As the ocean changes in response to anthropogenic influences, it is important to understand the impacts on the cycling and distribution of dissolved gases such as oxygen (O₂), carbon dioxide (CO₂) and methane (CH₄) Talley et al. (2016), Halpern et al. (2019), Hopkins et al. (2020), Levin (2018), Doney et al. (2009), Reay et al. (2018). Warmer ocean temperatures lead to larger deoxygenated zones, which impact ecosystems, and rising carbon dioxide levels lead to ocean acidification, which impacts ocean ecosystem health Levin (2018), Doney et al. (2009). To monitor dissolved gases, researchers use a combination of discrete bottle samples and in situ instrumentation. These measurements enable the observation of changing conditions over time, as well as model ground-truthing, which can lead to the prediction of future changes Takahashi et al. (2002), Talley et al. (2016), Wanninkhof et al. (2013).

Many techniques for measuring dissolved gases in situ involve using an equilibrator to extract gas from water, for analysis by gas phase instrumentation. Several reviews have compared equilibrator styles (Santos et al., 2012; Yoon et al., 2016; Webb et al., 2016), but most use direct air-water contact, which can only be used for surface measurements. For deep-sea applications, the only equilibrator style available is a membrane inlet, which uses a nonporous gas-permeable polymer membrane, supported mechanically to withstand hydrostatic pressure, to isolate water from an internal gas volume. Unfortunately, gas transport through nonporous membranes is slow, because of the solution-diffusion mechanism for mass transport through the polymer. Membrane inlets have been used successfully for a variety of deep-sea dissolved gas instruments (Tortell, 2005; McNeil et al., 2006; Bell et al., 2007; Wankel et al., 2010, Wankel et al., 2011, Wankel et al., 2013; Fietzek et al., 2014; Reed et al., 2018; Michel et al., 2018; Grilli et al., 2018).

Overcoming the low gas transport rate is a significant design challenge for deep-sea dissolved gas instrumentation. Several approaches have been used to improve time response, such as minimizing the internal gas volume (McNeil et al., 2006; Reed et al., 2018; Fietzek et al., 2014), pulling a vacuum (Tortell, 2005; Bell et al., 2007; Wankel et al., 2010; Wankel et al., 2011), or using a sweep gas (Grilli et al., 2018). While these approaches have been demonstrated in the field, the trade-offs between different design choices are not readily apparent. To the authors’ knowledge, the analytical tools to understand different membrane equilibration approaches are not available in the literature, making it impossible to make physics-informed design decisions. Here, we aim to fill this gap by deriving analytical models to describe membrane equilibration from first principles, and use those models to derive design rules and identify trade-offs. The goal is to provide the tools needed for future instrument developers to make more physics-informed design decisions.

We consider four approaches, or “operation modes”: (1) passive equilibration, where a fixed volume of gas is allowed to fully equilibrate with a flow of seawater, (2) active equilibration, where a vacuum is continuously pulled on the gas side of the membrane, (3) hybrid equilibration, a balance between passive and active where a vacuum is periodically pulled and released, and finally (4) exchange equilibration, where a fixed volume of water is allowed to equilibrate with a fixed volume of gas. Passive equilibration is used by many instruments reported in the literature [e.g., Reed et al. (2018) or Fietzek et al. (2014)]. Underwater mass spectrometers continuously pull a vacuum to extract gas and can achieve rapid time responses Chua et al. (2016), although the exact instrument control dynamics may differ from the dynamics derived here. The CADICA instrument, presented by Bass et al. (2012), uses exchange equilibration: it monitors total dissolved inorganic carbon (DIC) by acidifying and degassing an aliquot of water and monitoring the carbon dioxide in a fixed gas volume. Here, we introduce a new method, hybrid equilibration, that balances the trade-offs between passive and active equilibration.

In this work, analytical models are derived for different equilibration methods for membrane inlet dissolved gas instrumentation. Supplementary Material S1 lists all of the nomenclature used throughout this paper. We then use these models to understand the advantages, limitations, and design space for each method. We begin by deriving differential equations describing the dynamics of each method, then solve these for their steady state and step responses. Design relationships are called out to aid future instrument development, such as the dependence of time response on internal volume and the trade-off between sensitivity and speed. The resulting analytical models should also aid in future instrument designs, as they offer a way to parameterize expected ranges in the internal gas volume, allowing for informed design decisions for the instrumentation to measure composition of the equilibrated gas. With the analytical models and results, the instrument designer and field-going scientist will be able to make more informed design, calibration, and deployment decisions.

