Gravimetric deposition of microliter drops with radiometric confirmation

Abstract

A manual microliter gravimetric dispensing technique is demonstrated using a micropipettor modified for use with removeable microcapillaries. Liquid scintillation sources were prepared from a well-characterized ²⁴¹Am reference solution, providing a radiometric check of dispensed masses. Further experiments confirmed controlled dispensing of drops onto gold foils with losses ≤ 0.34(4) % of the total drop activity. A detailed measurement equation for the weighing technique, including the corrections for evaporation, is presented with a full accounting of associated uncertainties.

1. Introduction

Research in radionuclide metrology justifiably concentrates on the direct realization of the becquerel (Bq) for primary standards traceable to the International System of Units (SI). The dissemination of these standards through calibrations or reference materials requires a well-characterized link between the sources counted by primary methods and related materials. Mass measurements link individual sources to the bulk solution from which they are prepared, making massic activity (Bq/g) the critical measurand for calibrations, comparisons, and reference materials. Mass metrology rarely contributes significantly to source activity uncertainty while the quantities measured exceed 1 g. With decreasing mass, however, the challenges associated with weighing aqueous material grow increasingly important.

Growing interest in absolute decay energy spectrometry (ADES) with cryogenic calorimeters has spawned research into methods for the preparation of sources with very small quantities of radioactive solution (Kohler, 2021; Fitzgerald et al., 2021). The aspirating pycnometer method routinely used for gravimetric preparation of radioactive sources achieves relative combined standard uncertainties ≈ 0.05 % when several 10 mg to 20 mg drops of aqueous solution are dispensed (e.g., Sibbens and Altzitzoglou, 2007). For smaller masses (< 10 mg), relative uncertainty contributions from effects like balance linearity and repeatability become increasingly significant, as does solution concentration change due to solvent evaporation.

The “drop on demand” approach to depositing microliter quantities of solution using inkjet dispensing devices and ultramicrobalances (e.g., Verkouteren and Verkouteren, 2009; Verkouteren and Verkouteren, 2011; Liang et al., 2013; Trost, 2017) is promising if appropriate corrections for evaporation can be made and if stable jetting conditions can be achieved with the aqueous solutions in question. For some of the acids commonly encountered in radionuclidic solutions, material compatibility concerns arise.

Here, we focus on a “manual microdrop deposition” technique, wherein microliter drops are dispensed from a glass microcapillary (Shaw, 2022). The dispensed mass is determined by weighing the microcapillary before and after deposition and correcting for solvent evaporation. Essex et al. (2023) showed that manual microdrop deposition can achieve relative combined standard uncertainties of < 0.1 % on 4 mg to 5 mg drops of a solution of Gd in nitric acid. Here, we present the complete measurement model for manual microdrop deposition and use liquid scintillation (LS) counting of ²⁴¹Am as a radiometric confirmation of the mass measurements. For comparison, we also measured sources with larger masses prepared by the aspirating pycnometer method.

2. Methods

2.1. Am-241 solution

Sources were prepared from an ²⁴¹Am solution originally acquired from the Czech Metrological Institute (CMI) in 2004 with a relative combined standard uncertainty of 0.3 % on the certified activity. Gamma-ray spectrometry reports indicate no observed photon-emitting impurities above detection limits. Serial dilutions of the original material were carried out gravimetrically with 0.1 mol/L HNO₃ to achieve the desired massic activities for the present experiments.

2.2. Radiometric confirmation of microdrop masses

To achieve the first radiometric confirmation of manually-deposited aqueous droplets, a series of liquid scintillation sources were prepared according to the scheme in Figure 1. The contents of a 5 mL flame-sealed ampoule containing approximately 0.7 g of a solution containing approximately 145 kBq/g ²⁴¹Am were transferred to a polyethylene pycnometer. One drop (≈ 13 mg) of the solution was dispensed gravimetrically to each of two 20 mL scintillation vials containing 10 mL of liquid scintillation cocktail (Ultima Gold, PerkinElmer, Waltham, MA).¹ Pycnometer masses before and after each drop were read from a XSR-105 microbalance (Mettler Toledo, Columbus, OH) and recorded. The remaining solution was then dispensed into a 2 mL vial (9 mm Wide Opening SureStop Vial, ThermoFisher Scientific, Waltham, MA), which would serve as the reservoir for filling a microcapillary.

Figure 1.

Picture of the modified micropipettor used for microdrop deposition and source preparation scheme for radiometric confirmation of microdrop masses. Two LS sources and a reservoir were dispensed by aspirating pycnometer. From the reservoir, five LS sources were prepared by microdrop dispensing and the remainder from the reservoir was transferred to an aspirating pycnometer to prepare three more LS sources.

