity of this elution spectrum hinges on the absence of substantial secondary nonequilibrium effects (IO, II).) The above authors have applied this exponential form of programming to a large variety of colloidal materials (9). Unfortunately, no theory has been developed to examine resolving power in programmed FFF operation. Thus, few well-founded guidelines have emerged to assist in the choice of experimental parameters for programmed operation. Furthermore, it is not clear whether resolution is degraded in the course of normal programmed operation, and if degraded, by how much. Also, nothing is known about the way in which resolution depends on the form of programming. Thus there is presently no rational basis to search for optimal program forms. In this paper we take the first steps necessary to fill this knowledge gap. We do this by formulating a general theory for fractionating power valid for all field programmed FFF techniques. We then apply the theory to exponential decay field programmed sedimentation FFF. We examine a number of cases in which assorted parameters are systematically varied. The plots we have generated help explain the influence of instrumental and operational parameters on resolution in exponentially programmed sedimentation FFF systems. In a future publication we will extend this work to include other program modes. GENERAL THEORY For satisfactory particle fractionation or characterization, we require adequate resolving power. In this document, we will not attempt to determine the magnitude of resolving power necessary to reach different goals because this requirement varies widely with sample type, with circumstances, and even with outlook. However, we will formulate (or reformulate) a criterion for resolving power which can then be required to reach levels which depend on the unique circumstances of each case. This criterion must, of course, account for fractionation within the continuous particle distributions commonly treated in FFF systems. our criterion, the fractionating power Fd, is a dimensionless parameter which we here define by
where R, is the resolution, 6tr/4ut, between particles whose diameters differ only by the small relative increment 6d/d, and whose retention times consequently differ by 6tr, with ut being the mean standard deviation in retention time for particles of diameter d. In Figure 1 we illustrate the significance of Fd by reference to the resolution of two monodisperse particle populations for which 6d/d = 0.1. This figure depicts
I I
29
1 1
//
F, = 20
I
Figure 1. Relative outlet distribution of two monodisperse particle populations wRh particle diameters differing by 10% at different F , values. Unit resolution between these populations Is not obtained until F, 10.
the outlet distribution of two monodisperse particles whose diameters differ by 10% at some different Fd values. Unit resolution is obtained when Fd reaches 10. Hereafter we take Fd to be the limit of eq 1 as 6d 0
-
t , d In t , d _ dt, = -F d =_ 4ut d In d 4at dd Since the diameter-based selectivity of the fractionation is given by (12)
d In t, d In d
(3)
Fd = (t,/4at)Sd
(4)
Sd we have
We note that Fd resembles closely a mass-based fractionation power defined earlier by MIAM,where M is the molecular weight (or mass) and AM is an increment in M sufficient to give unit resolution (13). However, Fd is a function defined uniquely at any point along the elution axis without reference to finite increments and is thus a more satisfactory function for mathematical manipulation. While Fd is technically a size- or diameter-based fractionation power, we can replace MI AM by an equally well-defined mass-based fractionation power F,, in which particle mass m (or molecular weight M) replaces d. We thus obtain by steps parallel to those leading up to eq 4 where S , is the mass based selectivity
d In t , =d In m Both Sdand S , can be obtained readily from retention theory (which provides tr(d)or t,(m)),applied to either constant field (ismatic) or programmed FFF. Thus Fd and F, are described fully by retention theory except for the single parameter uv We will consider later the steps necessary to obtain ut. We also note that Fd may be defined in terms of the particle diameter-based specific resolution factor, Rs,(l+x), a term proposed by Yau and Kirkland (14). This Rs,(l+z)is simply the resolution predicted for particles differing by x in relative
s,
30
ANALYTICAL CHEMISTRY, VOL. 59, NO. 1, JANUARY 1987
diameter (Le., x = A d / d ) . The fractionating power Fd is therefore given by limx+,Rs,(l+x)/x.If band broadening is independent of d,then Fd is equal to Ra2. However, in FFF, bandwidth may change 20-fold over a particle diameter range of two. The mass-based fractionating power F,,, is likewise related to the mass-based specific resolution factor, except in this case x = AM/M. The particle mass discrimination capacity XM, as defined by Yau and Kirkland (14),is exactly equivalent to the fractionation coefficient, 8, previously defined by Giddings and co-workers (13). It is the minimum fractional difference in mass that a system is able to separate with unit resolution. In the special case in which band broadening is independent of particle mass, it is apparent that F , = 1 / x M . Similarly Fd becomes equal to the reciprocal of the particle diameter discrimination factor, xd. Specific resolution factors (for arbitrary x ) , fractionation coefficients, and discrimination capacities are not easily calculated for systems where band broadening depends on particle mass or diameter. Fractionating power, being a continuous function independent of arbitrary r i t e incrementa in mass or diameter, does not pose any such problem. It is thus preferable for the mathematical study of resolution for systems, such as FFF, size exclusion chromatography, and hydrodynamic chromatography, that fractionate continuous distributions of particles or macromolecules. For isocractic FFF, retention time t, is given by
