Kinetics of cooperative CO2 adsorption in diamine-appended variants of the metal–organic framework Mg2(dobpdc)†

Carbon capture and sequestration is a key element of global initiatives to minimize anthropogenic greenhouse gas emissions. Although many investigations of new candidate CO2 capture materials focus on equilibrium adsorption properties, it is also critical to consider adsorption/desorption kinetics when evaluating adsorbent performance. Diamine-appended variants of the metal–organic framework Mg2(dobpdc) (dobpdc4− = 4,4′-dioxidobiphenyl-3,3′-dicarboxylate) are promising materials for CO2 capture because of their cooperative chemisorption mechanism and associated step-shaped equilibrium isotherms, which enable large working capacities to be accessed with small temperature swings. However, the adsorption/desorption kinetics of these unique materials remain understudied. More generally, despite the necessity of kinetics characterization to advance adsorbents toward commercial separations, detailed kinetic studies of metal–organic framework-based gas separations remain rare. Here, we systematically investigate the CO2 adsorption kinetics of diamine-appended Mg2(dobpdc) variants using a thermogravimetric analysis (TGA) assay. In particular, we examine the effects of diamine structure, temperature, and partial pressure on CO2 adsorption and desorption kinetics. Importantly, most diamine-appended Mg2(dobpdc) variants exhibit an induction period prior to reaching the maximum rate of CO2 adsorption, which we attribute to their unique cooperative chemisorption mechanism. In addition, these materials exhibit inverse Arrhenius behavior, displaying faster adsorption kinetics and shorter induction periods at lower temperatures. Using the Avrami model for nucleation and growth kinetics, we determine rate constants for CO2 adsorption and quantitatively compare rate constants among different diamine-appended variants. Overall, these results provide guidelines for optimizing adsorbent design to facilitate CO2 capture from diverse target streams and highlight kinetic phenomena relevant for other materials in which cooperative chemisorption mechanisms are operative.


Introduction
Steadily rising atmospheric CO 2 levels and the associated increase in average global temperatures have created an urgent need to curb anthropogenic CO 2 emissions. 1 While long-term solutions to this challenge necessitate a shi to renewable energy sources, fossil fuels will continue to supply a major portion of global energy in the near future. 2 One proposed strategy to mitigate atmospheric CO 2 emissions in the short term is carbon capture and sequestration from major point sources, such as power plants. 3 To realize this strategy, carbon capture materials are needed that possess high selectivities and CO 2 capacities, as well as minimal energy requirements for CO 2 desorption. Rapid kinetics to bind and release CO 2 are also critical, because they can dictate bed utilization and thus impact the cost and efficiency of a carbon capture process. 4 Aqueous solutions of organic amines are a mature CO 2 capture technology, but they suffer from numerous drawbacks. For example, solutions with relatively low amine concentrations are necessary to minimize corrosive effects, thereby decreasing the solution CO 2 absorption capacities and increasing the energy required to heat the absorbent during temperatureswing cycling. 5 As a result, the implementation of aqueous amine scrubbers in a power plant places a parasitic load of 25-30% on the net power output. 6 Furthermore, aqueous amine solutions are prone to thermal and oxidative degradation. 3 As an alternative, porous solid adsorbents have been proposed as capture materials, owing to their lower heat capacities and high surface areas, which create the potential for high adsorption capacities and more efficient adsorption-desorption cycling. Nevertheless, many of these solid adsorbents fail to capture CO 2 selectively from humid gas streams. 7 Amine-functionalized solid adsorbents combine the advantages of aqueous amine solutions and porous solid adsorbents. Examples of these materials include amine-functionalized silicas, [8][9][10][11] porous polymers, 12,13 zeolites, 14,15 and metal-organic frameworks. [16][17][18][19][20] Owing to their high crystallinity and chemical adjustability, metal-organic frameworks possess ordered structures that can be tailored with respect to pore size, shape, and chemical environment. In particular, amine functionalities can be incorporated within the organic linkers of these materials, both during framework synthesis 21 or through postsynthetic modication. [16][17][18][19][20]22 Additionally, the high internal surface areas accessible with metal-organic frameworks can allow for rapid diffusion of CO 2 through the pores. 23 Diamine-appended variants of the metal-organic framework Mg 2 (dobpdc) [24][25][26][27][28][29][30][31] (dobpdc 4À ¼ 4,4 0 -dioxidobiphenyl-3,3 0 -dicarboxylate) represent a particularly promising class of aminefunctionalized frameworks, as they readily adsorb CO 2 through a chemically-specic, cooperative mechanism ( Fig. 1). 25 Here, the metal-bound amine reacts covalently with CO 2 to generate a carbamate while the pendent amine is concomitantly protonated. This process propagates through the material to yield chains of ammonium carbamate stabilized through ionic interactions along the pore axis. As a result of this unique capture mechanism, diamine-appended variants of Mg 2 (dobpdc) exhibit step-shaped CO 2 adsorption proles, which give rise to large CO 2 cycling capacities that are accessible with relatively small temperature swings. 25 Importantly, by varying the metal cation, 25 diamine, [26][27][28] or organic linker, 28 the adsorption step position can be tuned in pressure by over ve orders of magnitude (from $10 À5 to $1 bar at 40 C) to enable the precise targeting of specic CO 2 separation conditions. Moreover, these materials have also been shown to maintain high CO 2 working capacities aer 1000 adsorption/desorption cycles under humid gas streams. 27 While the thermodynamics of CO 2 capture in diamineappended variants of Mg 2 (dobpdc) are promising for numerous carbon capture applications, the kinetics also play a crucial role in the practical application of these materials. For porous solid adsorbents, small-scale breakthrough experiments are oen used to simulate a xed-bed adsorption process. In these experiments, shaped particles of the adsorbent are packed into a column, a CO 2 -containing gas stream is fed through the inlet, and the outlet composition and ow rate are measured as a function of time until CO 2 "breaks through." Multiple kinetic parameters can inuence the performance of an adsorbent in a xed bed, including interparticle, intraparticle, and micropore diffusional resistances, as well as potential reaction limitations for amine-based chemisorption of CO 2 . Critically, the overall CO 2 adsorption kinetics must be sufficiently fast to maximize bed utilization in the process. Promising initial results have been obtained for diamine-appended Mg 2 (dobpdc) variants in gram-scale breakthrough experiments for CO 2 capture from simulated coal ue gas (15% CO 2 in N 2 ) under both dry and humid conditions. 27 In addition, rapid cycle times have been employed for diamine-appended Mg 2 (dobpdc) variants in simulated temperature-swing experiments under 15% CO 2 with a pure CO 2 purge, and here cycle times were limited only by the temperature ramp rate of the thermogravimetric analyzer. 27,28 Given the promise of diamine-appended Mg 2 (dobpdc) for CO 2 capture applications, a detailed analysis of the kinetics of CO 2 adsorption in these materials is necessary for optimal process implementation. Toward this end, we herein utilize thermogravimetric analysis (TGA) to systematically investigate the dry CO 2 adsorption kinetics in diamine-appended variants of Mg 2 (dobpdc) under a range of adsorption conditions. Our results demonstrate the inuence of adsorption temperature, CO 2 concentration, and diamine structure on the rate of adsorption. On the basis of these correlations, we conclude with guidelines for the optimization of adsorbent structure and process parameters in CO 2 capture applications.