The microcapillary dispensing method was described recently by Shaw (2022). Drops are dispensed from a micropipettor (Pipet-Lite SL 200, Rainin) modified by drilling a vent hole that allows a full microcapillary to be mounted onto a septum (diaphragm) without expelling its contents (Figure 1). The dispensing procedure, as applied in the present work, is given here. For each drop:

1. A LabVIEW (National Instruments, Austin, TX) routine for communication with a calibrated ultramicrobalance (XPRCU, Mettler Toledo, Columbus, OH) was initiated, commencing acquisition of one mass reading every 0.5 s

2. Using balance tweezers, an empty 10 μL glass microcapillary (Drummond Scientific, Broomall, PA) was placed on the ultramicrobalance for > 1 min

3. The microcapillary was then removed using balance tweezers and mounted onto the septum of a vented dropper

4. The reservoir was opened and the microcapillary was inserted into the liquid; it was allowed to fill partially by capillary action, leaving an air gap of a few millimeters at the top; using the LabVIEW interface, a flag was placed in the balance data to mark the time when the microcapillary was filled ($t_{\mathrm{f}}$)

5. The sides of the microcapillary were carefully wiped with a lint free wipe to remove any liquid from the exterior and the clean microcapillary was transferred back to the ultramicrobalance using balance tweezers

6. Masses were recorded for > 1 min to allow equilibration and a good measure of the evaporation rate of the solution in the microcapillary

7. The microcapillary was removed from the balance and mounted onto the septum of a vented micropipettor

8. The micropipettor was positioned over the target, the vent hole was covered with a gloved finger, and most of the solution in the microcapillary was ejected by rapidly depressing the plunger; using the LabVIEW interface, a flag was placed in the balance data to mark the time when the drop was ejected from the microcapillary ($t_{\mathrm{d}}$)

9. The microcapillary was returned to the ultramicrobalance using balance tweezers

10. Masses were recorded for > 1 min to allow equilibration and a good measure of the evaporation rate from any residual solution in the microcapillary

11. The microcapillary was removed from the balance and discarded as radioactive waste

In this experiment, the targets referred to in Step 8 were five LS vials to which 10 mL of Ultima Gold and 10 μL of 0.1 mol/L HNO₃ had been added for composition matching with the pycnometer-prepared sources. After these sources were prepared, the solution remaining in the reservoir was transferred to a new polyethylene pycnometer, which was used to prepare an additional three LS sources with compositions matching the other pycnometer-prepared LS sources (≈ 13 mg of solution added to 10 mL of Ultima Gold in a 20 mL scintillation vial). These last three sources were intended to provide confirmation that the massic activity of the solution, to be determined by LS counting, did not change due to evaporation from the reservoir during the experiment.

The sources prepared by pycnometer before and after the sources prepared by manual microdrop deposition (Figure 1) are atypical due to the small mass of ²⁴¹Am solution added. Typically, when preparing LS sources using the aspirating pycnometer method, weighing uncertainties are minimized by dispensing many drops to achieve larger masses. In this study, a separate dilution of the ²⁴¹Am solution was prepared, with a gravimetric dilution factor of 76.13(8). From this solution, 6 LS sources were prepared by pycnometer with approximately 0.5 g of the ²⁴¹Am solution each, achieving activities ranging from 0.81 kBq to 1.0 kBq.

All of the LS sources were counted, along with composition-matched blanks, on the NIST triple-to-double coincidence ratio (TDCR) liquid scintillation counter (Zimmerman et al., 2004) and on a Packard Tri-Carb 4910 (PerkinElmer, Waltham, MA, USA) liquid scintillation counter. Since the massic activities of the ²⁴¹Am solutions were known, a comparison of the massic activities determined for sources prepared by manual microdrop deposition and by pycnometer would provide a radiometric check of the measured masses.

2.3. Confirmation of controlled drop dispensing

The experiment described in Section 2.2 was designed to confirm that the manual microdrop deposition technique could achieve mass uncertainties appropriate for radionuclide metrology. For the technique to be useful in the preparation of deposits on small metal foils, it is also important that drops are controllably dispensed onto the target without losses from spray or aerosolization at the tip of the microcapillary. Such losses would not be detected when depositing directly into LS vials (as in Section 2.2) since the tip of the microcapillary was kept below the mouth of the vial so that any stray solution would still be collected. To investigate possible losses from spray or aerosolization, drops were deposited onto foil targets resting on “placemats”.

Figure 2A shows the source scheme employed in this experiment. The contents of a 5 mL flame-sealed ampoule containing approximately 0.51 g of the 145 kBq/g ²⁴¹Am solution were transferred to a polyethylene pycnometer. The contents of the pycnometer were transferred gravimetrically to a 2 mL vial, which would serve as the reservoir for filling a microcapillary. Microdrops were dispensed manually according to the procedure described in Section 2.2. The targets (Step 8) were:

Figure 2.

(A) Source preparation scheme for confirmation of drop integrity. All sources were prepared using the microdrop method. See Section 2.3 for details. (B) Photograph of a nanoporous gold foil being removed from a gel placemat. The gel square rests on a glass microscope slide. Blue tape is used to improve visual contrast during drop deposition.

