Acid–base concentration swing for direct air capture of carbon dioxide

Abstract

This work demonstrates the first experimental evidence of the acid–base concentration swing (ABCS) for direct air capture of CO₂. This process is based on the effect that concentrating particular acid–base chemical reactants will strongly acidify solution, through Le Chatelier's principle, and result in outgassing absorbed CO₂. After collecting the outgassed CO₂, diluting the solution will result in a reversal of the acid–base reaction, basifying the solution and allowing for atmospheric CO₂ absorption. The experimental study examines a system that includes sodium cation as the alkalinity carrier, boric acid, and a polyol complexing agent that reversibly reacts with boric acid to strongly acidify solution upon concentration. Though the tested experimental system faces absorption rate and water capacity limitations, the ABCS process described here provides a basis for further process optimization. A generalized theoretical ABCS reaction framework is developed and different reaction orders and conditions are studied mathematically. Higher order reactions yield favorable cycle output results, reaching volumetric cycle capacity above 50 mM for third-order and 80 mM for fourth-order reactions. Optimal equilibrium constants are determined in order to guide alternative chemical searches and synthetic chemistry design targets. There is a substantial energetic benefit for reaction orders above the first, with second- and third-order ABCS cycles exhibiting a thermodynamic minimum work for the concentrating and outgassing steps around 150 kJ per mole of CO₂. A significant advantage of the ABCS is that it can be driven through well-developed and widely-deployed desalination technologies, such as reverse osmosis, with opportunities for energy recovery when recombining the concentrated and diluted streams, and extraction can occur directly from the liquid phase upon vacuum application.

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1 Introduction

Carbon dioxide removal has emerged as a key component of addressing global climate change, necessary for compensating for hard-to-avoid emissions from industries such as agriculture and aviation.^1,2 With global warming exceeding 1.3 °C,³ and already harming the most vulnerable and underprivileged populations on this planet, carbon removal will likely need to reach a scale of gigatonnes of CO₂ per year. However, the international scientific community is in agreement that reducing greenhouse gas emissions must be prioritized to avert the worst harms of climate change.⁴ Some have called for decoupling emission reduction and carbon dioxide removal targets, to avoid the dangers of mitigation deterrence and the associated moral hazard, which follow from setting massive carbon dioxide removal targets of 10–20 GtCO₂ per year.⁵

1.1 Direct air capture

Carbon dioxide removal spans a broad range of approaches, from land and ocean-based methods, such as reforestation, shifts in agricultural practices, and the addition of alkalinity into the ocean, to industrial methods, such as enhanced mineralization and direct air capture with carbon storage (DACCS).^6,7 DACCS, in particular, is a highly energy-intensive chemical process for capturing atmospheric CO₂, concentrating and compressing it, and then durably storing it in geological reservoirs or materials. Although exceptions exist, DAC methods are broadly split into two categories: one that is based on a solid sorbent and another that relies on solvent contacting. Each approach confers its own advantages. Solid sorbents, which often rely on amine polymers to react with CO₂, require only low-grade heat or moisture for regeneration, and rely on minimal water usage.⁸ Solvent-based approaches, on the other hand, require large amounts of water, but have the advantage of decoupling contacting from regeneration allowing for continuous absorption.⁹

In particular, electrochemical aqueous solvent DAC approaches are being investigated extensively.^10–14 One type of electrochemical DAC can be driven by changing the charge state of a redox-active molecule to either directly bind with dissolved inorganic carbon (DIC) or indirectly control the hydroxide concentration in solution. Such systems have the advantage of feasibly reaching low energy of capture, below 100 kJ mol⁻¹; however, the proposed approaches face a range of implementation challenges, such as chemical degradation, membrane stability and fouling, low kinetics of reaction.¹⁵ Other electrochemical approaches are driven through electrolysis reactions to create a stream of base and acid, for CO₂ capture and subsequent neutralization, respectively.¹⁶

In contrast, the approach explored in this work, the acid–base concentration swing (ABCS), is based on a new driving method that controls the concentration of solutes in solution. No redox reaction is involved, nor is the addition or removal of acids or bases from solution. Osmotic pressure drive, such as through reverse osmosis, can be used to control the concentration of reactants in solution. In this way, though finding reversible and stable candidate reactions, as well reaching high absorption kinetics, pose challenges, the ABCS paradigm opens the exploration of new aqueous DAC chemistry.

1.2 Enhancing the alkalinity concentration swing with acid–base reactions

Our previous work, characterizing the alkalinity concentration swing (ACS), reported a process driven by concentrating an alkaline solution loaded with DIC captured from atmospheric CO₂. Increasing the alkalinity and DIC concentrations together increases the solution partial pressure of CO₂ and allows for CO₂ collection upon vacuum application. However, the ACS, without modification, requires excessive amounts of water and has an extremely slow characteristic absorption timescale due to the requirement that it operate close to equilibrium with atmospheric CO₂.

We have since improved on the ACS process, by proposing the addition of a step that selects between bicarbonate and carbonate ions, which increases the volumetric cycle capacity and the absorption rate.^17–19 In this work, we propose yet another modification to the ACS, the acid–base concentration swing, which introduces additional acid and base molecules into the solvent to enhance the cycle output. The relation between the alkalinity carrier, the DIC system, and the acid–base system through reaction with water, protons, and hydroxides allows for a wide range of solution state control that can be driven through osmotic pressure. The fundamental concept underlying the ABCS enhancement is the concentration-dependent charge competition between DIC anions and the added acid anions in solution. The non-conservative nature of DIC species and the ability for carbon species to be uncharged in the form of aqueous CO₂ allows for shifting the equilibrium between DIC species in a favorable way as the solution is concentrated and diluted. At dilute concentrations the added acid is un-dissociated, and thereby uncharged, and the DIC anions, bicarbonate and carbonate, dominate. As the solution is concentrated, the acid anions replace bicarbonate and carbonate ions as negative charge carriers, shifting the DIC species to aqueous CO₂, which therefore increases the solution partial pressure of CO₂.

1.3 Non-linear chemical reactions

To develop a theoretical framework for the ABCS, we relied on past literature of non-linear chemical reactions.^20,21 The ABCS efficiency increases as cooperativity and non-linearity of the underlying chemical reactions increase. Other chemical systems, such as pH oscillators,^22–24 which can persist in isolated systems driven solely through chemical potential, and which exhibit complicated spatial and temporal behaviors, have been studied extensively and also rely on non-linear, cooperative reactions. The existence and wide-range of such non-linear chemical interactions suggests the potential of identifying and engineering chemical systems optimal to the ABCS.

Most famously, Belousov–Zhabotisnky reactions have been studied as a model chemical oscillator based on bromine chemistry;²⁵ however its reaction occurs at highly acidic conditions, making it inapplicable to high pH capture. The boric acid and polyol system, which we discuss later in this work, is better matched to the ABCS due to higher pK_a values, and it exhibits the desired qualitative behavior required for the ABCS.²⁶ Polyols, such as Mannitol or Sorbitol, react with boric acid reversibly to increase its acidity, or lower its effective pK_a. Concentrating these molecules together will, therefore, strongly decrease the pH of the solution and enhance CO₂ outgassing. Though the equilibrium constants of the boric acid and polyol system are not in the optimal range for the ABCS, the system serves as a first existence proof for non-linear acid–base systems that can be further developed or synthetically designed.