Results and discussion
Experimental setup for a thermogravimetric assay to study CO 2 adsorption kinetics Diamine-appended variants of Mg 2 (dobpdc) were synthesized using our previously reported procedure. 26 Specically, methanol-solvated Mg 2 (dobpdc) was soaked in a 20% (v/v) toluene solution of the diamine of interest, and subsequent ltration, washing with toluene, and activation yielded adsorbents with one diamine per metal site. To ensure internal consistency for this study, all experiments were performed with the same batch of Mg 2 (dobpdc). The rod-shaped crystallites in this batch exhibited an aspect ratio of $10 and were heterogeneous in size, with widths ranging from $60 nm to >1 mm (Fig. S1 †). Measurements in this study were carried out on assynthesized diamine-Mg 2 (dobpdc) samples in powder form; future studies will explore adsorption/desorption kinetics in structured forms of these materials.
We developed a TGA assay at atmospheric pressure to evaluate the CO 2 adsorption kinetics of diamine-appended Mg 2 (dobpdc). In this assay, samples were rst activated in the TGA furnace under owing N 2 to remove any captured CO 2 , solvent, or excess diamine present in the pores. Next, samples were cooled to a temperature of interest, the gas ow was switched to a CO 2 -containing stream, and the change in mass was monitored as a function of time (Fig. 2). Because N 2 adsorption is negligible at and above room temperature in these materials, 26,27 we approximated that all mass increase was due to CO 2 adsorption. In addition, mass changes due to buoyancy effects upon switching the gas stream from N 2 to CO 2 were negligible compared to the mass of CO 2 adsorbed by the standard sample size of 3 mg of diamine-Mg 2 (dobpdc) used in this work (Fig. S2 †).
This TGA-based assay, which has been reported for fundamental kinetics characterization of other solid CO 2 adsorbents, 11,32-35 is advantageous given its simple setup, the small sample mass requirement, and the fact that adsorbents can be activated in situ and rapidly tested under many temperatures and partial pressures of CO 2 . However, this assay also has some limitations. For example, because TGA detects gas adsorption through a change in sample mass, it cannot discriminate between CO 2 , H 2 O, and N 2 adsorption (although, as mentioned above, N 2 adsorption is negligible for diamine-appended Mg 2 (dobpdc) variants under the conditions relevant for CO 2 chemisorption). As a result, determining CO 2 adsorption kinetics under humid conditions remains challenging, and here we discuss only dry adsorption/desorption kinetics proles. In addition, the rst few points of kinetics data from a TGA experiment are obscured by exchange of the initial gas in the furnace with the CO 2 -containing stream in adsorption experiments or the N 2 purge stream in desorption experiments.
As has been previously noted, 11 differences in ow rate and sample mass can also greatly affect TGA adsorption kinetics proles. For example, in the case of m-2-m-Mg 2 (dobpdc) (m-2m ¼ N,N 0 -dimethylethylenediamine), we found that the sample mass has a substantial impact on adsorption kinetics from a simulated coal ue gas stream of 15% CO 2 in N 2 , 36 with larger samples displaying slower overall adsorption kinetics (Fig. S3 †). Therefore, to maintain consistency across all samples in this study, we used a sample mass of 3 mg and ensured the powder was evenly distributed across the surface of the TGA pan. Also consistent with previous reports, faster adsorption kinetics were observed with faster ow rates (Fig. S3 †). This ow rate effect could be due to the time required to completely exchange the initial gas in the furnace and/or to the observed kinetics being inuenced by a mass transfer resistance related to diffusion. As a result, the kinetics of adsorption presented in this work likely represent lower bounds on the intrinsic adsorption kinetics of these materials. For the fastest ow rates tested (>100 mL min À1 ), adsorption was essentially complete within the rst few data points collected on the TGA, making it difficult to quantitatively compare the kinetics among different adsorbents under these conditions. We therefore used a consistent ow rate of 25 mL min À1 for all experiments to facilitate quantitative comparisons among the diamine-appended variants. Under these conditions, a consistent delay of 19 s was observed before the sample mass increased, aer switching the TGA valving to the analysis gas. This delay time corresponds to the time required for CO 2 to reach the sample in the TGA furnace. Additionally, given a ow rate of 25 mL min À1 , we approximate that at least 14-23 s are required at temperatures ranging from 120 to 30 C for complete exchange of gases in the TGA furnace, beyond the initial 19 s delay (Table S1 †). By accounting for these considerations in the measurements detailed below, we were able to compare the effects of temperature, CO 2 partial pressure, and diamine structure on the CO 2 adsorption kinetics of these materials.
