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6490738ba2c387fa9a92d863 | 12 | 2 Model and Method To represent the interpolyelectrolyte complex in equilibrium with supernatant in silico, we opt for the approach used in our previous works on multi-phase polyelec-trolyte systems. We independently simulate a concentrated solution of polyelectrolytes, representing the bulk of the interpolyelectrolyte complex phase, and a salt solution, representing the bulk of the supernatant phase as captured in Figure . The two phases are indirectly coupled, as detailed in the following subsections. |
6490738ba2c387fa9a92d863 | 13 | Instead, we choose the parameters in a reasonable range, based on semi-empirical estimates within the range used for similar models in the literature, roughly matching the key features of some common experimental systems. In this section we only list the default values of model parameters. Later, in Sec. 3.4 we present a systematic analysis and a detailed discussion of how these parameters affect our numerical results. |
6490738ba2c387fa9a92d863 | 14 | The phase of polyelectrolyte complex, denoted by superscript pec , consists of N A = 32 polyanions and N C = 32 polycations of length M = 32 units each. This system is enclosed in a cubic simulation box of size L with periodic boundary conditions. Monomeric units of the polyelectrolyte chains are modelled as point particles characterized by effective size, σ mon , and fixed charge number, z mon . Unless stated otherwise, σ mon = 0.426 nm for both polycation and polyanion, and z = ±1 respectively. Each pair of monomeric units interacts through the non-bonded pairwise potential introduced by Weeks, Chandler and Andersen (WCA): 112,113 |
6490738ba2c387fa9a92d863 | 15 | where x is the instantaneous distance, ϵ = k B T = 1/β and λ is the parameter controlling the hydrophobicity. Further details are provided in Sec. S1.1 of the ESI, where we also plot the potential in Figure and relate the values of λ to the second virial coefficient, B 2 . 114 We always use the truncated and shifted form 115 of the potential in Eq. 5 with the cutoff of 2.5σ mon = 1.065 nm. Unless stated otherwise, we use λ = 0, yielding a purely repulsive WCA potential which represents good (athermal) solvent conditions. |
6490738ba2c387fa9a92d863 | 16 | where K bond = 827 k B T /nm 2 is the spring stiffness and R 0 = σ mon = 0.426 nm, unless stated otherwise. The combination of potentials yields the average bond length b ≈ 0.438 nm ± 0.025 nm, where the first value is the estimated mean and the second value is the estimated standard deviation. |
6490738ba2c387fa9a92d863 | 17 | Additional solutes are present in some cases, such as generic multivalent ions, denoted as M 2+ , X 2 -, or a small diprotic weak acid (succinic acid) in all of its ionization forms H 2 SuA, HSuA -, SuA 2 -. All solutes are being exchanged with the supernatant as described in Sec. 2.2, hence their numbers fluctuate. They are all represented as point particles with λ = 0, an effective diameter σ i = 0.355 nm instead σ mon and the respective charge number z i , as depicted in Figure and listed in Tab. S2 in the ESI. |
6490738ba2c387fa9a92d863 | 18 | where l pec B = e 2 /4πε 0 ε r k B T = 0.89 nm is the Bjerrum length of the implicit solvent, which is represented as a dielectric continuum. This corresponds to the relative permittivity of the polyelectrolyte complex phase ε r ≈ 62. The electrostatic potential and forces are evaluated using the particle-particle particle-mesh (PPPM) method, 116 |
6490738ba2c387fa9a92d863 | 19 | The phase of of the supernatant, denoted by the superscript sup , contains only small ions and other solutes in implicit solvent but it does not contain any polymer. Unlike the polyelectrolyte complex phase, we set the Bjerrum length in the supernatant to l sup B = 0.71 nm, which corresponds to the permittivity of pure water at room temperature, ε r ≈ 78. Otherwise, all other interactions in the supernatant phase are the same as in the polyelectrolyte complex phase. |
6490738ba2c387fa9a92d863 | 20 | Absence of polymer in the supernatant phase We assume that the supernatant phase does not contain any polymer. Ref., and also our experiments in Sec. 3.2, show that this is indeed a reasonable assumption for chains as short as M = 50, provided the two phase system is far enough from the critical point. Admittedly, because of this assumption, our method is unable to reconstruct the full phase diagrams, leaving the regions around the critical point unrendered. Nevertheless, as we show later, our method allows self-consistent assessment of its validity, and hence the requirement of sub-criticality can be tested. |
