Anal. Chem. 1984, 56, 2481-2484
2481
Focusing in Field-Flow Fractionation Josef JanEa* and Josef Chmelik Institute of Analytical Chemistry, Czechoslovak Academy of Sciences, 611 42 Brno, Czechoslovakia
A new separation prlnclple, belonging to the class of field-flow fractionation methods, Is proposed. It uses the focusing of the solute under the Influence of the drlving forces. Previously described as a special method called sedlmentatlon-flotation focusing field-flow fractlonatlon, this prlnclple Is now theoretically generalized. Relatlons descrlblng the fundamental characterlstlcs of the separation process, Le., retention and efficiency, have been derlved. A theoretlcal analysis Is carried out for the bask slmple geometrlcal shapes of fractlonatlon channels. The advantages of varlous shapes of fractlonatlon channels allowing the formatlon of axlally symmetrical and axlally nonsymmetrlcalflow veloclty profiles In the fluid flowlng lnslde the channel are dlscussed. Axially nonsymmetrical flow velocity proflles are formed, to an advantage, lnslde channels wtth modulated cross sectional perms ablllty. The principle proposed can be used especially for the analytical separatlon of macromolecular and particulate samples as well as for contlnuous preparatlve fractionation.
At present, field-flow fractionation (FFF) becomes a very effective method of choice for the separation and characterization of samples of macromolecular and particulate character. In the classical arrangement (I),the forces of an external field act perpendicularly to the direction of the flow of fluid inside a channel having usually a rectangular cross section. These forces make the solute (originally distributed homogeneously over the cross section of the channel) move toward one of the channel walls. The concentration gradient formed causes a reversed flux of the solute, Le., a flux against that due to the external field. After a certain time, an almost stationary state is reached, and the concentration distribution of the solute across the channel thickness can be described by an exponential function with a maximum concentration at the accumulation wall and decreasing in the direction toward the opposite wall. The exponential concentration profile is combined with a parabolic (or nearly parabolic) flow velocity profile of the fluid moving inside the channel. Consequently, the solute is transported along the channel with a velocity corresponding approximately to the streamline velocity at the coordinate of the mass center of the exponential concentration profile. Provided different solutes show different exponential distributions of concentrations across the channel thickness, thus having different mass center coordinates, they are carried along the channel at different velocities and may be separated. This method, previously developed in only a few laboratories (I, 2) and called alternatively "one phase chromatography" or "polarization chromatography" (3) has recently become more extensively investigated. At the present time, a number of laboratories in the world are engaged in the further development of this technique (see ref 4 for review). In contrast to the classical FFF techniques, with the new focusing FFF methods the solute migrates under the influence of the driving forces into the position a t which the intensity of these forces is zero. Starting from this position, the intensity of the driving forces increases in the direction toward the opposite channel walls. This phenomenon is combined with a flow velocity profile. 0003-2700/84/0356-24S1$01.50/0