Adsorption kinetics and temperature dependence for m-2-m-Mg 2 (dobpdc) We utilized our optimized TGA assay to characterize the CO 2 adsorption kinetics of m-2-m-Mg 2 (dobpdc), the rst reported diamine-appended variant of Mg 2 (dobpdc), 24 at selected temperatures. Extensive gas adsorption, structural, and spectroscopic data have been previously reported for this material, [24][25][26] and an overview of the adsorption mechanism is shown in Fig. 3a. We rst investigated the CO 2 adsorption kinetics of m-2-m-Mg 2 (dobpdc) from a pure CO 2 stream at atmospheric pressure over a range of temperatures below the adsorption step Fig. 2 Overview of TGA adsorption kinetics assay. The diamineappended metal-organic framework sample is activated within a thermogravimetric analyzer under flowing N 2 to remove solvent molecules and non-metal-bound diamines from the pores. The sample is then cooled under N 2 to the temperature of interest, the gas flow is switched to a CO 2 -containing stream, and CO 2 adsorption is monitored as a function of time.
temperature (T step ) of 127 C, 26 dened here as the onset of the step-shaped adsorption isobar (Fig. 3b). Rapid uptake of CO 2 was observed below T step , as is evident in a plot of fraction of diamine sites occupied (Q t ) vs. time (Fig. 3c). Interestingly, adsorption of CO 2 is faster at lower temperatures and follows an inverse Arrhenius behavior (Fig. 3d). Such inverse Arrhenius behavior has been observed previously for a polyethylenimineappended mesoporous silica 11 and amine-functionalized carbon nanotubes, 37 whereas normal Arrhenius behavior has been observed for other amine-appended solid adsorbents. 34,35,38 Our results suggest that although the equilibrium capacity of m-2-m-Mg 2 (dobpdc) is relatively insensitive to temperature below T step , 26 lower adsorption temperatures promote more rapid saturation with CO 2 . A more in-depth discussion of this phenomenon is provided below (see "Inverse Arrhenius behavior").
In addition to this inverse Arrhenius behavior, we also observed an induction period between the time at which the mass begins to increase, which corresponds to CO 2 rst entering the furnace, and the time associated with the fastest rate of CO 2 uptake (Fig. 3d). This effect is particularly pronounced at temperatures close to T step . To elucidate whether the observed induction period is intrinsic to the CO 2 adsorption kinetics of m-2-m-Mg 2 (dobpdc) or is an artifact of the experimental setup, we investigated CO 2 adsorption in the bare framework material with no appended diamines, Mg 2 (dobpdc) ( Fig. 3e and f). In Mg 2 (dobpdc), CO 2 binds to open metal coordination sites exposed upon solvent removal from the framework, leading to a typical Langmuir adsorption prole (Fig. 3e). 24 Importantly, because CO 2 adsorption in Mg 2 (dobpdc) does not involve a chemical reaction, the adsorption kinetics are likely diffusion-limited. Consistently, Mg 2 (dobpdc) reaches its equilibrium CO 2 adsorption capacity within a similarly short time for all temperatures examined ( Fig. 3g and h), with the maximum rate of adsorption occurring at earlier times as the temperature is increased, in stark contrast to the behavior exhibited by m-2-m-Mg 2 (dobpdc). For example, at 120 C, approximately 51 s elapses before the maximum rate of adsorption is achieved in m-2-m-Mg 2 (dobpdc), whereas the maximum rate is reached in only 14 s for bare Mg 2 (dobpdc) under the same conditions (see asterisks in Fig. 3d and h). Note that comparison of the absolute rate of adsorption is complicated by the lower equilibrium capacity of Mg 2 (dobpdc) at the low partial pressures present in the TGA furnace during the initial mixing period (Fig. S4 †). Furthermore, the apparent kinetics of Mg 2 (dobpdc) include competition of N 2 and CO 2 for the same binding sites, and thus CO 2 adsorption requires displacement of any adsorbed N 2 molecules. Nevertheless, only a small fraction of metal sites ($2-10%) is expected to be occupied by N 2 under the conditions examined here, based on previously reported N 2 adsorption isotherms for the isoreticular smaller-pore framework material Mg 2 (dobdc). 39 In addition to the marked differences in the kinetics of CO 2 adsorption, the bare framework Mg 2 (dobpdc) and amineappended m-2-m-Mg 2 (dobpdc) also exhibit distinct trends in their variable-temperature equilibrium CO 2 adsorption capacities (Q e ). Whereas the CO 2 saturation capacity of m-2-m-Mg 2 (dobpdc) is similar across nearly all temperatures investigated, decreasing only near T step (Fig. 3b), the capacity of Mg 2 (dobpdc) varies substantially with temperature ( Fig. 3f). This difference arises due to the different adsorption proles of these two materials and is best demonstrated through a van't Hoff plot (Fig. 4a). To quantify the kinetics in these two materials while accounting for variations in equilibrium capacity, we also analyzed the percent of adsorption complete (Q t /Q e ) vs. time ( Fig. 4b and c). This analysis again reveals that adsorption is faster at lower temperatures for m-2-m-Mg 2 (dobpdc), with induction periods observed at temperatures close to T step . In contrast, the adsorption proles are similar at all temperatures for Mg 2 (dobpdc), with the highest temperatures exhibiting the fastest initial progress toward equilibrium. Overall, these ndings corroborate that the chemisorptive mechanism operational in m-2-m-Mg 2 (dobpdc) leads to an unusual induction period that is not observed in the physisorptive mechanism in Mg 2 (dobpdc).