6490738ba2c387fa9a92d863 | 21 | In the current simulations, we assume that the charges on the polymers are fixed, independent of the pH, as if the polymers were strong polyelectrolytes. In contrast, the polymers used in our experiments, PAA and PAH, are weak polyelectrolytes, which can attain different degrees of charging, depending on pH. However, the experiments were carried out at pH ≈ 7, where the Henderson-Hasselbalch equation predicts that both ideal polycation and polyanion are fully charged, hence effectively quenched. Even though, the real system is non-ideal, both experiments on interpolyelectrolyte complexes and simulations of electrostatically cross-linked gels 111 suggest that charge regulation of interpolyelectrolyte complexes should be close to the ideal one, described by the Henderson-Hasselbalch equation. Arguably, the charge regulation can become a non-trivial actor, especially if the polyanion-polycation mixing ratio is not unity, due to the emerging Donnan potential between the phases. In principle, the G-RxMC method used here is able to quantitavely capture these affects, as repeatedly proven in simulations of gels and peptide solutions. 120-122 Herein, we focus only on the charge regulation of a small solute in the presence of interpolyelectrolyte complex. The effects of charge regulation and pH on the formation of interpolyelectrolyte complexes will be addressed in detail in our follow-up publication. |
6490738ba2c387fa9a92d863 | 22 | ) and ⟨Y p i (t)⟩ = 0, where p, q ∈ {x, y, z} are Cartesian coordinates, δ is the Kronecker delta and F i is the deterministic force acting on the particle, originating from the gradient of potentials, given by Eq. 5, Eq. 6 and Eq. 7. We set the mass of each particle to m = 1, which in conjunction with σ = 1 and k B T = 1 defines the internal unit of time as τ = (mσ 2 /k B T ) 1/2 = 1 ≡ 1/γ = 100δt, where δt = 0.01τ is the integration time-step. Notably, the choice of the particle mass is arbitrary because it has no effect on the partition function and hence it does not affect any thermodynamic properties. It only affects dynamical properties of the system which we did not study here. |
6490738ba2c387fa9a92d863 | 23 | where ν i is the stoichiometric coefficient and µ i the chemical potential and µ ⊖ the standard chemical potential of species i. The constant Γ (H + ,OH -) = 10 -14 is the ionic product of water. Two out of the remaining three equilibrium constants, Γ (Na + ,Cl -) , Γ (H + ,Cl -) , Γ (Na + ,OH -) are linearly independent and can be chosen independently as input parameters which define the salt concentration and pH of the supernatant solution. |
6490738ba2c387fa9a92d863 | 24 | The third constant cannot be chosen independently because its value is determined by the electroneutrality constraint (see Sec. S1.2). The exact salt concentration and pH in the supernatant are determined from an auxiliary simulation using the chosen values of equilibrium constants. Therefore, the reported values of salt concentration usually are not round numbers, as expounded in Ref. When considering a weak diprotic acid as an additional solute, we add acid-base dissociation reactions of the solute to the above set of reactions: |
6490738ba2c387fa9a92d863 | 25 | where K A,• are the respective acidity constants. For simplicity, we refer to this acid as succinic acid and accordingly, we use pK A,1 = 4.2 and pK A,2 = 5.6. To account for exchange of the solute between the supernatant and polyelectrolyte complex, we add further reactions, analogical to the above |
6490738ba2c387fa9a92d863 | 26 | To perform a single Monte Carlo step, we first randomly select one of the considered reactions, each of them with equal probability. Next, we choose whether to carry out the reaction move in the forward or backward direction, also with equal probabilities. For a chosen direction, a trial reaction move is carried out by removing particles, by changing the particle identities, inserting new particles at random positions or deleting old ones, as prescribed by the stoichiometry of the corresponding reaction. The proposed new trial state (n) is accepted with the probability Ultimately, the input of the method is the set of equilibrium constants of all reactions, Γ. The output is the composition of the system, determined by the average concentration of the exchangeable species in the simulation box. One can perceive this method as a combination of Monte Carlo in reaction ensemble, and semi-grandcanonical ensemble, where we fix the linear combinations of chemical potentials of species because the chemical potentials of individual ionic species are ill-defined. 105 |
6490738ba2c387fa9a92d863 | 27 | which is our input parameter. Our typical volume fractions ϕ ≈ 15% correspond to the reduced particle densities ρ ⋆ σ 3 ≈ 0.2 and box sizes L ≈ 30σ ≈ 7R g , where R g is the radius of gyration of a single chain. These volume fractions are still low enough to assure that the chains do not interact with their own periodic images through the boundary conditions. Upon reaching the final box size, we eventually re-tune the parameters of the PPPM method to the relative accuracy of 10 -3 and we equilibrate the system for 10 7 steps of Langevin dynamics (10 5 τ ) and 10 5 steps of Monte Carlo particle exchanges until time drifts in energies and pressures cease to be measurable. |
