Temperature and wall coating dependence of alkali vapor transport speed in micron-scale capillaries

Atomic spectroscopy with alkali vapor is useful for numerous applications, including laser stabilization, ultraprecise metrology, and creating optical quantum interference devices based on slow light or extremely large nonlinearities.^1–3 For applications, miniaturizing and integrating these alkali vapor cells is highly desirable.¹ Examples of such miniature vapor cells include optical fibers,² on-chip glassblown cells,³ evanescent wave coupled waveguides,⁴ and ARROW devices.^5,6 These vapor cells allow experimentation with Rydberg states and quantum interference in linear and nonlinear optics.⁷ However, in such tightly confined geometries, a fast transport speed of the alkali vapor is critical to rapidly distribute the atoms throughout the optical path. As such systems are typically at low pressures (10⁻³–10⁻⁶Torr), the Knudsen model applies and wall collisions are the dominant transport mechanism.⁸ Wall interactions with alkali vapor lead to problematic chemisorption and phase depolarization.⁹ These effects can negatively impact the transport speed of rubidium through the capillary and reduce the overall device lifetime.¹⁰ However, these negative effects can be mitigated through the application of a variety of coating layers such as paraffin,¹¹ polydimethylsiloxane,¹² dimethyldichlorosilane (DMDCS),¹³ and others. It is well documented that these layers help reduce the phase depolarization but their impact on transport speed has not been fully quantified.

In order to better understand the relationship between transport speed and the presence of wall coatings, a testing platform was built and tested over time. The relationship between temperature and transport speed was also investigated to complete our understanding of the capabilities and limitations of microcapillary based atomic spectroscopy cells. The results were then compared to existing Knudsen diffusion models. Surprisingly, our experimental results indicate that movement of Rb through a confined space matches a theoretical model for diffusion in which the diffusion constant does not depend on geometry—specifically, the diameter of the channel. Instead, the diffusion constant is a coefficient dependent on storage temperature and wall coatings. This would argue against using a classical gas kinetics model to describe the behavior of the Rb atomic vapor even if long wall adsorption times are taken into account.

The testing platform was built using three separate components: (1) glass capillaries with inner diameters of 100–200 μm and lengths varying from 8 to 38 mm with thick side walls (Wilmad-Labglass), (2) Pyrex glass chromatography bottles with a diameter of 1 cm, and (3) cold weldable oxygen-free high-conductivity copper tubing with outer diameter of 6.4 mm (Sequoia Brass Copper). The capillaries were heated on a lathe, and a stainless-steel needle was used to slightly taper out the openings to prevent collapse during assembly. The chromatography bottles were heated on the same lathe, and the base of the bottles was opened and glassblown to the capillaries. This was done on both extremities of each capillary to create an all-glass system with two reservoirs. The copper stubs were then cut into 3 cm sections, and one end of each stub was tapered on a belt sander. The copper stubs were cleaned in dilute sulfuric acid and oxidized in a tube furnace at 100 °C to remove contaminants and build a clean copper oxide layer. The tapered ends were then inserted tightly into the openings of the chromatography bottles and sealed in place using epoxy (Aremco-Bond 2310). The complete system was then placed in a vacuum oven at 150 °C for 24 h to drive out contaminants and solvents in the epoxy and other surfaces. The testing system was then ready to be coated and for rubidium loading. At the completion of an experiment, the units can be recycled. This is done by placing the whole system in Piranha (a 50:50 solution of sulfuric acid and hydrogen peroxide) to dissolve the epoxy and clean the glass. The glass is rinsed with acetone and isopropyl alcohol and placed in a vacuum oven at 150 °C for 24 h to evaporate solvents. Finally, new copper stubs can be reattached with epoxy, and the system is once again ready to be loaded. A diagram and photograph of the complete testing platform are shown in Fig. 1.

In order to determine the influence of the capillary surface on atom transport and spectroscopy performance, the inside surface of the platform was coated by filling one side with a 10% solution of DMDCS in cyclohexane and pulling the solution through the capillary by attaching the other side to a vacuum. The solution was flowed through the capillary for 1 min and then rinsed with pure cyclohexane for another minute before being heated to 150 °C for an hour to evaporate any solvents. The validity of the DMDCS coating was tested prior to the experiments by measuring water contact angles on a planar silicon substrates using a goniometer.¹⁴ The substrates had a variety of glasses deposited (i.e., plasma-enhanced chemical vapor deposition) or grown (thermal oxidation) on their surfaces. For each type of glass, the goniometer tests confirmed a large increase in contact angle upon application of the DMDCS solution, indicating an altered, more hydrophobic surface. The rubidium (Rb) loading was done in a nitrogen glovebox to avoid contamination. The testing platform was loaded by applying a small piece of natural Rb to the inside wall of one side of the system. Both sides of the platform were then connected to a vacuum and allowed to reach <1 mTorr before being hermetically sealed with cold-weld pliers. At this point, the loaded platforms were stored in a tube furnace and monitored daily.

