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Anal. Chem. 1986, 58,735-740
Simplified Nonequilibrium Theory of Secondary Relaxation Effects in Programmed Field-Flow Fractionation J. Calvin Giddings Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
Prlmary and secondary relaxation effects in field-flow fractionation are described. I t is shown that secondary relaxation processes invariably cause a departure from steady-state migration when field programming is used. To treat this rather complicated phenomenon in a simple manner, nonequilibrium theory has been applied. With this approach, first-order approximatlons have been obtained for the departure from equilibrlum of the concentration across the channel. Following this, correctlon terms have been derived for the corresponding increments in particle migration rate, retention time, and derived physicochemical properties such as particle mass. These corrections are valid for small departures from equilibrium for any subtechnique of FFF and provide a simple criterion for the magnitude of equilibrium departure.
The basic mechanism of field-flow fractionation (FFF) entails the formation of a thin cloud of particles near the accumulation wall of the FFF flow channel. When flow is initiated in the channel, the cloud is displaced downstream at a velocity roughly proportional to the cloud thickness, 1. Since particles of different sizes and types form clouds of different thicknesses, the downstream displacement velocities differ for different kinds of particles, thus ensuring separation. This basic mechanism of separation has been elaborated many times and will not be further detailed here (1-3). In this paper we focus on changes that are induced in the distribution of particles within a cloud due to changing conditions, particularly changes in the strength of the field applied across the channel. Changes in field strength in the course of a run constitute the basis of field-programmed FFF ( 4 , 5 ) . Corresponding to each specific value, S, of the field strength, there is a unique equilibrium or steady-state distribution of particles near the accumulation wall for any given particle type. When the field strength changes from one value t o another, a corresponding change occurs in the steady-state distribution. However, the particle cloud requires a finite time to adjust from one steady-state condition to another. The process of changing from one steady-state distribution to a subsequent one is termed relaxation. Similar relaxation processes occur throughout physics and chemistry; each relaxation process can be characterized by a relaxation time constant, T,which is generally the time required for reaching the new steady-state distribution within a certain fraction (6). All FFF procedures are subject to a primary relaxation process (7). At the beginning of the FFF run, particles are generally distributed evenly over the thickness of the FFF channel as a consequence of the injection process. This initial particle distribution in a short segment of an FFF channel is illustrated in Figure 1A. With the application of the external field, the particles are displaced toward the new steady-state distribution illustrated in Figure 1B. This primary relaxation process approaches completion in one relaxation time, T ~ where , T~ is given by 71
=
W / V l
(1)
where w is the channel thickness and IUl is the absolute value of the displacement velocity of the particles due to the applied field. Clearly, in the course of the primary relaxation process, particles are scattered to varying degrees over the cross section of the channel. Under such circumstances, the initiation of channel flow will cause band broadening and distortion of the original narrow pulse of injected particles (7). In order to avoid such complications, a stop-flow procedure is often used in which flow at the beginning of the run is halted for a time equal to or greater than T ~ of, sufficient duration to establish the steady-state condition of Figure 1B. It has been proposed that a split-inlet system could be used to eliminate most of the adverse effects of primary relaxation without stop flow (8).
