Anal. Chem. 1880, 60, 2364-2374
2364
Automatic0 ed Informatica Applicata (C.I.C.A.I.A.) of Modena University for the computing facilities. Registry No. 3-Nitropropanoicacid, 504-88-1;2-nitrobenzoic acid, 552-16-9; 3-nitrobenzoic acid, 121-92-6;4-nitrobenzoic acid, 62-23-7; 4-nitrophenol, 100-02-7; picric acid, 88-89-1; 2-methoxvethanol. 109-86-4:ethane-1.2-diol, 107-21-1:. N,”-diphenvl. - guanidine, 102-06-7.. LITERATURE CITED (1) Chantooni, M. K., Jr.; Kdthoff. I. M. J. Phys. Chem. 1978. 82, 994-1000. and references therein. (2) . . Kolthoff. I. M.: Chantooni. M. K.. Jr. Anal. Chem. 1978, 5 0 , 1440-1446, and references therein. (3) Kratochvll, 8. Anal. Chem. 1982, 5 4 , 105R-121R. (4) Preti, C.: Tosl, 0. Anal. Chem. 1981, 5 3 , 40-51. (5) Pretl, C.; Tessl, L.; Tosi, 0.Anal. Chem. 1982, 5 4 , 796-799. (6) Neviani Qillberti, E.: Preti, C.; Tassi, L.; Tosi, G. Ann. Chlm. (Rome) 198% 7 3 , 527-532.
(7) Preti, C.; Tassi, L.; Tosi, 0.Ann. CMm. (Rome)1985, 7 5 , 201-206. (8) Handbodc of Chemlshy and Physlcs, 66th ed.; Weast, R. C.. Ed.; The Chemical Rubber Co.: Cleveland, OH, 1985 p C-293. (9) Van Meurs, M.; Dahmen, E. A. M. F. Anal. Chim. Acta 1958, 79, 64-73. (lo) Van M. Anal, chim, Acte 19gg, 2 , , 443-455. (11) Kolthoff, I. M.; Bruckenstein, S.; Chantooni, M. K., Jr. J. Am. Chem. S0C.1961.83.3927-3935. (12) Franchini, G. C:: Ori, E.; Preti, C.; Tassi, L.; Tosi, G. Can. J. Chem. 1987, 6 5 , 722-726. (13) Franchini, G. C.; Tassi, L.; Tosi. G. J . Chem. SOC.,Farady Trans. 7 1987, 83, 3129-3138. (14) Franchini, G.C.; Preti, C.; Tassi, L.; Tosi, G., submitted for publication in Can. J . Chem .
RECEIVED for review January 21, 1988. Accepted June 10, 1988* The deuaPubblicaIsh’~ione(MpI) of is gratefully acknowledged for the financial support.
Analytical Solution for the Ideal Model of Chromatography in the Case of a Langmuir Isotherm Sadroddin Golshan-Shirazi 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
An exact solutlon of the Ideal model of chromatography for one compound Is derived in the case of a Langmulr isotherm (C,= aC,,,/(l bC,), where C, Is the amount of solute sorbed per unit mass of packing material, In equllibrium wlth a concentration C , In the mobile phase). A set of reduced coordinates can be chosen that permits the representation of the elution profiles as a set of curves that are a function of only the reduced sample she. The reduced time Is ( t to)/(tR,o to),whkh provides a t h scale bnlependent of the first Isotherm parameter, a. The reduced ordinate Is bC,. The reduced sample size Is the loading factor, Le., the ratlo of the sample size to the amount of sample that would completely saturate the cdumn. WHh the parameters of the Isotherm known, It Is easy to predlct the elution band profile for any sample size. With the sample slze known, it Is easy to calculate the parameters of the “best” Langmulr Isotherm. From these the prolile ot the elutlon band of any sample slze can be predicted. The bsnds eluted from real cokmne having several thousands of theoretical plates are so sharp that the effects of axial diffusion and of the radlal resistance to mass transfer Introduce only very smaii devlatlons that can be corrected eadly by d n g a dlmenslonless plot. This method is IJgorous. It Is mole general than the mplrkai approaches recently wggested. It Is easier to use and provides more 8CCW8te W e d k t h S . It IS bound t0 f8l4 hOWeVBI, WheIWVW the Langmulr model does not account correctly for the adsorption behavior of the compound under study In the concentratlon range Investigated. I n this way, It provldes 8 sensltivo test of the SuItabUIty of a Langmulr isotherm to represent adsorption In a glven system.
+
-
-
There is a considerable interest in the prediction of band profiles in chromatography. Although this is a very old
* Author to whom correspondence should be sent. 0003-2700/88/0360-2364$01.50/0
theoretical problem, originating with the work of Wilson (1) and De Vault (2),it has not been solved rigorously yet, in spite of many efforts, notably by Glueckauf ( 3 , 4 ) ,Guiochon and Jacob (5, 6) and Rhee et al. (7, 8). Recently, a numerical solution has been published, which permits the accurate prediction of the elution band profile of a pure compound when its equilibrium isotherm in the chromatographic system used is known (9). An experimental demonstration of the validity of this numerical solution has been published (10). The difficulty in applying this solution is that the exact equilibrium isotherm should be known. The elution profile depends very strongly on this isotherm, and it is very sensitive to small changes of the numerical values of its constants. Moreover, the determination of equilibrium isotherms, even with the simple, rather straightforward,and fast ECP method, is time-consuming (10). Besides the theoretical interest in solving well-formulated problems, such as the prediction of the elution band profile of a pure compound, there is an important practical interest in being able to optimize experimental conditions in preparative liquid chromatography. This method is largely used in the pharmaceutical and biotechnological industries for the extraction and purification of valuable intermediates or of finished products, to be used as drugs. The elimination of trace impurities is required. Rapid optimization procedures are sorely needed. Recently an empirical method has been suggested (11,12). It seems to be valid essentially in the case of a Langmuir-type isotherm. This is certainly an important restriction from a theoretical viewpoint. In practice, however, it would be potentially very useful, because the equilibrium isotherms of pure compounds in most adsorption systems can be fairly well approximated by a Langmuir-type equation. Although the experimental and the predicted results seem to agree reasonably well regarding the retention times of the band maxima, the agreement is merely fair for band widths or column efficiencies (12). Furthermore, as all empirical results, the method does not give much of a clue concerning the physical phenomena involved and cannot be extended to other related problems. Especially taxing is the optimization 0 1988 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 80, NO. 21, NOVEMBER 1, 1988
of the experimental conditions for the separation of two or more compounds: the aim of chromatography is not to produce single compound bands but to separate the bands of the components of a mixture. The prediction of the band profie of a single compound is of interest only as an intermediate step toward the understanding of the mechanism of band separation in chromatography. Theoretical investigations of the relatively simple problem of the prediction of the band profile of a pure compound in the case of a Langmuir isotherm could illuminate the problem of the separation of two or several compounds as well. The potential simplicity of the former problem stems from the fact that a Langmuir isotherm has two parameters only. It should be rather easy to find reduced coordinates that would eliminate the influence of the retention time (first coefficient, a) and of the isotherm curvature at the origin (equal to -ab). The sample size is the natural parameter in the series of profiles. The column efficiency seems to be a significant correction factor, but only when it is small, and it could be dealt with separately. The influence of the second coefficient of the Langmuir isotherm could be parametricized, as shown by previous results (9). Rhee has already shown that, in the case of a Langmuir isotherm, it is possible to derive an analytical solution of the Riemann problem for the partial differential equation of chromatography describing the elution of a single compound (13). In chromatographic terms, the Riemann problem is the derivation of the signal obtained at the column outlet, as a response to the injection of a concentration step. There is no solution available, however, for the signal obtained as response to a pulse injection. Our aim is to study the theoretical relationships between the elution band profile of a large sample of a pure compound and the experimental conditions in the case of a Langmuir isotherm and to derive simple rules that could be used for the prediction of band profiles, with a minimum amount of measurements and calculations.