Chemical potential gradients drive the mass transport of gases in membrane applications, and the chemical potential is continuous at material interfaces, such as the water-membrane and gas-membrane interface Wijmans and Baker (1995), Baker (2004), Nagy (2019); Cussler (2009), Wijmans and Baker (2006). We can model the transport across a membrane inlet by considering it as a series of “layers”: (1) a well-mixed water layer, (2) a water-side stagnant boundary layer, (3) a solid, nonporous polymer membrane, (4) porous membrane support structure(s), (5) a gas-side stagnant boundary layer, (6) and a well-mixed gas layer. At each interfacial transition, such as the water-membrane interface, the chemical potential is continuous, and constitutive equations like Henry’s Law and the ideal gas law can be used to relate the chemical potential to gas partial pressures (Wijmans and Baker, 1995; Baker, 2004; Nagy, 2019; Cussler, 2009; Wijmans and Baker, 2006). In this dissolved gas monitoring application, we seek to make measurements of the partial pressure of gas in the well-mixed gas layer, P g to infer the partial pressure of gas in the well-mixed water layer, P w . We use a lumped-parameter model to describe the mass transport across the overall membrane inlet: J = k eff P w − P g (1) where J is the flux of an individual gas (e.g., CO₂) from the water-side to the gas-side of the membrane inlet and k eff is an effective mass transport coefficient. The total gas flux across the membrane inlet can be computed by adding up the fluxes of each individual gas. To use this lumped-parameter model, we must make the assumption that the timescales of changes to the overall system (e.g., equilibration of the instrument), are much slower than the timescales of changes within each “layer” (e.g., concentration within the membrane itself) Nagy (2019); Cussler (2009); Baker (2004). From Equation 1, it is clear that gas flux can be increased by minimizing P g . This can be done by pulling a vacuum, using a sweep gas, or otherwise scrubbing the gas of interest.

The effective mass transport coefficient (Equation 1) can be computed using a resistance in series model Baker (2004), Nagy (2019), Wijmans et al. (1996): 1 k eff = 1 k b l + 1 k m (2) where k m is the mass transport coefficient for the membrane and k b l is the mass transport coefficient for the stagnant boundary layer. We neglect the mass transport resistances on the gas-side of the membrane due to the significant differences in diffusivity in gas versus water, but could be included as an additional term if warranted (e.g., if long tubing is used to connect the membrane to the internal gas volume).

The mass transport coefficient for the membrane can be expressed in terms of the membrane thickness, δ m and the membrane permeability, P , or equivalently with the solubility, S m and diffusivity, D m , of the gas in the membrane material (Nagy, 2019; Wijmans and Baker, 2006; Baker, 2004): k m = P δ m = S m D m δ m (3) Similarly, mass transport coefficient within the stagnant boundary layer can be expressed as: k b l = D w K H δ b l (4) where D w is the diffusivity of the gas in water, K H is the solubility of the gas in water, and δ b l is the thickness of the stagnant boundary layer (Sarmiento and Gruber, 2006; Cussler, 2009; Nagy, 2019). Many fields use mass transport coefficients with different variable naming conventions, so care is required to ensure the continuity of chemical potential and match units.

Considering Equations 2-4, the stagnant boundary layer can be thought of as second “membrane”, with effective permeability equal to the product of the diffusivity and solubility of the gas in water. If flow speed is increased significantly, eventually the stagnant boundary layer will be very thin, both through the increase in flow velocity and the triggering of turbulence. In this case, mass transfer resistance due to the stagnant boundary layer may be negligible. Once the flow has reached a point where the stagnant boundary layer can be ignored, the “best” possible mass transport has been reached, and increasing flow rate after this point will not appreciably impact the instrument response.

We compare four operational modes for an underwater membrane-equilibrated dissolved gas instrument (Figure 1). In all modes, gas-side instrumentation (e.g., a laser spectrometer) is used to monitor the partial pressure of gas(es) inside an internal gas volume, with the overall goal of using those measurements to estimate the water-side partial pressure(s). The four modes are:

1. Passive Equilibration, where a fixed volume of gas is allowed to equilibrate with flowing water.

The performance of each equilibration mode was assessed to provide insight into their theoretical performance characteristics and to guide future designs. First, the differential equations describing dynamics of each mode were derived starting from first principles using a mass balance approach. Key assumptions used in the derivations are listed in Supplementary Section S2. Then, where possible, the differential equations were solved analytically for the steady state and step responses. A computational approach was used for equilibration modes where a direct solution of the differential equations was not straightforward. Finally, design insights were drawn from the analytical results, leading to conclusions about optimization for each mode.

Membrane properties and dissolved gas conditions were chosen to simulate oceanographic conditions and are provided in the Supplementary Section S3. Representative operational settings (e.g., vacuum level) used for the four methods are also provided. Computations were performed using MATLAB 2020a.