2 Acid–base concentration swing (ABCS): process description

The CO₂ separation processes explored in this work is based on the principle that the concentration and dilution of an alkaline solution shifts the solution partial pressure of CO₂. Here, we expand on principle of the previously reported alkalinity concentration swing,²⁷ and evaluate the addition of various weak acid and complexing agents to enhance the concentration swing mechanism.

Different acid–base reaction orders will exhibit different pH scaling properties, based on Le Chatelier's Principle, as solutes are concentrated or diluted. This pH control can be combined with an alkaline aqueous system to drive CO₂ absorption and outgassing. We find that the addition of higher-order acid–base reactions could be used to enhance the outgassing of CO₂ by rapidly decreasing the pH of the solution as the ions in solution are concentrated. To evaluate the system analytically, we separate the system into three components (Fig. 1).

Fig. 1 Acid–base concentration swing schematic. (A) Top panel: The feed condition at 20 mM is depicted corresponding to State 1 in (C). At dilute concentrations, no weak acid anions are present and solution partial pressure of CO₂ is low. The dashed box represents the charge concentration of carbonates (at −2 charge per ion) to illustrate that charge neutrality is maintained. Bottom panel: The post-concentration solution at 2 M is depicted corresponding to State 2 in (C). Π represents the osmotic pressure necessary for solvent–solute separation. The weak acid is the dominant anionic species, which results in non-charged DIC to be present in the form of aqueous CO₂. (B) A DIC-to-alkalinity schematic depicted modeled solution states with only alkalinity and DIC present. (C) A DIC-to-alkalinity schematic depicting a modeled 3rd-order acid system with an overlayed ABCS cycle.

First, the alkalinity carrier: the alkalinity carriers in our system are equivalent to strong base cations (denoted by [B⁺] in this work), such as sodium or potassium. The alkalinity provides a positive charge in solution, but does not directly interact with any other component of the system.

Second, the acid–base system: the addition of a weak acid and/or bases, as well as non-charged complexing agents (denoted by L), can be added to solution to add proton and hydroxide sinks and sources for the DIC species to exchange with. For simplicity, we only consider acid–base species that mediate interactions through the aqueous solvent, and do not interact directly with DIC species (e.g. such as binding of CO₂ and amines to form carbamates).

Third, dissolved inorganic carbon (DIC): aqueous CO₂, bicarbonate, and carbonate concentration and speciation are determined for different conditions, either at fixed DIC or fixed CO₂ partial pressure.

When no additional acid–base species are introduced, the system is equivalent to the alkalinity concentration swing process (Fig. 1B). There is no feasible DAC cycle possible under these conditions.

The interactions between the three components above are subject to the following two conditions: (1) charge neutrality and (2) reactions between systems are mediated through H⁺ and OH⁻ or, in other words, protons and hydroxide ions are the only species exchanged between the DIC and acid–base systems.

The states of the ABCS cycle are best represented by the arrows overlaid on the DIC-to-alkalinity diagram in Fig. 1C. This is because in a plot of DIC-to-alkalinity vs. alkalinity, concentrating and diluting steps are represented simply by horizontal lines. The ABCS cycle is described succinctly as follows:

• Step 1 → 2: concentrating acids and bases – alkalinity, DIC, and acid–base system are concentrated together. Alkalinity is used as a reference concentration, and all other species concentrations are defined in relation to alkalinity.

• Step 2 → 3: CO₂ outgassing – vacuum is applied and the solution CO₂ partial pressure is taken to p_out, thereby outgassing and collecting CO₂. The collected CO₂ quantity, or the volumetric cycle capacity, C_out, is defined in terms of moles of CO₂ collected per volume of feed solution.

• Step 3 → 4: diluting acids and bases – the concentrated solution, having lost the extracted CO₂, is diluted to its initial concentration.

• Step 4 → 1: absorbing atmospheric CO₂ – diluted solution with depleted DIC concentration is exposed to atmospheric CO₂ until a solution partial pressure of p_in is reached.

Finally, the main driving mechanism explored in this system is osmotic pressure, resulting in concentrating and diluting the solution, or equivalently adding and removing water (Fig. 1A). This can be done through a number of mechanisms, such as reverse osmosis or capacitive deionization,^28–30 which are not explored in detail here. Our previous study evaluates the work necessary to drive the concentration process.^18,19,27,31 In this work, a lower bound energy analysis is conducted to establish feasibility and evaluate parameter trade-offs.

In general, alkalinity carriers can also be entirely removed if the acid–base system includes some weak base species, however this is not considered in this work. Additionally, the charge state of the acid–base system species can also be used as a basis for separation. For example, nanofiltration membrane separation can be used to separate uncharged and charged species.^32,33

This work reports the first proof of principle experiment demonstrating the ABCS through a boric acid and polyol experimental system. The system is characterized and its limitations are discussed. A general ABCS theoretical framework is developed to explore requirements for a solution that absorbs CO₂ in a diluted state, and then outgasses CO₂ when the solution is concentrated. Requirements for the acid–base system are identified to reach adequate acidification as the solution is concentrated. In general, this work builds on the large body of research of non-linear chemical reactions,^24,34 and outlines pathways for theoretical and experimental improvement of the ABCS.

2.1 ABCS chemistry: boric acid and polyol

One experimental candidate system exhibiting desired properties is a boric acid and polyol system. Boric acid, H₃BO₃, is a very weak acid, with an ionization constant around pK_a = 9.20.²⁶ Boric acid does not react in aqueous solution as a Brønsted acid, but instead, it behaves as a Lewis acid, binding a hydroxide to form the B(OH)₄⁻ anion:A longstanding experimental observation states that the strength of boric acid can be increased through the addition of organic compounds with at least two hydroxyl groups (polyol), such as mannitol, sorbitol, glycerol, and others. In fact, boric acid reacts with polyols and other polyalcohols, either in a 1 : 1 or 1 : 2 boron to polyol molar ratio, represented by the following two reactions:

B(OH)₃ + H₂L ↔ B(OH)₂L⁻ + H₂O + H⁺ (2)

B(OH)₃ + 2H₂L ↔ BL₂⁻ + 3H₂O + H⁺ (3)

Here, L corresponds to the polyol species. The boric acid disassociation reaction is a first order reaction, while the two reactions above are second and third order reactions, respectively, corresponding to the generalized reaction schemes detailed in the ESI,† in Table C1.

The modified boric acid equilibrium constant, replaces K_a in eqn (1) and, as a function of the added polyol, takes the form of ref. 26:

Here, β₁ and β₂ are experimentally measured equilibrium constants, with log(β₁) = 3 and log(β₂) = 5,²⁶ relating to the following reactions (with mannitol as the polyol) where the anion species is treated as the reactant:

This reaction scheme can be converted to a system of equations and applied to the ABCS, and all the reactants in this system are commercially-available chemical species that can be studied experimentally. However, the following realizations makes this particular chemical system likely impractical to be realized in a scalable DAC process. First, the DIC system at atmospheric CO₂ partial pressure buffers the pH around 9 to 10. And second, the pK_a of boric acid means that much of the acid will be in a disassociated state in this pH range. This negates the cooperative effect from the polyol in the operating regime of the ABCS. In other words, if the post-absorption pH is greater than 9.3 (the boric acid pK_a), then a majority of the boric acid is already in the anionic form, and the interaction with polyols will not shift its charge state significantly. To make use of the polyol and boric acid reaction as a driver of the ABCS, the CO₂ absorption step would need to bring the pH of the solution below 9, which implies extremely slow absorption kinetics.