THEORY I. The Ideal Model of Chromatography. Under the assumptions that the effects of axial dispersion on the band profile are negligible and that the kinetics of mass transfer in the chromatographic column are infinitely fast, the system of partial differential equations that describes the band profile simplifies considerably (1-10). The concentration in the mobile phase a t the column outlet is given by the following two equations:
dCm + dz
?(1+
dC, dC, = 0 F=)-
u
dt
(1)
where C , is the solute concentration in the mobile phase, C , is the solute concentration in the stationary phase, a t equilibrium (eq 2), F is the phase ratio, u is the linear velocity of the mobile phase, and z and t are the abscissa along the column and the time, respectively. The first equation is a simplified mass balance equation for the compound studied (with D = 0, no axial diffusion). The second equation is the equilibrium isotherm of the compound. This system assumes that the solvent is not sorbed by the stationary phase. A proper convention for the definition of the adsorption quantities can be chosen (14), but this also requires that the mobile phase be a pure solvent. The validity of the single component approach in the case of a binary solvent has been discussed (10). It was shown that the approximation is excellent when the adsorption Henry constant of the compound studied in the pure weak solvent exceeds at least 5 times that of the strong solvent. It is still reasonable when the ratio exceeds five, but becomes rapidly
2365
unacceptable for smaller ratios. The system of eq 1and 2 was f i t discussed by Wilson ( I ) , who discovered that a smooth injection profile could give rise to a discontinuity of the elution band profile and that the speed of migration of the band cannot be derived simply from the mobile phase velocity and the column capacity fador, but depends also on the solute concentration. A rigorous study of the properties of this system enabled De Vault (2)to present a full qualitative description of the propagation of a large concentration band. Equation 1is a quasi-linear, hyperbolic, first-order partial differential equation. Its properties have been extensively discussed by mathematicians, especially with regards to its ability to propagate discontinuities. A numerical algorithm for its solution was discussed by Courant et al. (15). By assuming the existence of generalized Riemann invariants, Lax (16) was able to extend the results of the simple wave theory and to develop a general theory of discontinuities and of their properties and behavior. Except for the theory of characteristics (5-81, which is directly derived from the theory of linear hyperbolic equations, however, the theory of nonlinear hyperbolic systems of partial differential equations is, for the most part, still incomplete. Some results can be obtained in simple cases, such as when eq 2 is linear (the elution profile is identical with the injection profile) or when eq 2 is a Langmuir isotherm. In the next section we summarize the results of the theory of propagation of discontinuities (i.e., shocks), which are relevant to our purpose. 11. Propagation of Discontinuities in Ideal, Nonlinear Chromatography. Discontinuities, or shocks, are part of a profile where the function instantaneously jumps from one fiiite value to another one, for an infinitesimal change in the variable. The system of nonlinear hyperbolic partial differential equations that describes the migration of large concentration bands in ideal chromatography can propagate such discontinuities (5, 17). The origin and properties of these discontinuities have been recognized first by De Vault (2). They have been discussed by Guiochon and Jacob (5) and by Rhee et al. (7,8),using the method of characteristics, which explains the appearance, growth, decay, and collapse of the discontinuities (17). De Vault (2) has shown that a propagation velocity along the column can be associated to every concentration. The solute propagation velocity in ideal, nonlinear chromatography, u,, is given by u, =
U
1 + F dC,/dC,
(3)
where C, is given by eq 2 and F is the phase ratio of the chromatographic system. In linear chromatography, eq 3 reduces to the classical u, = u / ( l + Fa),with Fa = kd. The concentration profile can thus be seen as propagating along straight lines, of slope u,,in the ( z , t ) plane. The lines are all parallel in the case of a linear isotherm. Otherwise, the different concentrations move a t different velocities. For a convex isotherm, such as a Langmuir isotherm, dC,/dC, decreases and u, increases with increasing concentration of the solute in the mobile phase. Therefore the velocity associated to a certain solute concenbration increases with increasing values of this concentration. However, it is not physically possible for the higher concentrations, which move faster than the lower ones, to pass them. Otherwise,the profile would adopt an "S" shape, and, as pointed out by De Vault ( Z ) , it is not possible to have two or three different values of the concentration of a compound at any point in space. The physically meaningful solution is the appearance of a discontinuity (2, 5-8, 18). We can express this in another way. The trajectory of a certain concentration is a straight line of slope u,. We can draw from the injection profile as many straight lines as
2368
ANALYTICAL CHEMISTRY, VOL. 60, NO. 21, NOVEMBER 1, 1988
needed, starting from points of this profile, with slopes u,. As long as they do not intersect, the solution of eq 1 and 2 may be obtained from these lines (with the additional condition of peak area conservation). If two straight linea intersect, this procedure does not give the solution, since a different value of the concentration is associated to each line. Thus, the concentration C, should have two different values at this intersection point. The solution to this problem is obtained by allowing discontinuities to be considered as part of the solution. A t a discontinuity, eq 1 is no longer valid. It must be replaced by another conservation equation, expressing propagation of the discontinuity at a speed such that there is no accumulation of matter or of energy at the discontinuity. This mass balance then tells us that the rate of propagation of the discontinuity is given by U u, = 1 + F AC,/AC,