The loaded capillary systems were tested daily and monitored for Rb spectra in the unloaded side. The devices were placed on a platform heated to 85 °C and the unloaded side was interrogated with a 780 nm laser (ThorLabs ITC4001). The laser light was collimated and focused before being sent through the glass chamber and hitting a photodetector (ThorLabs PDA36A). The laser was tuned to sweep across the Rb D2 absorption line transitions. The geometry of this experimental setup is shown in Fig. 2.

Absorption was monitored by measuring the optical depth of the Doppler broadened peak extracted from the D2 ⁸⁵Rb F = 3 -> F′ = 2, 3, 4—the largest of the four absorption peaks. A cut-off value of 2% was chosen as the absorption percentage necessary to claim successful transport. This value was chosen because it was the lowest value safely above our system noise. In other words, transport time was defined as the time between initial Rb loading and the first instance of >2% absorption on the unloaded side of the testing platform. This transport time was the metric used to compare the different systems. Figure 3 shows a filtered sample spectrum obtained from the unloaded side of a capillary system after the Rb has successfully reached that side.

Previous studies with uncoated capillaries have demonstrated that the transport time required to observe absorption in the unloaded compartment is well described by a model taking the following form:

where L and d are the length and diameter of the capillary in micrometers, respectively. It is well known that a seasoning time is required for the wall–vapor reactions to stabilize and the data to be reliable.^15,16 This seasoning time has also been incorporated into the transport equation. The data obtained in these experiments were fit to this transport equation so that the different tests could then be compared with their fitting parameters. A total of five tests were run. Two sets of 200 μm diameter systems (a total of eight capillary cells) were tested with and without a DMDCS coating at a storage temperature of 90 °C. Three sets of uncoated 100 μm diameter systems (a total of seven capillary cells) were tested at storage temperatures of 90, 110, and 130 °C. The results are shown in Figs. 4 and 5. In each case, the transport equation above is fit to the data. Figure 6 compares the transport time to storage temperature for the capillaries with different diameters and lengths.

The presence of a DMDCS wall coating and a higher storage temperature both led to a shorter transport time. The best fitting parameters to the transport equation is shown in Table I. While in both cases, the overall transport speed is increased, the effect on seasoning time is different in each case. The seasoning time is a factor of the wall–vapor interactions in the reservoirs on both sides of the capillary. As a higher temperature does not affect the chemistry or volume of the reservoirs, the seasoning time of nine days remains unchanged. However, since the DMDCS coating is applied to the entire inside surface of the system, the coated cells have a seasoning time that is approximately 40% shorter.

Using the computed values for the η parameter in Table I, the uncoated samples can be fit to the following linear equation:

where T is the storage temperature in Kelvin. This equation should be valid for temperatures close to the range tested in these experiments (90–130 °C).

When considering the form of Eq. (1), we should be able to relate it to the well-known diffusion equation for gas atoms, namely,

where J is the atomic flux per unit area, D is the diffusion coefficient, and C is the concentration of atoms along the capillary tube. We can apply this equation to the situation in which a capillary is joining two gas reservoirs,¹⁷ the first with atomic concentration C₁ (loaded reservoir) and the second with concentration C₂ (unloaded reservoir). While C₁ is much greater than C₂ (meaning there is very little Rb in the unloaded reservoir) and after the space inside the capillary has been filled with Rb, we can assume that the atomic flux into the unloaded reservoir will be proportional to the difference between the concentrations in the two reservoirs. This nondepletion approximation of Eq. (3) can be written as

where L is the length of the capillary. Given our assumption that C₂ ≪ C₁, we can simplify this equation for flux as

Equation (5) should be valid as long as C₂ ≪ C₁ (very little Rb population in the unloaded reservoir). Given our test procedure outlined above, in order to detect strong enough absorption spectra in the unloaded reservoir to confirm the presence of Rb, we must have a minimum number of atoms N_min. To achieve this minimum concentration, the atomic flux, J_filling, must add the necessary atoms into the reservoir, which will take a time given by

Equation (6) has the same geometric dependence as our transport equation [Eq. (1)] if D does not depend on the diameter or length of the capillary. However, for classical Knudsen diffusion of a gas through a long tube^17,18 of diameter d, taking into account possible adsorption of the gas molecules on the tube wall, the diffusion constant D_kn usually takes the form

where $v¯$ is the average atomic speed and τ_a is the average time atoms would spend on the tube wall due to adsorption.