Alternatively, cycle output would improve if synthetic chemistry modifications to boric acid complexes were made to increase its pK_a, preferably to 10 or higher. Ideally, however, the pK_a of the weak acid is as high as possible, such that deprotonation can occur only through reaction with a complexing agent.

3 Materials and methods

3.1 Measuring solution states

To characterize the alkalinity, DIC, and boric acid and polyol system, solution states were determined by first measuring the pH of a concentrated solution prepared by mixing dry chemicals in different ratios with diH₂O. Then, taking each concentrated solution starting point and measuring the pH of the dilution series, which keeps the chemical ratios constant but reduces the concentration, allows for mapping out the pH-concentration parameter space (Fig. 2A; black dots). An Orion ROSS Micro Electrode was used to measure pH with a Thermo Orion Star A211 Benchtop pH Meter. In this work, alkalinity was introduced as sodium cations. Sodium hydroxide, sodium carbonate, sodium bicarbonate, boric acid, L-sorbitol were all purchased from Sigma-Aldrich.

Fig. 2 Experimental ABCS cycle demonstration. (A) pH-concentration plot, with each point corresponding to an equilibrated solution state locked at a 1 : 1 : 1 sodium : boric-acid : sorbitol concentration. Dashed black lines connect dots at the same DIC-to-alkalinity ratios, with alkalinity being equivalent to [Na⁺]. (B) Absorption of 10 mM sodium, boric acid, and sorbitol solution with no pre-loaded DIC from 400 ppm CO₂ in N₂ balance. (C) Direct measurement of CO₂ in gaseous phase during outgassing from State 2 into 50 mbar CO₂ in N₂ balance. (D) Visual confirmation of outgassing. Sorbitol solution was added to a sodium, DIC, and boric acid solution, which triggers CO₂ outgassing due to the acidification reaction between sorbitol and boric acid. Image contrast was modified to highlight bubble generation.

To map the pH-concentration parameter space, each prepared concentrated solution, at either 0.1 or 1 molal, was measured immediately after mixing, as well as 30 minutes after mixing to ensure that no significant CO₂ outgassing takes place at the given concentration ratio. This ensures that the CO₂ partial pressure of the prepared solution is sufficiently close to equilibrium with atmospheric conditions.

Solutions were all prepared with a fixed ratio of sodium, boric acid (B), and sorbitol (S) of: Na⁺ : B : S = 1 : 1 : 1. Four starting points were chosen with a different DIC-to-alkalinity ratio of 0, 1/8, 1/4, and 1/2. From each starting point, each solution was diluted in series by factors of 1/3 by adding diH₂O.

3.2 Absorption and outgassing

To study absorption and outgassing rates, a stirred-cell reactor (SCR) setup was assembled (ESI,† Fig. A1). Three mass flow controllers (Sierra SmartTrak 50) were installed in parallel, each connected to its own gas tank: 100% N₂, 100% CO₂, 0.4 mbar CO₂ in N₂ balance (gas tanks were purchased from AirGas). Gases were combined and mixed at a known ratio and flow rate, and bubbled through deionized water to moisten the gas feed and reduce evaporative loss. The gas line was then fed into the SCR, which was installed with conductivity, temperature, and pH probes (ThermoFisher). The SCR was placed on a stir plate and a medium-sized stir bar was spun at 200 rpm. This rpm was chosen so as not to significantly deform the solution surface, but still be fast enough to achieve a well-mixed solution condition and reach the pseudo-first order absorption regime. At 50 mL of solution loading, the gas–liquid surface area was measured to be 0.0029 m².

For each absorption experiment, 50 mL of solution was loaded into the SCR, and temperature was controlled at 25 C using a water jacket connected to a temperature-controlled water bath. Measurements were continuously recorded onto a computer for analysis. The known solution alkalinity allowed for using pH to calculate CO₂ absorption. Gas flow rates were always high enough to deviate in CO₂ concentration by <5% due to absorption. Conductivity measurements were used to ensure that evaporative loss was within <5%.

For outgassing experiments, the gas outlet was passed through a 10% infrared CO₂ sensor (CO₂Meter K30 Sensor) to measure the change in CO₂ partial pressure in nitrogen balance. Outgassing into vacuum was simulated by applying a 50 mbar CO₂ in N₂ stream to the solution. Outgassing experiments were conducted with 10 mL of solution loaded.

3.3 Modeling ideal chemical reactions

To study the relationship between DIC, alkalinity, the weak acid system, and partial pressure, we developed an Aqueous DIC Model (ADICM), which solves the system of equations detailed in Section 4.2.1 and ESI,† Section B. In this theoretical study, we assume dilute solutions in which the activity of any non-solvent species is proportional to its concentration, and therefore chemical relations can be evaluated through concentrations and fixed equilibrium constants. We also assume that there are no solubility constraints in the tested regimes, with final alkalinity conditions being studied up to 3 M. The Python package, SciPy, was used to solve the nonlinear system of equations using the “fsolve” function.

4 Results and discussion

4.1 Experimental ABCS analysis

To demonstrate the ABCS cycle, the absorption, concentrating, outgassing, and diluting steps were characterized through a combination of 400 ppm CO₂ absorption experiments, solution pH measurements, and direct CO₂ outgassing measurements into a headspace of 50 mbar of CO₂ partial pressure.

First, a solution without any pre-loaded DIC prepared at 10 mM sodium hydroxide, boric acid, and sorbitol, was exposed to 400 ppm CO₂ and allowed to reach equilibrium in the absorption module (Step 4 → 1). The pH dropped from 10.27 to 8.98 (Fig. 2B). Using the mapped out pH-concentration space as reference (Fig. 2A; black dots), we infer that the DIC-to-alkalinity ratio increased from 0 to 1/4. This is equivalent to 2.5 mM of DIC loading.

For the concentrating step (Step 1 → 2), a solution was prepared at 1 molal alkalinity concentration, with a 1/4 DIC-to-alkalinity ratio or 0.25 molal DIC, to effectively simulate a concentrating step of 100×. The concentrated solution was loaded into the outgassing module and exposed to 50 mbar CO₂ in N₂ balance. Outgassed CO₂ was measured directly in gas phase to confirm that CO₂ can be extracted from this solution condition (Fig. 2C). Initial pH was measured at 7.50 and increased to 8.02 upon equilibration with 50 mbar in the outgassing module (Step 2 → 3).

The post-outgassing solution (State 3) was taken from the outgassing module and the pH of its dilution series was measured (Fig. 2A; blue dots). Based on the pH-concentration parameter space, the DIC-to-alkalinity dropped from 1/4 to 1/8, which means that roughly 0.125 molal of DIC was outgassed. The directly measured gaseous CO₂ quantity was found to be within 10% of the solution measurement.

That the four steps enclose a finite area in the pH-concentration space demonstrates the first experimental evidence of the ABCS cycle. The full cycle, at the tested parameters, would have a volumetric cycle capacity, or C_out, of 1.25 mM extracted CO₂ relative to the feed solution. Under these unoptimized conditions, outgassing into 50 mbar of CO₂ occurred on a timescale of minutes, whereas absorption required hours at a measured average absorption flux of 0.40 μmoles m⁻² s⁻¹. In this work, because the chemical conditions are far from optimal for the absorption and cycle capacity, more detailed kinetic analysis was not pursued. However, preliminary findings suggest that absorption promoters are required to make this chemical system competitive with incumbent aqueous DAC technologies.

