Anal. Chem. 1988, 6 0 , 2647-2653
2647
Shock Effects in Nonlinear Chromatography Bingchang Lin, Sadroddin Golshan-Shirazi, Zidu Ma, and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
Shocks are dlscontlnultles of a functlon that appear under certain sets of experimental condltlons, such as a shock wave. Thls Ideal concept has to be completed by the more reallstlc notlon of shock layer. I n a shock layer the functlon (pressure for a shock wave, concentratlon In chromatography) varles very steeply, and all the polnts In the shock layer (e.g., In a concentratlon proflle) move at almost the same veloclty as the shock Itself would In the Ideal case. I n nonIlnear, Ideal chromatography, the behavlor of elution band profiles starts to devlate markedly from what takes place In linear chromatography as soon as a concentratlon shock f m . Under the Influence of the axlal dlffuskn and the mass transfer klnetlcs the actual band profiles devlate from those predlcted by the Ideal model. The shocks are replaced by shock layers, but thelr posltlons and mlgratlon rates are modlfled only very sllghtty. The Importance of these shocks on the shape and mlgratlon rate of the bands Is dramatlc. Classlcal concepts such as those of retentlon tlme, column capaclty factor, column efflclency, and resolutlon must therefore be analyzed wlthln the framework of the shock theory.
The fact that a self-sharpening of certain parts of the band profiles takes place, while other parts spread and become continuous, smooth arcs, is an essential character of the development of these band profiles in nonlinear chromatography (I, 2). This self-sharpening effect is a direct result of the underlying forces that drive the elution of a pure compound band or the separation between the components of a mixture. With the highly efficient modem columns currently available, very steep, almost vertical fronts are often observed, as if concentration discontinuities were trying to build up and nearly succeeding to do so. The concept of shock or discontinuity is not natural in experimental sciences. Its origin seems to be related mainly to the study of catastrophic events, such as the fall of a wall or the breaking of a mechanical part. It appears in the realm of the physics of continuous media through the study of the mathematical properties of a type of nonlinear hyperbolic partial differential equation used in the modeling of the propagation of waves. Indeed, the propagation of a shock wave, such as those generated by the bow of a solid moving faster than the speed of sound in air, is very well represented by the propagation of a shock, as described by the wave equation. Air is compressible and is heated by the passage of a compression wave. The wavelets of sound, which would tend to propagate faster than the shock, penetrate into cold air where they are slowed down and overtaken by the shock. Those that would move more slowly drag into warm air,where they accelerate, since the speed of sound increases with temperature. So the shock is stable and can propagate over very long distances before fading away. Similarly, the notion of shock was introduced in chromatography by De Vault in 1943 (31, to solve an apparent paradox *Author to whom correspondence should be addressed at the
University of Tennessee.
0003-2700/88/0360-2647$01.50/0
resulting from the theory of the ideal model developed by Wilson (4). The ideal model assumes the column efficiency to be infinite. During the migration of the band, the mobile and the stationary phases are constantly in thermodynamic equilibrium. The study of the properties of the mass balance equation written for a pure compound shows that a migration rate can be associated to each concentration. But this rate is not constant, it depends on the concentration. Consequently, some parts of the profile move faster than others and either the front or the rear of the profile tends to get steeper. Since there is no axial diffusion to relax the concentration gradients that build up progressively, it will happen sooner or later that some small concentration either will pass larger ones on the rear of the profile or will be passed by them on the front. This would result in either case in a profile having three values of the concentration at the Same time, in the same place in the column, clearly a physical impossibility. Instead, a discontinuity appears (see Figure 1). It can be shown that one of the fundamental properties of equations of the class of the mass balance equation of the eluite in the case of the ideal moel of chromatography is that, whatever the profile of the input function (i.e., the injection profile), and except in some special cases (e.g., linear isotherm), the