6490738ba2c387fa9a92d863 | 28 | The mass of the systems in vials was weighed at several points: empty, filled with all components, with the supernatant removed, and dried. By comparing the mass of the systems with the supernatant removed and when dried, we obtain the water content of the PEC. Drying is verified via evaluation of the cobalt colour; hydrated wet cobalt is bright pink, while dehydrated cobalt is dark blue. Sufficient cobalt was present after drying to visually verify successful dehydration. |
6490738ba2c387fa9a92d863 | 29 | where c sup Co 2+ is the concentration of Co 2+ in the supernatant, normalized for the polyelectrolyteabsent control C, which is defined as 100%. The mass of the hydrated complex is m pec , while m sup is the mass of the supernatant. An assumption was made that the density of (hydrated) PEC was approximately equal to that water. 3 Results |
6490738ba2c387fa9a92d863 | 30 | To determine the composition of the coexisting phases, we expand on the approach used in our previous simulations of polyelectrolyte gels. 111,120,125 First, we select a supernatant phase of a given composition, and then try to identify, whether there could ) are snapshots of the three selected systems, using the same the color as in Figure . Green ticks mark a stable single-phase system, whereas red crosses mark systems with undesirable phase separation within the box. |
6490738ba2c387fa9a92d863 | 31 | pressures of both of the phases are equal. The first condition is automatically fulfilled by the G-RxMC algorithm. To meet the second condition, we utilize the pressurecomposition protocol. We carry out a series of simulations of the interpolyelectrolyte complex phase at various polymer volume fractions, determined by setting the simula-tion box length. In each simulation we measure the osmotic pressure difference between the interpolyelectrolyte complex and the supernatant, ∆P (ϕ mon ) = P pec (ϕ mon ) -P sup , as a function of the polymer volume fraction. Then we fit the data using a smooth function to find the polymer volume fraction, ϕ 0 mon , such that ∆P (ϕ 0 mon ) ≈ 0, as illustrated in Figure ). |
6490738ba2c387fa9a92d863 | 32 | A typical example of the pressure-composition protocol is represented by the cyan dataset in Figure ), corresponding to c sup NaCl ≈ 156mM. We observe that ∆P (ϕ mon ) ≫ 0 at high polymer volume fractions, mainly due to the steric repulsion. As the density of the PEC phase decreases, the ∆P (ϕ mon ) dependence eventually crosses the zeropressure baseline, indicating that the conditions for phase coexistence of supernatant and interpolyelectrolyte complex can be fulfilled at a certain volume fraction, ϕ 0 mon , shown on the snapshot in Figure ) . While volume fractions at ϕ mon > ϕ 0 mon correspond to the states of interpolyelectrolyte complex under compression, we have to be cautious about interpreting the compositions of ϕ mon < ϕ 0 mon , because they no longer correspond to a single homogeneous phase. At ϕ mon < ϕ 0 mon , the system can undergo a phase separation within the simulation cell, forming a set of polymer-rich and polymerpoor domains, as exemplified in Figure ). In such cases, the actual monomer density of the polymer-rich domains is not equal to the mean monomer density. Naturally, such a separation within the box is affected by massive finite-size effects and leads to large fluctuations of pressure. Therefore, we fit the P (ϕ mon ) dependence only in the range where ∆P ≳ 0, in order to avoid the region of possible phase separation and simultaneously obtain a reliable estimate of ϕ 0 . We employ a phenomenological fitting function, ∆P (ϕ) = a 0 + a 1 / tan(ϕ -a 2 ) where a 0 , a 1 , a 2 are the adjustable parameters. |
6490738ba2c387fa9a92d863 | 33 | This function is monotonic and qualitatively accounts for the shape and curvature of the P (ϕ mon ) dependence, as explained in Ref. The final step of the protocol is the assessment of self-consistency. As noted above, we require the interpolyelectrolyte complex at ϕ 0 mon to be a homogeneous single-phase system. For that reason we analyze the spherically averaged structure factors, S(k), between monomeric units, displayed in Figure ). If the system should demix to form polymer-rich and polymer-poor domains, the structure factor would diverge in the limit of small wavevectors, k → 0. 113 In Figure ) for c NaCl ≈ 156mM we indeed see such demixing peak in S(k) of the system with mean volume fraction 0.60ϕ 0 mon , indicating the separation. On the contrary, S(k) of the interpolyelectrolyte complex with mean volume fraction of ϕ 0 mon does not feature such a peak, indicating a stable homogeneous phase, coexisting with the supernatant. We can observe similar hallmarks of demixing at very high salt concentrations, c NaCl ≈ 1529mM, as testified by the peak in Figure ) |