Finally, a visual inspection of outgassing from the concentrated solution (State 3) was performed in a 25 mL vial (Fig. 2D). Small bubbles were detected within seconds of adding sorbitol to the sodium, boric acid, and DIC solution. We note that outgassing into 50 mbar under flowing headspace exhibits different kinetics than rapid mixing in a vial under ambient conditions.

4.2 Generalized ABCS theory

Though the addition of the boric acid and polyol system represents an experimentally feasible ABCS cycle, its cycle capacity and absorption rate are impractical for a scalable DAC process. We therefore develop a theoretical model to identify optimal ABCS chemical properties. This analysis is intended to guide synthetic chemistry work and the search for different applicable chemistry systems.

4.2.1 Incorporating a weak acid system. We build on an aqueous DIC model that characterizes the CO₂ partial pressure based on alkalinity and DIC concentrations (detailed in ESI,† Section B). We introduce a generalized theoretical weak acid system that allows for evaluating reaction order, concentration, and pK_a. This framework allows for mathematically studying arbitrary reaction orders through the following chemical reaction system:

(n − 1)L + AH ↔ L_n−1A⁻ + H⁺ (7)

Here, L is a non-charged complexing species that reversibly interacts with the weak acid, AH, to deprotonate it. An nth order reaction means that n reactant species need to react together to form a product and a proton. Therefore, in this scheme, one AH molecule and n − 1 L molecules constitute an nth order system. We define K_a,_n as the equilibrium constant for the corresponding nth-order reaction. A generalized equilibrium relation can be written in the following way:Evaluating the concentration-pH scaling differences helps reveal the strength with which different reaction orders drive the ABCS process. Taking the nth order case as a generalizable example, we study a simplified system where that all protons come from AH (i.e. there is no additional proton-based reaction or buffering, and the water dissociation reaction is negligible). This corresponds to a solution without DIC, and implies that the two products in eqn (7) must be in a 1 : 1 ratio, so [L_n−1A⁻] = [H⁺].

Because the driver of the ABCS is a concentrating step that increases the concentration of all the non-solvent species together, we define a concentration factor, χ, to characterize the pH-concentration scaling. Mathematically, this is expressed by: ([AH]₂⁺ [L_n−1A⁻]₂ = [AH]₀χ). The subscript “0” represents that the total product and reactant concentration in the pre-concentrated solution (State 1) of the given species, such that: [AH]₀ = [AH]₁ + [L_n−1A⁻]₁. Equivalently, [AH]₀ is the initial weak acid concentration dissolved necessary to prepare State 1. The subscripts 1 and 2 correspond to the pre- and post-concentrating condition, respectively. Similarly, the initial complexing agent concentration is defined as follows: [L]₀ = [L]₁ + [L_n−1A⁻]₁. We further simplify the relation by assuming that the acid and complexing species concentrations are at a ratio of θ = [L]₀/[AH]₀. Taking eqn (8) and incorporating the concentration factor perturbation, we derive the following relation:

Rearranging and taking the log of both sides results in the following scaling relation:

pH = −(n/2)log₁₀(θχ) + const (10)

This relation reveals that the acidification pH increases more sharply with concentration factor at higher reaction orders, suggesting that higher reaction orders are favored in the ABCS. The concentration ratio of complexing agent to weak acid enhances acidification logarithmically. However, we note that this concentration-pH scaling relation will be modified depending on which other acid–base species are added to the solution, and at different molar ratios. A detailed scaling analysis is reported in ESI,† Section C. The concentration-pH scaling for the combined strong base and weak acid system is plotted in Fig. 6: without DIC in panel A and B, and with DIC in panel C and D, revealing the concentration-dependent pH effects that are the basis of the ABCS. ESI,† Table C1 lists the reactions and corresponding equilibrium relations, for a range of conditions, as well as the associated concentration-pH scaling of that particular reaction scheme.

4.2.2 Optimization analysis. With a theoretical basis developed, parameter optimization analysis reveals equilibrium relations and initial concentrations that maximize the ABCS cycle capacity, allowing for conditions yielding high C_out, high output partial pressure and high post-outgassing pH (labeled with a 4 subscript). High C_out values reduce water-handling requirements. High output partial pressure increases the outgassing rate and reduces vacuum pumping costs. Higher pH₄, which corresponds to higher hydroxide concentration, corresponds to higher absorption rate, which is evident through the dominant absorption reaction pathway: CO₂ + OH⁻ → HCO₃⁻.³⁵ Higher absorption rates reduce air–liquid contacting infrastructure costs and required land area. A detailed optimization analysis is conducted in ESI,† Section C.1, finding that higher reaction orders yield more favorable ABCS cycle outputs.

Fig. 3 plots the optimal points for each reaction order, maximizing C_out at different strong base ([B⁺]₀) to weak acid ([AH]₀) ratios. Non-intuitive K_a,n relations are revealed as reaction order and base-to-weak-acid ratios are varied. For all conditions, at the optimal point, higher reaction order and higher weak acid to base ratios increase the cycle capacity (C_out), maximum solution CO₂ partial pressure post-concentration but before any extraction and pre-absorption pH (pH₄). The output from the optimal C_out analysis is used to study ABCS operating conditions in the next section.

Fig. 3 ABCS optimization analysis. The optimal nth-order equilibrium constant (K_a,n) and initial strong base concentrations ([B⁺]₀) are reported for each reaction order (1, 2, 3, and 4), and for four strong base (or alkalinity) to weak acid ratios: 2 : 1, 1 : 1, 2 : 3, and 1 : 2. The corresponding C_out, , and pH₄, at each optimal point is plotted. Each optimal point corresponds to the red dot in ESI,† Fig. C2.

Whereas completely optimal conditions are unlikely to be found in existing synthetic or naturally-occurring chemical systems, this analysis serves as a guide for chemical species exploration, as well synthesis targets for chemical design.

4.2.3 ABCS optimal cycle output. Given the optimization results from the previous section, we evaluate the ABCS process output as a function of reaction order and starting concentrations. We find a key trade-off such that at lower initial concentrations (alkalinity around 10 mM), higher CO₂ partial pressure and higher post-outgassing pH is reached, whereas the cycle capacity output is lower compared to that from higher initial concentrations (alkalinity around 50 mM or more).

Table 1 reports sample process conditions for different [B⁺]₀, [AH]₀, and [L]₀ concentrations, and optimal pK_a,n values. For a second order reaction, optimal initial concentration of 50 mM base can reach C_out around 15 mM and pH₄ of 9.78. Decreasing the initial concentration to 10 mM decreases C_out to 5.8 mM, but increases pH₄ to 10.38 and more than quadruples the CO₂ partial pressure (compare Table 1 Examples 3 and 4).

