Anal. Chem. 1989, 6 1 , 11-18
problems. First, certain bonded phases change in nature as well as in the amount of stationary phase; both of these effects result in irreproducible separations. Second, the degradation products from the bonded phase can contaminate a fraction isolated during a separation; such adulteration is of special consideration in preparative chromatography. We have developed two new classes of bonded phases that result in significantly increased stability of HPLC column packings during use. The first class uses bifunctional (or bidentate-type) silanes, which contain one reactive site on each of two silicon atoms of the silane. These two silicon atoms are connected by a bridging group, such as -0- or -(CH2)n-. The type of bridging group can be varied, to change the spacing between the silicon atoms of the silane for the most favorable reaction with the SiOH groups on the surface of silica support^. Bonded-phase packings with methyl, vinyl, phenyl, isopropyl, and tert-butyl functional groups show similar properties-but greater stability at low pH-than their corresponding dimethylsilyl derivatives. A second class of more stable bonded-phase materials uses monofunctional silanes, which contain one or two bulky groups (e.g., isopropyl or tert-butyl) on the silicon atom of the silane. These bulky groups provide steric protection to the Si-0-Si bond on the surface of the silica supports against hydrolysis at low pH, while still providing equivalent retention and column efficiency, compared to those of conventional monofunctional silane bonded-phase packings.
ACKNOWLEDGMENT Special thanks are given to R. F. Carver for performing the solid-state NMR experiments. We also thank G. R. Wooler and J. B. Marshall for their work in performing many of the syntheses and chromatographic experiments.
11
LITERATURE CITED (1) Majors, R. E. LC 1984, 2 , 660. (2) Unger, K. K.; Becker, N.; Roumellotis. P. J. Chromatogr. 1978, 725, 115. (3) Majors, R. E.; Hopper, M. J. J. Chrometogr. Sci. 1974, 72. 767. (4) Bennett, H. J. P.; Hudson. A. M.; McMartin. C.; Purdon, 0. E. Bioahem. J. 1977, 768, 9. (5) Glajch, J. L.; Kirkland, J. J.; Kijhier, J. J . Chromatogr. 1887, 384, 81. (6) Blen-Vogelsang, U.; Deege, A.; Figge, H.; Kahler, J.; Schomburg, G. Chromatographla 1984, 79, 170. (7) Figge, H.; Deege, A.; KGhkr, J.; Schomburg, G. J. Chromatogr. 1988, 357, 393. (8) Kohier, J.; Kirkland, J. J. J. Chromatogr. 1987, 385, 125. (9) Snyder, L. R.; Kirkland, J. J. An Introduction to Modern LquM Chromatography, 2nd ed.; Wiley-Interscience: New York. 1979; Chapter 5. (10) Frye, J. S.; Maciel, G. E. J . M g n . Reson. 1982, 48, 125. (1 1) Lippmaa, E.; Samosan. A. Bruker Rep. 1982, 7 , 6. (12) Kijhier, J.; Chase, D. B.; Farlee, R. D.; Vega, A. J.; Kirkland. J. J. J. Chromatogr. 1888, 352, 275. (13) Slndorf. D. W.; Maciel, G. E. J. Am. Chem. SOC.1983, 705, 3767. (14) Bein, T.; Carver, R. F.; Farlee, R. D.; Stuckey, G. D. J . Am. Chem. SOC. 1988, 770, 4546-4553. (15) Cwke, N. D.. Beckman Instruments, Fullerton, CA, private communication, July 1987. (16) k y e r , E.; Albert, K.; Reiners, J.; Nieder. M.; Muller, D. J. Chromatcgr. 1983, 264, 197. (17) Glalch, J. L.; Gluckman. J. C.; Charlkofsky. J. G.; Minor, J. M.; Kirkland, J. J. J. Chromatogr. 1985, 378. 23. (18) Glajch, J. L.; Kirkland, J. J. US. Patent 4,748,572, 1988. (19) Markiewicz, W. T.; Wiewiorowski, M. Nuchic Ac& Res., Spec. Pub/. 1978. No. 4. 185. (20) Cheng, W.; McCown, M. J. Chromatogr. 1985, 318, 173. (21) Nice, E. C.; Capp, M. W.; Cooke, N.; O’Hare, M. J. J. Chrometogr. 1981, 278. 569. (22) Pearson, J. D.; Lim, N. T.; Regnier, F. E . Anal. Bbchem. 1982, 724, 217. (23) GlaJch, J. L.; Kirkland, J. J. U.S. Patent 4705725, 1987. (24) Glajch. J. L.; Klrkland, J. J. U.S. Patent Appllcatlon, 1987. (25) Glajch, J. L.; Kirkland, J. J.; Minor, J. M. J. Liq. Chromatogr. 1987. 70, 1727.
