Anal. Chem. 1988, 6 0 , 2147-2152
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Diffusional Microtitration: Stationary or Nonstationary Reagent Delivery Miklds Gratzl' Institute for General and Analytical Chemistry, Technical University of Budapest, Gelldrt te'r 4, Budapest, H-1111, Hungary
Ultramlcrosamples in the mlcrollter and submicrollter range can be easlly tltrated by employing only dlffuslon and caplllary forces. Such dlffuslonal mlcrotltrations can be performed elther wlth a stationary reagent transport or by using inltial condltlons in which the reagent reservolr and the membrane, used for dlffuslve reagent dellvery, are In equllibrlum wlth each other when the tltratlon Is started. This theoretically nonlinear technique is descMed and compared with the ilnear technique of stationary dlffuslon, reported earlier. Experlmental examples of potentlometrically lndlcated preclpltate tltratlons prove the standard error of determlnatlons to originate mostly In uncertainties In the sample slze at both technlques. Ease of operation renders the employment of Inltlal equllibrlum more attractlve when a sufflciently thln dlffuslon membrane Is used, requlrlng only short pauses for concentratlon equlllbratlon between subsequent tltratlons.
Titration is still one of the most reliable and precise methods of quantitative analysis. The smallest solution volumes that can be titrated with miniaturized versions of the ordinary equipment are generally not less than about 100 p L ( I ) . With more sophisticated mechanical (2) or coulometric (3) reagent delivery systems it is possible to reduce this lower volume limit to a certain extent, but simultaneously other difficulties arise, as e.g., excessive complexity of the devices (2,3),unstable reagent delivery rate (21, or the handicap that it is impossible to accurately know the volume of solution that is finally titrated (3). The demand for titration of even smaller samples, however, keeps increasing, e.g., in biological research, in production control of microtechnologies, or in analyzing precious materials. Recently, a solution to the problem of reducing titration sample size down to the ultramicro range has been shown to be possible by employing diffusion processes and capillary forces in a proper way (4). With the new technique, called diffusional microtitration, samples of microliter and submicroliter volumes have been titrated with an accuracy being determined mostly by that of sample sizes. Thus, overall errors were determined finally by the uncertainties of syringes used for sample application. Quality of titration curves is generally good for this technique, as demonstrated in Figure 3 of ref 4 for titration of a 0.1-pL silver nitrate droplet. As the reagent in diffusional microtitration is delivered from a reservoir into the sample droplet via a membrane by diffusion, it is essential to provide strictly reproducible initial and boundary conditions for the titration of each individual sample. This requirement can be fulfilled basically in two different ways: either a stationary transport can be used for reagent delivery with linear concentration profile in the diffusion membrane or the reestablishment of equilibrium in the system reservoir/membrane must be ensured before the ti-
'Present address: University of Utah, Department of Materials Science & Engineering, Center for Sensor Technology, Salt Lake City, UT 84112. 0003-2700/88/0380-2147$01.50/0
tration of each sample. The former stationary technique, resulting in a linear calibration graph, has already been discussed and successfully used for determination of samples with acid/base, complexometric and precipitate titrations using both visual and instrumental (potentiometric) end point indication (4). In the present work the technique of equilibrium initial conditions, being characterized in principle by nonlinear relationship between the end point time and sample amount, will be outlined and compared with that of stationary diffusion.