Table 1 Acid–base concentration swing ideal examples. For all conditions: p_in = 0.3 mbar; p_out = 50 mbar; final alkalinity is set to 2 M. Concentrations and pK_a,_n values chosen based on optimization results from Fig. 3

Example	Reaction order	[B^+]0 : [AH]0	[B^+]0 (M)	[AH]0 (M)	[L]0 (M)	pKa,n	Cout (mM)		pH4	ECO2 (kJ mol^−1)
1	1st order	1 : 1	0.01	0.01	0	9.0	0.29	60	8.71	1110
2	2nd order	1 : 1	0.05	0.05	0.05	8.0	9.5	270	9.62	165
3	2nd order	1 : 2	0.05	0.1	0.1	9.0	15.6	320	9.78	173
4	2nd order	1 : 2	0.01	0.02	0.02	9.0	5.8	1400	10.38	134
5	3rd order	1 : 1	0.1	0.1	0.2	8.5	23.3	540	10.21	157
6	3rd order	1 : 2	0.15	0.3	0.6	9.5	46.0	570	10.18	165
7	3rd order	1 : 2	0.01	0.02	0.04	9.5	7.1	4700	11.67	145

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This trade-off is more substantial for third order reactions. Initial concentrations of 150 mM base outgasses 46 mM, reaches partial pressures above 500 mbar, and post-outgassing pH of 10.18. An initial strong base concentration of 10 mM, however, results in 7 mM of outgassed CO₂, but more than 4.5 bar of partial pressure and pH₄ of 11.67 (compare Table 1 Examples 6 and 7).

Fig. 4 plots C_out, , and pH₄, as functions of final alkalinity for a range of reaction orders. Each reaction order is modelled with the corresponding optimal initial strong base concentration and K_a,n, determined by maximizing C_out. Parameter map and optimization analysis is reported in ESI,† Fig. S3.

Fig. 4 ABCS process output. C_out (left), (middle), and pH₄ (right) are plotted as functions of final alkalinity. (A) [B⁺] : [AH] = 1 : 1; (B) [B⁺] : [AH] = 1 : 2. Each reaction order is modelled with the corresponding optimal initial concentration and pK_a,n, taken from Fig. 3. For all points, p_in = 0.3 mbar; p_out is chosen at 50 mbar and denoted with a black dashed line in the middle plots.

For each reaction order, C_out becomes positive at a final alkalinity around 1 M, suggesting that final alkalinity must at least exceed that concentration. Outputs of higher reaction order conditions increase relatively faster as a function of final alkalinity. C_out can reach 50 mM for reaction order 4 and 30 mM for reaction order 3, at a 1 : 1 base to weak acid condition (Fig. 4A). When the acid concentration is doubled (Fig. 4B), C_out is also roughly doubled for comparable parameter choices.

When optimizing on cycle capacity, the pre-absorption pH falls between 9 and 11 (Fig. 8). Hydroxide-based absorption pathways are still relatively slow for a scalable DAC process in this pH range, suggesting that CO₂ absorption promoters, such as carbonic anhydrase, will likely need to be added to the system to reach incumbent technology absorption rates that reach high pH (>12) through alkaline solutions.^36,37 It is also possible that certain chemical species that exhibit cooperative acid–base reaction properties will also act as promoters. For example, even boric acid has been shown to have modest CO₂ absorption enhancement properties.³⁸

4.2.4 Thermodynamic minimum work for concentrating and outgassing. To evaluate how the reaction order affects the minimum cycle capture energy, which we define here as the required minimum work for the concentrating and outgassing steps of the ABCS cycle, we apply a thermodynamic minimum model that we derived in previous work:²⁷Here, C₀ is the total initial solute concentration in State 1, which includes all the non-solvent species: strong base, weak acid, complexing agent, and DIC. The output is reported in units of kJ per mole of captured CO₂. And, w_vac(p_out) is the work necessary to maintain a vacuum at a given outgassing pressure.

Table 1 reports the required minimum energy for seven representative conditions. We find that the thermodynamic minimum energy is approximately 150 kJ mol⁻¹ for reaction orders two and three, for the example parameters and with p_in = 0.3 mbar, p_out = 50 mbar, and final alkalinity set to 2 M. The first order reaction, or the base ACS case, requires upwards of 1000 kJ mol⁻¹. This suggests significant benefits when implementing a multi-order system, as compared to a first order chemical reaction. Though the cycle capacity and maximum CO₂ partial pressure increase at higher reaction order, there is no significant energy difference across higher reaction orders or specific parameter choices. Based on energy costs of implementing physical reverse osmosis systems, we expect physical energy values to be at least twice the thermodynamic minimum.³⁹

For comparison, bench scale redox electrochemical DAC processes have been shown to reach 100 kJ mol⁻¹,¹⁵ and industrial solid sorbent and calcium-looping processes can reach required energy lower than 300 kJ mol⁻¹.⁴⁰

Importantly, whereas the values determined in this section represent an energy minimum, they also do not account for the possibility of energy recovery through the entire thermodynamic cycle. Specifically, during recombination of the concentrated and diluted streams, opportunities for energy recovery exist through forward osmosis or reverse electrodialysis,^41,42 as discussed in previous works evaluating the ACS process.^18,31

4.3 More effective potential acid–base systems

In Section 4.2, we discussed a theoretical framework for a cooperative acid–base system that will outgas CO₂ upon concentration, and absorb atmospheric CO₂ upon dilution. While the focus of the analysis was on a weak acid and uncharged complexing species, in general, other acid–base schemes can reach the same effective result.

Although we do not explore this theoretical chemical scheme, the cooperative reaction between the weak acid molecules directly, without the need for a complexing agent, would result in a stronger acidification driving force per reactant (for example, AH + AH ↔ A₂²⁻ + 2H⁺). In this case, the alkalinity carrier must still be a strong base, such as sodium or potassium.

Alternatively, an intermediate strength base and a weak acid system could achieve desired effects, while reducing the overall concentration of necessary species in solution by eliminating the need for alkalinity carriers, as well as a complexing agent. The following is an example of a second order reaction with such properties: B⁺ + AH ↔AB + H⁺. Here, B⁺ may come from an undissociated molecule, BOH, that is strong enough to be disassociated at pH 9–11, but also has affinity to the weak acid.

In general, although different chemical reaction pathways can be envisioned, when searching for desirable candidate chemistry schemes the following chemical and reaction properties should be satisfied: reversible reactions; solubility above 1 M; no DIC interaction; non-volatile; and non-toxic.

5 Conclusion

This experimental and theoretical analysis reveals the possibility of the ABCS as a potential approach for direct air capture of CO₂. The first-ever proof of principle experiment reported here demonstrates the feasibility of using osmotic pressure and exploiting cooperative acid–base chemistry to drive a DAC cycle. However, the low cycle capacity and slow absorption kinetics in the boric acid and sorbitol system suggest that different nonlinear chemical reactants must be explored to find a potentially scalable regime.

Given the guiding principles outlined above, if other reaction schemes can be identified and engineered to have high-order cooperativity then such a process could be competitive with incumbent DAC approaches. Importantly, the ABCS can be driven through reverse osmosis or capacitive deionization modules, which are technologies that are already well developed and benefit from decades of large-scale deployment. The thermodynamic minimum energy analysis suggests that whereas first-order reaction energies are exceedingly high, second- and third-order ABCS cycles are around 150 kJ mol⁻¹. Physical implementations of these systems will likely be a factor of two or more higher, which suggests the need for energy recovery modules, such as forward osmosis, if energy consumption is to be reduced.