RECEIVED for review June 6, 1988. Accepted September 23, 1988.
Flow Field Flow Fractionation in Hollow Cylindrical Fibers Jan Ake Jonsson* and Alf Carlshaf Department of Analytical Chemistry, University of Lund, P.O. Box 124, S-221 00 Lund, Sweden
A technkally simple version of flow field flow fractionation Is presented. The separatlon takes place In a porous hollow flber instead of in a flat channel. Two syringe type pumps are used, one to provide an axial carrier flow and one to control the radial (cross) flow. The pumps are connected to a computer control unit. This arrangement gives a precise and Independent control of the flows and permits the sample InJectlon, relaxation, and elution phases to be performed successively by smooth varlatlon of the pump flows. A theory that describes the retention and peak broadening, as well as the relaxation process, has been developed and experimentally verlfled by model separations of polystyrene latex beads and vlruses.
INTRODUCTION Field flow fractionation (FFF), pioneered by Giddings (I), is a versatile family of separation methods related to liquid chromatography. Since none of the subtechniques utilize a stationary phase for separation and therefore do not depend on an equilibrium process like classical chromatography, FFF 0003-2700/89/0361-0011$01.50/0
is not in a strict sense a member of the family of chromatographic techniques. FFF involves the application of an external force field to a solution, causing a migration of its constituents toward the separation channel wall. Depending on the magnitude of the force field and on chemical/physical properties, a certain solute will equilibrate into an exponentialdistribution, forming a zone with a specific average thickness close to the separation channel wall. This process is usually termed relaxation. If the liquid in the channel at this point is caused to move forward in a laminar way, a parabolic flow profiie will develop. An individual particle will move forward with the axial velocity corresponding to the actual distance of the particle from the wall. The resulting velocity of a sample zone will be determined by its average thickness. The applied field may be e.g. thermal gradients (thermal FFF), centrifugal forces (sedimentation FFF),electrical forces (electrical FFF), transverse or lateral flow (flow FFF), and transverse pressure gradients (pressure FFF). The different field types in FFF have recently been described by JanEa (2). The most universal FFF technique is flow FFF (FFFF), which has been explored for a wide range of characteriztion problems, involving virus samples (3),proteins (4),and silica 0 1988 American Chemical Society
12
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, 1989
HF
Flgure 2. Coordinate system used. Figure 1. Schematic diagram of the principle of hollow fiber FFF. For explanation of symbols, see the text.
sols (5), as well as synthetic polymers of both lipophilic (6) and hydrophilic (7) nature. Significant technical improvements in flow FFF have been made in recent years by Wahlund and Giddings with coworkers (8-10) who obtained shorter elution times and better resolution. Pressure FFF (FFFF) is a subtechnique that is closely related to flow FFF. In both FFFF and PFFF a cross flow represents the lateral field and the separation channel is almost always of the parallel plane membrane type. The difference between FFFF and PFFF is that in FFFF, the flow field is applied externally across the channel by a separate pump, whereas in PFFF the lateral flow is created by a pressure drop over the semipermeable membrane. In 1974, Lee et al. (11) published a paper where the first experimental work with PFFF in a cylindrical channel was presented. Doshi et al. (12)have shown that this geometry in theory is expected to give better resolution than both FFFF and PFFF with parallel plane membranes. In spite of this, the method has to our knowledge not been further developed up to now. In this paper we present a simple and modern technical variation of FFF in a hollow fiber, called hollow fiber flow FFF, characterized by the use of a second pump to suck liquid through the fiber wall. This permits the lateral (radial) flow to be much more precisely controlled than in the conventional PFFF technique, where the radial flow has to be controlled by variable restrictors. The lateral flow is also independent of the forward (axial) flow. The flow conditions in a hollow fiber flow FFF system are relatively complicated, due to both the cylindrical geometry and the successive loss of liquid through the wall. In this paper is also presented a theoretical discussion of the operation of such a system, including expressions for retention time and efficiency.