L
t, = R(u)
(7)
where L is channel length, ( u ) is the cross-sectional average flow velocity, and R is the retention ratio, equal to the ratio of the displacement velocity of the particle band to ( u ) . For programmed FFF, in which R or ( u ) varies during the run, t , is given by the integral equation ( I )
L = S f0 ' R ( u ) dt
(8)
equals k T / F w (2), the constant A can be expressed as A = - kTd" =-
kTd" 4
F/S
where F is the force exerted on a particle of diameter d by the field and 4 = F / S is the field-particle interaction parameter. For sedimentation FFF 4 is equivalent to the effective mass (particle mass less buoyant mass), or expressed in terms of d
where Ap is the density difference between the particle and the carrier. We now turn our attention to the band-broadening term ut. It has been shown that the variance u2 (with u in units of distance) at the end of the column in programmed temperature chromatography is given by (15) u2 =
s,
L
H dz
where the integration of plate height H follows the path of the band center as it proceeds along axis z toward the end of the column at z = L. This equation, which allows for the variable H typical of programming (but which works also for nonprogrammed runs made at constant H)applies equally well to FFF. Once a2 is obtained by this integration, the timebased variance u t is acquired in the usual way by (16)
where R, is the retention ratio R at the point of elution, z = L, and ( u ) is again the mean cross-sectionalvelocity (assuming constant flow conditions). The substitution of eq 16 into eq 17 yields
L L H dz
In the case of field programming, in which R changes but ( u ) remains constant, we have
ut2
=
R,2(u)2
(18)
or where t o is the elution time of a nonretained component. To obtain the derivative oft, with respect to d , needed to evaluate Fd from eq 2, we write
dt, -ato/ad -=-dd ato/at,
-1
If we replace the incremental zone migration distance dz by R ( u) dt and change the variable of integration to time t, we
traR;;,t) dt
-
R,
(10)
Retention ratio R in FFF is a function of field strength S; desired variations in R are thus induced by changes in S. The relationship between R and S, however, is expressed through the intermediate term A, known as the retention parameter. The dependence of R on X (ignoring steric effects) is (2) R = 6X(c0th (1/2X) - 2X) (11)
have
The latter form is valid only when ( u ) is held constant. The plate height H in FFF can be expressed by (17, 18)
or, for small X values
R = 6X The retention parameter X for spheres can be generally expressed by X = A/Swd"
where x is a complicated function of X and D is the particle diffusion coefficient. When this expression is used in the final form of eq 20, we get
(13)
where w is the channel thickness and n is an exponent that depends on the type of field employed (e.g., n = 3 for sedimentation FFF and n = 1for flow FFF). Because X in general
When this and eq 10 are substituted into eq 2, we obtain for Fd
ANALYTICAL CHEMISTRY, VOL. 59, NO. 1, JANUARY 1987
31
t
t o = R o J ’ d t = Rotr 0
Fractionating power Fd can either be obtained from this expression or from eq 2 by using eq 20 and the derivative dt,/dd. The evaluation of this expression (or eq 2) requires, first, that we obtain t, from eq 9. To get both t, and Fd, we must know how S varies with t as expressed by some function S(t),how h varies with S as given by eq 13, and how R depends on A, as was discussed in connection with eq 11and 12. To get Fd, or ut, we must also know x(X). The dependence of x on X is very complicated when expressed rigorously, but various approximations can be used. In the important case in which h 0, x assumes the form
-
(17,18)
x
= 24X3
(24)
When X from eq 13 is substituted into this expression, we get X=-
24h3
S3~3d3n
where Rois the retention ratio for the component at the initial field strength. Substituting for Rousing eq 27 and remanging, we get an expression for the retention time
where X,is the value of h for the component at the initial field strength and is obtainable by appropriate substitution into eq 13. Substitution for X,in eq 31 using eq 13 allows us to determine the derivative of t, with respect to d. On reintroducing Xo, this reduces to dt, nto -=-
This equation shows that H and thus ut decrease dramatically with increasing field strength S and particle diameter d. The use of the approximations expressed by eq 12 and 24 simplifies the integration of the two key integrals found in eq 23 providing the program is of a simple form. However, these approximations are useful only for rather small X values, which may not always apply in field programmed FFF. When field strength S ( t ) is reduced rapidly in a field program, X, which is inversely proportional to S as shown by eq 13, increases rapidly. For this reason, X is easily carried beyond the range where eq 12 and 24 can be applied. However, use of the rigorous equations relating R and x to X makes the integrations extremely difficult. In these cases we compromise by using higher order approximations which are still tractable but which are valid over a larger range of X than eq 12 and 24. For R we use the expression (17) R = 6h(1 - 2h) (27) which is valid within 0.37% up to X = 0.15. For x we use the approximation (17)
+
24h3(1 - 1OX 28X2) X = (28) 1 - 2X which is valid within 6.2% up to X = 0.15. For programmed operation, S and/or ( u ) are varied with time. In this development, we will emphasize the case of field programming in which S is time dependent and ( u ) is constant. We will consider the specific case of exponential field decay following a time lag of arbitrary duration in which the field strength is held constant.
