Anal. Chem. 1994,66, 1976-1982
Diffusional Microtitration: Reagent Delivery by a Diffusional Microburet into Microscopic Samples Chen Yi and Mlkl6s Gratzl’ Department of Biomedical Engineering, Case Western Reserve Unlversity, Cleveland, Ohio 44 106 Diffusional microtitration can be used to precisely titrate extremely small samples from tens of microliters to a few hundred femtoliters. A cylindrical diffusion membrane of macroscopic dimensions is used for reagent delivery into macroscopic samples (>100nL). Microscopic samples ( Oo, it is practical to assume that the dynamic solution may contain again a steady-state term and a transient one. The transient term may be conceived as a perturbation to the steady-state solution that vanishes as time passes. The steady-state term is given by eq 4. The transient term for a conical membrane can then be obtained by modifying the transient term of the cylindrical diffusion problem in such a way that the following requirements are met: (A) the modified term must degenerate to the transient term of the cylindrical problem when the taper angle becomes zero, and (B) the sum of the steady-state term (eq 4) and the new transient term must be equal to cr for t = 0, as required by the (homogeneous) initial condition. The equation fulfilling these requirements is a solution to the dynamic problem of diffusion through a conical membrane:
RESULTS AND DISCUSSION Steady-StateConcentration Profde. Curve s in Figure 1C (solid line) displays the final concentration distribution for a typical acid/base titration such as those discussed in ref 3. Membrane dimensions close to those specified in Figures 2 and 3 of the cited reference were used in the equations here, with a diffusion coefficient valid for the employed KOH reagent.'-3 The inner diameter of a fine pulled capillary is significantly less than its 0.d. In this example, i.d. was estimated to be about 0.4 pm, while the (outside) taper angle of the DMB tip was about loo. The internal taper angle, 24, must have been obviously smaller: an estimated value of 24 = 6 O was used for computing the curves in Figure 1. While the steady-state profile is linear for a cylindrical membrane (see curve s in Figure lB, ref l), the conical shape of the membrane in a DMB apparently causes the steadystate concentration profile to become curved. Dynamic Concentration Profiles. Concentration profiles computed with eqs 4-6 for different times are shown in Figure 1C. Figure 1D displays the corresponding transient terms C m ( X , f ) = C m S ( X ) + c,'(x) (5) (eq 6) separately. The gradual decrease of this transient term corresponds to the quickvanishing of the initial "perturbation" of the steady-state profile which is represented by the transient where the first, steady-state, term is given by eq 4 and the term. The sum of the steady-state profile (curve s) and any second, transient, term is transient term in Figure 1D results in the corresponding c,'(x) = - (2/r)cFJm/ [rolm transient solution (dotted lines in Figure 1C). Analogous curves for a cylindrical membrane are shown (ro- r,)x] C[(-l)"/n] s i n [ ( n r / l , ) ~ ] e - ~( ~6 )~ ~ " ~in~Figure 2B in ref 1 and in Figure 2 in ref 2, respectively. n=l Both the steady-state and transient profiles show a markedly curved character in a conical membrane, as compared to the Reagent Delivery by a Diffusional Microburet. The amount corresponding profiles in the cylindrical problem. Dynamic and Steady-State Characteristics of Reagent of reagent, R, delivered by the microburet into the sample Delivery by a Diffusional Microburet. Concentration distrifrom the beginning of titration up to time, t , can be obtained as follows: (1) integrating the flux density (eq 2) at the very bution of the reagent within the diffusion membrane in Figure tip of the pipet (x = l m 4 ) with respect to time and (2) 1C appears to be very close to the steady-state profile only 0.2 multiplying the resulting expressionby the cross-sectionalarea s after the beginning of titration. The corresponding reagent of the DMB tip, A(lm): delivery rate at this instant is only 5.8% higher than at steady m
1978 Ana!YtkalCh"Stty,
Vol. 66, No. 13, Ju& 1, lQQ4
80
I
0
0
25
R0
7.5
5
R (fmol) Figure 2. DMB calibrationcurves as predictedby theory.(A) To obtain the curves, eqs 7 were used wlth parameters ldentlcei to those used in Figwe 1, except for the membranethickness which was varted from 10 to 200 pm (marked for each curve in the upper dght comer).The inflnlte sum In eq 7c was approximated with 25 terms. The (inverse) R(t) curve computed with I,, = 80 pm simulates the experimental callbration curve in Figwe 3, ref 3. (B) MagniRed view of this latter curve during the first 0.5 s of titration, with the graphlcai interpretatbn of RO and f. / I
4-1
~,
016, 0
I
I
I
,
500
,
I
I
I
I
I
I
lo00
,
I
1500
sample amount, S (fmol) Figure 3. Calibrationof a DMB wtth mlcroscoplc samples of different concentrations.x, 0.03 M, 0,0.015 M nitric acld samples (for other data and detalied statlstlcs see Table 1, rows 2A and 28).