1. Two LS vials containing 10 mL of Ultima Gold

2. Two 8 mm × 8 mm gel squares (cut from Gel-Pak boxes, Ted Pella, Inc., Redding, CA) centered on glass microscope slides

3. Six 3 mm × 6 mm sections of nanoporous gold foil centered on 8 mm × 8 mm gel squares centered on glass microscope slides

4. Another two LS vials containing 10 mL of Ultima Gold

The LS sources a & d were prepared as in Section 2.2, providing a check that the massic activity of the solution did not change due to evaporation from the reservoir during the experiment. Drops deposited onto gel squares and foils were allowed to dry before all the foils and gels were transferred to LS vials containing 10 mL of Ultima Gold. The drops deposited directly onto gel squares (b) were intended to check for any change in the LS counting efficiency arising from the gel. The gel squares under the foils in (c) were the “placemats”, intended to catch any spray from the microcapillary tip during deposition. The photograph in Figure 2B shows a foil being removed from a placemat with tweezers. In this experiment, after each foil was removed from the placemat, the tweezers were checked for contamination using a Ludlum 43-93 (Ludlum Measurements, Sweetwater, TX) in alpha-only mode. After the gel placemat was removed, the tweezers and glass microscope slide were also checked. Based on the manufacturer’s specifications, the detector should have been sensitive to approximately 0.7 μg of the ²⁴¹Am solution in use, or approximately 0.01 % of a 5 mg drop; in no case was contamination detected on tweezers or glass slides.

All LS sources were counted on the NIST TDCR liquid scintillation counter (Zimmerman et al., 2004) and on a Packard Tri-Carb 4910 (PerkinElmer, Waltham, MA, USA) liquid scintillation counter.

3. Results

3.1. Measurement equation

Figure 3 shows typical balance data from a manual microdrop deposition, as described in Section 2.2. The first time window (${\mathrm{t}}_0$) represented in the data corresponds to weighing the empty microcapillary (Step 2 in Section 2.2). The mass of the empty microcapillary, $m_{\mathrm{e}}$, is defined as the average mass reading during ${\mathrm{t}}_0$ and will be used in later calculations.

Figure 3.

Ultramicrobalance data acquired during a microdrop deposition. The blue open circles represent mass readings from the balance, with the thin blue line connecting the points. Time windows when the microcapillary was on the balance are denoted with gray bars labeled ${\mathrm{t}}_0$, ${\mathrm{t}}_1$, and ${\mathrm{t}}_2$. The time at which the microcapillary was filled ($t_{\mathrm{f}}$) and dispensed ($t_{\mathrm{d}}$) are shown as orange and green dotted lines, respectively. See text for additional details and mass term definitions. Supplemental Figure 1 gives a closer look at regions ${\mathrm{t}}_1$ and ${\mathrm{t}}_2$.

The microcapillary is filled at $t_{\mathrm{f}}$ and its time-dependent mass ($m_{\mathrm{f}}(t)$) is a function of the evaporation rate, $E_1$:

$$m_{\mathrm{f}}(t)=E_1\cdot (t-t_d)+m_{\mathrm{f}}(t_{\mathrm{d}})$$ (1)

A drop is dispensed from the microcapillary at $t_{\mathrm{d}}$, leaving behind some residual solution with time-dependent mass ($m_{\mathrm{r}}(t)$) a function of the second evaporation rate, E2:

$$m_{\mathrm{r}}(t)=E_2\cdot (t-t_d)+m_{\mathrm{r}}(t_{\mathrm{d}})$$ (2)

The mass of the dispensed drop, $m_{\mathrm{d}}$, is:

$$m_{\mathrm{d}}=(m_{\mathrm{f}}(t_{\mathrm{d}})-m_{\mathrm{r}}(t_{\mathrm{d}}))∕b_f$$ (3)

where $b_f=1\text{-}{\rho }_{\mathrm{a}}∕{\rho }_{\mathrm{f}}$ is the buoyancy correction using the density of air (${\rho }_{\mathrm{a}}$) and the fluid (${\rho }_{\mathrm{f}}$). The evaporation constants are estimated from the slope of a linear fit to the data acquired in ${\mathrm{t}}_1$ (for $E_1$) or ${\mathrm{t}}_2$ (for $E_2$). In the present studies, observed values for $E_1$ ranged from 0.1 μg s⁻¹ to 0.5 μg s⁻¹ and values for $E_2$ ranged from 0.03 μg s⁻¹ to 0.9 μg s^-1.

Evaporation during drop deposition changes the activity concentration in the microcapillary and creates a concentration gradient. Appendix 1 considers the concentration gradient in the “instantaneous mixing” and “no mixing” limits discussed by Essex et al. (2023) as “Model 1” and “Model 2”. Appendix 1 also presents a more general approach from which these specific cases can be derived. These models provide a method to calculate the massic activity of the reservoir solution, $A_{\mathrm{m}}^0$.