For a situation without adsorption or with very short adsorption times, and given our capillary diameters, we would expect a diffusion coefficient, D_kn, of greater than 100 cm²/s, leading to very small t_min. The largest values of t_min calculated using the expected coefficient D_kn and the worst-case-scenarios for N_min and C₁ were, at most, a few minutes. These times were much smaller than the transport times we measured empirically. Consequently, we hypothesized that wall adsorption times may be skewing t_min heavily.

To investigate whether long adsorption times on the walls might explain our large measured transport times, we can rewrite Eq. (7) for the case where τ_a is large compared to d/$v¯$⁠. The Knudsen diffusion equation then simplifies to

Substituting Eq. (8) into Eq. (6), we obtain a t_min relation with the following geometric dependence:

This would lead us to believe that if long adsorption times for Rb on the glass capillary walls could explain our overall long transport times, our transport time should be proportional to 1/d⁴. Our experimental data, however, show no such dependence, and in fact, there is excellent correlation with a form of the diffusion equation solution [Eq. (6)] in which the diffusion coefficient, D, is a constant [comparing Eq. (1) with Eq. (6)]. Based on our experimental results, we must conclude that in our case D does not have a geometric dependence on the diameter of a channel because devices of two different geometries exhibited similar diffusion coefficients. This result is somewhat surprising given the expectation that the Rb atoms would behave according to the classical Knudsen diffusion model.

If we assume that the geometric dependencies in Eqs. (1) and (6) do match, we can then derive a relationship for our expected diffusion coefficient, D, as a function of η. Given that N_min can be written as a concentration C_min times the volume of the reservoir (V_R), we can write a relationship for D given by

Given the known volume of the reservoir is 1.5 ml and that the ratio C_min/C₁ is approximately 0.05 when the signal is visible in the unloaded reservoir (based on comparing the strengths of the absorption signals in the loaded and unloaded reservoirs), the diffusion coefficient can be approximated as

where η is in units of micrometer-days as shown in Table I. Recalling that η is not geometry dependent, the approximate effective diffusion constants for our experiments are found in Table II.

The data shown in Figs. 4 and 5 indicate that raising the storage temperature and applying wall coatings can speed the transport of reactive vapors through highly confined geometries. This effect is most likely due to an increase in the effective diffusion constant for our system. This should be expected because diffusion coefficients in any system typically have temperature dependence. A second possible explanation for why the higher temperatures may decrease transport time is that raising temperatures increases the density of alkali vapor atoms on the source side of the capillary system. If a larger number of atoms were introduced at the source side, we would expect a larger number to transition through the capillary and reach the collection chamber. Given that $ttransport$ is measured at the point optical absorption reaches a specific threshold, we would expect larger initial alkali populations to thus result in faster $ttransport$ times because the necessary alkali population density required to reach that threshold could be supplied sooner. We have observed, however, that the strength of the optical absorption signal versus time rises over several days and then decreases (most likely due to interactions of the alkali vapor with the epoxy used to seal the copper tubing to the glass collection chamber). The shape of this signal intensity versus time curve is similar for all samples tested, with a peak in intensity occurring several days after it crosses the minimum detectable threshold.

Given the nature and overall duration of the signal intensity versus time curves, we would expect the $ttransport$ difference between high population (high temperature) samples and low population (low temperature) samples to be no more than two days for a cell of the same geometry. In our experiments, we saw much larger differences. For instance, the longest length cells reported in Fig. 5 showed $ttransport$ time differences of almost 20 days. This leads us to conclude a temperature induced change in the diffusion constant is the most likely explanation for the pronounced changes in $ttransport$ as temperature is raised. This temperature dependence is apparent in the η fitting parameter as seen in Eq. (2). Therefore, combing Eqs. (2) and (11) gives an approximate temperature dependence for D where, unsurprisingly, D increases with temperature.

While raising the temperature in a structure with confined geometries may appear like an obvious way to decrease alkali vapor transport times and push alkali atoms throughout the volume of a structure, there are additional considerations. Higher temperatures may pose problems for device lifetime and integrity. Specifically, adhesives may outgas or become structurally compromised at higher temperatures while the thermal expansion of metals may lead to strain and stress in the device.

Alkali vapors, such as rubidium, do not appear to follow classical, or modified, Knudsen models for transport. Higher storage temperature leads to lower transport times for alkali vapors through confined spaces by, most likely, altering the effective diffusion constant for a given system. Additionally, wall coatings also have the potential to accelerate alkali vapor transport in confined spaces along with their well-documented ability to reduce unwanted phase depolarization.

Further investigations into other coatings and a wider range of temperatures may lead to additional insight but require very long testing periods. Additionally, light induced atomic desorption experiments will be interesting for testing desorption of Rb from capillary walls and its impact on overall transport.^19,20

The authors would like to thank Kevin Teaford for assistance with glass bonding. They would also like to acknowledge the financial support from the National Science Foundation: ECCS-1101801 and ECCS-1101902.