6490738ba2c387fa9a92d863 | 34 | Despite the above qualitative analysis, we emphasize that our approach cannot properly identify the critical point because the assumption of no polymer in the supernatant phase breaks down in that region. Furthermore, as shown in Sec. S2.1 in the ESI, we see that in our finite simulation box, the scattering peak at k ≈ 0 gradually emerges as c NaCl is increased. Therefore, we need to introduce an arbitrary discrimination criterion as a limit of validity of our approximations. In our case, we consider the system to be demixing, if the peak at k ≈ 0 is the global maximum of the S(k) curve. |
6490738ba2c387fa9a92d863 | 35 | Accordingly, as the c NaCl increases, so does also the systematic error of our approach to identifying phase coexistence. Nevertheless, constructing the full phase diagrams is not our objective. Instead, we focus on the two-phase regions far from the critical point, where our method can capture ample features of the experimental systems, as will be demonstrated in the following subsections. |
6490738ba2c387fa9a92d863 | 36 | In Figure ), we plot a phase diagram, where we used the pressure-composition protocol from Figure to relate the composition of the supernatant and the co-existing interpolyelectrolyte complexes. The phase diagram manifests some typical features of polyelectrolyte complexation. First, the polymer volume fraction determined from our simulations is between ∼ 10% and ∼ 15% which is within the rather broad range reported by various experiments and simulations. Next, the interpolyelectrolyte complex density decreases with increasing c NaCl , as expected due to stronger electrostatic screening. This works well far enough from the critical point, when it is reasonable to assume that there is no polymer in the supernatant. However, our model cannot accurately predict that the interpolyelectrolyte complex eventually dissolves when the critical c NaCl reached. Nevertheless, our predictions of the phase coexistence remained self-consistent up to c NaCl ≈ 532mM. Accordingly, the critical salt concentration would be greater than 532mM, which is consistent with critical point reported in various studies, anywhere between 500mM and 3000mM, depending on specific parameters of the systems. Finally, in Figure ) we show that partition coefficients of NaCl between the two phases, obtained by dividing the salt concentrations at the end-points of the tielines in Figure , are commensurate with partition coefficients in interpolyelectrolyte complexes formed by poly(L-lysine)/poly(D,L-glutamic acid) polypeptides of length 50, measured in Ref. In the limit of low c NaCl , the partition ratio is slightly above unity, but it becomes smaller than one if c NaCl is increased, pointing to the preferential partitioning of monovalent ions into the supernatant phase. This behavior is in agreement not only with experiments but also with recent liquid state theories, which attribute it to steric repulsion and correlations due to the chain connectivity, both of which are explicitly included in our simulation model. The above studies also observe that at high c NaCl the partition ratio starts to increase and approaches unity close to the critical point. We do not observe this regime, as our model loses validity close to the critical point. Nevertheless, in Figure ) we see that the experimental critical c NaCl ≈ 850mM for the system reported in Ref., hence our simulations are likely to be self-consistent up to the roughly 532/850 ≈ 60% of the salt resistance. |
6490738ba2c387fa9a92d863 | 37 | showing that water comprises ∼ 90% of the complex. In our simulations, mass fraction is not well defined because we are using an implicit solvent. Therefore, we used volume fractions, which can be assumed proportional to the mass fractions by a multiplicative factor given by ratio of mean mass densities of water and the wet complex. As wet complex is mostly composed of water, the above factor should be close to unity, although its accurate value cannot be determined. Given this inherent uncertainty, we interpret the comparison of partition coefficients in Figure as a sufficient concurrence between the experiment and simulation. We reiterate that absolute quantitative agreement between the two can be in principle reached by fine-tuning of model parameters, but this is not the main objective of our study. We will comment on this point in more dedail in Section Sec. 3.3. Finally, the experiments also show, that the ratio between the polymer concentrations in the interpolyelectrolyte complex and supernatant is ≈ 200 (Sec. S3 in the ESI), hence validating our assumption of near-to-none polymer in the supernatant. |
6490738ba2c387fa9a92d863 | 38 | Albeit the ionic-specific effects known in the context of Hoffmeister series certainly can play a role in experimental systems, we emphasize that in our simulations the only difference between Co 2+ and Na + ions is their charge, while all the other interactions are kept the same. Therefore, our simulations clearly show that a change in the valency alone is sufficient to ensure preferential accumulation of multivalent ions in the PEC phase, yielding an almost quantitative agreement with the experiments. |