band profile will always tend to exhibit a discontinuity on one side or the other of the profile ( 1 , Z ) . The appearance, growth, decay, and collapse of discontinuities is a property of hyperbolic, nonlinear partial differential equations ( 4 4 ) . The properties of these discontinuities have been studied by Rhee et al. (5) and by Jacob and Guiochon (6). The concept of shock as a discontinuity (of pressure in the case of the shock wave and of concentration in the case of nonlinear chromatography) is very useful for explaining clearly the mechanism of formation of the shock, its stability, and the characteristics of its migration. This concept is too theoretical, however, to completely account for the experimental facts. Indeed, in the derivation of the partial differential equation that propagates the discontinuities, a drastic simplification has been made: dispersion has been neglected. This is to some extent self-contradictory, because at a discontinuity the concentration gradient and hence the diffusive flux are infinite. There should be a dynamic equilibrium between the effect of the concentration-dependent migration rate, which tends to build up the shock, and the effect of the axial dispersion, which smoothes it out. The concept of shock should not be dismissed on the grounds that the column efficiency is never infinite and the ideal model is unrealistic. Modern chromatographic columns are highly efficient and it has been shown that the difference between the profiles predicted by the ideal model and the real profiles is very small at efficienciesin excess of 5000 theoretical plates or for plate heights less than 50 Mm, no big feat by present high-performance liquid chromatography standard (7). The reason is that, as long as the dispersion is not very large, i.e., as long as the kinetics of mass transfers between phases is fast but axial diffusion remains slow, the thermodynamics drives the band profile, its shape, and migration rate. It merely seems that the discontinuity has become blurred; the shock has acquired some thickness. In other fields of physics or physical chemistry where similar equations are 0 1988 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
2848 a
c
I
same speed as the shock, which is its ideal representation. A true physical discontinuity is hard to imagine: the concentration jump should take place over a distance of a few molecular diameters. A shock layer has the same propagation properties as a shock, but a finite thickness, function of the mass transfer kinetics, and for practical purposes in preparative chromatography behaves much the same. The purpose of this paper is to apply the shock theory to chromatography, to discuss the properties of the concentration shocks and shock layers in chromatography, to show how shock layers are formed, grow, and propagate and how their properties permit a better understanding of the band broadening phenomena and of the separation between different bands.
THEORY
b
Figure 1. Origin of the Concentration Shocks. (a) Propagation of a continuous profile and shock formation. A continuous profile (0123456) is injected at the column origin (plane Cot). Each concentration moves along the characteristic line correspondlng to this concentration (00‘, aa’, bb‘, cc‘, dd’, ee’, 66‘, i.e., cc’ for point 3 of the input profile). On the part 3’4’5’6’ of the profile the characteristics do not intersect. On the part 0’1’2’3’, they do. The front arc is a physical impossibfity, since there would be two values of the concentration at the same time in the same location. Note also that with this scheme, the band width at any concentration remains constant and its height would not change. There is no dilution. (b) Propagation of a rectangular pulse. A rectangular pulse (01020,0,)is injected in the column (plane C o t ) . The front is a stable shock (see text). It moves at a constant velocity (eq 4), along OIA,B,, as long as its height is constant. The back of the injection pulse is not a stable shock. Each concentration propagates at its characteristic velocity (eq 3), along the corresponding characNow, we consider teristic line (see wlculp,X, and w2cu2p2A2.O,A,B,). the profile as it passes at point B, inside the column. The points B, and 8, coincide. Point 4,at the back of the plateau, which propagates as a point of a continuous profile along the associated characteristic, has just reached point B,, which propagates at the shock velocity. Profile A is intermediate,with a front shock, a plateau, and a continuous rear. Between B and L, at column exit, the front shock slows down, as height decreases (eq 4). It foltowsthe curved path B,L,. The decay of the band height follows the trajectory B,L,.