aC m c, = 1 + bC,
(5)
The coefficient b is positive. The coefficient a is equal to the ratio of the classical column capacity factor, k,,', to the phase ratio. The ratio a / b is the saturation column capacity, assuming a monolayer formation. The retention time of the elution band corresponding to the injection of a narrow, rectangular pulse of width t, and initial concentration Cm0can be calculated, using the method of characteristics (20). The injection pulse front is a shock, with C, = C, and C, = C,, where C, is given by eq 5 when C, = Cm0. This shock is stable in the case of a Langmuir isotherm, because the high concentrations of the shock move faster than the low ones. Accordingly, it proceeds immediately, moving along the column at a velocity that is given by eq 4 and in this case becomes
Since the injection pulse is rectangular, its rear is a shock too, but this one is unstable, since its top moves faster than its bottom. It collapses into a flight of characteristics and gives rise to a continuous profile. At time t , the rear part of the profile has moved a distance z,, so that t = t, + z,/u, (7) where u, is given by eq 3, with C, given by eq 5 , hence
where k( is the column capacity factor for an infinitely small sample size (linear chromatography), while the front part has moved a distance zf, such that
+
1 k"'bC,o
t,(l
)
(9)
+ bC,o)(l + bC,o + kef)
ti =
(10)
kofbCmo
and
ut,(l zi
=
+ bCmo)2 (11)
ko'bcmo
Until this point is reached, the band has a shock front with height Cd and a flat top of decreasing width, becoming naught a t this point, beyond which the shock height decreases and becomes C,, function o f t . The slope of the shock path, i.e., its velocity, is given by eq 4. Thus we may write for the shock
(4)
where AC, and AC, are the differences in the concentrations of the solute in the stationary and the mobile phase on both sides of the discontinuity, respectively; i.e., AC, is the difference between the concentrations of the solute in the stationary phase immediately before and after the passage of the discontinuity. This had already been noted by Guiochon and Jacob (5), Rhee et al. (6, 7), and Rouchon et al. (19). 111. Derivation of the Retention Time for a Langmuir Isotherm. The Langmuir isotherm relates the concentration of the solute at equilibrium in the stationary phase, C,, and in the mobile phase, C, as
t = Uq l +
Comparison between eq 8 and 9 shows that the front moves more slowly than the rear (since b > 0), and is overtaken at the point I, of time and space coordinates ti and zi,respectively
dt
dz
u
1
+ bC,
which we rearrange as
d ko' (t - t, - z / u ) = d(z/u) 1 + bC,
(13)
On the other hand, eq 8 may be rewritten:
k d ( t - t, - Z / U )
= -
(1
z/u
+k:cmy
(14)
Combination between eq 13 and 14 gives, after rearrangement and integration
t = t,
+ (z/u)+ [ ( k { z / u ) 1 / 2- (tpbCm0)1/2]2(15)
This equation gives the time, t, at which the shock front passes at any point, z, of the column. It has been derived by Aris and Amundson (20). Consequently, the shock front exits at time t f ,corresponding to an abscissa L
tf = t,
+ to + ko'to[l
- (t,bC,o/ko'to)1/2]2
(16)
Equation 16 gives the elution time of the band front for a compound with a Langmuir isotherm. IV. Reduced Band Profile for a Compound with a Langmuir Isotherm. In the case of a Langmuir isotherm (see eq 5))the retention time of a narrow, zero-sample size band, i.e., the elution time of the rear end of a large concentration band, depends on the first coefficient, a, while the shape of the elution profile depends essentially on the coefficient, b, of the isotherm. More specifically,we are going to show that the profile is entirely described by the value of the loading factor, Lf, the ratio of the sample amount to the amount of sample required to saturate the column
where V is the volume of sample injected (Cmois the concentration), c is the fraction of void volume in the column, S is the geometrical cross-section area of the column, L is the length of the column, and Q, (=a/b,for a Langmuir isotherm) is the saturation concentration of the stationary phase. The volume of sample injected, during a time t,, at a velocity u is
v = &ut,
(18)
So, since the column capacity factor at zero sample size, kd, is equal to the product Fa (with F = (1 - e)/€), the loading factor becomes
ANALYTICAL CHEMISTRY, VOL. 60,NO. 21, NOVEMBER 1, 1988
\
where tRois the retention of a zero-sample size pulse, tois the solvent hold-up time, N , is the amount of solute injected (mole), and F, is the volumetric flow rate of mobile phase. Combination of eq 16 and 19 gives tf =
tp
+ t o + (tR,O - to)(l - L f ' / 2 ) 2
(20)
The retention time of the band front depends only on the loading factor, on the corrected retention time at zero-sample size, and on the width of the injection band. The retention time of a concentration C, on the continuous part of the profile is given by eq 8 , where z is substituted by L and z l u by to t=t,+to
I
1+
=1[
b
(
tR,O
\
k0'
(1
+ bC,)2
This equation or similar ones have been previously derived in the literature and used by Aris and Amundson (20) and Knox and Pyper (21). Solving eq 21 for the concentration, Cm,gives
c,
2367
-
6
5 4
)"' - 1]
t-t,-to I -r
I
0
if tf < t C te
=0
(23)
if t C tf or t , C t
tp
240
260
280
IO
Flgure 1. Band profiles derived from the analytical solution of the ideal model of chromatography (eq 20 and 22) for various sample sizes of a compound having a Langmuir equilibrium isotherm between the mobile and the stationary phase: equilibrium isotherm, C, = 25Cm/(1 0.25C,), where C, and C, are the concentrations (molar) of the compound at equilibrium in the mobile and statlonary phases, respectively; phase ratio, 0.25; column length, 25 cm; column efficiency, infinite (ideal model); mobile phase Row rate, 5 mUmln; solvent hold-up time, 40 s; sample sizes, (1)0.083, (2) 0.415, (3) 0.83, (4) 1.66,(5) 2.49, (6)4.15 mmol.
+
where tf is the retention time of the shock front and t, is the time when the concentration returns to zero. Equation 22 gives the elution profile of the rear part of the band. It is valid until C, becomes equal to 0, in which case it gives t e = ~ R , O+
220
TiME .(set>
and
c,
200
(24)
Langmuir isotherm chromatographic profile, under a dimensionless form.