Assuming instantaneous mixing of the solution in the microcapillary (i.e., evaporation creates no concentration gradient across the microcapillary), $A_{\mathrm{m}}^0$ can be expressed in terms of measured quantities as

$$A_m^0=\frac{A_{\mathrm{d}}{b}_{\mathrm{f}}}{m_{\mathrm{f}}(t_d)-m_{\mathrm{r}}(t_{\mathrm{d}})}\cdot \left(1-\frac{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{f}}(t_{\mathrm{d}})}{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{e}}}\right)$$ (4)

where $A_{\mathrm{d}}$ is the activity of the dispensed drop.² Alternatively, if no mixing of the solution in the microcapillary is assumed (i.e., evaporation occurs only at the drop end of the microcapillary, creating a concentration gradient), then $A_{\mathrm{m}}^0$ can be expressed in terms of measured quantities as

$$A_{\mathrm{m}}^0=\frac{A_{\mathrm{d}}{b}_f}{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{r}}(t_{\mathrm{d}})}$$ (5)

where all terms have been previously defined.

In practice, the models yield similar results because nearly all of the liquid in the microcapillary is ejected when the drop is dispensed. In the case where all of the activity in the microcapillary is dispensed in the drop, the evaporation model should not affect the result. Setting $m_{\mathrm{r}}(t_{\mathrm{d}})=m_{\mathrm{e}}$, Equations 4 and 5 become equivalent.

3.2. Determination of massic activities

The principal goal of this study was to compare sources prepared with the different methods, so LS count rate was used as a proxy for activity and only counting uncertainties were considered. For alpha-emitting ²⁴¹Am, this is a good first approximation. For comparisons with certified activities, or with non-LS methods (e.g., DES), small corrections and additional uncertainties would be considered.

For the microdrop sources, the calculated drop masses ranged from 5.0 mg to 5.8 mg. There was some drop-to-drop variation in the relative contributions of the mass uncertainties and LS counting uncertainties to the combined standard uncertainties on massic activities. A representative uncertainty budget is given in Table 1.

Table 1.

Representative uncertainty budget for a microdrop deposition. These data correspond to the third data point in Figure 4.

Term	value		u
${\mathit{m}}_{\mathbf{f}}({\mathit{t}}_{\mathbf{d}})$	31.8863	mg	0.0012	mg
${\mathit{m}}_{\mathbf{r}}({\mathit{t}}_{\mathbf{d}})$	26.1471	mg	0.0042	mg
${\mathit{m}}_{\mathbf{f}}({\mathit{t}}_{\mathbf{f}})$	31.9088	mg	0.0040	mg
${\mathit{m}}_{\mathbf{e}}$	24.9354	mg	0.010	mg
${\mathit{b}}_{\mathrm{f}}$	0.99890		0.00013
${\mathit{u}}_{\text{bal}}$			0.0006	mg
${\mathit{A}}_{\mathbf{d}}$	831.60	Bq	0.59	Bq
${\mathit{A}}_{\mathbf{m}}^0$ (Eq. 4)	144.27		0.17	Bq/mg
${\mathit{A}}_{\mathbf{m}}^0$ (Eq. 5)	144.17		0.18	Bq/mg

The overall contributions of the mass uncertainty and LS counting uncertainty to the combined standard uncertainty on the massic activity were of similar magnitude. The main contributors to the mass uncertainty were the extrapolated $m_{\mathrm{r}}(t_{\mathrm{d}})$ and $m_{\mathrm{f}}(t_{\mathrm{f}})$ terms. These terms appear in both Equation 4 and Equation 5, so the uncertainties on the massic activities determined with both approaches were similar. The uncertainties on the extrapolated mass values were estimated using the procedure (5-4.1.2.1) described in NBS Handbook 91 (Natrella, 1966). Since the same balance data are used to calculate extrapolated values and uncertainties for $m_{\mathrm{f}}(t_{\mathrm{f}})$ and $m_{\mathrm{f}}(t_{\mathrm{d}})$, the Equation 4 uncertainties are correlated. Correlations were handled using the NIST Uncertainty machine (Lafarge and Possolo, 2015) and were found to have negligible impact, which can be expected due to the relatively low sensitivity to the $m_{\mathrm{f}}(t_{\mathrm{d}})$ uncertainty.

The uncertainty on the buoyancy correction was estimated based on conservative limits for the environmental conditions influencing air density. The balance uncertainty ($u_{\text{bal}}$) was estimated from the calibration certificate issued when the balance was installed (approximately 1 month before these measurements). As mentioned above, the LS count rate was adopted as a proxy for activity and only the LS counting uncertainty (estimated as the standard uncertainty of repeat counts) was considered. LS count rates from the TDCR and TriCarb measurements were consistent. Only the TriCarb results are presented here since longer counting times and more repeat counts resulted in smaller uncertainties.

For pycnometer-prepared LS sources, masses were taken as the buoyancy-corrected difference between masses observed before and after dispensing drops. The sources prepared from the same solution as the sources prepared by manual microdrop deposition received one drop each from the pycnometer, with masses ranging from 12.5 mg to 18.3 mg. The weighing uncertainty for these small masses on the XSR-105 microbalance was estimated from the balance specifications for repeatability and linearity. Consistent with standard practice with aspirating pycnometers, no corrections or uncertainties were estimated for evaporation.