state. This means that the duration of the initial transient before the establishment of steady-state delivery is extremely brief. The parameters used in calculating the profiles in Figure 1 are similar to those of the acid/base titrations reported in ref 3. This explains why no effect of any initial transient was found in the experimental results discussed there. From this it also follows that the delivery rate must be virtually constant during the entire titration. This explains the other earlier experimental findings, i.e., that DMB calibration is characterized by good linearity (r = 0.99 in Figure 3 in ref 3), and an absence of any significant offset term. Linearity and Offset dTheoretical DMB Calibration. (A) Diffusion C o n f d in the Membrane. Equations 8 predict a slope (i.e., steady delivery rate) that is proportional to the
square of the geometric mean of the two radii involved, (rVm)’/’. Thus, diffusional “through-put” is determined by the average cross-sectional area of the membrane. This is a generalization of the simple transport equation of a cylindrical (or rectangular) membranelJ for conical geometry. The formulas also predict a non-zerobias for the (linearized) DMB calibration curve. Intercept with the R axis should occur at Ro which represents an excess amount of reagent that ultimately ends up in the sample,in addition to what is delivered by the steady-state concentration profile. Ro is part of the reagent excess that is present within the membrane at the initial uniform concentration distribution, with respect to the steady-state concentration profile. Just as the transient term in eq 6 may be conceived as a perturbation of the steady-state profile, Ro represents a (positive) “perturbation” in the delivered reagent amount. This corresponds to a negative intercept with the time axis at -to in the inverse calibration curve (i.e., time versus reagent, or tq versus titrated sample, amount). In the case of the example of the acid/base titrations discussed in ref 3 and here, R(t) has been computed with eqs 7 and displayed in Figure 2. The expected “bias”, Ro = 0.67 fmol (with 1, = 80 pm), is clearly negligible with respect to the 1000 fmol sample range. The predicted bias on the time axis, -to = -49 ms, is similarly negligible with respect to the minute time scale of practical titrations. Thus, both the reagent excess (RO)and theduration oftheassociated transient (in the order of a few times t o ) , are insignificant. The linearized eqs 8 therefore represent a very good approximation of DMB calibration for real titrations. Exactly two-thirds of the “excess” reagent stored initially in the membrane would ultimately end up in the sample in the case of a cylindrical membrane, where Bi = Oo. This equals to one-third of the total reagent contained in the cylindrical membrane at the beginning of the titration.2 Ro is much smaller than one-third of the same quantity in a conical membrane (solely due to peculiarities of mass transport within conical constraints). This explains why DMB calibration is linear with a virtually zero offset. Operation of a DMB can therefore be fully characterized by the steady-state term alone in eq 7b. This can be further simplified by realizing that ro = 1, tan(0i) + rm. For diffusion membranes thicker than a few tens of a micrometer, ro >> rm. (In the example discussed,ro = 4.4 pm >> rm = 0.2pm.) Thus, to a very good accuracy R ( t ) a Dmcfmutan(@$
(9)
Figure 2A displays theoretical calibration curves for different membrane thicknesses. If 1, is larger than a few tens of a micrometer, membrane thickness has clearly no significant effect on reagent delivery, as correctly predicted by eq 9. The slope of the calibration curve (i.e., the inverse of steady-state delivery rate) changes only by 2.7% when 1, changes from 80 to 200 pm. The reason for this is apparent in Figure 1C: in the case of the steady-state concentration profile for a 200-pm-thick membrane (dashed line), about 90% of the total concentration drop occurs within 15% of membrane thickness near the pipet tip while the profile becomes virtually flat at distances beyond 50 pm from the tip.