encountered, such as in hydrodynamics, the notion of shock or discontinuity has been completed by the concept of shock layer (8). A shock layer is a thin region of space where the property studied (air speed, eluite concentration, etc.) varies continuously, but very, very rapidly, and propagates at the
The cause of the self-sharpening of the band profiles during their migration along the column is that the propagation velocity of each concentration wavelet is a function of the concentration of the eluite in the mobile phase. The origin of the association of a velocity to each value of the eluite concentration can be traced to a property of the mass balance equation of that compound (1-4). For a solute dissolved in the mobile phase and susceptible of interacting with the stationary phase or one of its component, a mass balance can be written as follows, assuming that the axial dispersion is negligibly small:
where x and t are the two variables, the abscissa along the column and the time, respectively, c(x,t) is the concentration of the solute in the mobile phase, at time t and abscissa 3c, q(c) is the concentration of the solute in the stationary phase (in an appropriate unit), at equilibrium with a mobile phase where the solute concentration is c[q(c) is the equation of the equilibrium isotherm], u is the velocity of the mobile phase (We can assume u to be constant along the column, since the compressibility of liquids is very small and the partial molar volumes of the solute in both phases are very close], and F is the phase ratio. Rearrangement of eq 1, by observing that aq f at = (dq/dc) (&/at), gives ac
-d t+
U
1
+ F dq/dc
-ac= o ax
The differential of the isotherm, dq/dc, is a function of the solute concentration in the mobile phase, c, and so is the velocity of propagation of this concentration uz = 1 F Udq/dc
+
(3)
where u is the mobile phase velocity. In the case when the isotherm is convex toward the axis of concentrations in the stationary phase (Le., the q axis), the concentration in the stationary phase at equililbrium increases more slowly than the concentration in the mobile phase. The derivative dq f dc decreases with increasing concentrations c, and the velocity u, associated to c increases. The higher a concentration, the faster it migrates along the column. This concentration dependence of the propagation velocity results obviously in a self-sharpening effect of the band front (see Figure la). The peak maximum travels faster than any other point on the front profile and tends to pass them all. The self-sharpening eventually ends up with the formation of a shock after a certain relaxation time. In the framework of the ideal model, a true concentration discontinuity is formed. In practice, the axial dispersion and the mass transfer kinetics are finite, as well as the column efficiency, and a shock layer
ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
is formed. It is remarkable, however, that in a shock layer, the migration velocity is the same for all the concentrations involved. Aris and Amundson have shown that the relaxation time of the formation of a shock is proportional to the rising time of an injection profile, i.e., to the reverse of the slope of the injection profile (1). If the rising time of a pulse or step injection can be considered as negligible, so can the relaxation time. In this case the formation of the shock, or shock layer, takes practically no time. The propagation of a continuous concentration profile through a chromatographic column can be considered as the result of the propagation of each concentration along a straight line of slope u, (eq 3), going through the corresponding point of the injection profile (see Figure la). These straight lines on which the concentration remains constant are called characteristics. If the isotherm were linear, all the characteristics would be parallel and the injection profile would propagate unchanged. Real isotherms are not linear, however, and the slope of each characteristic depends on the concentration to which it is associated. As long as the characteristics that are associated to different concentrations of an arc of the profile do not intersect (Figure la, lines dd’, ee’,66’))the profile remains continuous and propagates as such. A solution of eq 1can be obtained merely by drawing as many characteristics as necessary. Note that in the case of a continuous profile, since the characteristics are not parallel and do not intersect, they diverge. The profile widens and the band dilutes. When two characteristics intersect (Figure la, lines 00’)aa’, bb’, cc’), however, no regular solution of eq 1can exist: since the concentration is different on each characteristic, it cannot be defined as a continuous solution at the intersection point. The only way out of that apparent contradiction is to allow discontinuities to be part of a generalized solution, called the weak solution by mathematicians (9). The velocity of the discontinuity is not given by eq 3. It has been shown by Aris and Amundson (1)that, in the ideal case, the shock velocity is given by the following equation:
u, = 1 + F UAq/Ac
(4)
where Aq and Ac are the concentration amplitudes of the shock in the stationary phase and the mobile phase, respectively. If a positive step injection (concentration Co) is performed in a chromatographic system, with a compound that has a Langmuir-type equilibrium isotherm between the two phases ( q = a c / ( l bc)), the velocity u, increases with increasing concentration (see eq 3)) so a stable shock is formed. The shock amplitude is Co. It moves at a constant velocity, given by eq 4, and its retention time is given by
+
where a and b are the coefficients of the Langmuir isotherm, Cois the concentration of the step, and L is the column length. If a negative step is injected when the shock has been eluted (e.g., in frontal analysis followed by frontal analysis by characteristic point), a continuous profile appears, when the amount of material sorbed by the column is purged. Since the equilibrium isotherm is Langmuir, and the velocity of each concentration on a continuous profile is given by eq 3, the equation of this elution profile is t,=-
L U(
l+F (1
If a narrow pulse injection is made (width t,, concentration
2649
Co, see Figure Ib) with a compound that has a Langmuir equilibrium isotherm between the mobile and the stationary phase, we observe that the front discontinuity of this pulse (Figure lb, 0,OJ is a stable shock and will move at the velocity U,.On the opposite, the rear discontinuityof the pulse (Figure lb, 0304) is not a stable shock. It will give rise to a continuous rear part of the profile. It is important to compare the velocities of the top points of each part of the profile (t = 0, c = C,, and t = t,, c = Co, respectively). The front point moves at a velocity given by eq 4, the rear point at a velocity given by eq 3. Since bc is positive, the rear point overtakes the front one (Figure lb, profile B). When the width of the pulse top has been reduced to zero, the front shock decreases in amplitude and slows down, while the band broadens. Eventually, the peak is eluted as a shock appearing at a time given by
and followed by a continuous profile, which is an arc of hyperbole given by eq 6. The band ends for c = 0, at the classical retention time of an analytical size band, tR = to(l + a n , as shown by eq 6. The properties of this profile have been discussed (7). It has been shown that the profile can be represented on a universal diagram as a function of one single parameter, the loading factor, or ratio between the actual sample size and the column saturation capacity. The theoretical results obtained have been compared to experimental data and excellent agreement has been demonstrated for columns having an efficiency larger than a few thousand theoretical plates (IO). In practice, however, there is no column with an infinite efficiency. In all real cases, the thermodynamical equilibrium can never be achieved in a chromatographic column, although it is often approached very closely. Even if the isotherm is linear, or the sample size so small that a linear isotherm is a satisfactory approximation, the band broadens under the influence of axial diffusion and a finite kinetics of mass transfer between phases. The true mass balance equation, taking the influence of these phenomena into account, is written
with
swat = m ( c ) - Q)
(9)
where Q is the actual concentration of solute in the stationary phase, now different from the equilibrium concentration, q(c), D is the coefficient of axial dispersion, and K is a kinetic constant, the mass transfer coefficient. Equation 9 is the simplest form of mass transfer kinetics between the two phases. In most cases, much more complex equations should be written. A t the difference of the system accounting for the ideal model, this system of partial differential equations does not propagate discontinuities. The mathematical properties of the second-order partial differential equation 8 are very different from those of the first-order equation 1. The ideal shock (i.e., the true concentration discontinuity) vanishes as soon as there is a second-order term in eq 8 and the thermodynamic equilibrium is never reached. The system tends toward equilibrium (see eq 9), however, which means that there is a strong trend of the band profile toward selfsharpening. Furthermore, while in the ideal model, both the concentration and the propagation velocity are discontinuous at the shock; in the nonideal model, the propagation velocity remains discontinuous. Only the concentration profile has become continuous (a situation somewhat analogous to a phase
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