The sources prepared from a diluted ²⁴¹Am solution received 427 mg to 542 mg from the pycnometer. The weighing uncertainty in this mass range was estimated as 0.05 %, in keeping with routine practice in our laboratory when using the aspirating pycnometer method. This estimate seems to be relatively conservative, based on typical differences between observed dispensed and received masses. In fact, basing the uncertainty estimate on balance specifications (as for the single-drop sources in the preceding paragraph) would result in a relative uncertainty of approximately 0.01 %.

Figure 4 shows that the massic activities determined for sources prepared by manual microdrop deposition and by pycnometer are consistent. The ratio of the average values for $A_{\mathrm{m}}^0$ determined for the microdrop- and pycnometer-prepared (0.5 g only) sources was $A_{\mathrm{m},\text{microdrop}}^0∕A_{\mathrm{m},\text{pycnometer}}^0=0.9973(29)$ using Equation 4 and $A_{\mathrm{m},\text{microdrop}}^0∕A_{\mathrm{m},\text{pycnometer}}^0=0.9965(30)$ using Equation 5, where the stated uncertainties are from the microdrop values only. The apparent bias is on the order of the estimated uncertainties.

Figure 4.

Massic activities determined for LS sources prepared by manual microdrop deposition using Equation 4 (blue open circles) and Equation 5 (orange filled circles) compared with massic activities determined for pycnometer-prepared (red triangles and black diamonds) LS sources. Uncertainties for pycnometer-prepared sources with < 20 mg (red triangles) are much larger than for pycnometer-prepared source with ≈ 0.5 g (black diamonds). Uncertainties were estimated as described in Section 3.2 and uncertainty bars shown here are expanded uncertainties ($k=2$).

3.3. Confirmation of controlled drop dispensing

Figure 5A shows the gross LS count rates for the ²⁴¹Am sources prepared in Section 2.3 (a to d). Since the principal goal of this study was to determine whether activity was lost to the gel placemats under the gold foils (Section 2.3, c), the weighing procedure was not carried out for all drops. The drops deposited directly into LS vials were weighed and the massic count rates confirm that evaporative losses from the reservoir during the experiment were not significant (compare the LS sources prepared before and after the gels and foils in Figure 5). For the subset of drops deposited on foils that were weighed, the massic count rates in Figure 5B indicate reduced LS counting efficiency. While the detergency of Ultima Gold was expected to result in most of the ²⁴¹Am dissolving into the LS cocktail, it is possible that some material was left in the nanopores; if the experiment is repeated, some water will be added to the cocktails to hopefully improve detergency and, perhaps, ²⁴¹Am counting efficiency. The two gel sources with weighed drops (closed diamonds in Figure 5B) do not provide sufficient information for comment on any effect on the counting efficiency.

Figure 5.

Liquid scintillation results from the placemat experiment. Blanks are solid black squares. For the other LS sources, the sample order reflects the order in which they were prepared (See Figure 2). The green open diamonds correspond to the drops deposited directly into LS vials, the red circles to the drops deposited directly onto gel squares, the orange diamonds to the nanoporous gold foils, and the blue open circles to the gel placemats under the foils. The symbols for the placemats are shown at the same Sample # as their associated foils. Panels A and C show results as gross LS count rate, whereas B shows massic gross count rate. Uncertainty bars reflect the standard deviation on repeat measurements and are in many cases smaller than the symbols.

In Figure 5C, the LS count rates for the vials containing the gel placemats are higher than background. The highest LS count rate on a placemat corresponds to 0.34(4) % of the activity deposited onto the associated gold foil. The average activity on the placemats was 0.13(11) % of the activity on the foils.

4. Discussion

Using the aspirating pycnometer technique, the relative combined standard uncertainties on the massic activities determined for sources prepared with 0.5 g of solution were approximately 0.1 %. The microcapillary dispensing method (“microdrop method”) was used to prepare sources with approximately 100x smaller masses. For these, the relative combined standard uncertainties on the massic activities ranged from approximately 0.1 % to 0.2 %. The single-drop sources prepared with the aspirating pycnometer technique had about twice the mass of the microdrop sources and the relative combined standard uncertainties on the massic activities determined for these were approximately 0.4 %.

Comparing the massic activities determined for the microdrop sources and for the 0.5 g sources, there appears to be a bias. The mean activity for the five microdrop sources is slightly lower than the lower (k = 2) uncertainty bounds from the 0.5 g sources. This may indicate that the microdrop masses are overestimated by the current models. This possible bias is only discernible when comparing with the 0.5 g sources; the uncertainties on the < 20 mg pycnometer-prepared sources are too large to detect the effect. Essex et al. (2023) used the same manual microdrop deposition method to prepare samples in a recent isotope dilution mass spectrometry (IDMS) study. They achieved relative combined standard uncertainties in the microdrop masses of 0.06 % to 0.13 % and found no consistent bias relative to a pycnometer-prepared sample. However, the pycnometer-prepared sample carried a large relative combined standard uncertainty (0.52 %) on the 8.7 mg single drop. Further study is needed to determine whether the small apparent bias observed here indicates a systematic problem with the manual microdrop dispensing technique as-implemented or the model described in Section 3.1.