From eq 20 and 24 we can derive the base-line bandwidth, which is given by
w=
te
- tf = (tR,O - to)[l - (1 - Lf1/2)2]
w = (tR,O - to)(2Lf'/2- L f )
(25)
(26)
and
and the maximum concentration of the band, which is obtained as the concentration of the continuous, rear part of the profile at the retention time of the front
Combination of eq 20 and 27 reduces to
Lp2 CmMax =
b ( l - Lf'l2)
(28)
Finally, a combination of eq 22 and 28 gives the reduced, general profile of a large concentration band, when the compound considered has a Langmuir isotherm
The eq 20, 26, and 28 permit the normalization of the
It is important to observe that the dimensionless retention time of the band maximum, the dimensionless band width, and the dimensionleespeak height depend only on the loading factor. Conversely, it is possible to derive the loading factor from the band profile and, from the sample amount, it must be possible to calculate the column saturation capacity and to predict the band profile for any other sample size.
RESULTS AND DISCUSSION I. Comparison between Band Profiles Predicted by the Ideal and the Semiideal Models. Figure 1 shows a set of elution profiles calculated by using eq 20 and 22, for increasing sample sizes of a given compound, under constant experimental conditions. The general form of the profile series
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 2 1 , NOVEMBER 1, 1988
.--
F!
43 0 2
e.
m 0’ n
1 0
E=?
-0
ci z
8 ?
0
N 0
2 0
200
220
240
260
TIME
280
300
320
(sec)
Figure 2. Comparison between the band profile derived from the analytical solution of the ideal model of chromatography and the band profiles obtained with columns of finite efficiency: same experimental conditions as for Figure 1, except sample size, 4.15 mmol; number of theoretical plates, ( 1 ) 1250, ( 2 ) 2500, (3) 5000, (4) 10000, (5) infinite.
is representative of the results obtained with Langmuir isotherms (7-10,21)and is classical. Increasing the sample size results in a decrease of the retention time and an increase of the bandwidth since the final elution time of the profile remains constant. The band shape is triangular, but the hypotenuse of the triangle is not a straight line. As predicted by eq 22, it is hyperbolic. This hyperbole has a vertical asymptote at t = to + t,. Equations 20, 26, and 28 predict that the retention time of the band maximum decreases and that the band height and the bandwidth increase with increasing loading factor. The exact dependence of the retention time, of the band maximum concentrations, and of the bandwidth on the loading factor are not simple, however (see eq 20, 26, and 28), and they have not been elucidated yet by an empirical approach. In order to assess the potential applicability of our results to experimental band profiles, we have first to investigate the effect of departure from the ideal model, since, in practice, the column efficiency is finite. The extent of the deviation of real profiies from those predicted by the solution of the ideal model is determined by employing the computer program previously described (9) and whose ability to predict exactly the experimental profiles has been demonstrated (10). We have simulated a series of bands of decreasing efficiencies, obtained under experimental conditions identical with those for which the largest band profile on Figure 1has been obtained. The results are shown on Figure 2, for columns having 10000, 5000, 2500, and 1250 theoretical plates. As has been reported many times, in the case of a Langmuir equilibrium isotherm, the band profiie has a very sharp front. For the most efficient column the deviation between the prediction of the ideal model and those of the more exact and realistic semiideal model is extremely small. It would probably be negligible compared to experimental errors, especially to the ones in-
troduced by the detector time constants, which has to be very small for the elution profile not to be distorted. The simulated band profiles are in agreement with the profiles predicted by the ideal model and also with those recorded experimentally (10). They show the propagation of a very sharp front. This propagation takes place at a velocity that does not depend on column efficiency but only on the system studied (i.e., nature of the solute studied and the stationary phase, composition of the mobile phase). This can be explained by the “shock layer” theory, first evolved in the study of flow problems in compressible fluids. A full discussion of the shock layer in chromatography has been presented by Rhee et al. (22), in the case of frontal analysis. These authors have shown that in a real chromatographic system, with a finite mass transfer resistance and a nonzero longitudinal diffusion, the self-sharpening of the front results in the formation of a thin transition zone, which propagates at the same speed as the corresponding discontinuity of the ideal model. This transition zone, which is the real phenomenon corresponding to the ideal shock, is called a shock layer. The thickness of the shock layer is proportional to the apparent diffusion coefficient of the solute. If the apparent diffusion coefficient tends toward zero and the column tends toward an ideal column, the shock layer approaches the corresponding shock wave. Our results (Figure 2) show that the results of Rhee et al. (22) apply also to elution chromatography and may be used to describe the elution of a large concentration band. The shock layer propagates at a constant velocity, independent of the column efficiency. The profiles are smoothed by axial diffusion and resistance to mass transfer, the extent of this smoothing being a function of the column efficiency. Since the thickness of the shock layer increases with the apparent diffusion coefficient, so does the retention time of the band maximum. The band also becomes broader and its tail smoother and longer (see Figure 2). The shock layer theory permits the extension of the analytical solution of the ideal model of chromatography to the case of a finite column efficiency. 11. Dimensionless Plots of Chromatograms. If one assumes that the equilibrium isotherm is of Langmuir type, which can often be done as a first approximation, it is possible to use eq 20 or 28 to derive the value of the parameters a and b of the isotherm. The determination of the retention time, tR,O,of a very small pulse gives the first coefficient
(33) where k,,’ is the column capacity factor and F the phase ratio ( F = (1- €)/E). The determination of either the retention time or the height of a large concentration band and the use of eq 20 or 28 permit the calculation of the coefficient b. The analytical solution of the ideal model shows that, for a given compound and chromatographic column (i.e., constant k,,’), the retention time of the band maximum and the bandwidth depend only on the loading factor (see eq 20 and 26), while the maximum concentration, i.e., the peak height, depends on the loading factor and the coefficient b of the isotherm. The profiles shown in Figure 3 are numerical solutions of the semiideal model (9) which correspond to different Langmuir isotherms, all with the same a coefficient, Le., the same band limit retention time for a zero sample size, the same loading factor, and the same column efficiency. The retention times and the bandwidths are independent of b, while the peak maximum concentrations are inversely proportional to b. Since the second coefficient of the isotherm, b, is different for the four profiles, the sample sizes required to achieve that same loading factor are different, which is why
ANALYTICAL CHEMISTRY, VOL. 60, NO. 21, NOVEMBER 1, 1988
2369
4
\ 4
150
200
250
300
TiME.(sec)
Figure 3. Influence of the second coefficient of the Langmuir isotherm on the band profile, at constant value of the loading factor and of the limk column capacity factor (k,'): column capacity factor, k,' = 6.25; number of theoretical plates, 2500; loading factor, L , = N,,,b/Fv(tR,o - to) = 0.05; Langmuir isotherm, C, = 25C,/(1 bC,); (1) sample size, 51.9 mmol, b = 0.02; (2) sample size, 10.38 mmol, b = 0.10; (3) sample size, 4.15 mmoi, b = 0.25; (4) sample size, 0.415 mmol, b = 2.5.