This focusing principle was originally described by one of the authors (5, 6 ) for a special case of focusing in a density gradient formed across the channel during rotation in the centrifuge. The method has been called sedimentation-flotation focusing field-flow fractionation (SFFFFF). The new SFFFFF method was further improved with respect to a more advantageous shape of the flow velocity profile formed inside the channel (7). Just recently, Giddings (8)described a method equivalent to SFFFFF, giving it the name sedimentation hyperlayer FFF. The concentration profile in the direction of the driving forces (across the cross section of the channel) in focusing field-flow fractionation (FocFFF) can be described by a Gaussian distribution (or by a function very similar in shape to a Gaussian distribution). This is due to the mentioned fact that the solute inside the channel is focused, under the influence of the primary external field, into the position where the intensity of the driving forces is zero. Diffusion (Brownian migration) works opposite to the driving forces concentrating the solute around the equilibrium position. As soon as dynamical equilibrium between the concentrating and dispersive processes has been established,narrow focused zones of solutes are formed. In this paper, a generalization of the focusing principle in FFF is developed. THEORETICAL ANALYSIS Retention. The solute concentration profile in the direction of the focusing forces can be described with a fair approximation by a Gaussian distribution, C(CP)
C(@) = C(@maJ e x p H @- @ m a J 2 / 2 ~ 2 1
(1) where C(9) is the solute concentration at the coordinate 9 in the direction of the focusing forces, C(CP,) is the maximum concentration in coordinate Om,, and Q is the standard deviation expressed as a fraction of the total width of the channel in the direction of coordinate a. The coordinate CP is defined in such a way that it lies in the plane of the cross section of the channel, the origin CP = 0 is in the center of the channel, 9 = -1 being at one side wall of the channel and CP = 1at the opposite side wall. In these coordinates, eq 1 is defined in the range (-1 + 3u) I Om, I(1- 3u). The Gaussian distribution is generally unlimited in coordinate 9. The differences due to the integration within the above limits are practically negligible for the following derivations (see the following text). This type of concentration profile has been described for the special cases of sedimentation in density gradient (9) or isoelectric focusing (IO) as well as for a general case of equilibrium gradient methods of separation (11). When focused zones having Gaussian concentration distributions but different 9- values occur in the fluid moving in the direction perpendicular to the direction of the focusing forces, and providing the flow has a suitable velocity distribution, then the focused zones migrate along the channel at different velocities and are longitudinally separated. Such a situation is shown schematically in Figure 1, where the focused zones occur in the fluid having a parabolic axially symmetrical distribution of streamline velocities. In order to demonstrate the basic differences between the classical FFF methods and FocFFF ones, the exponential course of the 0 1984 American Chemical Society
2482
9
ANALYTICAL CHEMISTRY, VOL. 56, NO. 13, NOVEMBER 1984
'm External Field
CHAWNEL CROSS-SECTIONS
VELOCITY PROFILES
aa8a
F
~
~
O
a
W
Centra I Axis
.-
External Field
Velocity Pdile
FocFFF
FFF
c7
Flgure 1. Focusing principle in field-flow fractionation compared with classical field-flow fractionation
b
concentration profile of the solute in the case of FFF is also shown in Figure 1. The use of the axially symmetrical parabolic flow velocity profile has originally been proposed in our paper (46).Such a velocity profile occurs in a hypothetical channel formed between two planes (when the influence of the side walls is not taken into account) or in a real channel of a rectangular cross section. The parabolic flow velocity profile is then formed in the plane parallel to the shorter side walls, as demonstrated in Figure 2a. It is obvious from Figure 1that the use of an axially symmetrical flow velocity profile (e.g., a parabolic one) is not very advantageous. Two solutes focused at the same distances from the central axis of the channel (even when in the opposite parts of the channel) move along the channel with identical average velocities. At the end of the channel the solutes are not separated in the longitudinal direction and, consequently,they leave the separation channel at the same moment. It can, of course, be suggested that the channel be divided a t the end by a longitudinal partition in the axis, in order to conduct each half of the fluid via a