THEORY Components of Flow Velocity. Consider the following arrangement (see Figure 1): A hollow fiber (HF), i.e. a cylindrical tube with porous walls, is connected to a pump (PI), which pumps liquid axially through the fiber with the volumetric flow rate F1.The fiber is enclosed in a mantel, to which a second pump (P2)is connected. This pump creates a flow rate F2 of liquid radically through the fiber walls. Consequently, the axial flow within the fiber decreases from the value F1at the fiber inlet to a lower value, Fo,at the outlet. Obviously, Fl = Fo+ F2. To describe the flow conditions in the hollow fiber, a cylindrical coordinate system will be used, with an axial coordinate, z, and a radial coordinate, r (see Figure 2). The radius and length of the fiber are R and L , respectively. The two components of the linear fluid velocity inside the tube are designated u, and u,, respectively. Kozinski and co-workers (13) have developed complete formulas for the variation of u, and ur,describing the flow dynamics in a hollow fiber. First consider the radial flow velocity at the fiber wall,
u,(R,z). Due to the pressure drop along the fiber, this velocity decreases successively with z: u,(R,z) = u,(R,O) exp(-az)
(1)
where a is a constant, characteristic of the fiber. It is given by a = (q/3/R3)'t2where /3 is the permeability of the fiber (the flow rate through 1 cm2 of the fiber wall at 1-atm pressure difference), and q is the viscosity of the liquid. The axial flow velocity also decreases with z as liquid is constantly lost through the fiber wall:
Here, ii,(z) is the mean axial velocity, averaged over the cross section of the fiber according to R
a,(z) = LRru,(r,z) dr/
0
r dr
(3)
Assuming laminar flow conditions, a parabolic flow profiie will develop across the fiber for uz(r,z):
u,(r,z) = u,(O,z)(l - x 2 )
(4)
where x = r/R. Averaging this, we obtain ii,(z) = '/Zu,(O,z)
(5)
The radial flow velocity profile will be given by
u,(r,z) = ur(R,z)(2x- x 3 )
(6)
To relate the flow velocities to volumetric pump flow rates we observe
F, = ii,(O)rRZ
= 2aRu,(R,O)(L- CYL2/2
+ . ..)
Thus the velocities ii,(O) and u,(R,O) can be calculated from
F1,F2, and the fiber parameters L, R, and CY. This permits the calculation of u, and u, at any point in the fiber. The dead time to, i.e. the transport time of an unretarded compound through the fiber, can be calculated by integration of the transport time through an infinitesimal portion of the fiber (considering eq 2):
Retention of Sample in an Idealized Fiber. Consider a cloud of particles in the fiber. Because of the radial flow they will migrate by convection toward the tube wall. The particle flux created by this process is given by ur(r,z)C(r,z) where C is the local particle concentration. The flux creates a concentration gradient of particles. Due to this gradient particles will diffuse in the opposite direction of the field, following Fick's law. A steady state will establish where the convection flux equals the flux generated by diffusion. This is the classical description of flow FFF (14), which leads to the differential equation
ANALYTICAL CHEMISTRY, VOL. 81, NO. 1, JANUARY 1, 1989
u,(r,z)C = D dC/dx
(9)
Considering eq 6 and defining the PBclet number Pe as P e = u,(R,z)R/D
(2x - x3)C(Pe) = dC/dx
(11)
with the solution (a is an integration constant)
c = a exp(Pe(x2
-
:))
Equation 12 describes the radial distribution of sample. The mean velocity of the sample particles through the fiber can be calculated from this distribution and from the parabolic flow profile (eq 4). By combining eq 4,5, and 12, we obtain
L( U,
is effectively concentrated as liquid constantly is filtered through the fiber wall. Disregarding this effect, which only very slightly influences the peak shape, we find
(10)
we obtain
4 (exp(Pe/4) - exp(Pe)) (Per)ll2 (erf Pe112 - erf (Pe/4)'12)
- 2 )
(13)
The overscore signifies averaging over the cross section of the fiber (cf. eq 3). For large values of the PBclet number (Pe > 50), eq 13 is reduced to
L Pe tR = -uz 4 Equation 13 was given by Lee et al. (11). It is an exact mathematical solution to the pfoblem if Pe is treated as a constant. However, from eq 1and from the definition of Pe (eq 10) it is obvious that in a real case, Pe will vary along the fiber. This question will be addressed below. Peak Shape and Dispersion in an Idealized Fiber. A separation system is characterized not only by retention time equations as given above, but also by equations for the dispersion of peaks. To accomplish this, we make use of the work by Doshi et al. (151, who derived the following peak shape equation for essentially the same system as described above, assuming an ideal pulse injection.