Table 1. CalikaHon Charact.rtrHcr
of DHiwonl DMB’m F l W wHh 0.2 M KOH 4- 0.1 M KNOt
sample range
calibration graph
concn (M) amt (fmol)
expt
vol (pL)
l(3) 2A
0.7-16.3 3.2-11.8
0.06 0.03
42-980 96353
2B
1.9-94.9 14-103
0.015
29-1420 841-6210
t , range (s)
(m$gol)
IS71 21-67
70.5 169
3
0.06
4.5-243 58442
10.9
biasa a 9 5 k confidence intervals 5.5
* 2.6
full-scaleb error re ression coeficient (9) raw correctedd (%)
4.1 f 8.3
0.99
3 s
1.7
.oo 1.oo
2.4
1.9
3.1
2.8
1
3.4 f 22.9
* The reported bias values are raw data that do not include the additional random delay caused by the 4 s/frame data acquisition rate. This delay is due strictly to the low fr uency of data acquisition and is, on the average, 2 s. The true offset values are then 2 s larger than the raw values reported in Table 1. b Full scale is d%ed here as the range of end point times encountered and mapped onto the fitted re ression line. This is why the random error given here, 3.51, is less than what was reported in ref 3 (4%) where the actual1 obtained extremes in entpoint time were used to calculate full scale. d The 4 s frame sampling frequency is corrected for by subtracting the standard dleviation of a 4-9 wide uniform statistical distribution, Le., (2 3)’12 = 1.2 s, from t e raw random Errors similar to these corrected ones should have been obtained if a data acquisition of higher frequency & o u t 1-2 Hz or higher) had been used to record the titration curves. Therefore, the corrected values better represent the true random error of the technique.
x
The largest deviation from a zero offset is obtained for 1, = 200 pm when Ro = 1.7 fmol and to = 0.1 3 s. Even this bias is clearly negligible in real titrations. This (already negligible) offset further decreases with decreasing reagent concentration and, especially, tip i.d. While eq 7b degenerates to the correct solution for a cylindrical membrane when t9i = Oo, this is not true for eq 9 which becomes meaningless with taper angles close to zero. However, a half taper angleof 3O is still apparently not ‘close” to zero, because the delivery rate predicted by eq 9 (12.9 fmol/s) and eq 7b (13.5 fmol/s) are indeed close to each other. This can be rationalized by comparing the steadystate concentration profiles within a cylindrical versus a conical membrane. While this is linear for Bi = Oo, it becomes extremely curved for a half taper angle of 3O (see profiles in Figure 1B for 1, = 80 and 200 pm). This means that an angle as ‘close” to zero as 3O induces already profound changes in the pattern of diffusion. In other words, a conicity of only 3O is extremely significant for membranes whose thickness is much larger than the radius. The condition for a membrane to be “conical” is then
If this holds then eq 9 is valid and (steady-state) delivery rate of the DMB will not depend on the actual thickness of the membrane. Linearity and Offset of Theoretical DMB Calibration. (B) Diffusion Not Confined in the Membrane. In the theoretical model derived above, diffusion was assumed to be confined within the 50-200 pm thick membrane of the DMB. This situation can be realized if a membrane material with low diffusivity is used, Le., when Dm 0 everywhere within the membrane, as eqs 4-6 indeed predict (Figure 1C). The important consequence of this result is that no sample can get lost untitrated via the microburet. If Dm = Dr then the boundary between the membrane and reagent reservoir does become, however, ‘blurred”: thedomain of diffusion may well extend into the reservoir, effective ‘membrane thickness”becoming much larger than the physical thickness of the gel plug. In a steady state, the concentration gradient at this boundary is expressed for “conical”membranes (as defined by eq 10) in the following way: dc,(x
=: O)/dx
= -~~,/l,~tan(13,)