separate output. The necessity of using two independent detectors at the end of the separation channel constitutes experimental difficulties in analytical applications. Some benefit of such an arrangement follows from the increase of the final concentration of the solutes after separation, as proposed by Giddings et al. (12). Still, this arrangement can be applied advantageously to continuous preparative fractionation. In principle, it is possible to use only half of the channel for analytical Separation;Le., that between one wall and its central axis. However, a more beneficial solution to this problem is to form an axially asymmetrical flow velocity profile in the plane of the formation of focused concentration profiles. This situation is demonstrated in Figure 2b,c. First we define the coordinate Q in the plane of the cross section of the separation channel. This coordinate is perpendicular to the coordinate @ and its origin = 0 lies a t @ = 0 at the central longitudinal axis of the channel. The coordinate Q has the value Q = -1 and Q = 1,respectively, a t the opposite walls of the channel, regardless of whether these walls are plane and parallel or not. It means that even when the distance of these walls from the axis changes @(@I), the Q value of the wall position at each point of the 9 coordinate is constant = fl). Asymmetrical flow velocity profiles of convenient shapes are obtained when the cross section of the channel is trapezoidal or when the longer walls have a shape of parabolic sections in the plane of the channel cross section (see Figure 2b and Figure 2c, respectively). The important central part of the flow velocity profile is given approximately by a function
*
3u a v
C
f
average linear velocity at the coordinate \k from @ = -1 to @ = 1. a is the angle contained between the two opposite longer walls of the channel of trapezoidal cross section, and U(@)is the linear velocity of the fluid. In the case of the channel with parabolic shape of the walls, the flow velocity profile is approximately given, in its important central part, by a linear function
when \k = const (3) where c is the half-dimension of the channel, as measured in the direction of the @ axis, and d is the distance of the peak of the parabola from the side wall of the channel in the direction of the @ axis; the peak of the parabola lies in the CP axis. Here again, it is only in the proximity of the side walls of the channel that the flow velocity profile is steeply curved down to V ( @=) 0. Both types of separation channel mentioned, called the channel with modulated cross-sectional permeability, were proposed in our previous paper (7), where an exact theoretical analysis of the shape of the flow velocity profiles was given and the evolution of these velocity profiles was proved under real experimental conditions. The relations given here apply to channels for which the aspect ratio A = b / c is large enough (A > 20). The retention ratio R is defined as the ratio of the average linear velocity of the focused zone along the channel to the average linear velocity of the fluid
+ tg2a
when \k = const
a
Flgure 2. Schematic representation of various cross sections of the fractionation channels and corresponding flow velocity profiles
U(@)= -(1 + @tga)2 3
/
Fluid Flow
l : U ( @ )C(@)N O ) d@
(2)
for the first case of the channel of trapezoidal cross section. The velocity profile is steeply curved down to the linear velocity U(@)= 0 only very near to the side walls. Uavis the
R=
x:
s'
V ( @dQ, )
-1
(4)
C(@) b(@)d@
Solution for Axially Symmetrical Parabolic Flow Velocity Profile. The axially symmetrical parabolic flow
ANALYTICAL CHEMISTRY, VOL. 56, NO. 13, NOVEMBER 1984
velocity profile formed between two parallel planes is described by 3 uav U ( @ )= -(1- @2) (5) 2
It holds for this channel b(@)/b(O)= (1 + tga@)
H=
1’ - a2) 1: -1
+
when b = const
(6)
By using the substitution and per parts methods, a partial solution is obtained in the form
- a2) + (3a2[(amS+ 1) exp(-(1 @maJ2/2a2) - (arn,- 1) exp(-(-l -
R = (3/2)(1 -
-
1
@max)2/2~2)])/(21 -1 expi-(@ - @m,)2/2u2] d@) (7) The analytical solution of the integral in the denominator on the right-hand side of eq 7 is not known. However a fair approximation is given by