It is worth noting, however, that even optimal ABCS cycles still face absorption rate challenges. For third order reaction processes, absorption pH levels can reach above 11, but not above 12, which establishes an upper bound on absorption rate. By comparison, incumbent technologies have engineered their absorption modules around pH closer to 13 or 14.⁹ This limitation can be overcome through CO₂ absorption promoters.^36,43 Despite this challenge, a significant advantage of the ABCS is its simplicity. The ABCS is driven by a single concentrating step followed by an application of vacuum to extract CO₂.

Data availability

Experimental measurement data for this article is plotted in its entirety in Fig. 2. The code associated with this modeling analysis in this work can be found in Anatoly Rinberg's PhD thesis (2024), Concentrating Alkalinity for Direct Air Capture of Carbon Dioxide: Using Osmotic Pressure for Concentration and Separation, at https://dash.harvard.edu/handle/1/37377926.

Conflicts of interest

There are no conflicts to declare.

Appendices

A Absorption and outgassing setup

A setup to measure absorption and outgassing from a candidate solution interacting with a controlled headspace gas was assembled and used for testing absorption from 0.4 mbar of CO₂ and outgassing into 50 mbar of CO₂ (Fig. 5).

Fig. 5 Absorption and outgassing setup. (A) Schematic of mass flow controllers and reactor vessel. (B) Image of setup.

B Dissolved inorganic carbon equilibrium relations

First, it is worth understanding the pH dependence of the ACS, without any added acid–base species. The charge neutrality condition necessitates:

A = b + 2c + K_w/h − h (B1)

Here A, b, c, and h are the alkalinity, bicarbonate, carbonate, and proton concentrations, respectively, and the equilibrium relations between the DIC species are defined as follows:

H_cp = a/p_CO2 (B2)

where a corresponds to the aqueous CO₂ concentration and p_CO2 is the CO₂ partial pressure of the gas in equilibrium with the solution. For an accurate treatment, the activity of the species must be considered, however, in this work, we assume dilute solution conditions to simplify the analysis. The conditions for CO₂ capture in the ABCS take place roughly with pH between 9 and 11. Therefore, the hydroxide and proton concentrations are much smaller than the rest of the alkalinity and DIC species. Plugging in the DIC equilibrium relations, the above equation can be rewritten as a function of DIC concentration, C_DIC:Here, f_DIC(h) is a function specific to the DIC system that relates alkalinity to C_DIC through the proton concentration. This allows for investigating the concentration perturbation, which can be mathematically written through the concentration factor, χ. The concentration factor applies to all non-aqueous species in solution together: C_i = χC_i,0, where C_i is the concentration of species i in solution. Applying the concentration factor perturbation to the DIC system gives:

This reveals that, because the χ terms cancel out, pH is invariant to concentration factor in this regime. In fact, the solution CO₂ partial pressure will increase linearly with χ, but will not be further enhanced due to a shift in pH. The black line in Fig. 6C confirms this derivation. In reality, the equilibrium relations, K₁ and K₂, depend on the ionic strength and will result in a shift in pH as DIC solution is concentrated. Past analyses reveal that the ionic strength increases the outgassing efficiency by a factor of 30–50% depending on the regime,²⁷ however, these effects are not considered in this work for simplicity.

Fig. 6 Modelled pH as a function of fixed concentration. Abscissa corresponds to the concentration of the strong base in solution, and is locked to the weak acid concentration based on the corresponding ratio. (A) pH of a solution with no DIC and a base to weak acid ratio of 1 : 1 is evaluated for different reaction orders (Table 2). (B) Same conditions, but at a base to weak acid ratio of 1 : 2 is plotted. (C) Same conditions as (A) are plotted, but with the addition of 66% DIC relative to strong base concentration. (D) Same conditions as (B) are plotted, but with the addition of 66% DIC relative to strong base concentration. Optimal K_a,n is picked for each reaction order based on optimization analysis in subsequent sections.

C Weak acid scaling relations

When the ratio of strong base to AH initial concentration is 1 : 1 (Fig. 6A), the pH increases for all reaction orders, but at varying rates, decreasing with higher reaction order. When the ratio of strong base to AH initial concentration is 1 : 2 (Fig. 6B), the pH first increases and then decreases as cooperative concentration effects begin to dominate. This non-monotonic effect is present only for reaction orders above 2. It further reveals that is possible to reach high pH regimes at lower concentrations, and then acidify a solution upon concentrating, which corresponds to the operating principle of ABCS.

Fig. 6C and D plots the concentration pH dependence of a 1 : 1 and 1 : 2 strong base to weak acid solution with the DIC system added. This shows that at a given solution loaded with DIC at lower concentration, can be concentrated and acidified, thereby increasing the partial pressure of CO₂. Higher weak acid to base ratios increase acidification as a function of concentration, but also lower the concentration point at which the pH deviates from the strong base line. In general, this deviation is due to the fact that at low concentrations AH is entirely associated and uncharged, and so there is no anion to negate the strong base.

Finally, as with the alkalinity and DIC analysis in the previous section, a charge neutrality condition with the weak acid system can be analyzed:

[B⁺] = [HCO₃⁻] + 2[CO₃²⁻] + [L_n−1A⁻] + [OH⁻] − [H⁺] (C1)

In the conditions where hydroxide and proton concentrations are significantly smaller than those of the other non-conservative ions, the equation above simplifies to:

A = f_DIC(h)C_DIC + f_a,n(h)αⁿ (C2)

Here, f_a,n(h) is a function specific to the nth order condition relating the charge state of the acid to the pH of the solution, and α is the total concentration of the weak acid, assuming that the non-charged species, L, is of equal concentration. Adding the concentration factor perturbation, a pH-χ dependence is revealed:

Inspecting the equations confirms the concentration-dependent pH effects that are the basis of the ABCS. Fig. 6 plots the solution to the equation above for various parameter choices.

Table 2 Generalized reaction schemes

Condition	Base	Acid	Reaction	Equilibrium relation	χ scaling at high conc.
Strong base	B^+	—	—	—	ΔpH ∼ log(χ)
1st order acid	—	AH	AH ↔ A^− + H^+		ΔpH ∼ −(1/2) × log(χ)
2nd order acid	—	AH	L + AH ↔ LA^− + H^+		ΔpH ∼ −log(χ)
3rd order acid	—	AH	2L + AH ↔ L2A^− + H^+		ΔpH ∼ −(3/2) × log(χ)
nth order acid	—	AH	(n −1)L + AH ↔ Ln−1A^− + H^+		ΔpH ∼ −(n/2) × log(χ)
Carbonate system	HCO3^−	CO2(aq)	CO2(aq) + H2O ↔ HCO3^− + H^+		ΔpH ∼ −(1/2) × log(χ)
	CO3^2−	HCO3^−	HCO3^− ↔ CO3^2− + H^+		ΔpH ∼ −(1/2) × log(χ)

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C.1 Parameter space analysis and optimization. DIC-to-alkalinity diagram analysis in Fig. 7, relating the DIC-to-alkalinity ratio, the alkalinity, CO₂ partial pressure, and reaction order, allows for graphically understanding the possible concentration swing improvements at higher reaction orders. Each x-axis alkalinity value corresponds to an equivalent strong base concentration, and a fixed ratio of added weak acid (denoted in the bottom of each DIC-to-alkalinity diagram). In other words, at a 1 : 1 base to weak acid ratio, alkalinity of 0.1 M corresponds to 0.1 M of weak acid. A 1 : 2 ratio would imply that 0.1 M of alkalinity corresponds to 0.2 M of weak acid. This graphical representation projects to an ideal concentration step when moving right in a fixed DIC-to-alkalinity line, or dilution when moving left.