transition). A shock layer may form, if a dynamic equilibrium may arise between this self-sharpening trend and the dispersive effect of the kinetics of mam transfers between phases and of the axial diffusion. Then the front of the peak will not be a discontinuity, but it will be very steep, and the velocities of all the concentrations that it contains will be the same. This velocity will be almost exactly equal to the velocity of the shock of the ideal model. In frontal analysis, for example, the velocity of the different points of the self-sharpening front is the same. This results from eq 4,with Aq = (qo- q-1 and Ac = Ic0 - c-I. Although the front is not a true discontinuity, but just a very steep change of the concentration, it is a shock layer, because all the points travel at the same rate as a true shock would. This conclusion is derived from a mathematical analysis of the properties of the solutions of the system of partial differential equations 8 and 9 (11),which shows that the solution is of the form c ( x , t ) = C(E) = c(x - Ut) (10) This equation implies that the velocity of all the points involved in the shock layer is the same. This phenomenon and the existence of shock layers have very important consequences which force a total reexamination of all traditional concepts of chromatography, such as the retention time of a sample pulse, the column capacity factor, the resolution between bands and its relation to column efficiency, and the influence of the diffusion and dispersion coefficients and of the rate constant on the resolution. We shall now discuss these different points. I. Retention Time of a Pulse Injection. In the previous section, we have presented a derivation of the retention time of a narrow sample pulse in the case when the equilibrium isotherm of the sample between the stationary and the mobile phase is described by a classical Langmuir isotherm. In this section we present a more general discussion, valid whatever analytical function is used to account for the isotherm, provided this function can be differentiated at least once. When a pulse is injected, two shocks, a positive and a negative one, are injected close to each other. One of them only is stable (see above discussion). The other gives rise to a continuous profile. For a while, the band has a flat top at concentration cot with a shock on one side and a continuous profile on the other. It is easy to see (compare eq 3 and 4) that the propagations of the discontinuity and of the maximum point of the continuous profile are such that the width of the flat top decreases constantly, while the band profile widens (in agreement with the second law of thermodynamics). It is easy to calculate the profile as long as the maximum concentration remains equal to co. When the flat top collapses, the maximum concentration starts decreasing and the shock velocity decreases (see eq 4). The determination of the profile in this case is the aim of the second part of this section. Let us consider eq 2. When q ( c ) is convex, the velocity of the shock in the ideal model is dx/dt = U,
(44
It remains constant as long as the maximum concentration is equal to co. Hence, the position of the shock is x, =
ust
(11) On the other side of the peak, the velocity of a wavelet associated to a concentration c is given by eq 3. The position of the point of concentration c on the continuous (rear) part of the band profile is given by x , = u,(t - t,)
(12)
At the peak maximum, we have on the one side (front) a concentration discontinuity at the abscissa x , where the
concentration jumpts from 0 to the band maximum concentration, cot and on the other side (rear) a concentration equal to co that is part of a continuous profile, but happens to be at the same place x&o) = d c o ) (13) Combining eq 11to 13 and solving for t give the time when the continuous part of the band profile is maximum t,(l + Fqo/co) tM(C0) = (14) F(qo/co - (dq /dc)lco)
where qo = q(co). The top point of the continuous profile propagates faster than the shock. When they meet, the shock begins to erode and to slow down. In order to calculate analyticallythe band profiie when the maximum concentration of the band decreases, one needs to solve the following two equations: dx/dt = U, x , = u,(t - t,)
This system can be rewritten as
dt - 1 + Fq/C _
dx t - t,
--
U
- 1 + F dq/dc
X,
U
or
d(t - t, - X/U) d(x/u)
= F-q C
This system of differential equations can be solved easily if dq/dc = !b(q/c) (17) Then, we have d(t - t, - X/U)
t - t , -x/u =
d(x/u)
+-I(
x/u
)
(18)
Combining eq 14, 15, and 17 gives the general equation for the retention time of the maximum of the band profile, when c = co tM =
t,(l
+ F!b-l(qo/co))
W-'(qo/co) - qo/co) and its position in the column =
(19)
t PU
(20) F(!b-'(qo/co) - QO/CO) The solution of the problem depends on the successful integration of eq 18, an ordinary, homogeneous differential equation. It is well-known that its solution is XM
Ln(t)=
s
d[(t - t , - x/u)/(./u)l
rC.-'[(t - t , - x/u)/(x/u)l - [(t - t , - x/u)/(x/u)l + constant (21) Unfortunately, eq 21 cannot be integrated in the general case, i.e., for any isotherm function. There are a few isotherm
ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
2651
functions for which this is possible, however. The classical case is when
$ ( q / c ) = ( q / c ) " (and thus, #-'(q/c) = (q/c)'/") (22) with n equal to 1 or 2. Then the isotherm function can be solved. In the first case, the isotherm is linear ( q = aC), and the solution, derived directly from eq 16, is tR = tp to(1 a') (23)
