Recommended Volumetric Capacity Definitions and Protocols for Accurate, Standardized and Unambiguous Metrics for Hydrogen Storage Materials
Applied Physics A
Philip A. Parilla1†, Karl Gross2, Katherine Hurst1, Thomas Gennett1
1National Renewable Energy Laboratory, Golden CO 80401
2H2 Technology Consulting LLC, Alamo, CA 94507
† corresponding author, , 303-384-6506
Online Resource 1
This document contains supplementary materials relevant to the journal article as listed above. It contains two derivations to show the relationship between absolute capacity and excess capacity and between total capacity and excess capacity. There are also two examples. The first shows how the total number of moles is determined for a specific manometric instrument and associated measurement protocols. The second example illustrates how the recommendations as given in the main paper are implemented in practice.
S1. Absolute Capacity
We now show the mathematical relationship between the absolute capacity and the combined excess capacity plus a free-gas component. However, first for the sake of clarity and scientific rigor, a few definitions and assumptions must be stated explicitly. The molar density of gas near the surface of an adsorbent can be described by the unknown function where is the position vector normal to the surface. Far away from the adsorbent’s surface, the gas density approaches the free-gas molar density, (Eqn. 6b, main paper). The location where defines the boundary between the volume of gas near the sample surface that experiences adsorption forces Vad and the free-gas in the sample, Vfgs. The following equation provides the absolute capacity of the sample through an integration of the gas density over the entire adsorption volume, Vad:
By adding and subtracting the free gas that would normally be present in the same volume
the equation can be re-written as:
which provides the relationship between “absolute” and “excess” capacity:
More specifically, Eqn. S1a represents all the gas moles in the region defined by Vad, the adsorption volume. In Eqn. S1b, we add and subtract the free-gas moles that could reside in Vad as shown by the right two integrals. By reorganizing the terms in Eqn. S1c as shown in Eqn. S1d, we recognize that the first integral is just the definition of the excess capacity, producing Eqn. S1e. Note, that even though the function ⍴ex(x) is not explicitly known, the value of its integral is what is determined in the capacity measurements. Finally, if the pressure and temperature are uniform within the sample volume, then the molar free-gas density is constant and can be taken outside the integral and results in Eqn. S1f. This last equation just states that the absolute capacity is just the sum of the excess capacity plus the free-gas capacity in Vad, just as was illustrated in Fig. 1 of the main paper. Unfortunately, the volume of gas that experiences adsorption forces (Vad) is ill defined and therefore difficult or impossible to calculate. Because Vad is not well defined then the absolute capacity is also not well defined.
S2. Total Capacity
An almost identical approach to absolute capacity can be taken with the total capacity calculations, but this time there are well-defined volume definitions that can be used to allow real values to be determined and used for practical materials performance evaluations. The same notation as in the absolute capacity discussion above is used and we emphasize that there is a free-gas volume associated with the sample, Vfgs. The total extent of the free-gas region is defined to represent the intrinsic free gas in the sample. For example, it could be defined as the free-gas volume within the packing volume, Vpk (Fig. 2, main paper); other volume definitions are possible as well. We choose Vpk as a typical representative volume. The total volume associated with the sample that is accessible to the gas, both adsorbed and free is Vgs tot and is equal to:
The total molar capacity is then given by the following volume integrals which demonstrates the mathematical equivalence progressing from ill-defined volumes to well-defined volumes and concepts:
Adding and subtracting the free gas within the adsorption volume gives
which can be reorganized to give
the first term of which is the excess capacity
If the pressure and temperature are uniform within the sample volume, Vgs tot, then the molar free-gas density is constant and can be taken outside the integral and results in Eqn. S4f.
This mathematical progression is very similar to that followed by the absolute capacity equations.
S3. Example of derivation for total amount of moles in an instrument
We now give an example that demonstrates how the total amount of sorbate moles is determined at all times in a manometric instrument measurement. Although it is well known that the total number of moles in the instrument is needed for the mole-balance equation, most references in the literature either take it as a given or the complete mole-balance equation is derived, but the total number of moles in the instrument is not identified explicitly within the equation. We explicitly identify the expression for the total number of moles in the instrument. A basic manometric experimental setup is shown in Fig. S1. It consists of a calibrated reference volume monitored by a pressure sensor and connected to two valves. Valve 1 allows the insertion or extraction of a gas to the reference volume and valve 2 allows the gas in the reference volume to access the sample. By knowing the volumes, their temperatures, and the gas pressures before and after exposure to the sample, the amount of gas sorbed or desorbed from the sample can be inferred on the basis of mole balance. For this example, we will assume an isothermal system. During the measurement cycle, the apparatus can be considered to be in one of two states: closed and opened. In the ‘closed’ state (Fig. S1a), valve 2 is closed and the pressure sensor only monitors the reference volume. In the ‘opened’ state (Fig. S1b), valve 2 is opened and the pressure sensor monitors gas pressure in the total volume of both the reference and sample volumes.