To further understand the induction period exhibited by m-2-m-Mg 2 (dobpdc), we characterized the adsorption kinetics from a 15% CO 2 stream (Fig. 5), corresponding to the approximate partial pressure of CO 2 in coal ue gas. 36 In this case, the investigated temperature range (45-100 C) is lower than that used in the experiments with 100% CO 2 (75-120 C), reecting the lower adsorption step temperature under 15% CO 2 (102 C) compared to 100% CO 2 (127 C) (Fig. S8 †). At temperatures below 75 C, adsorption was nearly complete in less than 1 min. At temperatures near T step , however, the induction period was even more pronounced than in the analogous experiments with 100% CO 2 (Fig. 5b vs. Fig. 3d). The effect was particularly dramatic at 100 C (Fig. 5c)-a small amount of CO 2 was rapidly adsorbed (6% of the diamine sites occupied within 1 min), followed by a period of 4 min during which almost no additional adsorption occurred. Aer 5 min, the rate of CO 2 adsorption accelerated, and a substantial CO 2 occupancy was ultimately reached, corresponding to occupation of approximately half of the diamine sites. The fast capture of CO 2 at $6% of the m-2-m sites, as shown in Fig. 5c, is consistent with the previous nding that m-2-m-Mg 2 (dobpdc) adsorbs a small amount of CO 2 even at pressures below the step. 24,25 Spectroscopic studies indicate that this pre-step chemisorption arises due to ammonium carbamate species that form without metalamine insertion. 7 We hypothesize that the pre-step ammonium carbamate species forms rapidly, followed by slower cooperative adsorption of additional CO 2 via the formation of metal-bound ammonium carbamate chains. Qualitatively similar results were obtained using a 5% CO 2 stream, for which the induction period is even more pronounced (Fig. S6 †).
Taken together, these results suggest that the sigmoidal kinetics prole of m-2-m-Mg 2 (dobpdc) is directly related to its cooperative CO 2 adsorption mechanism. This material exhibits a high degree of cooperativity in equilibrium gas adsorption isotherms, with its Hill coefficient of $11 indicating that, from a thermodynamic standpoint, capturing one CO 2 molecule facilitates the adsorption of subsequent CO 2 molecules. 25 The pronounced sigmoidal kinetics prole exhibited by m-2-m-Mg 2 (dobpdc) likewise suggests that initial capture of a CO 2 and Mg 2 (dobpdc), respectively. Samples were activated under flowing N 2 at 130 C for 5 min (c) or at 310 C for 5 min (g) prior to CO 2 adsorption at the indicated temperatures. "Fraction CO 2 sites occupied", or Q t , indicates the ratio of adsorbed CO 2 molecules to m-2-m in the material (c) or the ratio of adsorbed CO 2 molecules to Mg 2+ sites in the material (g). Panels (d) and (h): rate of CO 2 adsorption vs. time corresponding to the data shown in (c) and (g), respectively. The asterisks in panels (d) and (h) indicate the times at which the maximum rate of adsorption was observed during the 120 C adsorption experiments. In panels c-d and g-h, a 19 s delay is observed prior to the sample mass increasing, corresponding to the time required for CO 2 to reach the sample in the furnace.
This journal is © The Royal Society of Chemistry 2020 Chem. Sci., 2020, 11, 6457-6471 | 6461 molecule also enhances the adsorption kinetics of subsequent CO 2 molecules. Similar sigmoidal kinetic proles have been reported previously for autocatalytic 40 and autoinductive 41 chemical reactions. While a recent TGA-based assay previously revealed that a polyethylenimine-appended mesoporous silica also exhibited sigmoidal CO 2 adsorption kinetics, 11 amineimpregnated clays did not exhibit sigmoidal uptake kinetics when evaluated using comparable TGA equipment and procedures. 34 These precedents establish that not all amine-based CO 2 adsorbents exhibit sigmoidal kinetics. As noted in a recent study, 42 a long induction period is undesirable for implementation in a practical process because it can lead to lower bed utilization. Specically, in breakthrough experiments simulating direct air capture with m-2-m-Mg 2 (dobpdc), it was found that aer rapid partial breakthrough of CO 2 , the CO 2 concentration at the outlet decreased before full breakthrough eventually occurred, consistent with a delayed onset of adsorption. 42 Hence, it is critical to identify appropriate diamine variants and/or CO 2 adsorption conditions that minimize this induction period (see below).