THEORY OF EXPONENTIAL FIELD DECAY
(32)
An expression for ut may be obtained from eq 22. Here, both x and R are constant throughout the elution since we are still considering only t I t,; we therefore assign values attained at initial field strength So. Substituting for R and x using eq 27 and 28, respectively, gives us ut
24A3(u )
DwS3d3”
d 6X0(l - 2h0)2
dd
and thus H in this high-retention limit becomes
H=
1 -4ho
12wt,‘/2 =X2(1 - 10x0 D1/2Rr
+ 28h2)ll2
(33)
and since, for this case, R, = Ro,we get 2wtr1l2 ho(1 - 10x0
ut =
D’/2
+ 28X2)1/2
1 - 2ho
(34)
We are now able to substitute for dt,/dd and ut using eq 32 and 34 into eq 2 to give the following expression for Fd: n t o ( l - 4h0)D’/~
Fd =
- 10x0 4 8 ~ t , ’ / ~ X o ~-( 2ho)(1 l
+ 28&2)1/2(35)
We can in turn substitute for t, using eq 31 to obtain F,.j =
We shall now consider the elution of a component for which t, > t,. Retention time may again be obtained from eq 9. Here to =
Xt’Ro d t
+
st’R tl
dt
(37)
From eq 13, 27, and 29 we find
R = 6x0 exp((t - tl)/7’) - 12Xo2exp(2(t- tl)/7’)
(38)
Evaluation of the integrals of eq 37 results in a quadratic in exp((t, - t l ) / T ’ ) , the meaningful root of which is given by exp((t, - t 1 ) / 7 ’ ) = 2x0 [1-[1-:+4ho(>-l)+
Hence
From the initiation of flow (subsequent to primary relaxation), which marks the beginning of the run,the field strength is held constant at So for an arbitrary time t,, at which point it begins to decrease exponentially according to the program S ( t ) = So exp(-(t - tl)/7’) (29)
where
where S ( t ) is the field strength at time t (t 2 t l ) and 7’is the exponential time constant for the program. We shall consider first the isocratic elution of a component within the period t,. From eq 9 we have
It is apparent from eq 13, 29, and 40 that we can also write
32
ANALYTICAL CHEMISTRY, VOL. 59, NO. 1, JANUARY 1987
B = (1 - 2&)2
Retention Parameter, h
(42)
where A, is the value of X at the point of elution. Substitution for X,in eq 40 using eq 13, differentiating with respect to d, reintroducing X,and reducing the result with the aid of eq 41 and 42 gives us
An expression for
ut
may be obtained by substituting for
R and x in eq 22 using eq 27 and 28, respectively, and solving the integral. After collecting terms we have T'ho4
ut
= - t1h4(1 - loxo + 28XO2)- -(3
12
7%:
- 24X0
+ Flgure 2. Plots showing relative change in selectivity Sd and nonequilibrium coefficient with variations in retention ratio Rand retention parameter A.
x
- 24X, + 56X):
56x2) + -(3 12 In this case, R, is given by
R, = 6&(1 - 2 4 )
(45)
From eq 2,43,44, and 45 we obtain the following expression for F d
system temperature of 293 K is assumed. Once a particle diameter is encountered for which t, > tl, it is known that it and all remaining particles will be eluted during the exponential phase. For these particles B is determined via eq 41, A, is obtained from eq 47, and F d is in turn calculated via eq 46. A limiting form for F d obtained from eq 46 and valid when Xo