+
+
where a' is constant. This is the classical equation of linear chromatography. If n is an integer different from 1,we obtain by integration of eq 21
Ln
(t)
=
n n-1
-Ln (1 -
(
t
- t,- x / u
x/u
) )+ 1-w")
constant (24)
If n is equal to 2, this reduces to
( t - t, - X / U ) ' / ~- (FUX/U)'/~= constant
(25)
Equation 25 has already been derived directly in the case of the Langmuir isotherm (1). The constant in eq 25 is determined by using eq 19, 20, and 22, and the final result is eq 7. In the case of an anti-Langmuir isotherm, the isotherm function q is concave. The injected pulse, being rectangular, has a front and a rear shock. The front shock is not stable and turns into the continuous front part of the profile. The rear shock is stable and propagates. Equations 3 and 4 are valid, but now for the opposite parts of the profile. If the isotherm were approximated, in a certain concentration range, by a Langmuir type of equation ( q = a c / ( l + bc), with b < 0) the retention time would be given by A comparison between eq 7 and 26 shows that in the case when b is positive (convex isotherm), the retention time decreases with increasing sample concentration, while in the case when b is negative (concave isotherm), it increases. These results (see Figure 2) are consistent with those of the perturbation analysis made in the case when the concentration is small (12). The two most important parameters in eq 7 and 26, which determine the position of the band maximum at the elution of the column ( x = L)are the slope of the isotherm, a, and the specific stationary phase saturation capacity, a f b, which appears to be the natural unit to quantify the column loading. 11. Column Capacity Factor. The column capacity factor, k , is traditionally defined in linear chromatography as
where G is the slope of the isotherm. In nonlinear chromatography, we have shown above that, for a pulse injection, the elution band profile contains two parts. One of these parts is a shock, the other is a continuous, smooth profile, which is nonsharpening. If the isotherm is convex, the front of the peak is a shock; it is a continuous arc if the isotherm is concave. For a convex isotherm, all concentrations on the rear of the peak travel at a velocity given by eq 3, including the maximum concentration of the elution band, hence
Therefore
Figure 2. Chromatograms obtained successively by numerical simulation for three different compounds having the same slope for their isotherm at the origin: same column and sample size in the three cases; column length, 25 cm, 4.6 mm i.d.; phase ratio, F = 0.20; flow rate, 5 mL/min; sample size, 4.15 mmoi. Isotherms: 1, 9 = 25c/(l
+ 0.25C); 2, 9 = 2 5 ~ 3,; 9 = 2 5 ~ / ( 1- 0 . 2 5 ~ ) .
The isotherm is obtained, in classical thermodynamics, by writing that, at equilibrium, the chemical potential of the compound studied is the same in both phases. If the phases in equilibrium are an ideal gas and an ideal sorbed partial monolayer, we obtain a Langmuir isotherm. In other cases, a Langmuir isotherm does not have much theoretical justification but may be a very convenient equation to fit experimental data on, because it accounts for the general features of adsorption on a quasi-homogeneoussurface. If we take into account the existence of molecular interactions between the molecules of the solvent and those of the compound studied, we obtain the Improved Langmuir isotherm ac exP(Pi,t/kT)
= 1
+ bc exp(bint/kT)
(30)
where wint is the chemical potential contribution due to molecular interactions between solvent and solute at finite concentration. Preliminary results obtained by using Monte Carlo (13)simulations have shown that if the interaction is not strong and the concentration is dilute, the corresponding chemical potential can be replaced by a two-term expansion as a function of the concentration where a. and al are constants. In the case of nearly infinite dilution, we obtain Log k ' = A
+ Bc
(32)
where A and B are constants. 111. Influence of t h e Diffusion and Mass Transfer Coefficients. Only in ideal chromatography, do we have true ideal shocks, i.e., concentration discontinuities. In practice the column efficiency, how large it can be, is finite. The way in which this affects the shocks and the degree to which it does it are very important questions, which are extremely difficult to discuss from a mathematical point of view.
2652
ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1 , 1988
7
4
6
a TIME
10
I
12
Figure 3. Elution profiles obtained by numerical slmuiatlon for a constant sample size of a pure compound: column length, 25 cm, 4.6 mm i.d.; flow velocity, 1 cm/s. Isotherm: 9 = 4c(l c). Axial diffusion coefficients: 1 , D = 0.005 cm2/s; 2, D = 0.01 cm2/s; 3, D = 0.02 cm2/s; 4, D = 0.03 Cm2/s; 5, D = 0.05 cm2/s; 6, D = 0.10
+
10
Figure 4. Influence of the mass transfer coefficient on the band profile: linear isotherm; kinetics after eq 9; sample size, 4.15 mmol; flow rate, 1 mL/min; column, 25 cm long, 4.6 mm i.d. Isotherm: 9
= 25c (1) K = 0.004, (2) K = 0.01, (3) K = 0.1, (4) K = 0.5, (5) K = 1 , (6) K = 5.