Avrami model of the adsorption kinetics of m-2-m-Mg 2 (dobpdc)
We next sought to model the CO 2 adsorption kinetics of m-2-m-Mg 2 (dobpdc) from a gas stream of 15% CO 2 in N 2 . The fraction of sites occupied at time t (Q t ) was t at each temperature using either a pseudo-rst order model (eqn (1)) or Avrami's kinetics model (eqn (2)), which was originally developed to model nucleation-growth kinetics 43 and has recently found application as a model for chemisorption in an amine-appended mesoporous silica. 11 Note that in both models, the constant partial pressure of CO 2 (P/P 0 ¼ 0.15) is embedded in the rate constant k 1 or k A , respectively, and that n A in the Avrami model is also likely a function of P/P 0 . As expected, a pseudo-rst order model using rate constant k 1 (eqn (1)) failed to capture the induction period at temperatures near T step (Fig. 6a). In contrast, the Avrami model ts the data well, particularly for temperatures just below the step. We attribute the deviation from the model in the rst 1.5 min to incomplete gas mixing in the furnace and to pre-step chemisorption, as discussed above. This model incorporates the Avrami parameter, n A , as well as the Avrami rate constant, k A , to produce sigmoidal kinetics proles, with larger n A values leading to longer induction periods. For example, n A ¼ 2 corresponds to the sigmoidal kinetics prole of the growth of a one-dimensional crystal. 44,45 A plot of the Avrami rate constant k A vs. T, which is useful for visualizing trends in these data, is presented in Fig. 6b. This plot indicates that the Avrami rate constant becomes progressively larger with decreasing temperature, reecting the inverse Arrhenius behavior of m-2-m-Mg 2 (dobpdc) (Fig. 6b). At temperatures below 70 C, the slope of k A vs. T substantially decreases, possibly due to the intrinsic properties of the material or to instrumentation limitations associated with the high rate of adsorption at these colder temperatures. Unexpectedly, k A vs. T follows a linear trend from 70 to 95 C, inconsistent with standard Arrhenius behavior (see "Inverse Arrhenius Behavior" below). Notably, the x-intercept of this plot (k A ¼ 0) should correspond to T step ; indeed, the x-intercept of a linear t to the high-temperature data (70-95 C) of 100 C is close to the step temperature determined by cooling a sample of m-2-m-Mg 2 (dobpdc) under 15% CO 2 (102 C; see Fig. S8 and Table S2 †). In addition, the parameter n A progressively decreases from an initial value of 1.7 at 95 C to a value close to 1 at the lowest temperature of 55 C, reecting the longer induction periods near T step . Therefore, the parameters from these Avrami ts Fig. 4 (a) Van't Hoff plot depicting the equilibrium CO 2 adsorption capacity with varying temperature for m-2-m-Mg 2 (dobpdc) and Mg 2 (dobpdc) from a 100% CO 2 stream (based on the data shown in Fig. 3c and g). K eq is defined as Q e /[(Q max À Q e )pCO 2 ], where Q e is the equilibrium adsorption capacity, Q max is the theoretical maximum CO 2 adsorption capacity, and pCO 2 is the partial pressure of CO 2 , defined as p/p 0 , with p ¼ p 0 ¼ 1 atm. We note that this van't Hoff analysis assumes an equilibrium exchange. Although the chemisorbed CO 2 in saturated ammonium carbamate chains does not appreciably exchange with free CO 2 , 52 the changes in adsorption capacity for m-2-m-Mg 2 (dobpdc) at temperatures below the step are likely due primarily to CO 2 physisorption, which should be reversible. The enthalpy of CO 2 adsorption, Dh, can be estimated for bare Mg 2 (dobpdc) based on the slope of a linear fit to the data points (Fig. S7 †). (b) and (c): plots of % adsorption complete (Q t /Q e ) vs. time corresponding to the data shown in Fig. 3c and g, respectively. Adsorption was monitored for 30 min (b) or 10 min (c) to confirm mass equilibration, but only the first 2 min of adsorption are shown.
successfully reect the experimental observations of slower CO 2 adsorption kinetics (smaller k A ) and longer induction periods (larger n A ) at high temperatures close to the step temperature in m-2-m-Mg 2 (dobpdc). The smaller values of n A at lower temperatures reect shorter induction periods and demonstrate an advantage of maintaining a buffer between the adsorption temperature and the step temperature.

Inverse Arrhenius behavior
The inverse Arrhenius behavior observed here can be rationalized using a reaction coordinate diagram derived from several previous investigations of reactions with apparent negative activation energies (Fig. 7). 40,[46][47][48][49] In general, inverse Arrhenius behavior requires the reversible formation of an intermediate species with equilibrium constant K eq that (i) is lower in energy than the reactant(s), with energy difference Dg int ; and (ii) proceeds to product formation through a transition state, with kinetic barrier Dg ‡ rxn , that is also lower in free energy than the reactant(s), such that |Dg ‡ rxn | < |Dg int | (see Fig. 7). In addition, the product formation must be an effectively irreversible process, with negligible conversion back to the intermediate species under the reaction conditions. This scenario leads to inverse Arrhenius behavior because the rate of product formation is dependent on the concentration of the intermediate species. Critically, for an adsorption process, this key intermediate species is thermodynamically disfavored at higher temperatures due to the greater entropic penalty associated with removing CO 2 from the gas phase. In other words, increasing the temperature drives the intermediate species back toward the reactant(s) more than it promotes overcoming the  (a)) with the x-axis extended. A delay of approximately 10 min was observed before the maximum rate of adsorption was reached. In this plot, data are normalized such that t ¼ 0 is after the 19 s delay prior to CO 2 reaching the sample in the furnace. Fig. 6 Models for the kinetics of CO 2 adsorption from a 15% stream by m-2-m-Mg 2 (dobpdc). (a) CO 2 adsorption vs. time (black) at 95 C from a 15% CO 2 stream for m-2-m-Mg 2 (dobpdc). Pseudo-first order and Avrami fits to the data are shown as green and magenta lines, respectively. In this plot, data are normalized such that t ¼ 0 occurs after the 19 s delay prior to CO 2 reaching the sample in the furnace. (b) Avrami rate constants (k A ) and parameters (n A ) for CO 2 adsorption by m-2-m-Mg 2 (dobpdc) from a 15% stream at temperatures ranging from 55 to 95 C.
This journal is © The Royal Society of Chemistry 2020 Chem. Sci., 2020, 11, 6457-6471 | 6463 barrier to form the product. In m-2-m-Mg 2 (dobpdc), reversible adsorption of CO 2 to form a labile intermediate, such as a physisorbed or weakly chemisorbed species, likely precedes an exothermic rate-determining chemisorption step. The nal product of chemisorption is an ammonium carbamate chain, 26 which forms with a thermodynamic free energy change of Dg ads . However, C-N bond formation, proton transfer, and metaloxygen bond formation must all occur between the starting material and the nal ammonium carbamate chains; the number of steps in this reaction pathway and the ratedetermining step remain points of ongoing investigation. Nevertheless, the temperature dependence of the kinetics of CO 2 adsorption in m-2-m-Mg 2 (dobpdc) is consistent with an overall mechanistic model involving the reversible formation of an entropically disfavored, CO 2 -bound intermediate followed by a rate-determining chemisorption step. A similar model can be invoked to explain the observed inverse Arrhenius behavior of CO 2 adsorption in other amine-functionalized adsorbents. 11,37 To further understand the behavior of m-2-m-Mg 2 (dobpdc), we constructed an Arrhenius plot using the Avrami rate constants given in Fig. 6b. As expected, ln(k A ) increases with 1/T (Fig. 8), and the sharp curvature occurring near 1/T step is consistent with the unusually linear behavior of k A vs. T near T step (Fig. 6b). The data can be t well using a logarithmic function with a vertical asymptote corresponding to $97 C, close to the T step of 102 C (see Fig. S8 †). The steeper slope of the Arrhenius plot near 1/T step indicates that the temperature dependence of the rate of adsorption in m-2-m-Mg 2 (dobpdc) is greatest close to T step .