Fig. 7 DIC-to-alkalinity diagram for different reaction orders. DIC-to-alkalinity is plotted as a function of initial strong base concentration, or equivalently [B⁺]₀. The alkalinity to weak acid ratio is specified for each plot. Lines correspond to iso-pCO₂ values, with the red line corresponding to 0.4 mbar, and the green line corresponding to 50 mbar. Black lines from bottom up correspond to: 1, 10, 100, and 1000 mbar. Red region is eligible parameter space for absorption, and green region is eligible parameter space for outgassing (given the arbitrary choice of p_out = 50 mbar). Optimal K_a,n are selected based on optimization analysis in Fig. 3.

For all reaction orders, increasing alkalinity concentration results in increasing CO₂ partial pressure (Fig. 7). However, higher order reactions compress the iso-pCO₂ lines as a function of concentration, which results in a relatively higher partial pressure increase. At higher alkalinity (above 0.1 M), higher reaction orders lowers the DIC-to-alkalinity ratio at 50 mbar (green line), which implies that even more CO₂ can be outgassed upon concentration if the same amount of DIC is loaded into solution. Higher weak acid to alkalinity ratios result in further compressing of the iso-pCO₂ lines and lowering the DIC-to-alkalinity ratio at 50 mbar (green line) at high concentrations. The pK_a,n values for the DIC-to-alkalinity diagrams in Fig. 7 were chosen based on an analysis of optimal C_out conditions.

Fig. 8 plots C_out, , and pH₄ (pre-absorption pH), as functions of K_a,n and initial strong base concentration, [B⁺]₀. By evaluating this parameter landscape, an optimal pK_a,n and [B⁺]₀ can be chosen to maximize the cycle capacity, C_out. Plots A, B, C, and D in Fig. 8 correspond to first, second, third, and fourth order reactions at a 1 : 1 base to weak acid ratio. The red dot corresponds to the optimal parameter point where the process has the largest cycle capacity, revealing the trade-off between maximum C_out and absorption rate, which is captured by the pH₄ metric. [B⁺]₀ values that are lower than the optimal red point, can increase the post-outgassing pH by a unit or more, while decreasing the C_out value.

Fig. 8 ABCS output parameter plot. Each point calculates the output from a theoretical ABCS process with corresponding parameters. Contour plots report the C_out (left), (middle), and pH₄ (right), as a function of K_a,n and initial strong base concentration, [B⁺]₀. For each point, p_in = 0.3 mbar, p_out = 50 mbar, and final alkalinity is 2 M. A, B, C, and D correspond to first, second, third, and fourth order reaction conditions; the red dot in each plot corresponds to the optimal K_a,n and [B⁺]₀ value that maximizes C_out.

Acknowledgements

This research was supported by the U.S. Department of Energy Office of Fossil Energy and Carbon Management through contract DE-FE000331964 and the Harvard Climate Change Solutions Fund. We acknowledge helpful discussions with Andrew Bergman, Tommy George, Eric Fell, and Dawei Xi, as well as Professors Daniel P. Schrag, David Keith, and Jarad Mason.