+
the peak heights are different. They are inversely proportional to the value of this second isotherm coefficient. This is another way of saying that when the curvature of the isotherm increases (it is equal to -ab), the column overloads much more easily, Le., with much smaller samples. So the profiles predicted by the general model of chromatography support the conclusions drawn from the analytical solution of the ideal model. Similarly, the profiles on Figure 4, all derived again from the numerical solution of the general model of chromatography for the Langmuir isotherm, show the effect of a change in the slope of the isotherm at the origin (coefficient a), at constant value of the coefficient b, the loading factor, and the column efficiency. The band height is nearly independent of the sample size and of a as predicted by eq 28, but the retention time and the bandwidth are proportional to the coefficient a. So is the sample size required to achieve constant loading factor under the conditions required. Figure 5 shows dimensionless profiles obtained by plotting the product C,b, of the band ordinate and the second coefficient of the Langmuir isotherm, versus the corresponding value of the ratio k'/kd, of the column capacity factor at which the concentration C, is eluted to the column capacity factor for a zero sample size. Different profiles have been reported, corresponding to different values of the isotherm coefficients, of the sample sizes, and of the column lengths and flow rates but to the same value of the loading factor. The profiles obtained depend only on the loading factor and the column efficiency, so all these profiles are superimposed. Figure 6 shows a plot of the reduced concentration, bC,, versus the reduced retention time, (t - tO)/(tRO - to),for a series of values of the loading factor, between 1 and 25%. This universal reduced chromatogram may be used to predict the elution profile of a large concentration band, knowing the equilibrium isotherm (which has to be Langmuir) or, con-
x 0
100
I 300 400 500 600 700 800 900 1000 1100 1:
TIME.(sec) Figure 4. Influence of the first coefficient of the Langmuir isotherm on the band profile, at constant value of the loading factor and of the second coefficient of the isotherm (6 ): second isotherm coefficient, 6 = 0.25 (see Figure 3, curve 3); number of theoretical plates, 2500; loading factor, L , = N,b/FV(tR,,- t o ) = 0.05; Langmuir isotherm, C, = aC,/(l O.25Cm);phase ratio, F = 0.25; (1) sample size, 1.66 mmol, k,' = 2.5; (2) sample size, 4.15 mmol, k,' = 6.25; (3) sample size, 8.3 mmol, k,' = 12.5; (4) sample size, 16.6 mmol, k,' = 25.
+
versely, to determine the coefficients of the "best" Langmuir isotherm, from the profile of a large concentration band, obtained for a sample of known amount (see next section). Finally, Figure 7 shows a reduced plot of the apparent column capacity factor, where k '/kd has been plotted versus the loading factor, and Figure 8, the corresponding plot of the reduced maximum band concentration (product bC& versus the loading factor. In both cases, there is a slight effect of the column efficiency, the retention time decreasing more slowly and the peak height increasing more slowly with increasing loading factor when the column efficiency is lower. 111. Use of the Dimensionless Plots To Determine the Coefficients of the Langmuir Isotherm. If the equilibrium isotherm of the solute studied is of Langmuir type, and the coefficients are known, the calculation of the profile of the band obtained for any sample amount is easily carried out as follows. First the loading factor is derived, using eq 19. The reduced chromatograms shown on Figure 6, obtained for a series of values of the loading factor covering most cases of practical interest and between which interpolation is easy, are used to determine the reduced chromatogram corresponding to that loading factor. The actual band profile can be easily derived from the dimensionless profile by multiplying the ordinate by l / b and the abscissa by kdtO,then adding toto the abscissa. The derivation of the parameters of the "best" Langmuir isotherm, i.e., the inverse operation, is hardly more complex. The procedure is as follows: A. From the chromatogram obtained for an analytical scale injection, the retention time, tR,O, the dead time, to, and the column efficiency,N , are derived. The limit column capacity
2370
ANALYTICAL CHEMISTRY, VOL, 60, NO. 21, NOVEMBER 1, 1988 9
05
06
07
03
09
10
, l
T- TO/TRO -TO
Flgure 5. Comparison between the dimensionless profiles derived from four different Langmuir isotherms: loading factor, 0.05; number of theoretical plates, 2500; phase ratio, F = 0.25; (1) k,' = 6.25; b = 0.25; sample amount, 4.15 mmol; flow velocity, 0.625 cm/s; column length, 25 cm; (2) k,' = 4.0; b = 8.0; sample amount, 0.0833 mmol; flow velocity, 0.125 cm/s; column length, 25 cm; (3) k,' = 5.0; b = 1.0; sample amount, 0.833 mmol; flow velocity, 0.125 cm/s; column length, 25 cm; (4) k,' = 3.0; b = 2.0; sample amount, 0.50 mmol; flow velocity, 0.125 cm/s; column length, 50 cm. The four curves cannot be distinguished on the original drawing.