The manual microdrop dispensing method described herein is a modification of the method first described by Shaw (2022). Critical differences include omission of the reference weight and a quicker dispensing motion. Shaw forced fluid to the tip of a microcapillary and then touched the nascent drop to a surface to achieve a transfer. Whereas a 12 mg drop will detach from a pycnometer tip due to gravity, a 5 mg drop is not heavy enough to overcome the force of surface tension. Shaw’s contact transfer approach was appropriate for dispensing a drop onto a solid surface but could not be implemented when dispensing a drop into a vial containing LS cocktail. Instead, a quick depression of the micropipettor plunger was used to rapidly expel the drop from the microcapillary. This approach has the benefit of mitigating evaporation of the drop during dispensing, which Shaw considered as a major source of uncertainty, referred to as “excess evaporation.” The approach also resulted in less material remaining in the microcapillary (smaller $m_{\mathrm{r}}$), meaning that the evaporation rate after dispensing ($E_2$) is less well matched with the evaporation rate before dispensing ($E_1$). This mismatch would result in large uncertainties if the time at which the drop was dispensed ($t_{\mathrm{d}}$) was not well-known; in the experiments described herein, $t_{\mathrm{d}}$ was defined with a flag in the mass data so that it was known to approximately ± 0.5 s.

There is another concern with the measurement model that arises from the treatment of $E_1$ and $E_2$ as stable. In practice, small variations in the evaporation rates or small instabilities in the ultramicrobalance may invalidate linear extrapolations. For $E_2$ especially, any smaller residual droplets will have high and rapidly changing surface area to volume. If, for example, a series of small droplets evaporate to complete dryness, the overall evaporation rate from the microcapillary will vary wildly over time. The relatively small evaporation rates (< 0.001 mg/s) mean that such departures from linearity in the datasets will not dramatically affect the estimated masses; however, an improved measurement model might incorporate “medium-term” uncertainty components of the type used in estimating half-life uncertainties (Pommé 2007; 2015). These problems can also be mitigated by minimizing the extrapolations by reducing as much as possible the time between weighing, dispensing, and weighing again (Steps 7 to 9 in Section 2.2).

Finally, the experiment with placemats showed that drops deposited with the microdrop method reach their targets. LS measurements indicated losses ≤ 0.34(4) % of the total dispensed activity, so that > 99.6 % is consistently placed on the intended 3 mm × 6 mm target. This approach was therefore considered promising for the preparation of foil sources for absolute decay energy spectrometry with cryogenic transition edge sensors.

5. Conclusions

Liquid scintillation counting of ²⁴¹Am sources prepared with a manual microliter dispensing method confirmed that gravimetric source preparation was possible. Massic activities of sources prepared with approximately 5 mg of aqueous solution were determined with estimated relative combined standard uncertainties of 0.1 % to 0.2 %. The measurement model and uncertainty budget were presented in detail, and additional experiments confirmed that the dispensing method can be used to reliably deliver 5 mg drops to a 3 mm × 6 mm target.

We have demonstrated an approach to the preparation of nanoporous gold foils with embedded activity traceable to primary methods (4π liquid scintillation counting). Such sources will allow a direct comparison of ADES-determined activities with other primary methods, an important step in establishing ADES as a viable approach to the primary realization of the becquerel.

The manual microliter dispensing method described herein is labor intensive. Handling the microcapillaries is challenging and dropping a microcapillary containing a radioactive solution could result in contamination of expensive equipment (including the ultramicrobalance). Alternative approaches for the quantitative preparation of sources for decay energy spectrometry are needed. Preliminary experiments with inkjet dispensing of ²⁴¹Am solutions in our labs have been very promising but have revealed some technical hurdles that must be overcome. Until then, the microdrop method described herein provides 5 mg drops with accuracy and precision suitable for radionuclide metrology.

Appendix

Here, the effect of evaporation during drop deposition on the massic activity deposited is considered.

The activity in the full microcapillary, $A_{\mathrm{f}}(t_{\mathrm{f}})$, is

$$A_{\mathrm{f}}=A_{\mathrm{m}}^0(m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{e}})∕b_{\mathrm{f}}$$ (A1)

where

$A_{\mathrm{m}}^0$ is the massic activity of the reservoir solution $b_{\mathrm{f}}$, $m_{\mathrm{f}}(t_{\mathrm{f}})$, and $m_{\mathrm{e}}$ are defined as the buoyancy correction, mass of the full microcapillary, and mass of the empty microcapillary, as in the main text.

A.1. Instantaneous mixing

In the instantaneous mixing model (referred to as “Model 1” in Essex et al. 2023), we assume that as the solution in the microcapillary evaporates, it uniformly increases in massic activity.