Using the mathematics derived in Section 4, we can compare the performance characteristics of the four operational modes by computing a step response, focusing on a single gas for simplicity. Figure 2 shows the step response for each mode corresponding to a 20% increase in water-side pCO₂. Passive equilibration follows the canonical first order step response and at equilibrium the gas-side pCO₂ matches the water-side. Active equilibration is more complex because its response depends on the levels of all other dissolved gases (pN₂, pO₂, pH₂O, etc.). We computed the step response with all other gases varying within oceanographically-relevant ranges (Supplementary Table S3). The shaded region in Figure 2b is used to indicate the range of possible gas-side pCO₂ responses. This range exceeds the magnitude of the step response, indicating that active equilibration is highly cross-sensitive. With a mass-flow meter, it would be possible to compensate for the variation in other gases and avoid this cross-sensitivity. The equilibrium gas-side pCO₂ level for active equilibration is an order of magnitude lower than for passive equilibration, and a more sensitive gas-side instrument would be required. Active equilibration, however, is an order of magnitude faster than passive equilibration. Figure 2c shows that the hybrid equilibration step response follows active equilibration on average, both in terms of the magnitude and speed of the response. Hybrid equilibration oscillates around a steady state condition, which makes the data more difficult to interpret in real-time. Figure 2d shows the exchange equilibration response, under the condition that τ g = τ w . Its response is twice as fast as passive equilibration, and its steady state pCO₂ level is half that of passive.

Here, we derived analytical models for passive, active, hybrid and exchange equilibration to support the development of membrane-based dissolved gas instrumentation. The four operation modes were assessed for their steady state response and step response, and design rules and areas for optimization were identified.

Passive equilibration is straightforward: the differential equations are tractable, the steady state response is easy to interpret, there is no cross-sensitivity, and specific knowledge of membrane equilibration dynamics is not required to interpret results. The instrument design can be optimized for time response primarily by minimizing the gas volume, the membrane thickness and the water-side stagnant boundary layer and by maximizing the membrane area and the membrane permeability to the target gas. Passive equilibration is the slowest of all the techniques, and may not be useful if τ g cannot be reduced enough in design optimization.

Active equilibration, by comparison is complicated: the differential equations are nonlinear and coupled, the steady state response requires detailed knowledge of membrane properties and dissolved gas conditions to interpret, and it is cross-sensitive. Active equilibration is much faster than passive equilibration, and would be more suitable when an instrument requires as fast a response as possible. The operating pressure for active equilibration is a key parameter for optimization, as it determines the time response of the instrument. Active equilibration requires more sensitive gas-side instrumentation, and noise present in the measurements will be amplified when computing P w . The cross-sensitivity and required knowledge about the membrane dynamics will also introduce errors. Cross-sensitivity can be avoided by adding a mass flow meter.

Hybrid equilibration merges passive and active equilibration by alternating between the two. The dynamics during the equilibration phase of hybrid equilibration follow the straightforward passive equations. Hybrid equilibration matches the speed of active equilibration, and is not cross-sensitive. To interpret hybrid equilibration data, knowledge of the membrane properties for the gas of interest is required. Like active equilibration, the required dynamic range for the gas-side instrumentation is reduced. The main drawback of hybrid equilibration is the added complexity of interpreting the oscillatory data.

Exchange equilibration behaves like a discrete form of passive equilibration. While the equations that describe it are coupled, they are straightforward to solve analytically. Exchange equilibration can be optimized by reducing the water-side dissolved gas inventory, at the cost of reduced equilibrium gas pressure. To interpret the equilibrium value, knowledge of either the characteristic times or the inventories (i.e., volumes, solubility, temperatures, etc.) are required. There is no cross-sensitivity. Exchange equilibration performance can be improved using a vacuum or scrubbing the target gas. Exchange equilibation has higher mechanical complexity, as it generally requires active mixing and isolation valves.

Overall, we conclude that passive equilibration should be used if the design can be optimized to be fast enough for the intended application. Passive equilibration is the most straightforward to interpret, least complex, and does not require exact knowledge of the mass transport characteristics of the membrane in the system as implemented. If this is not possible, then time response correction, hybrid equilibration or active equilibration with mass flow monitoring are recommended for increasing response time. All do not incur cross-sensitivity as part of the cost of faster response. However, all amplify noise and may require much more sensitive gas-side instrumentation operating under vacuum. Exchange equilibration is more discrete in nature than the other methods, and should be chosen only for applications requiring this, such as a total dissolved inorganic carbon measurement, which requires in situ dosing of acid in a precise ratio to the sample.

We used a purely theoretical approach to derive the equations describing the dynamics of the equilibration methods, which represents a key limitation of this study. In the future, empirical evaluation of real dissolved gas instrumentation would be important to validate the results, particularly experiments to validate the design rules and cross-sensitivity observations. In addition, several key assumptions (Supplementary Section S2) were made in order to derive the analytical models in this study. These should be carefully assessed for specific designs, as these assumptions may not be valid for all instrument geometries.