13
p
= -KIT,
(17)
u2 = 2K2r
where p and 2 are the mean and variance of the Gaussian function. To fiid the retention time, we observe that the peak mean has reached the column end when p = X ( L ) , i.e. the value of X at z = L. After considering eq 15-17, this gives
which is a series expression alternative to eq 13. To obtain eq 18 we must also assume that u, is small, so that X ( L ) = DL/(R2ziz(0)), which is equivalent with the assumption that u, is constant throughout the fiber. For large Pe values, eq 18 also reduces to eq 14, as required. The variance of the peak described by eq 17 arises from variations in particle velocities caused by the parabolic flow profile, a phenomenon that is closely related to the dispersion in open tubes as described by Taylor (16). Another cause of peak broadening in the hollow fiber FFF system might be axial diffusion of sample particles. Due to the very low diffusion coefficients involved, this is, however, in practice insignificant. Therefore we may assume for the present that the band broadening is caused by the flow profile effect alone, and we can calculate the theoretical plate number n from its statistical definition: (1-L 2
+ 12-...)
pz
DPe2
a2
t R Z
n=-=
pe
Pe2
(1 - pe 24
+
...)
(19)
If Pe is large enough, eq 19 reduces to ii,2 =Z t R
(20)
Retention and Dispersion in a Real Fiber. In a real fiber, u,and u, vary along the fiber. To take into account this effect, we apply Doshi's theory to an infinitesimal portion of the fiber and obtain in analogy with eq 18
where
and A is proportional to the amount of sample injected, provided that the fiber wall is inpermeable to the sample particles. The parameters KO,K1, and K2 are functions of Pe only:
K1 = -4[ Pe
K2 =
"[
-
Pe4
-...I
4 + 72 Pe
Pe2
1-P24e
+ ...
Integrating over the entire length of the fiber (considering the variations of u,(z) and u,(R,z) and thus Pe) gives the total retention time. Assuming large Pe values, the integration leads to R2
tR = - In (U,(O)/U~(L))
(22) 80 For smaller Pe values, tR has to be calculated by numerical integration of eq 21. The plate number for a real fiber can be calculated in a similar way, by integrating the expression
(16)
1
These expressions are valid for large values of Pe. Again we assume a constant radial flow velocity u, along the fiber. Equation 15 is a Gaussian frequency function multiplied by an exponential. The latter corrects for the fact that the sample
which is obtained from eq 18 and 19. For large Pe values, the integration can be performed exactly, leading again to eq 20.
14
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, 1989
Summarizing,the retention time t, and plate number n can be calculated for the described hollow fiber PFFF system by numerical integration of eq 21 and 23, respectively, starting with numerical values for column parameters R, L, and a and the sample diffusion coefficient D. This takes into account the flow variations along a fiber with uniform porosity. Maximum Plate Number. Equation 19 suggests that, by selection of proper flow rate, conditions can be devised that would produce any number of theoretical plates. However, there is a limit: Increasing n also involves increasing the radial flow rate, thereby forcing the sample particles closer to the fiber wall. The theory described is only valid when the particle radii are small compared with the average thickness of the sample zone. If the particles are forced close to the wall, a quite different mechanism of separation, namely the so-called steric FFF technique governs the retention (10). This is a technique in iB own rights, which may be very useful, especially when one is separating very large particles. However, there is a transient area between the conditions of normal and steric FFF, in which the performance is considerably lower than for either technique separately. Assume that a particle with radius rpjust touches the wall. Then, the distance r from the tube center will be r = R - rp, i.e. x = 1 - r p / R . By expressing the travel velocity of the particle both by eq 14 and eq 4, we find that under these conditions
0 3
. ,
.
Flgure 3. Schematic diagram of the principle of the relaxation process.