The massic activity in the microcapillary at the time the drop is dispensed is

$$A_{\mathrm{m},\mathrm{f}}(t_{\mathrm{d}})=\frac{A_{\mathrm{f}}}{(m_{\mathrm{f}}(t_{\mathrm{d}})-m_{\mathrm{e}})}.$$ (A2)

So the activity of the drop is

$$A_{\mathrm{d}}=A_{\text{m,f}}(t_{\mathrm{d}})\cdot m_{\mathrm{d}}.$$ (A3)

Rearranging gives

$$A_{\mathrm{m}}^0=A_{\mathrm{d}}{b}_{\mathrm{f}}∕m_{\mathrm{d}}\cdot \frac{m_{\mathrm{f}}(t_{\mathrm{d}})-m_{\mathrm{e}}}{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{e}}}$$ (A4)

or

$$A_m^0=A_{\mathrm{d}}{b}_{\mathrm{f}}∕m_{\mathrm{d}}\cdot \mathfrak{E}.$$ (A5)

Where $\mathfrak{E}$ is defined as an evaporation correction term:

$$\mathfrak{E}=\frac{m_f(t_{\mathrm{d}})-m_{\mathrm{e}}}{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{e}}}.$$ (A6)

As expressed in the main text in terms of measured values,

$${\mathit{A}}_{\mathbf{m}}^0=\frac{{\mathit{A}}_{\mathbf{d}}\cdot {\mathit{b}}_{\mathbf{f}}}{{\mathit{m}}_{\mathbf{f}}({\mathit{t}}_{\mathbf{d}})-{\mathit{m}}_{\mathbf{r}}({\mathit{t}}_{\mathbf{d}})}\cdot \left(1-\frac{{\mathit{m}}_{\mathbf{f}}({\mathit{t}}_{\mathbf{f}})-{\mathit{m}}_{\mathbf{f}}({\mathit{t}}_{\mathbf{d}})}{{\mathit{m}}_{\mathbf{f}}({\mathit{t}}_{\mathbf{f}})-{\mathit{m}}_{\mathbf{e}}}\right).$$ (4)

Equivalently expressed in terms of the mass of the reservoir solution that contains the same activity as the dispensed drop, $m$’,

$$m^{\prime }=\frac{A_{\mathrm{d}}}{A_{\mathrm{m}}^0}$$ (A7)

which is also equivalent to

$$m^{\prime }=\frac{m_{\mathrm{d}}}{\mathfrak{E}}.$$ (A8)

Or, in terms of measured values to avoid additional correlations when estimating uncertainties,

$$m^{\prime }=\frac{m_{\mathrm{f}}(t_{\mathrm{d}})-m_{\mathrm{r}}(t_{\mathrm{d}})}{b_{\mathrm{f}}}\cdot \left(1-\frac{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{f}}(t_{\mathrm{d}})}{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{e}}}\right).$$ (A9)

A.2 No mixing, unidirectional concentration gradient

The model with no mixing (referred to as “Model 2” in Essex et al. 2023), assumes that all the evaporation occurs at the drop end of the microcapillary, so

$$A_{\text{m,f}}=A_{\text{m,r}}=A_{\mathrm{m}}^0.$$ (A10)

Therefore, the activity of the drop can be written as

$$A_d=\frac{A_{\mathrm{m}}^0\cdot (m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{r}}(t_{\mathrm{d}}))}{b_f}.$$ (A11)

If the activity of the drop can be measured, then

$${\mathit{A}}_{\mathbf{m}}^0=\frac{{\mathit{A}}_{\mathbf{d}}{\mathit{b}}_{\mathit{f}}}{{\mathit{m}}_{\mathbf{f}}({\mathit{t}}_{\mathbf{f}})-{\mathit{m}}_{\mathbf{r}}({\mathit{t}}_{\mathbf{d}})}.$$ (5)

Essex et al. (2023) found this expression to give a better estimate for $A_{\mathrm{m}}^0$ than Equation 4. As discussed in the main text, Equations 4 and 5 converge as $m_{\mathrm{r}}$ approaches $m_{\mathrm{e}}$; in the case where all of the solution is dispensed from the microcapillary, the evaporation model is irrelevant because all of the activity is in the drop.

The equivalent mass of the reservoir solution that contains the same activity as the dispensed drop, is again calculated by Equation A7 and can be rearranged here to give

$$m^{\prime }=\frac{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{r}}(t_{\mathrm{d}})}{b_{\mathrm{f}}}.$$ (A12)

A.3 A general model for evaporative concentration

A more general treatment of evaporative concentration should consider that evaporation may occur from both ends of a microcapillary, and at different rates depending on the distance between the liquid surface and the microcapillary end.

The main observables, discussed above, are $m_{\mathrm{f}}(t_{\mathrm{f}})$ and $m_{\mathrm{d}}(t_{\mathrm{d}})$. Additional terms are defined as:

$\delta =m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{f}}(t_{\mathrm{d}})$; magnitude of evaporation before dispensing

$f_{\mathrm{d}}$ = fraction of mixing of residual solution from evaporation at drop end of the microcapillary

$f_{\mathrm{b}}$ = fraction of mixing of residual solution from evaporation at back end of the microcapillary

$r_{\mathrm{b}}$ = fraction of evaporation that takes place at the back (not drop) end of the microcapillary.