factor, k,,', at zero sample concentration is calculated and a is derived by using eq 33. B. A chromatogram is run with a large, known sample size. From the retention time of the band maximum, its reduced retention time is derived, using the data determined during the first step. The values of the loading factor, hence that of the coefficient b, are calculated, using the graph on Figure 7. This procedure compensates for the slight influence of the column efficiency on the exact values of the retention time of the band maximum and its height. Alternately, if the column efficiency is large, this effect may be neglected, and the values of the loading factor, hence of the coefficient b, are calculated by using eq 20. If the maximum concentration is known, from prior detector calibration, b may be derived from the plot on Figure 8, but the procedure is less accurate and more cumbersome than step B above. We note in passing that the plot in Figure 8 can also be used for detector calibration purpose, when b is determined as described in step B. C. Knowing a and b permits the derivation of the loading factor and, from Figure 6, of the reduced, dimensionless band profile, which can be transformed into the actual profile. These two profiles are compared. A significant difference between them means that the Langmuir equation is a poor approximation of the experimental isotherm in the range of concentration investigated. IV. Column Bandwidth and Apparent Efficiency. Before discussing the apparent column efficiency, we want to emphasize first that the number of theoretical plates of an overloaded column is merely an empirical parameter that should be used with caution and restraint, for two main reasons. First, the plate number is defined by reference to Gaussian profiles, as the square of the ratio of the retention time to the peak standard deviation. An overloaded chro-
0.0
0.2
04
0.6
0.8
10
1.2
T-TO/TRO-TO Figure 6. Dimensionless chromatograms obtained with Langmuir isotherms: plots of bC, versus k'lk,', or (t - to)/(tR,o - to)for a series of values of the loading factor; number of theoretical plates, 5000; (1) 1, = 0.01, (2) 1, = 0.03, (3) 1,= 0.05, (4) L , = 0.10, (5) L f = 0.15, (6) L , = 0.20, (7) 1, = 0.25.
25
Figure 7. Dimensionless plot of the reduced retention time of the band maximum, ( t - fO)/(fR,O- t o ) = k'/ko', versus the loading factor, for columns having different efficiencies: (1) number of theoretical plates, 1250; (2) number of theoretical plates, 2500; (3) number of theoretical plates, 5000; (4) number of theoretical plates, 10 000.
ANALYTICAL CHEMISTRY, VOL. 60, NO. 21, NOVEMBER 1, 1988
..0
&)
2371
2
N = 5.54(
01 0
(34)
In order to relate N to the degree of column overload, we will assume with Knox and Pyper (21) that the column height equivalent to a theoretical plate, H, is the sum of two independent contributions
m 0
= Hkin + H t h e r (35) where H k i n and &er stand for the kinetic and the thermodynamic contributions to the height equivalent to a theoretical plate (HETP), respectively. The first term, HEn,is equal to the column HETP at zero sample size and accounts for the effect of a finite rate of mass transfer on the bandwidth. The second contribution, HWr,results from the nonlinear behavior of the equilibrium isotherm. We may derive this contribution from the analytical solution of the ideal model presented above, since &er is the HETP of the band profile supplied by the ideal model: its width depends only on the nonlinear behavior of the equilibrium isotherm. Given the classical definition of the HETP, eq 35 is equivalent to
h
0
1 0
-L?x m
Fo
0
d
N=
GOO
005
0 10,
0 15
0
20
0 25
Nm*b/ FydTRO-TO) ' Figure 8. Dimensionless plot of the reduced concentration of the band
maximum, bC-, versus the loading factor, for columns having different efflciencies: (1) number of theoretical plates, 1250; (2) number of theoretical plates, 2500 (3) number of theoretical plates, 5000; (4) number of theoretical plates, 10 000. matographic band does not have a standard deviation, even though it still has a variance. Under such conditions, the plate number becomes an empirical parameter, the value of which depends much on which fractional band height is used for the determination of the bandwidth. Its usefulness requires strict adherence to a detailed definition and is merely for comparison purposes. Second, and maybe more importantly, the real aim of preparative chromatography is to separate substances, not to supply single-component bands having a nice profile nor to characterize these profiles. It has been shown repeatedly that the elution band profiles of two compounds that interfere severely are not closely related to their individual profiles when they are injected separately on the same column (5, 23). Whatever the reluctance of the theoretician to discuss the apparent efficiency of an overloaded column for a single component band, he should not forget that experimentalists have found the parameter useful. Since the previous equations afford an easy way to describe the broadening of chromatographic bands with increasing sample sizes in terms of a parameter that is extensively used in practice, we do just that here. The following relationships should be used carefully, however, because they do not offer an easy way to the optimization of experimental conditions for maximum production rate. Eble et al. (11) have defined the plate number of an overloaded column as proportional to the square of the ratio of the retention time of the band maximum to its width at 61% of the band height. In a recent paper Snyder et al. (12) have preferred the use of the bandwidth at the base line, which is not very practical to measure because, in practice, overloaded bands tail always after the retention time at zero concentration. Most experimentalista seem to prefer the use of the width at half-height, the measurement of which is easy and precise. Then
N$\Tth
N O + Nth where N , No, and Nth stand, respectively, for the apparent column plate number for a certain loading factor, the plate number for a very small sample size of the same compound, and the plate number derived from the ideal model for the same loading factor. Within the framework of the ideal model, the base-line bandwidth can be calculated by using eq 26. The actual base-line bandwidth is equal to w(tR,O- to),where w is the bandwidth of the correspondingreduced chromatogram. The bandwidth at half height, Wo,s,can be derived from eq 29, by substituting Cm/CmMruwith 0.5, deriving t , and subtracting tf, given by eq 20. The result is
The retention time of the band is given by eq 20. Accordingly, the plate number, Nth, of an ideal model band corresponding to a loading factor Lf is given by
5.54
(2 - L p ) 4 (4Lf1l2- Lf)2
to
+ t,
1
+ kdto(1 - Lf/2)2
(38)