EXPERIMENTAL SECTION Essentially the same device that has been reported earlier (see Figure 1 in ref 4)vas used for diffusional microtitrations in this work, except that during longer series of titrations, the reagent in its reservoir was slowly stirred by a small magnetic stirrer. In some cases,instead of a loop of silver wire (4),a ring of silver plate surrounding the diffusion membrane was employed as an indicator electrode in argentometric titrations. Its larger surface area provided more reproducible end point indication, in spite of inhomogeneities in the sample droplets. Figure 1 in this paper reproduces only the most essential part of the device for diffusional microtitration. The dimensions of the agar gel diffusion membrane (Figure l),including its thickness, were varied according to the demands of the determinations. Appropriate changes could be done by varying the thickness of the Teflon plate used as a membrane holder. Preparation of this kind of membrane was described earlier (4). Membranes with sufficient mechanical integrity can be prepared in a broad range of dimensions (in both thickness and diameter): the thickest membrane used in this work was 1.8 mm thick, while the thinnest one had a thickness of 0.15 mm (with a diameter of 0.25 mm). Sample application was done by using microsyringes (Hamilton) of appropriate volumes. Preparation of solutions and of visual color indicators and details of potentiometric end point indication have been described earlier (4). Reagents were of analyticalgrade (Mallinkrodt, Merck, and Reanal). Proper washing of the indicator electrode and membrane holder after each titration is important in the technique of initial equilibrium, as any remnant of the overtitrated sample can shorten the detected end point time of the next titration. For washing, a droplet of a highly diluted solution of the titrated component was applied and sucked up by paper tissue before its titration end point was reached. THEORETICAL CONSIDERATIONS Stationary Diffusion or Equilibrium Initial Conditions. The reagent amount delivered within several minutes by only diffusion through an agar membrane of a thickness of some tenths of a millimeter into a sample droplet of microliter or submicroliter volume is sufficient to titrate components in such ultramicrosamples in ordinary concentrations. The simple arrangement shown in Figure 1 can be used for microtitrations on the basis of this fact, if some further conditions are also provided for ensuring reproducibility of the diffusional reagent transport process (4) during each titration. The diameter of the diffusion membrane, e.g., must be smaller than that of the smallest sample droplet applied, having the approximate form of a hemisphere, thus ensuring a constant cross-sectional area of diffusion fluxes for each 0 1988 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 19, OCTOBER 1, 1988
Figure 1. Schematic diagram of the arrangement used for diffusional microthation: (1) sample droplet; (2) reagent in reagent reservoir; (3) diffusion membrane made of a 10% agar gel; (4)holder of the d M i n membrane, made of Teflon plate; d,, diameter of the diffusion mem-
brane (and of the hole in the holder); I,, thickness of the membrane (and of the holder). determination. The concentration of the reagent in the reservoir is required to be much higher than that of the component to be titrated. Thus, the titration reaction zone, being characterized by the (generally very small) equilibrium concentration of both the reagent and the sample, is located approximately at the membrane/sample interface (see Figure 2B in ref 4). Homogeneity of the reagent in reservoir must be ensured either by a diffusion being more restrained in the membrane than in reservoir ( 4 ) or by mechanical stirring as was done in this work. Some further necessary conditions are detailed in ref 4. The most important of them, reproducibility
of the shape of reagent concentration profile within the diffusion membrane, and related problems, will be discussed in this work. If the titration of the sample droplet is started after a long pause in operation, the reagent concentration in the reservoir and that in the membrane are being practically equilibrated with each other when the sample droplet first touches the membrane. Then begins a reagent delivery by diffusion, which deforms the original equilibrium concentration profile. When a large sample is analyzed, after a considerable time a nearly linear concentration profile may develop within the membrane according to Fick's first law, with the original reagent concentration at the end contacting the reservoir and with zero concentration at the other end touching the sample. Between the original equilibrium state and this final linear (stationary) diffusion profile, all possible transient states (see Figure 2B in ref 4) may occur after the perturbation of the original concentration distribution, caused by a previous titration. In order to start each titration at identical concentration profiles, either it is necessary to wait sufficiently long after every titration for reestablishing the original equilibrium state or the samples have to be changed quickly, just after the chemical equivalence point of the actually titrated sample has been detected. The first technique utilizes equilibrium initial conditions while the second employs stationary diffusion. Linear calibration graph and simple evaluation are the advantages of the stationary technique, while the requirement of fast sample changes, done exactly at the time instants of chemical equivalence, and the technical difficulties associated with this demand are its disadvantages. On the other hand, with the technique of initial equilibrium, sample changes are easy to perform, but other disadvantages may arise. Namely, depending on the thickness of the membrane, each titration must be preceded by a shorter or longer waiting period, which may considerably slow down the frequency of analyses. The simplicity of linear calibration is also obviously lost in this case, at least in principle. Linear or Nonlinear Calibration Curve. If x = 0 at the reservoir/membrane and x = 1, at the membrane/sample interface, respectively, and x is a coordinate perpendicular
x 0 " Simulated reagent concentration profiles at equilibrium Initial condiilons, using eq 1: (s)concentratlon profile corresponding to term 1 In eq 1 (statlomy component);(t) concentrationPOWcorrespmdhg to term 2 in eq 1 at t +O (Initial concentration distribution of the transient component); (1-8) concentration profiles corresponding to the transient component at times, t = 0.03, 0.15, 0.3,0.6,1.2, 2.5, 5,and 10 s, respectively. The parameters used for simulation (I, = 0.4 mm and D, = 1.96 X lom5cm2/s)are identical with those of the tkration of HNO, droplets with a 0.1 M KOH reagent, discussed in ref 4 and used as example throughout this article. The superposition of the stationary component and any of the transient components provides the same concentration distribution as the corresponding profile in Figure 28 in ref 4 that have been calculated by a numerical solution of the identical diffusion problem. The reagent reservolr/membrane interface is at x = 0 while that of the membrandsample is at x = I,.
Flgure 2.