1:
exp[-(@ - @m,)2/2a21 d@=
dp =
1,
- a2 + a([(l +) ,@ ,
Similarly as in the case of axially symmetrical parabolic velocity profile, and taking into account the limitation for am= given by relationship 9, we obtained R = - 3 [(l + tga@m,)2 2tg2aa2 Ill (16) 3 + tg2a
+
+
Owing to its extreme complexity, expression IIon the righthand side of eq 16 is given in the Appendix. It should be noted, however, that with the majority of practical separations by FocFFF methods the value of Il is negligible. For a 5 0.1, eq 16 can be approximated again with a high accuracy by 3 R=[ ( l tga@,as)2 2tg2aa2] (17) 3 tg2a
+
+
+
- 2)
(3/2)(1 -
(12)
is valid. Equation 12 was also derived by Giddings (7),but in a different coordinate system. Solution for Asymmetrical Flow Velocity Profiles. The retention ratio R for the channel of a trapezoidal cross section can be expressed by substituting from eq 1and 2 into the general relationship
U(@C ) ( @ )(1+ tga@) d@
U(@)d@
1’C ( @ )(1+ tga9) d@ -1
Upon substituting from eq 1and 3 into eq 19 and rearranging we obtained
The solution in this case is
(11)
For an extremely narrow focused zone (a 5 0.01) the limiting relationship
R=
as
(10)
It has been found by numerical calculations that eq 10 can be approximated, with respect to the condition given by the relationship 9, with a relative accuracy of about 0.1% for a 5 0 . 1 by
R = (3/2)(1-
The retention ratio for the channel having the parabolic cross section is
exp(-(1 -
@mas)2/2a2)+ (1 - @mas) exp(-(l + @m,)2/2a2)I/(21r)1’2)I
Ji
d+
exp(-p2/2a2) dp = ~ ( 2 7 r ) l /(8) ~
The condition given by relationship 9 means that the total amount of the solute outside these limits (@- f 3a) and then the actual error involved is lower than 0.3% approximately. This solute amount, moving in the proximity of the side walls at a linear velocity different from that given by eq 5, is negligible with respect to the theoretically calculated R value compared to the real one within the range of usual experimental errors. The resulting relation for R is
R
+ tga@)exp[-(@ - +m,)2/2a2]
exp(-p2/ 2a2)
A numerical calculation has shown that the approximation used is very accurate for the given integration limits, providing amax is chosen within the limits -1 + 3a Iamax I1 - 3a (9)
R = (3/2)[1 -
tg2a 1 : ( l
and then for a 5 0.01 the limiting relationship is
1-0,
11-*-
1:(1 + t g ~ u @exp[-(@ )~ - @m,)2/2a21 d@
(15)
exp[-(@ - @m,)2/2a2] d@
2
3
exp[-(@ - @m,)2/2a2] d@
(1
(14)
Thus, we obtained after a rearrangement
By substituting from eq 1and 5 into eq 4 and by rearranging we obtained
3
2483
(13)
In the range of the validity given by relationship 9, for all real cases, it is possible to use the limiting relationship
R=1+-
Camas
(23) c+d This agrees with eq 22 to an accuracy of better then 0.05%, as the value of 1, is practically zero. Efficiency. The efficiency of the separation system in FocFFF, similarly as in chromatography, can be characterized by the height equivalent to a theoretical plate H. In our previous paper (7) we described the H in the channel with modulated cross sectional permeability (i.e., that having a trapezoidal or parabolic cross section) by a relationship that can be transformed into the present coordinate system as
2484
ANALYTICAL CHEMISTRY, VOL. 56, NO. 13, NOVEMBER 1984
H = [ g o / o ( a m a x ) I + [ w ( a m a x ) u(amax) / 1 0 5 ~ 1= u ; / L (24) where D is the diffusion coefficient of the solute, W(Om,) = 2b(@,,) is the thickness of the channel in the coordinate am= (in other words, it is the distance of the walls in the direction of the coordinate \k at a given coordinate a,,), L is the length of the channel, and u, is the standard deviation of axially dispersed zone as measured in the direction of the fluid flow along the T axis. O(ama)means the linear velocity averaged from = -1 to \k = 1 at a given coordinate am=.The first term on the right-hand side of eq 24 describes the contribution of axial diffusion and is usually negligible owing to the low values of D of macromolecular and particulate solutes and to the actual values of linear velocities O(@-). The second term on the right-hand