6490738ba2c387fa9a92d863 | 39 | To demonstrate the capabilities of our modeling approach, we use it to calculate the partitioning of a weak diprotic acid, i.e., a small solute which can undergo charge regulation. The charge states of this acid depend on the pH and are further modulated by intermolecular interactions, in particular electrostatics. Our model of the interpolyelectrolyte complex contains equal amounts of polycations and polyanions, therefore, the Donnan potential between the the complex and the supernatant phase is zero, causing that pH in both phases is the same. However, the ionic strength in the interpolyelectrolyte phase is much higher than in the supernatant, and simultaneously permitivitty in the interpolyelectrolyte phase is slightly lower than in the supernatant, causing that the electrostatic interactions are much stronger in the former phase. According to the Debye-Hückel theory, an increased ionic strength decreases the activity coefficients of charged species, causing that charged states of weak acids are more preferred than the uncharged ones. Consequently, charged states of the solute should be preferred in the interpolyelectrolyte complex, much more than in the supernatant. |
6490738ba2c387fa9a92d863 | 40 | In Figure we compare the populations of different charge states of the weak acid in both, supernatant and interpolyelectrolyte complex. As a reference, we use the Henderson-Hasselbalch equation, which neglects all interactions and predicts the ionization state in the ideal limit. In Figure ) we observe that the population of the uncharged state, H 2 SuA, in the supernatant phase is slightly suppressed, as compared to the ideal result at pH ≈ 4 ≈ pK A,1 . At this pH, the monovalent state, HSuA -, is favoured more than the uncharged one. The population of the monovalent state reaches a maximum at a pH slightly lower than the maximum of the ideal curve, which is at pH = (pK A,1 + pK A,2 )/2. At pH ≈ 5.5 ≈ pK A,2 , we observe that the population of the monovalent state is significantly suppressed in favour of the divalent state, SuA 2 -. These differences can be alternatively viewed as a shift of the ideal curves to lower pH values (dashed lines in Figure ). According to the Debye-Hückel theory (see Sec. S2.4), this shift should be proportional to valency of the respective ion. However, our simulations show that the population of the divalent species is shifted by about 0.3 units of the pH, more than twice as much as the population of the monovalent one, demonstrating the non-linear effect of electrostatic correlations. |
6490738ba2c387fa9a92d863 | 41 | The non-linear effect of electrostatic correlations gets further amplified when we examine the charge states of the same solute in the polyelectrolyte complex phase, shown in Figure ). Here, we observe that population of the uncharged state is suppressed much more than in the supernatant and it is shifted to lower pH values by about 0.5 unit. Population of the monovalent state reaches a maximum at pH < 4 < pK A,1 , which is more than 1 pH unit lower than position of the maximum of the ideal curve. Interestingly, population of the divalent ions starts to dominate already at pH ≈ 4, although the ideal curve predicts that this popuplation should be vanishingly small at the given pH. Again, the non-linear effect of electrostatic correlations causes that the divalent state of the diprotic acid is strongly preferred in the interpolyelectrolyte complex and starts to dominate already two units of pH below the value pK A,2 . |
6490738ba2c387fa9a92d863 | 42 | The strong preference of ionized states in the polyelectrolyte complex affects not only their populations within this phase but also the overall partitioning of the diprotic acid between the two phases. In Figure ) we show that the partition coefficient of the diprotic acid, K SuA,tot , strongly increases as a function of the pH. At low pH values, when the uncharged state dominates, the partitioning is governed by the steric repulsion, resulting in K SuA,tot < 1. As soon as the monovalent state starts to dominate, we observe K SuA,tot > 1 as a consequence of electrostatic interactions. As the pH is further increased, K SuA,tot continues to increase until it saturates at a value of K SuA,tot ≈ 5, which is slightly lower than the partition coefficients of divalent ions observed in Sec. 3.2. This difference can be explained by the concentration of the diprotic acid being about six times higher than the concentration of divalent salt in Sec. 3.2. This higher concentration was required to ensure good statistics on partitioning of individual ionization states. As a side effect, it caused that the fully ionized diprotic acid significantly affects the ionic strength in the supernatant, and thereby also the activity coefficients of all other ions. |