With increasing membrane thickness, the concentration gradient will decrease rapidly. This is another way of rationalizing why steady-state delivery by a DMB remains independent of l,, even when low density agar gel is used as membrane material. Thus, “blurred” boundary between reservoir and membrane cannot cause any anomaly in the steady-state operation of the microburet. The problem is, however, also related to the offset of DMB calibration. Experimentally found offset values are reported in Table 1: contrary to the predicted negative bias (-to), a positive one has been observed in each case. The offset was so small in each experiment that its very existence is questionable (as discussed later). Any negative offset predicted by theory must, however, be even “more” negligible than this experimentally found, always very small, positive offset. On the other hand, extension of the domain of diffusion far beyond the membrane into the rest of the pipet shank, the shoulder, and shaft should seriously affect the offset term of DMB calibration: both Ro and t o increase proportionally with 1, (since in eq 8c, ro also changes with lm). This would translate to long initial transients. This adverse effect, however, has never been observed
experimentally. Therefore, there must be a mechanism that effectively confines the domain of (conical) diffusion within the 100-200-pm shank of the DMB. Due to the special shape of a pulled glass micropipet (the taper angle of the shoulder region is much larger than that of the shank), diffusion efficiency in the shoulder abruptly increases with respect to theshank that contains the membrane. This situation is analogous to the boundary between a cylindrical membraneand the reagent reservoir: semispherical diffusion within the reservoir is far more effective than planar diffusion through the membrane.lS2 In the case of a DMB, eq 9 can also be used for the shoulder, with a taper angle larger than in the shank. This corresponds to confining almost all the concentration drop that occurs across the shoulder into its boundary region adjacent to the shank. Even though this gradient is extremely small in absolute terms, it can maintain a more efficient mass transport in the shoulder side of the boundary than diffusion can in the shank side. (Semispherical diffusion corresponds, in fact, to a "taper angle" of 24 = 180O.) On the other hand, in the narrow shank distances of the pipet walls are in the order of only a few micrometers. This is well within the typical extension of a stagnant diffusion layer at an electrode surface. Therefore, reagent delivery by a diffusional microburet must be limited by the (purely diffusive) domain of the entire, 100-200-pm, shank. Therefore, it is not necessary that a membrane of low diffusivity be used in order to confine rate-limiting diffusion into a finite (and small) domain. Experimental DMB Calibration: Validation of the Model. By inserting the respective actual parameter values of the example discussed throughout this report into eq 9 and taking the inverse of the delivery rate, 77.5 ms/fmol is obtained. This is close to the experimentally observed 70.5 ms/fmol slope of the calibration graph reported in Figure 3, ref 3. With eq 7b, which is the exact expression, 74.1 ms/fmol is calculated which is even closer to the experimentally determined value. An identical result can be obtained, of course, with "slope of calibration" = rO/Ro(eqs 8). Such a close agreement between theory and experiment is, of course, merely a coincidence because inaccuracies in the actual value of Dm and, especially, rm and Bi, are probably larger than a few percent. It is evident, however, that reasonable parameters yield a reasonableestimate for delivery rate of an actual DMB: a tip i.d. of 0.5 or 0.6 pm instead of the 0.4 pm used here would have still resulted in a good enough approximation (in view of the large uncertainties in rm and ei)
.
This and other evidence supporting the validity of the presented theory, and especially eq 9, are summarizedin Table 1. In another set of titrations with a different microburet (see experiment 2 in Table l), both the volume and concentration of the sampleswere varied. This also resulted in a calibration of good linearity (Figure 3), which proved that the end point times are proportional to the molar amount of sample and not to sample volume. In a third set of experiments (not included in Table l), the reagent in the DMB used in experiment 2 has been replaced with 0.1 M KOH. The end point times in the following titrations approximatelydoubled for similar sample amounts.