Fig. S1 shows a basic manometric instrument and its two states: a. (left) valve 2 closed and b. (right) open, respectively
The measurement protocol for this system is as follows for the ith step of the isotherm:
1) With valve 2 open, measure initial “equilibrium” pressure (= PEq i-1)
2) With valve 2 closed, measure “closed” pressure (=PCl i)
3) With valve 1 open, add or remove gas, then close valve 1
4) Valve 2 is still closed, measure “charging” pressure (=PCh i)
5) Open Valve 2 and let the gas reach equilibrium with the sample, measure final “equilibrium” pressure (=PEq i)
Note that valve 1 is always closed during a pressure measurement and that any pressure or temperature transients are allowed to settle before the pressures are measured. Also note that the final equilibrium pressure from the (i-1)th step is used as the initial equilibrium pressure in the ith step. This means that for an isotherm measurement with a total of n steps, 3n+1 pressure measurements need to be performed. The mole-balance equation for this sequence is given by the following:
where nads i is the number of moles sorbed or desorbed on the ith step, Vr is the reference volume, Vt is the total volume (valve 2 is open), and z is the compressibility factor as previously defined. If nads is positive, it corresponds to sorbate moles being sorbed on that particular step and if it is negative, then it corresponds to sorbate moles being desorbed.
It should be noted that this model differs from most in the literature in that most models develop their mole-balance equation in terms of a reference volume, Vr, identical to the above model, but with a sample volume, Vs, which is defined as the volume of the sample cell when valve 2 is closed. Although these models are conceptually correct, they do not take into account that the pressure in the sample cell cannot be measured with valve 2 closed (for a single pressure-sensor system). Additionally, the volume of the entire system changes slightly between valve 2 in the open and closed states, and often these models do not include that volume change in the equations. Finally, if valve 2 closes quickly such as occurs for pneumatic or solenoid valves, there is the possibility that it can “pump” moles from one volume to the other. The Vr : Vt model used here avoids all these issues.
To get the total number of moles on the sample, we must sum the moles per step so after the ith step:
Note that the PCh and PCl summations remain, but that the PEq summation only retains the first and last terms. Most often, the measurement begins at zero pressure so that
and the last term in parentheses in Nads is zero as well as the first term in the PCl summation. This yields:
This equation corresponds directly with Eqn. 6a and we can make the associations Vt = Vex cal and the total moles of gas in the instrument at the end of measurement step i is
This equation can be interpreted as follows. Whenever valve 1 opens before valve 2 with a pressure measurement in between (Pch) and with both valves closed, it is moles of gas in the reference volume entering the instrument; and whenever valve 2 opens before valve 1 with a pressure measurement in between (Pcl) with both valves closed, it is moles of gas leaving the instrument (hence the negative sign). Here “moles in the instrument” refers to moles being in either the reference volume or the sample volume and with valve 1 closed. This specific expression for ninst tot is derived for this particular apparatus configuration, measurement protocol sequence and mole balance equation; other expressions and instrument configurations are possible as well. This result provides an example to show how assertion #2 (and Eqn. 6a) for the manometric measurement is realized in practice, i.e., that the total number of moles in the instrument can be determined at each measurement step.
S4. Example of Reporting on Capacities
An example is now provided to solidify these concepts and demonstrate how to put the recommended protocols into practice.
4.1. Experimental Setup and Procedures
The material used for this example is the activated carbon Norit ROW (0.8mm pellets, steam activated; Alfa Aesar Product Number L16334); it is not a crystalline material and precludes calculating any FOMs based on the crystalline volume. The pellets are roughly cylindrical with 0.8 mm nominal diameters and varying lengths, typically 1 to 5 mm. The bulk density of an individual pellet was not determined nor was it listed in the manufacturer’s specifications. This material was measured at two temperatures: 303 K in a temperature-controlled water bath and in a liquid nitrogen bath (75.6 K). The sample was a commercial product, which prevents us from giving a thorough description of its synthesis and preparation into pellets. Had this example used a material that we had synthesized, we would describe the synthesis and preparation in detail. Samples were degassed using the following protocol:
- The sample material is loaded into the high-pressure sample vessel with manual isolation valve and sealed in open-air conditions.
- The sample vessel is evacuated at room temperature on a temperature programmed desorption (TPD) instrument, typically overnight, until the base pressure of the TPD is ~10-7 – 10-8 Torr. The TPD has a mass spectrometer (MS) to monitor the molecular species evolving from the sample material and to check for leaks in the sample vessel.
- While under active evacuation and monitoring with the MS, the sample vessel is pumped at room temperature for an hour then ramped to 100 C in 15 min. and held at 100 C for 4 hours then ramped to 300 C in 30 min. and held there for approximately 4 hours and then cooled while still under active evacuation to room temperature.
- The sample vessel’s manual isolation valve is closed while the sample is under vacuum and the sample vessel is transferred in vacuo to the manometric instrument.
- The instrument is fully evacuated up to the manual isolation valve of the sample vessel. The manometric instrument’s evacuation system consists of a turbo-molecular pump that can register a vacuum of 1 x 10-5 Torr (which is the minimum reading for the gauge, MKS 910). This “base” pressure is measured near the turbo-molecular pump.