For constant field or isocratic FFF, separation is achieved by use of only a single value of the field strength in which the particle distribution is determined by only one steady-state condition. It should be noted that slight departures of the actual distribution from the steady-state distribution have long been recognized to exist and to be responsible for the major part of band broadening in carefully executed FFF experiments (9, 10). In programmed field FFF (PFdFFF) the field strength, S, is decreased in the course of the run, creating a sequence of different steady-state distributions. With the decrease in field strength, the steady-state distribution expands, leading to a particle cloud something like that illustrated in Figure 1C. If the field strength in a programmed field experiment is decreased gradually, the actual particle distribution will follow very closely the steady-state distribution. However, if the field strength changes rapidly, the particle cloud will lag well behind its steady-state configuration. The calculation of particle migration rates and thus elution time based on the steady-state assumption will therefore incur significant errors. Whether or not the change is too rapid to maintain conditions near the steady state depends on the relaxation time for this secondary relaxation process. This relaxation time, equal to the time necessary to rearrange particles within a cloud of mean thickness I , can be equated to 72
= 2 P / D = t,
(2)
where D is the particle diffusion coefficient. It has been shown that this relaxation time, 212/D, is equal to the time, t,, necessary to generate 1theoretical plate in the FFF channel (11,12). In a recent paper, Yau and Kirkland have described (in other terms) both primary and secondary relaxation processes and have derived a rigorous correction term to adjust elution times for the departure from steady state caused by these processes (13). The correction term derived is rather complicated, involving one infinite series nestled within another. Yau and Kirkland imply that programmed FFF can be successfully run far from steady-state conditions by application of the correction term. This conclusion is undoubtedly true for spherical particles of constant density, although there may be a significant loss of resolution (see last section). However,
0003-2700/86/0358-0735$01.50/0 0 1986 American Chemlcal Society
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ANALYTICAL CHEMISTRY,
VOL. 58, NO. 4, APRIL 1986
srrong field
. 0
-
0
PRlMARY RELAXATION
SECONDARY RELAXATION
* *- * * * A
Field
1
no field
a
1
weak field
*a*
B
*** *** :**** C
Figure 1. Primary and secondary relaxation processes in which the
initial particle distribution (A) first responds to the initial application of a strong field by forming a thin steady-state particle cloud (E) and then responds to the programmed weakening of the field to form an expanded particle cloud (C). in that the departure from the steady state depends on the diffusion coefficient,D, as suggested by eq 2, a more complex sample will display a spectrum of D values that will complicate the correction and make the task of determining particle characteristics from FFF elution profiles even more difficult. It is likely that programmed runs made under conditions approaching those of the steady state will prove most convenient and definitive for most FFF applications. In this paper, we use a variant of a widely used nonequilibrium theory to derive equations that greatly simplify the calculation of departures from the steady-state condition and are applicable when those departures are relatively small. A single equation of only a few terms provides the particle distribution in the cloud for any continuous field program. From this, an expression for the departure of concentration from steady-state levels can be obtained. The concentration-departure expression can then be used to derive a simple equation for the departure of particle migration rate from its steady-state value. This, in turn, can be used to obtain a correction term for retention time. The equations derived can be used either to make corrections for nonsteady-state migration for small departures from the steady state or simply to provide a rapidly applicable criterion for the approximate magnitude of the departure and thus an assessment of the validity of the steady-state distribution.
NONEQUILIBRIUM THEORY The theory used here to follow small departures from the steady-state distribution during secondary relaxation is a special case of a rather general nonequilibrium formulation developed by this author ( 1 4 1 5 ) . This nonequilibrium approach has been used to describe departures from steady-state chemical kinetics with and without important diffusion effects (16, 17). Nonequilibrium theory has also been used widely in both chromatography and FFF to obtain meaningful plate-height expressions (15, 18). In both techniques, nonequilibrium theory has been used to relate plate height to departures from the steady state caused by the flow process. The general methodology of nonequilibrium theory can be applied equally well to the description of secondary relaxation with small departures from steady-state conditions. We start with the general equation of mass transport describing the rate of change of concentration at any position X
dc = -u-dc -dt dx
+ D-d2c
dx2
(3)
where c is concentration, t is time, and x is the coordinate position above the accumulation wall of the channel as illustrated in Figure 2. This figure illustrates the balance between the two mass transport terms in eq 3; the term containing U represents field-induced transport toward the wall, and the term containing D is the diffusion term repre-
Figure 2. Coordinate system of FFF channel and the opposing mass transport processes related to drift velocity, U , and diffusion coefficient, D.