The activity of the drop in terms of $A_{\mathrm{f}}$ and the activity of the remainder, $A_{\mathrm{r}}$ is

$$A_{\mathrm{d}}=A_{\mathrm{f}}-A_{\mathrm{r}}.$$

Before any evaporation, $A_{\mathrm{f}}$ is defined by Equation A1, restated here for simplicity without the buoyancy correction

$$A_{\mathrm{f}}=A_{\mathrm{m}}^0\cdot (m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{e}}).$$ (A1b)

Similarly,

$$A_{\mathrm{r}}=A_{\mathrm{m},\mathrm{r}}(t_{\mathrm{d}})\cdot (m_{\mathrm{r}}(t_{\mathrm{d}})-m_{\mathrm{e}}).$$ (A13)

The goal is to calculate $A_{\mathrm{m},\mathrm{r}}(t_{\mathrm{d}})$, the massic activity in the remainder at the time of dispensing. The average increase in concentration in the entire solution at $t_{\mathrm{d}}$ is $\frac{A_{\mathrm{m}}^0\delta }{m_{\mathrm{f}}(t_{\mathrm{d}})-m_{\mathrm{e}}}$.

The fraction of that increased concentration that takes place at the back end is $r_{\mathrm{b}}$, with the fraction occurring at the drop end of $1-r_{\mathrm{b}}$.

The concentration of the remainder solution immediately after dispensing is

$$A_{m,r}(t_{\mathrm{d}})=A_{\mathrm{m}}^0\left\{1+f_{\mathrm{d}}(1-r_{\mathrm{b}})-\frac{\delta }{m_{\mathrm{f}}(t_{\mathrm{d}})-m_{\mathrm{e}}}+r_{\mathrm{b}}{f}_{\mathrm{b}}\frac{\delta }{m_{\mathrm{f}}(t_{\mathrm{d}})-m_{\mathrm{e}}}+r_{\mathrm{b}}(1-f_{\mathrm{b}})\frac{\delta }{m_{\mathrm{r}}(t_{\mathrm{d}})-m_{\mathrm{e}}}\right\}.$$ (A14)

The three correction terms account for increases in the massic activity of the remainder due to evaporation occurring: at the drop end with mixing, at the remainder end with mixing, and at the remainder end with no mixing, respectively.

In this way, the measurand can be expressed as

$$A_{\mathrm{m}}^0=\frac{A_{\mathrm{d}}\cdot b_{\mathrm{f}}}{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{r}}(t_{\mathrm{d}})-c\cdot (m_{\mathrm{r}}(t_{\mathrm{d}})-m_{\mathrm{e}})}.$$ (A15)

And

$$m^{\prime }=b_{\mathrm{f}}^{-1}[m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{r}}(t_{\mathrm{d}})-c\cdot (m_{\mathrm{r}}(t_{\mathrm{d}})-m_{\mathrm{e}})]$$ (A16)

where

$$c=f_{\mathrm{d}}(1-r_{\mathrm{b}})\frac{\delta }{m_{\mathrm{f}}(t_{\mathrm{d}})-m_{\mathrm{e}}}+r_{\mathrm{b}}{f}_{\mathrm{b}}\frac{\delta }{m_{\mathrm{f}}(t_{\mathrm{d}})-m_{\mathrm{e}}}+r_{\mathrm{b}}(1-f_{\mathrm{b}})\frac{\delta }{m_{\mathrm{r}}(t_{\mathrm{d}})-m_{\mathrm{e}}}.$$ (A17)

With approximate form, useful for comparing to previous equations

$$A_{\mathrm{m}}^0\approx \frac{A_{\mathrm{d}}}{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{r}}(t_{\mathrm{d}})}\left(1+c\frac{m_{\mathrm{r}}(t_{\mathrm{d}})-m_{\mathrm{e}}}{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{r}}(t_{\mathrm{d}})}\right)$$ (A18)

The simplifying approximation $f_{\mathrm{b}}=f_{\mathrm{d}}=f$ is adopted in Table A1, which illustrates the recovery of Equations 4 & 5 (“Model 1” and “Model 2” from Essex et al., 2023).

Table A1.

Simplification of equation A18 according to the assumptions in Sections A.1 and A.2 to derive Equations 4 & 5.

${\mathit{r}}_{\mathbf{b}}$	$\mathit{f}$	${\mathit{A}}_{\mathbf{m}}^0$	Comment
0	0	$A_{\mathrm{m}}^0=\frac{A_{\mathrm{d}}\cdot b_{\mathrm{f}}}{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{r}}(t_{\mathrm{d}})}$	Equation 5; no mixing, all evaporation from the drop end.
0-1	1	$A_{\mathrm{m}}^0=\frac{A_{\mathrm{d}}\cdot b_{\mathrm{f}}}{m_{\mathrm{f}}(t_{\mathrm{d}})-m_{\mathrm{r}}(t_{\mathrm{d}})}\cdot \left(1-\frac{m_{\mathrm{f}}(t_{\mathrm{d}})-m_{\mathrm{f}}(t_{\mathrm{f}})}{m_{\mathrm{f}}(t_{\mathrm{f}})-m_{\mathrm{e}}}\right)$	Equation 4; instantaneous mixing.