High-pressure hydrogen capacity measurements were performed on a modified commercial manometric system (PCTPro-2000). One hardware modification consisted of adding a manifold to the high-pressure gas inlet that allows either hydrogen or helium to be introduced (with pump/purge cycles for gas switch-overs). In this way, exactly the same protocol could be used for the hydrogen measurements and for the helium calibration procedure (vide infra). Another hardware modification consisted of supplementing the as-received temperature-control system so that the temperature-controlled region was expanded to include the sample support arm and the sample vessel assembly (sample vessel, manual isolation valve, and 0.125 inch OD connection tubing) using temperature-controlled water circulating through copper components physically connected to the sample vessel assembly. The temperatures of the internal cabinet and the external circulator were equal and this modification greatly improved the overall temperature stability of the apparatus. For measurements at 303 K, the sample vessel was immersed in stirred water in a double-jacketed Dewar flask where the circulator water flowed through the jacket. For measurements at 75.6 K (the boiling point of liquid nitrogen at an altitude of 6000 ft. – Golden, CO), the sample vessel was immersed in liquid nitrogen in a glass-walled vacuum Dewar covered with a foam-insulating cap. An OFHC (Oxygen-free high thermal conductivity copper) split copper cylinder was clamped to the 0.125 inch tubing at the top of the sample holder so that the copper extended up to the foam lid and was always partially submerged in the liquid nitrogen; this copper piece helped to mitigate the effect of the falling liquid nitrogen level over time on the temperature profile and to extend the liquid nitrogen temperature up to the lid. Just above the foam lid was another split copper cylinder also clamped to the sample-holder tubing and the 303 K circulator water passed through this upper copper cylinder. In this way, the temperature gradient between 303 K and the lower copper cylinder was confined spatially to the thickness of the foam lid (~3.5 cm). The volume where the temperature gradient exists is estimated to be 0.084 ml while a typical total volume (Vmt cal) is in the range 12 to 16 ml depending on the sample vessel size. Finally, the liquid nitrogen level was monitored throughout the measurement and was maintained within a ~ 1.5 cm range ensuring that the lower copper cylinder was always at least always half submerged in the liquid nitrogen during the measurement. All these protocols allow the temperature profile of the sample volume to be very stable and reproducible. The manometric measurement steps were as follows:
- Measure hydrogen capacity of sample at 303 K.
- Warm to 303 K and pump off hydrogen (base pressure 10-5 Torr ) for 1 hour.
- Measure hydrogen capacity of sample at 75.6 K.
- Warm to 303 K and pump off hydrogen (base pressure 10-5 Torr ) for 1 hour.
- Perform helium calibration at 303 K with sample present.
- Open sample vessel, weigh bottom halve of vessel, remove sample material, weigh sample material directly, weigh cleaned vessel and reassemble clean empty sample vessel assembly.
- Perform helium calibration at 75.6 K on empty sample vessel.
- Perform helium calibration at 303 K on empty sample vessel.
For each measurement step, the pressure was held for 10 minutes to allow the sample to come to equilibrium, which is consistent with the physisorption mechanism expected in these materials. Pressure transients as a function of time confirmed that 10 minutes was sufficient for equilibrium. For the 303 K capacity determination, steps 1 and 5 are sufficient; while for the 75.6 K measurement, steps 3, 5, 7 and 8 are required. These steps allow calculation of the sample skeletal volume (steps 5 and 8) and the calculation of the total-warm and cold empty volumes (Vmt cal and Vmt cold respectively; steps 8 and 7). The sample skeletal volume can then be subtracted from the empty volumes to yield the cold volume with the sample present. The volume, VT, and gas density for the temperature gradient region is calculated directly using numerical integration, assuming a linear temperature gradient and using the geometrical dimensions of the tubing. The critical requirement needed for the above protocol is that the sample vessel volume be repeatable to a high degree upon disassembly and re-assembly as well as the repeatability of the temperature profile of the instrument at 303 K, any temperature gradients, and the volume at liquid-nitrogen temperatures. These requirements have been thoroughly verified through control experiments on empty sample vessels.
The above protocol avoids helium adsorption effects on the sample at low temperatures for the calibration steps when it is expected to be the most significant. Instead, the sample is only exposed to helium at 303 K where the effects are much reduced. The helium adsorption that occurs at 303 K is assumed to be negligible by necessity as it is very difficult to accurately determine this adsorption and compensate for its effects on the hydrogen adsorption determination. This is especially true for samples that cannot be heated to high temperatures such as MOFs. Not compensating for helium adsorption effects will yield capacity measurements that underestimate the hydrogen adsorption. As mentioned above, because of the modification that allows helium to be introduced into the high-pressure port of the instrument, the exact same measurement protocol used for hydrogen can also be done with helium. This provides a higher degree of confidence for the helium calibration and can also investigate any calibration effects dependent on pressure; none were found.