senting transport away from the wall. The figure shows that U is a negative term in the coordinate system used for the channel. Under steady-state conditions, the two terms on the right-hand side of eq 3 exactly balance one another, leading to zero net accumulation at any point dc/dt = 0
(4) Under these conditions, c acquires its steady-state value c*, which can be shown to equal (1) c* = c*oe-xIWD = c*oe-x/l (5) where c * is~ the steady-state concentration at the wall. Parameter l is the characteristic cloud thickness; it is essentially the mean altitude of particles in the cloud represented by the distribution of eq 5. Inspection of eq 5 shows that
1 = D/lUl
(6)
where the absolute value of U has been used to avoid dealing with a negative quantity. The basic assumption of nonequilibrium theory in the present context is that concentration is closely approximated by eq 5, and therefore the time derivative of concentration shown in eq 3 can be replaced by -dc= - dc* (7) dt dt Thus nonequilibrium theory allows for a finite accumulation rate at any point as opposed to the zero accumulation rate shown by eq 4 for strict steady-state conditions. The right-hand side of eq 7 is finite because c* changes as U changes, as shown by eq 5. Drift velocity, IUI, of course, changes in proportion to the field strength; in normal field programming, both IUI and field strength decrease in magnitude with time. The value of dc*/dt can be obtained by taking the derivative of c* in eq 5 . This procedure yields
In order to convert eq 8 into a useful form, we must find an expression for the wall concentration, c * ~ ,and its time derivative. To do this, we note that the integral of concentration (either c or c*) over the x coordinate yields the amount, A, of particulate material above a unit area of the accumulation wall. For simplicity in deriving the present limiting equations, we take the integral to infinity, which then assumes the form
La
c* dx = A
With the aid of eq 5 this integral can be evaluated, giving A = c*O D/lUl (10) When solved for c * we ~ get
ANALYTICAL CHEMISTRY, VOL. 58, NO. 4, APRIL 1986
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ming scheme, and far more complicated results, since it must account for all previous changes in IUl. The departure term of eq 19 can be expressed in another way by writing c = c*(l 4 (20)
The derivative of c * then ~ becomes
+
When the last two equations are substituted back into eq 8, we obtain
We now return to our basic differential equation of transport, eq 3; when this equation is combined with eq 7 and 13, it yields
Division of each term by the diffusion coefficient, D, provides the form
which is the basic differential equation of this treatment. Equation 15 admits of the general solution
We note immediately that the integration constant Bz is zero because concentration, c, must approach zero as x goes to infinity. . In order to evaluate integration constant B,, we reformulate the concentration integral of eq 9 using c instead of c*
Jmcdx=A
(17)
When eq 16 is substituted into eq 17 and the latter rearranged, we get
When this expression along with eq 10 is used in eq 16, we get the following concentration expression:
This equation shows that the concentration is composed of two terms: The first term is the steady-state concentration distribution expressed by eq 5, and the second term represents the departure from this steady-state profile. We note that the latter term is proportional to the rate of change of U, i.e., to the programming rate. Interestingly, this equation is applicable as long as Iq (and consequently the field strength) changes in a continuous (nonabrupt) way with time. In this case, the departure term in eq 19 depends only upon I and its instantaneous rate of change, but does not depend upon the “history” of U, that is, upon the prior values of U dictated by the particular form of programming. These results show that small departures from the steady state are independent of the form of programming at given values of IUl and dlUl/dt. A rigorous solution would, of course, depend on the exact form of the evolution of U. However, with relatively rapid relaxation (necessary to keep the departure term small), only the IU( values just preceding the time of calculation are important; these are adequately specified by the slope d(U(/dt. The rigorous approach, while needed for cases of larger departure from steady state, yields different results for each program-
in which e represents the fractional departure from steadystate concentration values. The substitution of eq 19 into eq 20 followed by rearrangement yields
This equation shows that the fractional departure, e, is positive a t the lower wall (recall that the time rate of change of lUl is negative), zero at the elevation x = 21/21,and negative for larger x values. This trend is expected on physical grounds: the lagging expansion of the particle cloud will leave somewhat more than the,steady-state concentration at the wall and will fail to fully populate the upper reaches of the cloud, giving positive and negative fractional departures, respectively.
PERTURBATION OF ZONE MIGRATION We now calculate how the steady-state departure influences the migration rate of the particle zone. The particle migration velocity is expressed by (1,3) v = R(u) (22) where R is the retention ratio and ( u ) is the mean flow velocity in the channel. The quantity R is expressed by
R = (c(x)~(x))/(c(~))(~(x))
(23)
where the triangular brackets represent cross sectional averages. For high retention, we need consider only the limited range x