-
to the plane of the diffusion membrane, then the development of reagent concentration profile within the membrane, c,, as a function of time, t , can be obtained as follows:
where c, and D,are concentration of the reagent in the reservoir and its diffusion coefficient in the membrane, respectively, and t = 0 a t the instant of sample application. The equation has been derived with the assumption of equilibrium initial conditions. Several equivalent numerical solutions to the corresponding differential equation are shown in Figure 2B of ref 4,where curve e represents the original concentration distribution of reagent at equilibrium initial conditions. As emphasized by Figure 2 in the present article, this concentration profile can be obtained also by superimposing two linear profiles of opposite slopes: curve s is the stationary part of the concentration distribution at any time, while curve t can be considered as responsible for its transient component. Accordingly, the first term in eq 1corresponds to curve s, while the second one describes the gradual vanishing of the initial perturbation represented by curve t, as a function of time. The concentration distribution of respective time in Figure 2B of ref 4 can be obtained by superimposing the stationary concentration profile and any of the transient components in Figure 2, according to the equivalence of eq 1 and the numerical solution used earlier. The amount of reagent, R, delivered through the membrane into the sample by diffusion as a function of time can be derived as follows:
where A , is the cross-sectional area of the diffusion membrane. By insertion of the first partial derivative of the function expressed by eq 1with respect to x, the characteristic equation of
diffusional microtitration at equilibrium initial conditions
ANALYTICAL CHEMISTRY, VOL. 60, NO. 19, OCTOBER 1, 1988
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can be obtained after integration
Because
at large times
-
where R, means reagent amounts delivered during long titrations (t =). Thus, when relatively large samples are analyzed, the delivered amount of reagent as a function of time becomes linear even in the case of equilibrium initial conditions. The characteristic equations of the two techniques differ only in the additive second term of eq 3 (or of eq 4 at large samples): this term does not appear at the stationary diffusion (see eq 1 in ref 4 for the basic relationship of the stationary technique), This additional term corresponds to the reagent amount by which the diffusion delivers more into the sample than it would at a steady-state linear concentration distribution. Its physical meaning can be better understood by realizing that the second term in eq 4 represents exactly two-thirds of the amount by which the diffusion membrane contains more reagent at the initial equilibrium state than at the final linear concentration distribution. By considering only the resultant concentration distributions (Figure 2B in ref 4), it is less plausible that only two-thirds of the reagent excess, being initially present in the membrane, can reach the sample. Taking into account, however, the transient component as introduced here (see profiles t and 1-8 in Figure 2), this phenomenon can be easily interpreted: its diffusion is facilitated in both directions by appropriate gradients. Due to steeper profiles (and thus, shorter distances) at the side close to the sample, only one-third of this reagent excess diffuses “back” into the reagent reservoir while two-thirds part can reach the sample. (Obviously, in fact, there is no diffusion leading “back” into the reservoir. Superposition of two concentration components as introduced here is only a mathematical tool that facilitates the derivation and understanding of eq 3 and 4. In the original representation, however, the “lack” of one-third of the initial reagent excess can also be a t least suspected by considering the very flat concentration profiles close to the reservoir, which may restrain delivery speed to a certain extent.) The inverse of function R(t),which is the theoretical calibration graph of diffusional microtitration a t equilibrium initial conditions, is simulated with eq 3 and shown for different physical parameters in Figure 3. Curve RE in Figure 3A corresponds to parameters being identical with those used for simulation of Figure 2B in ref 4; each individual point on curve RE belongs to a particular concentration profile, cm(x,t = t&,,), reported in ref 4. For comparison, curve Rs in Figure 3A represents the theoretical calibration graph a t the stationary technique, under otherwise identical circumstances. Every point on curve RS,calculated with the first term in eq 4 (or with eq 1 in ref 4), is related to the final stationary concentration profile of the KOH reagent used for titrating HNOB. This concentration distribution is marked by s in Figure 2B of the article cited. Sensitivity of the stationary technique with respect to reagent amounts (i.e. the slope of the calibration graph) is
R Inmol)
9
0
1
2
3
4
5
0
0
80 R (nmol)
160 ..