While the reaction coordinate diagram in Fig. 7 accounts for the inverse Arrhenius behavior of CO 2 adsorption in m-2-m-Mg 2 (dobpdc), it does not necessarily predict the extreme curvature of the Arrhenius plot near 1/T step . Because cooperative CO 2 adsorption does not occur above T step , the Arrhenius plot for the cooperative adsorption process should deviate to ÀN upon approaching T step from low to high temperature. Discontinuities have previously been observed in linear Arrhenius plots when a phase change occurs at a specic temperature in an enzyme. 50 However, the curvature exhibited in the Arrhenius plot of m-2-m-Mg 2 (dobpdc) is unusual and suggests a progressive decrease in the adsorption kinetics as T / T step . The curved Arrhenius plot can potentially be explained by considering the entropy change in the reaction coordinate diagram in Fig. 7, as has been previously described. 51 Moving from le to right along the reaction coordinate in Fig. 7 corresponds to a signicant decrease in degrees of freedom as gaseous CO 2 is immobilized and the initially dynamic diamines are locked into ammonium carbamate chains. As the temperature increases, the entropic penalty for immobilizing CO 2 increases and ultimately outweighs the enthalpic favorability of adsorption when T exceeds T step . Accordingly, the kinetic barrier of the ratedetermining chemisorption step should become increasingly Fig. 7 Proposed reaction coordinate diagram for CO 2 adsorption in m-2-m-Mg 2 (dobpdc) below T step . The inverse Arrhenius behavior suggests that CO 2 initially adsorbs in a reversible physisorption or weak chemisorption step, forming an intermediate species that is lower in energy than the starting materials (with energy difference Dg int ). The intermediate species is then converted to a stable CO 2 chemisorbed product in a highly exothermic step with activation energy Dg ‡ rxn . The inverse Arrhenius behavior suggests that Dg int is larger in magnitude than Dg ‡ rxn . We note that ammonium carbamate chains are known to be the final product of CO 2 chemisorption in m-2-m-Mg 2 (dobpdc), but the exact mechanism and rate-determining step leading to these chains remain unclear.
large at higher temperatures, consequently slowing the reaction kinetics. Therefore, increasing the temperature towards T step imposes two compounding deleterious entropic effects on the rate of adsorption in m-2-m-Mg 2 (dobpdc): it thermodynamically disfavors the formation of the intermediate species (as discussed above), and it increases the magnitude of the ratelimiting kinetic barrier. Together, these effects can potentially explain the observed progressively slower adsorption kinetics and resulting curved Arrhenius plot as T approaches T step . Determining the identity of the intermediate species and elucidating the mechanism of the rate-limiting chemisorption process remain active areas of investigation.
A structure-property kinetics relationship using a panel of ethylenediamine analogues To determine the generalizability of these characteristics for m-2-m-Mg 2 (dobpdc) to other diamine-appended frameworks, we evaluated the kinetics of CO 2 adsorption from a gas stream of 15% CO 2 in N 2 by a panel of Mg 2 (dobpdc) variants appended with structurally-diverse alkylethylenediamines. Notably, frameworks with 1 ,3 alkylethylenediamines 26 displayed minimal adsorption at temperatures $40 C (Fig. S12 †) and thus were not investigated further. In contrast, rate constants could be determined for frameworks appended with 1 ,1 ; 1 ,2 ; and 2 ,2 amines (see Fig. 9). We note that the diamines Nisopropylethylenediamine (i-2) and N,N 0 -diethylethylenediamine (e-2-e) exhibit two-step adsorption behavior, 28 and thus their kinetic proles are complex at temperatures at which both steps are operative. As a result, Fig. 9 depicts only the temperatures under which the less thermodynamically favorable step is not operative for these two diamine-appended frameworks.
A similar overall sigmoidal kinetic prole was observed for all ethylenediamine variants in Fig. 9, consistent with our previous ndings that these materials capture CO 2 by the same mechanism. 26,52 In each case, k A increases with decreasing temperature, indicative of inverse Arrhenius behavior, as observed for m-2-m-Mg 2 (dobpdc) (Fig. S13-S18 †). Furthermore, all of the variants exhibit linear plots of k A vs. T (Fig. 9b) and curved Arrhenius plots that could be t by logarithmic functions (Fig. S13-S18 †), suggesting that the complex behavior characterized for m-2-m-Mg 2 (dobpdc) represents a general feature of these materials. In Fig. 9b, the linear ts to the k A vs. T data yield different x intercepts for each diamine-appended framework, reecting their different adsorption step temperatures. The x-intercepts all match closely to step temperatures determined by cooling the frameworks under 15% CO 2 (Fig. S8-S11 and Table S2 †), except in the case of (AE)-transdach-Mg 2 (dobpdc), for which the k A vs. T plot deviates from linearity at high temperatures (Fig. S20 †).