This equation is very general, but it does not fully account for the influence of the width of the injection band (which is represented by t, in eq 38) when this width is large compared to that of the profile (volume overload). It relates the efficiency under overloaded conditions with a Langmuir isotherm to the loading factor, but it accounts only for the "thermodynamic" contribution to the bandwidth. The limit value of eq 38 at zero loading factor is infinite, which is expected since the calculation is carried out within the frame of the ideal model. Thus, the contribution of a finite injection volume has to be introduced separately (24). In many cases, t, is negligible, which permits some simplification of the last term of eq 38. When the loading factor is small compared to unity, eq 38 can be simplified considerably, by expanding its right-hand side in a power series of L$l2
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 21, NOVEMBER 1, 1988
I
-1
1
-6
-5
-4
-3
-2
-1
5
-7
0
Flgure 9. Plot of column apparent efficiency versus the loading factor (double logarithmic coordinates): plate number calculated from the width at haif-height (eq 38);influence of the limit column capacity factor; limit column efficiency, N = 10000 theoretical plates: (1) k,’ = 10,(2)ko’ = 5, (3)k,’ = 2, (4) ko’ = 1. If we return now to the definition of the plate number given by Knox and Pyper (21) or by Snyder et al. (12), we can use the same method and derive the different equation for the plate height of the ideal model band, calculated after the base width of the band, given by eq 26. The new plate number Nth’ is given by
Equations 38 and 40 give different results, which were expected since the band profile is not Gaussion. As we have done for eq 38, we can expand the right-hand side of eq 40 in a power series of Lf‘l’, which permits the derivation of a simplified equation, valid for small values of the loading factor
The limit thermodynamic plate number for very small sample sizes is 38% larger if it is defined from the width at half-height than if it is defined from the base width. Also the two plate numbers vary differently with increasing loading factors. This illustrates the earlier comment that strict adherence to a unique definition of the plate number is necessary. The limit of the plate number given by eq 40 when the loading factor tends toward unity is 16/kd2, while the limit for eq 38 is infinite. The reason is that when the loading factor becomes close to 1,the retention time of the front tends toward to + t,, while the band height increases indefinitely (eq 28) and the width at half-height tends toward 0 (eq 37). Finally, we note that the limit of eq 41 at very small loading factors is identical with the value derived by Knox and Pyper
-6
-5
-4
-3
-2
-1
LOG if
LOG Lf
Flgure 10. Plot of the column apparent efficiency versus the loading factor (double logarithmic coordinates): plate number calculated from the width at half-helght (eq 38); influence of the llmlt column efficiency; limit column capacity factor, k,’ = 5; (1) No = 1250theoretical plates, (2)N o = 2500 theoretical plates, (3)N o = 5000 theoretical plates; (4)N o = 10000 theoretical plates. (21),who assumed the band profile obtained with a Langmuir isotherm to be a rectangular triangle. This is also the relationship used by Snyder et al. in some of their work (12). Combination of eq 36 and either 38 or 40 permits the calculation of the variation of the actual plate number of a column with increasing loading factor. If, for example, we combine eq 36 with the limit of eq 41 at small sample sizes, we obtain
with
(43) Equation 43 is exactly the equation used by Snyder et al. (11, 12). The reason for the scatter of the data and the progressive deviation of experimental data from the prediction of eq 42 and 43 is that the value of N& used in eq 43 is approximate, and valid only a t low values of the loading factor. The only correct relationship, valid for all values of the loading factor, would be obtained by combining eq 36 with either eq 38 or 40, depending on the definition of the plate number selected. Then, deviations of experimental results from the predicted relationship could safely be ascribed to the equilibrium isotherm of the studied compound being significantly different from a Langmuir isotherm. Figure 9 shows a plot of N (eq 36,where N* is calculated by using eq 38) versus the loading factor for different values of kd (double logarithmic coordinates). In all cases, the plate number decreases rapidly with increasing loading factor as soon as the loading factor exceeds about 0.01% and is reduced
ANALYTICAL CHEMISTRY, VOL. 60, NO. 21, NOVEMBER 1, 1988
-6
-5
-4
-3
-1
-2
LOG Lf Flgure 11. Comparison between three equations predicting the variation of the column efficlency with increasing column loading factor: limit column efficiency, 10 000 theoretical plates; lmit column capacity factor, k,' = 5.0; (1) simplest equation, N = No/(l 0.25Ndf);(2) slmplified eq 43, N = No/(l 0.25N0L+k,'2/(1 k,')2); (3) exact eq 40.
+
+
+
by half for values of the loading factor between 0.25% and O.7%, depending on the limit column capacity factor, ko'. Figure 10 shows a plot of N (eq 36, where N b is derived from eq 38) versus the loading factor for different values of No, but the same column capacity factor (double coordinates). The results are very similar to those previously published by De Jong et al. (25) and Colin (26). Figure 11 compares the prediction of the three available equations for the determination of the variation of the column efficiency as a function of the loading factor for a compound having a relatively large value of the limit column capacity factor, kd, equal to 5. Figure 12 shows the same comparison for a compound having a value of kd of only 2. The first equation is the simplified one, derived by Knox and Pyper (21)and used by Snyder et al. (11,12),which neglecta the term (kd U/kd in eq 41 and 43, as well as the term in LIl2. The second equation is the one that only neglects the term in LfI2 in eq 41. The last equation is the exact eq 40. The first equation is a poor approximation, especially at small values of kd, unless the loading fador is very small, lower than 0.1% for kd = 5 and 0.01% for kd = 2. The second approximation is much better for values of the loading factor less than ca. 5-lo%, depending on kd. Of course, only the exact equation can account for the influence of the loading factor in the whole range-of the loading factor (27).
+
CONCLUSION As already suggested (21), the elution profiles of large concentration bands can be represented on a dimensionless plot. On this plot, the reduced abscissa is the ratio of the column capacity factor for the band maximum to the limit column capacity factor a t zero sample size. The reduced ordinate is the product of the eluite concentration and the
I
-6
I
-5
-4
I
-3
,
2373
I
-1
-2
LOG Lf Flgure 12. Comparison between three equations predicting the variation of the column efficiency with increasing column loading factor: limit column efficiency, 10 000 theoretical plates; limit column capacity factor, k,' = 2.0; (1) simplest equation, N = No/(l 0.25N0Lf);(2) simplified eq 43, N = No/(l 0.25NoL&,'*/(1 k;)'); (3) exact eq 40.