cm2/s. -' t
Numerical simulations show that when the axial dispersion coefficient is no longer zero, ideal shocks disappear, but they are replaced by shock layers. Shock layers are the result of the competition between the self-sharpening effect of the nonlinear isotherm and the dispersive effects of the axial diffusion and the mass transfer kinetics. All concentrations of factor, k', profiie in a shock layer move at the same velocity, which is nearly equal to that of the ideal shock. When the rate constant of mass transfers is infiiite, the thickness of the layer increases with increasing values of the diffusion coefficient. In fact the thickness of the shock layer is nearly proportional to the diffusion coefficient, as demonstrated by Whitham (14)and as illustrated Figure 3, at least as long as the diffusion coefficient is small enough. When the diffusion coefficient becomes too large, the shock layer disperses and disappears. The deviation of the retention time of the shock layer from the retention time of the ideal shock is of the same order as the thickness of the shock layer (14). Since the diffusion coefficient is always very small in high-performance liquid chromatography, the results concerning the shock layer can be applied directly. The retention time of the ideal shock can be derived easily by using the equations developed above. The thickness of the shock layer can be estimated from the value of the apparent diffusion coefficient, which gives an estimate of the delay between the elution of the real band maximum and that predicted for the true shock. If we assume a kinetic equation such as eq 9, we may wonder what effect a change in the value of the mass transfer coefficient K has on the shape of the elution band profile. The solution to this problem can be obtained by the numerical integration of the system of partial differential equations 8 and 9 (15). Typical results are shown on Figures 4 and 5. When K is zero, there is no mass transfer between phases which takes place during a finite time, so there is no retention, the velocity of the band and of all concentrations is equal to
-
N
l
0
6
-m $0
E u;
v
Z O
OC 0
4
0
N
0
0
a 100
150
200
250
300
350
400
450
'
0
TIME ( s e d
Figure 5. Influence of the mass transfer kinetics on the band profile: 0 . 2 5 ~ ) )kinetics ; after nonlinear (Langmuir) isotherm (9= 25c/(l eq 9; sample size, 4.15 mmol; flow rate, 1 mL/min; column, 25 cm long, 4.6 mm i.d.; phase ratio, F = 0.20; (1) K = 0.004, (2) K = 0.02, (3) K = 0.1, (4) K = 0.5, (5) K = 1 , (6) K = 50.
+
uo = L/to, and the band is eluted as a Gaussian profile (be-
cause of axial diffusion) at t~ = to. When K increases, in the case of a linear isotherm (Figure 4), t R increases progressively from t o ( K = 0)to t R = t o ( 1 + FG) (large values of K ) (16).Similar results have already been discussed by Giddings (17). In the case of a nonlinear iso-
Anal.
chin. 1988, 60, 2653-2656
therm, the resistance to mass transfer, which is represented by the reverse of the mass transfer coefficient (i.e., l / m , acta as a smoothing factor and in the same time it couples the nonlinear effects. Therefore, when K increases, the selfsharpening effect becomes more intense, and the smoothing effect decreases (Figure 5). When K is large enough (Klarger than ca. 0.1-0.2), the chromatographic process itself becomes really operative and the stationary and mobile phases achieve near equilibrium. The shock layer forms for K larger than ca. 1 and becomes thinner and thinner. Then, with K becoming very large (i.e., K larger than ca. lo), the peak becoming narrower and taller, the velocity of the shock increases, the retention of the band maximum decreases, until the ideal Chromatographyprofie is reached,for an infiitely large value of K (in practice for K larger than ca. 50), and a zero value of the axial diffusion. There is a maximum in the value of the retention time of the band maximum, for values of the rate constant of a few tenth of a unit. Then, when K exceeds about 10, the retention time is nearly constant (15). IV. Resolution between Bands and Column Efficiency. In linear chromatography, the resolution is defined by a Rayleigh equation, as the ratio of the distance between the maxima of the two bands to the average of their base width, i.e., twice the sum of their respective standard deviations
(33) This definition is absolutely unsuitable for applications in nonlinear chromatography, especially when shocks occur. The interactions between the components present in two bands which interfere (due to the competition between their molecules for adsorption sites on the adsorbent surface or, more generally, for interaction with the molecules or groups of the stationary phase) result in nonlinear effects. The only relevant parameter that can be used in this case is the yield of each of the mixture components at a stated degree of purity (18). This yield is an indication of the degree of interference between the bands. It is also directly related to the most important application of nonlinear chromatography, preparative separations, using overloaded elution.