side of eq 24 describes the dispersion due to nonequilibrium processes. Equation 24 can be used to characterize the dispersion inside the channel with modulated cross sectional permeability in those cases when the width of the focused zone in the direction of the CP axis is much lower compared to the thickness of the channel in the direction of the \k axis (Le., when b(a)>> 2u). When this condition is not fulfilled, we have to consider the contribution to axial dispersion due to the width of the focused zone proper (i.e., the value of u in the direction of the @ axis). Employing the random-walk theory, Giddings (8) derived a relation for H, applicable when only the nonequilibrium process, leading to axial dispersion, takes place in the direction of focusing (a axis). Upon the transformation into the present coordinate system, this relation is
*
H = 2m4d‘[dU(amax)/dam,12/[U(am,)Dl
UT,(P2
+ a,*
2
(26)
In other words, the total variance (second power of the standard deviation) of the zone is the sum of the variances due to the nonequilibrium processes in the planes of the axes @,T and \ k , ~ that act independently. It holds then that
Hb,l = H , , + H , ,
(27)
With respect to the above relations it can be written for the channel of trapezoidal cross section that
d u(@max) e---6 Uavtga damax
3 + tg2a
By substituting from eq 2 and 28 we obtain
(1 + @m=tga)
(28)
eq 25 and by rearranging
24m4u4tg2aUa,
H,,= (3 + tg2a)D
(@mm)o(@m,)
du(@max)
dam=
(29)
H,,,in this case is independent of the local velocity at the
cuav
=c+d
(30)
In this case, the substitution from eq 3 and 30 into eq 25 and the rearrangements give
H7ko
2m4u4c2Ua, = (c + d)[c(l + amax)+ d]D
(31)
For both channels with modulated cross sectional permeability, eq 24 can be simplified as
H , , = w(@max)o(@mJ / 105D
(32)
In both cases, the resulting width of the eluting zone (and/or the value of H) is given by the sum expressed by eq 27. For the channel in which an axially symmetrical parabolic flow velocity profile is formed it holds that (33) Finally, by substitution from eq 5 and 33 into eq 25 and by rearrangement we obtain (34) APPENDIX The expression Zl in eq 16 has the form
(25)
where m is the parameter of the order 2 (8). A hypothetical channel in this case has an infinite dimension in the direction of the \k axis. When applying the random-flight theory (13) to our case (Le,, when supposing the mutual independency of the random steps and, consequently, of the individual nonequilibrium dispersive processes in the plane of a, T and 9,T axes, respectively), the principle of the additivity of the variances due to the individual processes is valid. This fact is expressed by u2r,total =
position of focusing due to the fact that the product W in eq 24 remains constant along the 0 coordinate. It holds for the channel with parabolic cross section that
11
( 2 ~ t g a / ( @ a ) l(1 / ~+ tgaa,,,) + utgal‘exp ((I + @ m d 2 / 2 u 2 ) - exp((1 - amaJ2/2u2)I)){(1+ tga@max)2[exp(-(l+ @,,,)2/2u2) - exp(-(l @m,)2/2~2)]+ (3/2)tga(l + tga@max)[-(l + a m a x ) exp(-(l + (Pm,)2/2u2) (1 - amax) exp(-(l - amax)2/2u2)I + (tg2a/2)[(1 @ma)2 exp(-(l @ma,)2/2u2)- (1 @maxI2 exp(-(l - @m=)2/2a2)I+ (a/2)1/2utga(1 tgaa,,,))
+
+
+
LITERATURE CITED (1) Glddings, J. C. J . Chem. Educ. 1973, 50, 667. E. M. Proc. Natl. Acad. Sci. U . S . A . 1967, 58,
(2) . . Bera. H. C.; Purcell, 862:
(3) Reis, J. F. 0.; Lightfoot, E. N. AIChE J. 1976, 22, 779. (4) JanEa, J.; KlepBrnlk, K.; JahnovB, V.; Chmeik, J. J . Lip. Chromatogr. 1984, 7(S-1), 1. (5) JanEa, J. Makromol. Chem., RapldCommun. 1982, 3 , 887. (6) (7) (8) (9) (10) (11) (12) (13)
Jan& J. Makromol. Chem., Rapid Commun. 1983, 4 , 267. JanEa, J.; Jahnovi, V. J . Lip. Chromatogr. 1983, 6, 1559. Glddings, J. C. Sep. Sci. Technol. 1983, 18, 765. Meselson, M.; Stahl, F. W.; Vinograd, J. Proc. Natl. Acad. Sci. U . S . A . 1957, 43, 581. SvenSSOn, H. Acta Chem. Scand. 1961, 15, 325. Glddlngs, J. C.; Dahlgren, K. Sep. Sci. 1971, 6 , 345. Glddings, J. C.; Lln, Hao Chao; Caldwell, K. D.; Myers, M. N. Sep. Scl. Technol. 1983, 18, 293. Glddings, J. C. “Dynamlcs of Chromatography”; Marcel Dekker: New York, 1965.
RECEIVED for review April 6,1984. Accepted June 25,1984.