The differences in adsorption step temperatures complicate comparisons between materials because the free energy change associated with CO 2 adsorption (Dg ads ) is different for each material at a constant temperature. To account for this difference, we also plotted k A vs. T step À T to scale the x-intercepts of the linear ts to zero (Fig. 9c). Note that we do not consider T step À T to be a physically meaningful metric of chemical potential-rather, this plot is only intended to help visualize differences among diamine-appended frameworks. Importantly, the linear ts to the data in the corresponding plots all exhibit distinct slopes, reecting variability in the extent to which decreasing the temperature below T step increases the CO 2 adsorption kinetics. For practical applications, a steep slope in Fig. 9c is desirable to enable fast adsorption kinetics even at temperatures just below the step. In the plot of n A vs. T step À T (Fig. 9d), n A generally becomes smaller with decreasing temperature. Most of the materials exhibit values of n A between 1.3 and 1.8 at temperatures close to the step, decreasing to $1 at the lowest temperatures. Materials with smaller values of n A show less pronounced induction periods and are therefore preferable for implementation in a process.
Interestingly, the steepest slopes were observed for frameworks appended with the 2 ,2 diamines m-2-m and e-2-e. [24][25][26] These variants are notable for having weak metal-amine bonds, which serve to increase |Dh ads | for CO 2 and facilitate CO 2 insertion, thus likely contributing to their fast adsorption kinetics. 26 Despite these desirable kinetics properties, the weak metal-amine bonds lead to diamine volatilization under humid conditions, particularly at the high temperatures necessary for desorption under a pure CO 2 stream. 53,54 The second-fastest kinetics were found for variants of Mg 2 (dobpdc) functionalized with the 1 ,2 diamines N-methylethylenediamine (m-2), Nethylethylenediamine (e-2), and i-2. 26,28,55 We previously determined that the primary amine preferentially binds to the metal site and reacts with CO 2 , which may account for the slower adsorption kinetics in these materials compared to those functionalized with 2 ,2 diamines. 26 However, this effect also bestows 1 ,2 diamine-appended variants with enhanced stability toward diamine loss. 26,28 The compounds with the least sterically-encumbered 1 ,2 diamines exhibit slightly faster kinetics, as indicated by steeper slopes in the plots of k A vs. T (Fig. 9c). Overall, these results suggest a tradeoff between stability and adsorption kinetics for frameworks appended with 1 ,2 diamines compared to those bearing 2 ,2 diamines.
Finally, we studied the behavior of Mg 2 (dobpdc) variants appended with the 1 ,1 diamines ethylenediamine (en) and (AE)-trans-diaminocyclohexane (dach), 26,29,31,56 which have previously been shown to exhibit signicant adsorption/desorption hysteresis and unit cell contraction upon CO 2 adsorption. 56 Interestingly, the CO 2 adsorption kinetics of en-Mg 2 (dobpdc) are extremely similar to those of i-2-Mg 2 (dobpdc)-the 1 ,2 variant with the slowest CO 2 adsorption kinetics-while (AE)-trans-dach-Mg 2 (dobpdc) exhibits much slower kinetics. In addition, (AE)-trans-dach-Mg 2 (dobpdc) displays an unusually high n A of $3 at temperatures close to the step, corresponding to a very long induction period (Fig. S19-S21 †). As with the other diamines, the adsorption kinetics in (AE)-trans-dach-Mg 2 (dobpdc) become much faster and the induction periods become much shorter at lower temperatures (n A ¼ 1.3 at 60 C; see Fig. S21 †), and use of a fast ow rate (100 mL min À1 ) also shortens the induction period for this material (Fig. S22 †). One possible explanation for the large n A value and slower overall kinetics of (AE)-trans-dach-Mg 2 (dobpdc) is the unit cell contraction that occurs upon CO 2 adsorption in conjunction with crystallographically characterized ion-pairing interactions between neighboring ammonium carbamate chains in the ab plane. 56 Intriguingly, n A ¼ 3 for the Avrami model corresponds to the growth of a crystal in two dimensions, whereas n A ¼ 2 corresponds to growth in one dimension. 44,45 Accordingly, cooperative CO 2 adsorption in (AE)-trans-dach-Mg 2 (dobpdc) may be akin to two-dimensional sheet growth, whereas cooperative CO 2 capture by the other variants may be more akin to onedimensional chain growth. Elucidating the potential correlation between ab plane contraction and the CO 2 adsorption kinetics for 1 ,1 diamine-Mg 2 (dobpdc) variants remains a subject of investigation. In addition to ethylenediamines, Mg 2 (dobpdc) has also been appended with diaminopropanes to yield cooperative adsorbents. 27 For example, dmpn-Mg 2 (dobpdc) (dmpn ¼ 2,2dimethyl-1,3-diaminopropane) has a favorable adsorption step temperature for CO 2 capture from coal ue gas and maintains a high working capacity over 1000 cycles under humid conditions. 27 Notably, the complex CO 2 chemisorption mechanism in dmpn-Mg 2 (dobpdc) ultimately leads to the formation of carbamic acids interacting with ammonium carbamates, and this material exhibits a mixture of $4 chemisorbed species in the early stages of CO 2 adsorption. 52 Given this complexity and the promising attributes of dmpn-Mg 2 (dobpdc) for implementation in practical processes, we sought to investigate its adsorption kinetics from a gas stream of 15% CO 2 in N 2 .
Consistent with previous ndings, the 15% CO 2 adsorption isobar for this material exhibits a step around 60 C that is more broad than those of other diamine-appended Mg 2 (dobpdc) variants, reecting its complex adsorption mechanism (Fig. 10a). 27,52 Similarly, the kinetics prole of dmpn-Mg 2 (dobpdc) is distinct from that of the other diamine-appended variants ( Fig. 10b and c), with no induction period observed even at temperatures close to the adsorption step. At all temperatures below 55 C, dmpn-Mg 2 (dobpdc) exhibits a very rapid initial uptake, followed by slower uptake until reaching equilibrium. This unusual curve shape results in Avrami ts with small n A ($0.4-0.7) and k A ($0.003-0.006 s À1 ) values (Fig. S23 †). The fastest initial uptake in dmpn-Mg 2 (dobpdc) occurs at 35 C-the lowest temperature investigated-but only up to an occupancy of $0.6, aer which the kinetics slow (Fig. S24 †). We note that this slower adsorption at low temperatures and high occupancies was not observed in a comparable temperature range for the other diamine-appended variants, but a similar effect was reported previously in "molecular basket" polyamine-appended mesoporous silicas. 10 Overall, the lack of induction period for dmpn-Mg 2 (dobpdc) is promising for implementation in a CO 2 capture process.