-B Flgure 3. Simulated characteristic curves of diffusional microtitration at different parameters. (A) Curves corresponding to circumstances given in Figure 2 8 in ref 4: c, = 0.1 M, D , = 1.96 X cm2/s,I , = 0.4 mm, and A , = 0.79 mm2 (this corresponds to a membrane diameter of 1 mm. used in the acidlbase titration discussed in detail in ref 4); RS, calibration graph, t ( R ) , at the stationary technique, as simulated with eq 1 in ref 4, or wlth the first term in eq 3; RE, calibration graph, t ( R ) , at the technique of initial equilibrium, as simulated with eq 3; R,, straight line corresponding to long-term calibration characteristics at equilibrium initial conditions, as simulated with eq 4; ,:f reagent flux (see upper horizontal axis) at the stationary technique as a function of time, as simulated with eq 5; ,:f reagent flux at the technique of initial equilibrium as a function of time, simulated by using the expression for d R ( t ) l d t according to eq 2, from which eq 3 was derived by integration (fluxes are shown with thicker lines). Other symbols are explained in the text. Curves corresponding to the technique of initial equilibrium have been simulated at time instants being 0.6 s apart from each other. (B) Theoretical calibration curves, t ( R ) , corresponding to different membrane dimensions, as simulated with eq 3. The same parameters were used as in (A), except that for I , and A , varying values were employed, although with preserving a constant ratio of /,/A , (thus ensuring constant stationary sensitivltles or fluxes). Resolution of the time axis was 6 s. (0) limiting case at I,0. (1-5) I, = 0.2, 0.4, 0.8, 1.2, and 2 mm, respectively. A part of curve 2 is shown enlarged in (A), see the corresponding frame at the left lower corner in (B). Other symbols are explained in the text. All calculations involving an infinite series have been done by using 25 terms of the respective infinite sum.
while an identical expression can be obtained for the technique of initial equilibrium when analyzing relatively large samples: prE(t m) = @?, as shown by both eq 3 and 4. Equation 5 provides, e.g., a stationary sensitivity of 2.60 s/nmol for the parameters of the acid/base titration discussed as example in ref 4, which corresponds to a flux, f:, of 0.385 nmol/s (see curve f> in Figure 3A). As for this particular
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Table I. Relationship between the Dimensionless Time, t / t o ,and the Maximum Deviation from Straight Line Calibration at the Technique of Initial Equilibrium"
AS/S,,b % %
t / t o = 0.2
t / t o = 0.3
t / t o = 0.5
27.2 37.4
17.7 21.5
7.9 8.6
t/to
=1
1.2 1.2
t/to = 2
0.05 0.05
t/to
=3
0.02 0.02
t/to
=5
0.01 0.01
"Up to 800 terms in the respective infinite sum have been used for calculations. *Thevalues refer also to AR/R, and AR/R,respectively. titration a 0.1 M KOH solution was used, a reagent delivery rate of 231 nL/min (13.9 pL/h) would be required to perform an equivalent titration with a traditional mechanical microtitrator! So, clearly no equivalent titration is feasible with any traditional equipment. The calibration curve of the stationary technique is a straight line including the origin of the coordinate system, while the calibration curve of the other technique converges at relatively large samples to a straight line being parallel to the former one. It is shifted in the direction of smaller times, i.e. in negative direction, by
t o = lm2/3D,
(6)
and intersects the horizontal axis a t
R" = c,A,1,/3
s, e.g., AR already theoretically equals 0.24 nmol (or 1.2% only, as R,(to) = 2R0 = 21 nmol). For practice, the lower limit of linear section of the calibration graph can be considered to be 2R0, or t o if it is expressed in end point time. Above these limits systematic nonlinear deviations being certainly smaller than 1.2 % can be ensured. Finally, by substituting the definitions of Ro and t o into eq 3, its form analogous with eq 8 can be obtained
(7)
the physical meaning of which has been explained above. For the acid/base titration discussed so far, t o = 27.2 s, and R" = 10.5 nmol of KOH (see Figure 3A). By use of these quantities, eq 4 may be written in a simpler form as follows:
and the expression for stationary reagent flux, eq 5, becomes
When linear calibration characteristics are desired even with the technique of the initial equilibrium, the minimum reagent amount which already belongs to the linear range of the characteristics can be easily calculated. If the maximum tolerable deviation from linearity, AR, is known, then the difference of eq 3 and 4 may not be larger than this prescribed threshold value (see Figure 3A). The solution of the unequality
by a simple iteration for minimum necessary time, t-, ensures linearity with a tolerance equal to or less than AR. By insertion of this value into eq 3, an accurate estimate can be provided for the minimum reagent amount, Rmin, to be delivered for attaining stationary diffusion conditions (within the tolerance, AR, given). Under the same circumstances as used as example in ref 4 or in Figure 2 in this paper, e.g., for a tolerance of AR = 0.5 nmol, t- = 21.6 s, and R- = 18.26 nmol of KOH (see Figure 3A) can be calculated (in the iteration 200 terms of the respective infinite sum have been taken into account). Accordingly, in the sample range of about 20 nmol of H N 0 3 or more, the use of a linear calibration graph instead of the correct nonlinear one would cause a systematic error smaller than 0.5 nmol (