To derive an expression for the z coordinate for the relaxation point, {, we consider the two parts of the fiber on each side separately. (See Figure 3.) At the relaxation point, a,({) is zero. Thus, we obtain from eq 2 for the left-hand side of the fiber
Looking at the other part of the fiber, we obtain in the same way
The radial flow at the relaxation point u,(R,{) must be the same, irrespective of whether it is calculated from the left or the right end of the fiber. Thus, according to eq 1: ur(R,O) exp(-af) = ur(R,L) exp(-ru(L -
t))
(27)
Defining cp as the ratio of the flow rates into the fiber through each end, we obtain by combination of eq 25-27 if Pe is large. This gives a maximum value for the Pe number. If this is exceeded, the particle touches the wall and will be retarded by the steric FFF mechanism. In practice, the transient region where the steric effects become significant will start at lower Pe numbers than given by eq 24 because of a disturbance in the radial distribution of the sample particles. We may define a factor p as the number of particle radii that correspond to the smallest mean distance to the wall that can be tolerated. Thus Pe,, = R/prp,and a maximum number of theoretical plates can be estimated from eq 19 and 20. Unfortunately,this maximum value of n is very dependent on the unknown parameter p . Relaxation. When sample particles are introduced into the beginning of a FFF channel, they will be randomly distributed across the channel, and it takes some time until the proper concentration profile is developed and the FFF retention is effective. This time is typically considerably larger than the transport time of unretarded sample through the channel. Therefore, if the sample is introduced into flowing liquid, as in liquid chromatography, it will pass the channel with little or no retention. Thus, it is necessary to stop the forward flow for a time period that is long enough for the concentration profile to develop, before the separation can take place. This procedure is called relaxation. in PFFF, there are unique possibilities for combining the relaxation process with a concentration of the sample band, as was mentioned by Lee et al. (11). In the hollow fiber version of the technique, the procedure can be further developed. In reference to Figure 1, if the axial flow Fl is decreased so that it is less than F2 (which is kept constant with a separate pump), liquid will flow backward into the fiber. The outlet has to be connected to a separate solvent reservoir. At a certain point in the fiber, the axial flow is zero, while a flow is still maintained. The position of this point depends on the relation between Fl and F p Sample molecules injected into the fiber will eventually concentrate and relax at this point. A similar procedure for concentration and relaxation with the parallel plate configuration has been described by Wahlund and Giddings (9).
Realizing that CY generally is a small number, we may simplify the expression in eq 28 by McLaurin in expansions, which finally leads to
f=
LlP 1
+ cp + Lff(1 - (p)2
(29)
To obtain the relation between the radial flow velocity at the relaxation point and the pump flow F2,eq 1is integrated from z = 0 to { and from {to L , considering eq 27. The result is then
+ L
Fz = ur(R,f)27rR L
E(cs"
- (L - f)z)]
(30)
Thus, to calculate proper pump settings to achieve desired relaxation conditions ({and u,(R,[)),the first step is to use eq 30 to calculate F2 from {, u,(R,f),and the fiber parameters R, L, and CY. After that, eq 29 gives cp, from which Flis finally calculated (Fl = F,cp/(l + cp)). The time necessary for relaxation to be completed can be calculated as the time needed for the radial flow to transport a particle the maximum distance, i.e. from the center of the fiber to the fiber wall. This time, called t, is an upper estimate of the necessary relaxation time. Exceeding this time will not lead to a more complete relaxation (in the radial direction), but perhaps to a more concentrated sample zone (in the axial direction). An accurate quantitative description of the sample concentration process is very complex and is not attempted here. To calculate t,, we consider an infinitesimal radial distance dr. The time needed for a sample to migrate this distance by convection is dr/u,(r,z),and t, can be calculated by integration of this expression over the radius. Unfortunately, as the radial flow at the center of the fiber is zero, a particle that is initially positioned exactly at the center will never reach
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, 1989
,c
15
. HF
waste I
II
LLII
buffer
Figure 4. Experimental setup. For explanation of symbols, see the text. the fiber wall. Therefore, the integration cannot be carried out from r = 0, but from some small value rg.
If ro = R
X
time Figure 5. Pumping sequence for sample introduction (I), relaxation (II), and restoration (111) to elution conditions (IV).
the result is approximately
t.t = 5(R/ur(R,z))
(32) The fraction of particles that initially are found within the distance R x lo4 from the center is only In the experiments described below, eq 32 has been used (if not otherwise stated), which assures a sufficient relaxation time. When the relaxation is complete, the flows Fl and F2 are adjusted to desired values as given by eq 7 . The difference between F2according to eq 7 and to eq 30 will for all practical cases be negligible. Strictly, when one is calculating retention times and plate numbers, as described in the preceding sections, the fact that the elution starts from z = {instead of z = 0 should be considered. Normally, as {