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CONCLUSION This work deals essentially with the problem of the elution profile of a single, pure compound band. Although important, this problem is not the most critical one in chromatography. Since this method is a separation method, we need to study the elution profiles of mixed bands, where two or more compounds struggle to separate from each other and interact, as soon as the experimental conditions (essentially the total concentration) are such that nonlinear behavior takes place. In a further contribution we shall study the interactions between the shock layers corresponding to the bands of two different compounds (19). LITERATURE CITED Aris, R.; Amundson, N. R. In “Madhemetical Methcds In Chemlcal En@neefhg;Prentice Hall: Englewood Cliffs, NJ, 1973 Vol. 2. Lax, P. D. Commun. Pure Appl. Math. 1957, IO, 537. De Vault, D. J . Am. Chem. Soc. 1943, 65, 532. Wilson, J. N. J . Am. Chem. Soc. 1940, 62. 1583. Rhee, H. K.; Aris, R.; Amundson, N. R. PMlOs. Trans. R . Soc. London, A 1970, 267, 419. Guiochon, 0.; Jacob, L. Chromatogr.Rev. 1971, 14, 77. Golshan-Shlrazl, S.; Gulochon, 0. Anal. Chem. 1966. 60. 2384. Rhee, H. K.; Amundson, N. R. Chem. Eng. Sci. 1974, 29, 2049. Rouchon, P.; Schonauer, M.; Valentin, P.: Guiochon, G. In The Science of Chromatography; Bruner, F.. Ed.; Elsevier: Amsterdam, 1985; p 131. Golshan-Shirazi, S.: Guiochon, G.. In preparation. Rhee, H. K.; Amundson, N. R. Chem. Eng. Scl. 1074. 29, 2049. Lln, BingChang; Wang, J b ; Lln, BingCheng J . Chromarogr. 1968, 438, 171. Lln. B.; Ma, 2.; Guiochon, G. in preparation. Whitham. 0. B. In Llnear and Non-//near Waves; Wiley: New York, 1974. Lln, B.; Golshan-Shirazi, S.; Guiochon, G., accepted for publication In J . phvs. Chem . Seinfeld, J. H.; LapMus, L. In Mathemetlcel M e w h Chemlcal M gineerlng; Prentice Hail: Englewood Cliffs, NJ, 1974; Vol. 3. OMdings, J. C. In Dynamb of Chromatography;Marcel Dekker, New York, 1984. Guiochon, G.; Ghcdbane, S. J . Phys. Chem. 1966, 92, 3682. Lin. B.; Golshan-Shirazi. S.; Ma, 2.; Guiochon, G. in preparation.
RECEIVEDfor review June 17,1988. Accepted August 31,1988. This work has been supported in part by Grant CHE8715211 of the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory.
TECHNICAL NOTES Electron Mlcroscopy of Nanometer Partlcies in Freshwater Tsutomu Nomizu, Kenji Goto, and Atsushi Mizuike* Faculty of Engineering, Nagoya University, Chikusa-ku, Nagoya 464, Japan Nanometer particles, 1-100 nm, in freshwater play an important role to decrease the toxicity of heavy metals toward aquatic organisms by adsorption ( I ) . They are also known to contribute to sedimentation of heavy metals in estuarine regions (2). So far, however, little information has been available on the morphology and elemental compositon of individual particles. Scanning electron microscopy of particles smaller than 100 nm collected on a membrane fiiter does not give sufficient information, because of poor resolution in morphology and difficulty in X-ray microanalysis. An analytical electron microscope (AEM), a high-resolution transmission electron microscope (TEM) equipped with an X-ray microanalyzer, can be a powerful tool, although specimen 0003-270O/S6/0360-2653$0 1.50/0
preparation is more difficult. Direct or spray drying of the water sample on a film supported on a specimen grid is not ,applicable,because the concentration of nanometer particles is very low and dissolved salts crystallize out. Tipping et al. collected 0.1-0.5 pm hydrated iron(1II) oxide particles in lake water on a 5-nm-pore membrane filter by filtration, redispersed them in distilled water, and dried the suspension on a carbon film for observation and analysis with the AEM (3). Morphological modification as well as loss of particles may occur during the preparation, however. Biles and Emerson used centrifugation for collecting redispersed fiber aggregates, 0.5-2 pm, in water on a carbon film to observe fibers in beers with the TEM (4), but the specimen was not reproducible, 0 1988 American Chemical Society