Desorption kinetics
Desorption kinetics are also a critical consideration for the practical use of an adsorbent. In a typical xed-bed process, adsorption and desorption stages are operated simultaneously using a minimum of two beds. As a consequence, fast desorption kinetics are desirable to minimize cycle times. In a simulation of the desorption conditions typical of a temperature swing adsorption process, we previously demonstrated that CO 2 can be desorbed from diamine-appended Mg 2 (dobpdc) under a humidied stream of pure CO 2 within less than 10 min, as limited by the heating rate of the TGA furnace. 27,28 Alternatively, the introduction of dry N 2 as a purge gas can be used to simulate the application of a concentration-or vacuum-swing process. Because diamine-appended Mg 2 (dobpdc) variants can maintain high CO 2 capacities under pure CO 2 even at relatively high temperatures, these materials can be heated under CO 2 to a temperature of interest and allowed to equilibrate, aer which the gas stream can be switched to dry N 2 and the decrease in mass can be monitored.
We utilized this dry N 2 desorption assay to compare the desorption kinetics of dmpn-Mg 2 (dobpdc) and m-2-m- Fig. 10 Kinetics of CO 2 adsorption in dmpn-Mg 2 (dobpdc). (a) 15% CO 2 adsorption isobar for dmpn-Mg 2 (dobpdc) following activation at 120 C for 20 min. The gray dashed line represents the adsorption capacity corresponding 1 CO 2 per diamine (16.8 g/100 g). (b) Adsorption from a 15% CO 2 stream vs. time for dmpn-Mg 2 (dobpdc). Sample activation: 100 C for 5 min under N 2 . (c) Rate of CO 2 adsorption vs. time corresponding to the data shown in (b). A 19-s delay is observed prior to the sample mass increasing, corresponding to the time required for CO 2 to reach the sample in the furnace.
This journal is © The Royal Society of Chemistry 2020 Mg 2 (dobpdc) (Fig. 11). At the highest temperature investigated for each material (70 and 110 C for dmpn and m-2-m, respectively), nearly all of the CO 2 was desorbed within 1.5 min, and decreasing the desorption temperature progressively decreased the rate of desorption. For each adsorbent, the initial Q t values varied with temperature, consistent with the isobaric adsorption proles, and the larger Q t values at colder temperatures can be attributed to CO 2 physisorption. Overall, desorption from dmpn-Mg 2 (dobpdc) occurs in a lower temperature range compared to desorption from m-2-m-Mg 2 (dobpdc), reecting the lower desorption step temperature for the dmpn variant of 93 C 27 compared to 134 C for m-2-m, 26 as determined from dry 100% CO 2 desorption isobars collected using a ramp rate of 1 C min À1 .
As shown in Fig. S25, † the desorption kinetics for dmpn-Mg 2 (dobpdc) and m-2-m-Mg 2 (dobpdc) t well to the Avrami model across multiple temperatures. For both materials, desorption is initially slow, but is accelerated aer some CO 2 has been desorbed. Interestingly, the desorption kinetics behavior of dmpn-Mg 2 (dobpdc) differs from its adsorption kinetics behavior. While this material exhibits no induction period for adsorption (n A $0.5), a substantial induction period was observed for desorption (n A $2 for many of the conditions tested; see Fig. S25 †). These dissimilar kinetics proles may result from dmpn-Mg 2 (dobpdc) exhibiting a mixture of approximately four chemisorbed species in the early stages of CO 2 adsorption, while a 1 : 1 mixture of ammonium carbamates:carbamic acids is formed following saturation with CO 2 and long equilibration times. 52

Conclusions
The foregoing results describe a systematic investigation into the CO 2 adsorption kinetics of diamine-appended variants of the metal-organic framework Mg 2 (dobpdc) using a TGA-based assay. Sigmoidal kinetics proles were observed for all of the materials that exclusively form ammonium carbamate chains upon CO 2 adsorption, which we attribute to their cooperative adsorption mechanism. While these adsorbents exhibit long induction periods at temperatures near the adsorption step, the kinetics of adsorption can be accelerated and the length of the induction period minimized by decreasing the adsorption temperature. Additionally, the diamine structure can be optimized to yield adsorption thermodynamics and kinetics that are most favorable for a target gas separation involving CO 2 . Finally, no induction period was observed for the material dmpn-Mg 2 (dobpdc), which adsorbs CO 2 via a mixed ammonium carbamate/carbamic acid mechanism. Thus, accessing new chemisorption products within the diamine-appended materials may give rise to advantageous changes in kinetics as well as thermodynamics. Overall, this study provides key guidelines for utilizing diamine-appended variants of Mg 2 (dobpdc) in CO 2 separations and should prove more broadly useful for the design of new chemisorptive adsorbents for carbon capture applications. Fig. 11 Desorption of CO 2 under dry N 2 from (a) dmpn-Mg 2 (dobpdc) and (b) m-2-m-Mg 2 (dobpdc). In each experiment, the sample was allowed to equilibrate under pure CO 2 at the indicated temperature, after which the gas stream was switched to N 2 . In these plots, data are normalized such that t ¼ 0 corresponds to the data point immediately before mass decrease was observed due to CO 2 desorption (i.e., t ¼ 0 is after the 19 s delay prior to N 2 reaching the sample in the furnace).