+
+
+
second coefficient of the Langmuir isotherm. The profiles depend on a single parameter, the loading factor, or ratio of the actual sample size to the column saturation capacity. The suitable use of this representation permits the rapid determination of the band profile for any sample size. Only two independent measurements are needed. They require two successive injections, one of a very small sample size, the other of a large one. Conversely,and this is a very important result, the rapid determination of the "best" parameters of the Langmuir isotherm is easy. In this context, best is not related to a least-squares fit or similar procedure, but simply indicates that the process involving the use of experimental data introduces errors. Furthermore, in many cases, the Langmuir equation, although a reasonable first approximation, does not account perfectly well for the adsorption behavior of the compound studied. In such cases, nevertheless, the method described above will generate a reduced profile that will not coincide with the recorded band profile. Depending on the degree of accuracy required, on the precision of the experiment, and on the extent of the deviation, the Langmuir equation obtained could be considered as accounting properly or not for the adsorption behavior of the studied compound. But the Langmuir equation thus obtained will be the best one possible. Better results could be obtained only by resorting to the lengthy, tedious, and costly procedure of determining the isotherms, finding a proper equation to account for them, and calculating numerical solutions. Finally, insofar as the competitive Langmuir isotherm is an acceptable model for representing the adsorption behavior of several compounds, the method provides a rapid determination of the Langmuir isotherm of each compound, hence of the competitive Langmuir isotherm of the set of compounds studied. Depending on the system selected, this may be or
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Anal. Chem. 1988,60,2374-2379
may not be a satisfactory approximation. The analytical integration of the ideal model permits also the determination of the variation of the apparent column efficiency as a function of the loading factor in the whole range of sample size accessible. The experimental study of these relationships and examples of their application to the solution of practical problems will be published separately (27). LITERATURE CITED Wilson, J. N. J. Am. Chem. SOC. 1940, 62, 1563. De Vault, D. J. Am. Chem. SOC.1943, 65, 532. Glueckauf, E. Proc. R . SOC.London, A 1946, 186, 35. Glueckauf, E. Dlscuss. Faraday SOC. 1949, 7 , 12. Guiochon, G.; Jacob, L. Chromatogr. Rev. 1971, 14, 77. Jacob, L.; Vaientin, P.; Guiochon, G. Chromafograpbla 1971, 4, 6. Rhee, H. K.; Aris, R.; Amundson, N. R. Phllos. Trans. R . Soc. London A 1970, 267, 419. Rhee, H. K.; Aris, R.; Amundson, N. R . Chem. Eng. Scl. 1974, 29,
(14) Kovats, E. sz The Sclence of Chromatography; Bruner, F., Ed.; Elsevier: Amsterdam, 1985; p 205. (15) Courant, R.; Isaacson, W.; Rem, M. Commun. Pure Appl. Mfh. 1952, 5, 243. (16) Lax, P. D. Commun. Pure Appl. Mth. 1957, 70. 537. (17) Valentin, P.; Gulochon, G. Sep. Sci. Techno/. 1975, 10, 245. (16) Rouchon, P.; Schonauer. M.: Valentln, P.; Gulochon, G. S e p . Scl. Techno/. 1987, 2 4 , 1793. (19) Rouchon, P.; Schonauer, M.; Valentin, P.; Guiochon. 0. I n The Science of Chromatography; Bruner, F., Ed.; Elsevier:
1984: D 131.
Amsterdam,
(20) Ais, k:; Amundson, N. R. Mathemetlcal Metbds In Chemical Engineerlng; PrenticaHali: Englewood Cliffs, NJ, 1973. (21) Knox, J. H.; Pyper. H. M. J. ChrOmafogr. 1986, 363, 1. (22) Rhee, H.; Bodin, 8. F.; Amundson, N. R. Chem. Eng. Scl. 1971, 26, 1571. (23) Gulochon, G.; Ghodbane, S. J. M y s . Chem. 1988, 92, 3682. (24) Katti, A.; Guiochon. 0.. unpublished results. (25) De Jong, A. W. J.; Poppe, H.; Kraak, J. C. J. Cbromatcg. 1981, 209, 432. (26) Colin, H. Sep. Sci. Technd. 1987, 2 4 , 1933. (27) Golshan-Shirazl, S.; Guiochon, G., unpublished results.
2049. Guiochon, G.; Goishan-Shirazi, S.; Jaulmes, A. Anal. Chem., in press. Golshan-Shirazi, S.; Guiochon, G. Anal. Chem., in press. Ebb, J. E.: Grob, R . L.; Antle, P. E.; Snyder, L. R. J. Chromatogr. 1987, 384, 25. Snyder, L. R.; Cox, G. B.; Antle, P. E. Chromatographla 1987, 2 4 , 62. Rhee, H. K. Ph.D. Thesis, University of Minnesota, Minneapolis, MN.
1966.
RECEIVED for review May 12, 1988. Accepted July 15,1988. This work has been supported in part by Grant CHE-8715211 from the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory.
Direct Detection of Immunospecies by Capacitance Measurements Pierre Bataillard, Franqoise Gardies, Nicole Jaffrezic-Renault, and Claude Martelet*
Laboratoire de Physicochimie des Interfaces U A CNRS 404, Ecole Centrale de Lyon, B P 163, F 69131 Ecully Cedex, France Bruno Colin and Bernard Mandrand
Biomerieux, Marcy l'Etoile, F 69260 Charbonnicres les Bains, France
The basic feasibility of direct immunochemical detection is shown. The capacitance changes of the heterostructures (semiconductor/thln sliica layers wlth covalently bonded antlbodies/buffer) are measwed. A specific signal for the antigen-antibody Interaction, which is antigen concentration dependent, Is obtained for a-fetoprotein and IgE antigens. The kinetlcs of the interaction can be directly followed up, and after an acidic washing the system can be used again. The heterodructures keep their activity even after 18 months in buffer. The greatest sensitivity, 1 ngmL-', is obtained with a monoclonal antibody. I t should be noted that the use of antibody fragments and more accurate coupling conditions would improve this knmunosensor, which could be completely integrated in a complementary metal oxide semiconductor technology circuit. TMs system can also be used to determine size and to evidence conformational effects.
Biotechnology, in vitro diagnosis, and medical monitoring need devices that can continuously and specifically detect the main biological molecules involved. For immunological tests, the often-used IRMA (immunoradiometric assay) and ELISA (enzyme linked immunosorbent assay) are tedious and ex0003-2700/88/0360-2374$01 SO/O
pensive, so now considerable attention is given to microchemical sensors (l),biosensors (2-4),and particularly immunosensors (51, which allow the biomolecule concentration to be measured directly. The principle of such directly acting sensors is based upon modification of the local geometry, the dielectric constant, or the surface potential of an electrode due to the specific antigen-antibody interaction, this modification being detected by an optical (6,7) or an electric measurement. Electrically based systems have been studied most and can be divided into potentiometric, piezoelectric, and capacitive systems. The local variation in the surface potential of an electrode or in the drain current of a field effect transistor (immuno FET) has been measured in ref 8-12. Nevertheless, a nonspecific response was observed, due to the ionic buffer. Recently, by optimizing a differential method and using a LangmuirBlodgett technique to prepare an antigen-bound electrode, Katsube et al. (13)have developed a promising sensitive potentiometric type immuno (IgG) sensor. Piezoelectric systems are based upon a variation in the propagation speed of acoustic waves at the surface (SAW) or in the bulk (BW) of a quartz crystal, due to mass changes in the biomolecules bound to the coated layer (14). On IgG systems using a SAW technique, results have been obtained with a detection limit as low as 1ng, but such a method suffers 0 1988 American Chemical Soclety