Experiment 3 was performed with the same microburet as experiment 1 but 4 days later, and on a much broader sample range. The almost perfect identity of the delivery rates (Le., inverse slopes) obtained in the two sets of titration proves that a DMB, once prepared, is a remarkably stable device. The other conclusion is that no depletion or loss of reagent from this microburet could have occurred either during storage or when, relatively speaking, "very large" samples were titrated (such as those in experiment 3). A positive offset was found in the "end point time versus sample amount" calibration curves in each case (Table 1). This offset was, however, small with respect to the time scale of the titrations and entirely negligible in experiment 3. Apart from experiment 1, this bias is also small or negligible with respect to the width of the 95% confidence interval on the "end point time" axis. All the offset values were, however, remarkably close to each other. Because of the 4 s/frame sampling frequency, an average of 2 s should be added to these value^,^ resulting in an average offset of about +6.5 f 1.1 s. Thus, the consistently positive offset, despite being small, may reflect a real delay of a duration of a few seconds, occurring at the beginning of each titration. This may be due to the time needed to establish a hydrophilic connection between the DMB tip and the sample d r ~ p l e t . ~ Linearity was found to be good in all cases. The performance of the DMB used in experiments 1 and 3 is particularly remarkable, since it showed excellent linearity over a sample range of more than 3 orders of magnitude wide. This also proves that the acid sample droplets are stable during prolonged periods of time. These conclusions were further confirmed in a few titrations performed with a large tip DMB (- 10-pm tip0.d.) in the lower nanoliter sample range. (These results are not included in Table 1.) Full-scale errors, after subtracting the 1.2-s random delay3 introducedby the low data acquisition frequency (4 s/frame), are consistently around 2% which is extraordinary for the microscopic samples titrated. (Experiments not included in Table 1 comprised only a few titrations each, thus being insufficient for statistical analysis.) DesipCriteria for a Diffusional Microburet. An important practical consequence of eq 9 is that delivery rate does not depend on the thickness of the diffusion membrane as long as it is thick enough. On the other hand, due to the special shape of a pulled capillary pipet, rate-limiting diffusion is confined in the entire shank of the micropipet. Therefore, thickness of the agar gel membrane does not matter at all: it can even be extremely thin such as a few pm, or it could extend into, and fill a part of, the shoulder, without affecting operation and performance. Equation 9 thus proves to be valid for any membrane thickness, including very thick, or thin (a few micrometers) membranes. Therefore, there is no need for controlling the thickness of the membrane to obtain the desired DMB characteristics. This renders the fabrication procedure of a microburet simple, since relatively thick membranes are easy to make. The usual, 100-200-pm-long, micropipet shanks automatically ensure an effective "membrane" thickness that provides linear calibration with virtually zero offset and steady-state reagent delivery with an initial transient of about 0.1 s or shorter. While these conclusions are valid for the delivery of simple ions and small molecules, AmMicaiChembW, Voi. 66, No. 13, July 1, 1994
I981
similar charactersitics can be obtained for the delivery of larger particles, too, if a DMB with a shorter shank is fabricated: t o stays approximatelyconstant if 1, is varied proportionally to D,. The only remaining geometrical characteristics that influence delivery rate are (1) the (internal) taper angle of the shank, and (2) tip i.d. Reproducibility of these parameters is ensured by using identical glass capillaries and preset conditions for their pulling. The experimental results reported in this work show that it is indeed possible to adjust the parameters of a DMB in such a way that very different ranges of sample amount can be titrated with linear calibration and within reasonable time (Table 1). The diffusion coefficient of reagent in the membrane can only be varied by changing membrane material and/or composition. Reagent concentration, the size of the tip opening, and the internal taper angle are, however, parameters that are easy tovary to achieve the required delivery rate.
CONCLUSIONS With a device as simple as a diffusional microburet, remarkable performance can be achieved: the DMB used in experiments 1 and 3 (Table 1) delivered KOH from a 0.2 M solution into microscopic droplets at a rate of about 6 fmol/s. ~~
~~
(7) Gratzl, M. et al. Unpublished work, Case Western Reserve University, 1993.
1902 AnaIyticalChemishy, Vol. 66, No. 13, Ju& 1, 1994
This would correspond to a 1 pL/year volumetric flow rate if an equivalentmechanical delivery scheme were considered! On the other hand, another DMB could be used to titrate samples in the lower nanoliter volume range. The results of this work are useful in multiple ways: (1) they aid the design of titrations for chemical analysis of extremely small droplets, (2) the operational principles of a diffusional microburet can also be used for fine chemical manipulation of reactions and syntheses in microscopic volumes, and (3) continuous delivery of ions and molecules (such as drugs) into single biological cells became feasible with this new te~hnology.~ For realizing applications as complex as these, DMB and its operation had to be understood first on simpler examples. Such is titration of microscopic nonbiological droplets, as outlined in this paper.
ACKNOWLEDOMENT This work was partly supportedfrom the funds of the Elmer Lincoln Lindseth Chair of Biomedical Engineering at Case Western Reserve University. Received for review March 24, 1994. Accepted March 29, 1994.' ~
~~~
Abstract published in Advance ACS Abstracts, May IS, 1994.