References

1. Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change, ed. V. Masson-Delmotte, P. Zhai, A. Pirani, S. L. Connors, C. Péan and S. Berger, et al., Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, 2021.
2. A. Bergman and A. Rinberg, The Case for Carbon Dioxide Removal: From Science to Justice, in CDR Primer, 2021. Available from: https://cdrprimer.org/read/chapter-1.
3. R. A. Rohde and Z. Hausfather, The Berkeley Earth Land/Ocean Temperature Record, Earth System Sci. Data, 2020, 12(4), 3469–3479 . Available from: https://essd.copernicus.org/articles/12/3469/2020/.
4. IPCC, Global warming of 1.5 °C, 2018.
5. D. Lenzi, The ethics of negative emissions, Global Sustainability, 2018, 1, e7,  DOI:10.1017/sus.2018.5.
6. J. C. Minx, W. F. Lamb, M. W. Callaghan, S. Fuss, J. Hilaire and F. Creutzig, et al., Negative emissions—Part 1: Research landscape and synthesis, Environ. Res. Lett., 2018, 13(6), 063001,  DOI:10.1088/1748-9326/aabf9b.
7. S. Fuss, W. F. Lamb, M. W. Callaghan, J. Hilaire, F. Creutzig and T. Amann, et al., Negative emissions—Part 2: Costs, potentials and side effects, Environ. Res. Lett., 2018, 13(6), 063002,  DOI:10.1088/1748-9326/aabf9f.
8. E. S. Sanz-Pérez, C. R. Murdock, S. A. Didas and C. W. Jones, Direct Capture of CO₂ from Ambient Air, Chem. Rev., 2016, 116(19), 11840–11876,  DOI:10.1021/acs.chemrev.6b00173.
9. D. W. Keith, G. Holmes, D. St Angelo and K. Heidel, A Process for Capturing CO₂ from the Atmosphere, Joule, 2018, 2(8), 1573–1594 . Available from: https://linkinghub.elsevier.com/retrieve/pii/S2542435118302253.
10. M. Wang, H. J. Herzog and T. A. Hatton, CO₂ Capture Using Electrochemically Mediated Amine Regeneration, Ind. Eng. Chem. Res., 2020, 59(15), 7087–7096,  DOI:10.1021/acs.iecr.9b05307.
11. Y. Liu, H. Z. Ye, K. M. Diederichsen, T. Van Voorhis and T. A. Hatton, Electrochemically mediated carbon dioxide separation with quinone chemistry in salt-concentrated aqueous media, Nat. Commun., 2020, 11(1), 2278 . Available from: https://www.nature.com/articles/s41467-020-16150-7.
12. S. Jin, M. Wu, R. G. Gordon, M. J. Aziz and D. G. Kwabi, pH swing cycle for CO₂ capture electrochemically driven through proton-coupled electron transfer, Energy Environ. Sci., 2020, 13(10), 3706–3722 . Available from: https://pubs.rsc.org/en/content/articlelanding/2020/ee/d0ee01834a.
13. H. Seo and T. A. Hatton, Electrochemical direct air capture of CO₂ using neutral red as reversible redox-active material, Nat. Commun., 2023, 14(1), 313 . Available from: https://www.nature.com/articles/s41467-023-35866-w.
14. A. Sodiq, Y. Abdullatif, B. Aissa, A. Ostovar, N. Nassar and M. El-Naas, et al., A review on progress made in direct air capture of CO₂, Environ. Technol. Innovation, 2023, 29, 102991 . Available from: https://www.sciencedirect.com/science/article/pii/S235218642200414X.
15. T. N. D. Cao, S. W. Snyder, Y. I. Lin, Y. J. Lin, S. Negi and S. Y. Pan, Unraveling the Potential of Electrochemical pH-Swing Processes for Carbon Dioxide Capture and Utilization, Ind. Eng. Chem. Res., 2023, 62(49), 20979–20995,  DOI:10.1021/acs.iecr.3c02183.
16. G. Liu, A. Yang and R. C. Darton, Numerical Modeling and Comparative Analysis of Electrolysis and Electrodialysis Systems for Direct Air Capture, ACS Sustainable Chem. Eng., 2024, 12(10), 3951–3965,  DOI:10.1021/acssuschemeng.3c06259.
17. A. Rinberg and M. J. Aziz, Bicarbonate-Carbonate Selectivity through Nanofiltration for Direct Air Capture of Carbon Dioxide, ACS ES&T Eng., 2024, 4(8), 1959–1969,  DOI:10.1021/acsestengg.4c00150.
18. A. Rinberg, Concentrating Alkalinity for Direct Air Capture of Carbon Dioxide: Using Osmotic Pressure for Concentration and Separation, 2024, Available from: https://dash.harvard.edu/handle/1/37377926.
19. A. M. Bergman, Using Capacitive Deionization for Direct Air Capture of Carbon Dioxide: Theory and Demonstration of the Bicarbonate-Enriched Alkalinity Concentration Swing, 2024, Available from: https://dash.harvard.edu/handle/1/37377928.
20. I. R. Epstein and K. Showalter, Nonlinear Chemical Dynamics: Oscillations, Patterns, and Chaos, J. Phys. Chem., 1996, 100(31), 13132–13147,  DOI:10.1021/jp953547m.
21. F. Sagués and I. R. Epstein, Nonlinear chemical dynamics, Dalton Trans., 2003,(7), 1201–1217 . Available from: https://pubs.rsc.org/en/content/articlelanding/2003/dt/b210932h.
22. M. Orbán, K. Kurin-Csörgei and I. R. Epstein, pH-regulated chemical oscillators, Acc. Chem. Res., 2015, 48(3), 593–601.
23. E. Poros, V. Horváth, K. Kurin-Csörgei, I. R. Epstein and M. Orbán, Generation of pH-Oscillations in Closed Chemical Systems: Method and Applications, J. Am. Chem. Soc., 2011, 133(18), 7174–7179,  DOI:10.1021/ja2010835.
24. K. Kovacs, R. E. McIlwaine, S. K. Scott and A. F. Taylor, pH oscillations and bistability in the methylene glycol–sulfite–gluconolactone reaction, Phys. Chem. Chem. Phys., 2007, 9(28), 3711–3716 . Available from: https://pubs.rsc.org/en/content/articlelanding/2007/cp/b704407k.
25. Controlling chaos in the Belousov—Zhabotinsky reaction | Nature. Available from: https://www.nature.com/articles/361240a0.
26. M. C. C. Azevedo and A. M. V. Cavaleiro, The Acid–Base Titration of a Very Weak Acid: Boric Acid, J. Chem. Educ., 2012, 89(6), 767–770,  DOI:10.1021/ed200180j.
27. A. Rinberg, A. M. Bergman, D. P. Schrag and M. J. Aziz, Alkalinity Concentration Swing for Direct Air Capture of Carbon Dioxide, ChemSusChem, 2021, 14(20), 4439–4453 . eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/cssc.202100786.
28. Ion Selectivity in Brackish Water Desalination by Reverse Osmosis: Theory, Measurements, and Implications, Environ. Sci. Technol. Lett., 2020, 71, 42–47 DOI:10.1021/acs.estlett.9b00686.
29. Y. S. Oren and P. M. Biesheuvel, Theory of Ion and Water Transport in Reverse-Osmosis Membranes, Phys. Rev. Appl., 2018, 9(2), 024034,  DOI:10.1103/PhysRevApplied.9.024034.
30. M. E. Suss, S. Porada, X. Sun, P. M. Biesheuvel, J. Yoon and V. Presser, Water desalination via capacitive deionization: what is it and what can we expect from it?, Energy Environ. Sci., 2015, 8(8), 2296–2319 . Available from: https://pubs.rsc.org/en/content/articlelanding/2015/ee/c5ee00519a.
31. A. Rinberg, A. M. Bergman, M. J. Aziz and D. P. Schrag, Experimental Demonstration of Alkalinity Concentration Swing for Direct Air Capture of CO₂. Harvard Univ., Cambridge, MA (United States), 2023. Available from: https://www.osti.gov/biblio/2248073.
32. R. He, C. Dong, S. Xu, C. Liu, S. Zhao and T. He, Unprecedented Mg²⁺/Li⁺ separation using layer-by-layer based nanofiltration hollow fiber membranes, Desalination, 2022, 525, 115492 . Available from: https://www.sciencedirect.com/science/article/pii/S0011916421005634.
33. B. Van der Bruggen, A. Koninckx and C. Vandecasteele, Separation of monovalent and divalent ions from aqueous solution by electrodialysis and nanofiltration, Water Res., 2004, 38(5), 1347–1353 . Available from: https://www.sciencedirect.com/science/article/pii/S0043135403006420.
34. T. K. Gustafsson and K. V. Waller, Nonlinear and adaptive control of pH, Ind. Eng. Chem. Res., 1992, 31(12), 2681–2693 . Available from: https://pubs.acs.org/doi/pdf/10.1021/ie00012a009.
35. X. Wang, W. Conway, R. Burns, N. McCann and M. Maeder, Comprehensive Study of the Hydration and Dehydration Reactions of Carbon Dioxide in Aqueous Solution, J. Phys. Chem. A, 2010, 114(4), 1734–1740 , Available from: https://pubs.acs.org/doi/pdf/10.1021/jp909019u.
36. S. S. W. Effendi and I. S. Ng, The prospective and potential of carbonic anhydrase for carbon dioxide sequestration: a critical review, Process Biochem., 2019, 87, 55–65 . Available from: https://www.sciencedirect.com/science/article/pii/S1359511319310037.
37. A. de Oliveira Maciel, P. Christakopoulos, U. Rova and I. Antonopoulou, Carbonic anhydrase to boost CO₂ sequestration: Improving carbon capture utilization and storage (CCUS), Chemosphere, 2022, 299, 134419 . Available from: https://www.sciencedirect.com/science/article/pii/S0045653522009122.
38. U. K. Ghosh, S. E. Kentish and G. W. Stevens, Absorption of carbon dioxide into aqueous potassium carbonate promoted by boric acid, Energy Proc., 2009, 1(1), 1075–1081 . Available from: https://www.sciencedirect.com/science/article/pii/S187661020900143X.
39. M. Elimelech and W. A. Phillip, The Future of Seawater Desalination: Energy, Technology, and the Environment, Science, 2011, 333(6043), 712–717 . Available from: https://science.sciencemag.org/content/333/6043/712.
40. S. Pacala, M. Al-Kaisi, M. Barteau, E. Belmont, S. Benson and R. Birdsey, et al., Negative Emissions Technologies and Reliable Sequestration: A Research Agenda, The National Academies Press, Washington, DC, 2019 DOI:10.17226/25259.
41. Effect of Different Draw Solutions on Concentration Polarization in a Forward Osmosis Process: Theoretical Modeling and Experimental Validation. Industrial & Engineering Chemistry Research, Available from: https://pubs.acs.org/doi/full/10.1021/acs.iecr.2c03723.
42. R. A. Tufa, S. Pawlowski, J. Veerman, K. Bouzek, E. Fontananova and G. di Profio, et al., Progress and prospects in reverse electrodialysis for salinity gradient energy conversion and storage, Appl. Energy, 2018, 225, 290–331 . Available from: https://www.sciencedirect.com/science/article/pii/S0306261918306792.
43. P. C. Tseng, W. S. Ho and D. W. Savage, Carbon dioxide absorption into promoted carbonate solutions, AIChE J., 1988, 34(6), 922–931 . eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/aic.690340604.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4ya00251b

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