6490738ba2c387fa9a92d863 | 43 | To further elucidate the complexity of partitioning of multiprotic solutes, we plot in Figure ) the concentrations of various ionized stated of the diprotic acid in the supernatant. This figure shows that the concentration of the uncharged state in the complex is about two times lower than in the supernatant. This is because the interaction of the uncharged species is dominated by the steric repulsion which favours the more dilute supernatant over the highly concentrated interpolyelectrolyte complex. In contrast, the concentrations of monovalent species are almost the same in both, supernatant and the complex phase with a slight preference of the supernatant phase, in agreement with the partitioning of monovalent salt, discussed in Sec. 3.2. Finally, the concentration of divalent species is about six times higher in the complex phase than in the supernatant, in agreement with the partitioning of divalent salt. Thus, we can conclude that partitioning of a diprotic acid as a function of pH represents a gradual transition from the behaviour dominated by steric repulsion to that dominated by electrostatic interactions. However, this transition is strongly affected by deviations from the ideal behaviour, therefore, it cannot be correctly described by the ideal model of acid-base ionization. |
6490738ba2c387fa9a92d863 | 44 | In the preceding section, we showed that our generic coarse-grained model qualitatively mirrors multiple features of PEC phase separation, observed in experiments and predicted by other computational and theoretical models. Minor quantitative discrepancies between various experiments and theoretical models can be explained by differences in chemical details, such as the size of the monomeric units, permittivity of the PEC phase, charge density on the polymer chains, or specific polymer-polymer and polymer-solvent interactions. However, systematic and independent variation of these parameters is impossible in experimental systems or in models which have been tuned to reproduce one specific experimental system. In this section, we exploit the feature of our generic model that allows us to vary these parameters systematicaly, in order to explore how a quantitative agreement between simulation and experiment can be reached by tuning of the model parameters. Furthermore, it allows us to distinguish generic features of the PEC phase separation from those which depend on specific details of the model. |
6490738ba2c387fa9a92d863 | 45 | The bulkiness of the monomeric units and the asymmetry in size between the monomeric units and small ions affects the balance between steric repulsion and electrostatic interactions. The electrostatic interaction provides a cohesive force which favours phase separation whereas the steric repulsion has the opposite effect. Therefore, the choice of particle sizes in the simulation model affects the location of phase boundaries in the phase diagram. |
6490738ba2c387fa9a92d863 | 46 | We chose the size of monomeric units and small ions within a range commonly used in coarse-grained simulations of similar systems. For instance, the authors of Ref. proposed a bottom-up coarse-graining of PAA from atomistic simulations, where distances between different carbon atoms between neighbouring PAA groups ranged from 0.250 nm to 0.450 nm. Herein, we chose σ mon ≈ 0.426nm = 1.20σ to represent an effective size of whole PAA monomeric unit, modeled by a single particle. For simplicity, we chose the same size for the polycation, using the monomer-monomer distance from the coarse-grained model of PAH proposed in Ref. For the size of the ions we used a slightly smaller value, σ = 0.355 nm, matching our previous work, 125 where we demonstrated, that this ion size reproduces the experimentally determined activity coefficients in aqueous solutions of NaCl, 130 in a broad range of concentrations, up to the solubility limit of c NaCl ≈ 700mmol/L. |
6490738ba2c387fa9a92d863 | 47 | In Figure ) we explore the effect of the monomer bulkiness on the phase diagram of PEC in equilibrium with a supernatant containing NaCl only. Because our excluced-volume potential is much steeper at small distances than the bonding potential, an increase in size of the particle representing the monomeric units, σ mon , implies a simultaneous increase in the mean bond length to ⟨b⟩ ≈ σ mon , thereby decreasing the effective charge density along the chain contour. Lowering the charge density on the polymer decreases the entropic gain from the release of counterions and suppresses the elecrostatic interactions between the chains, thereby destabilizing the PEC phase. This decrease acts in synergy with an increased range of steric repulsion. Consequently, an increase in σ mon shrinks the co-existence region. Notably, if we use σ mon = 1.30σ ≈ 0.46 nm on Figure ), the pressure-composition protocol does not indicate a stable PEC phase at c NaCl ≈ 532mM, whereas a stable PEC phase exists at the same c NaCl if we use σ mon = 1.20σ ≈ 0.43 nm. Eventually, if we use σ mon = 1.40σ ≈ 0.50 nm, there is no phase coexistence at all and the phase diagram contains only a single-phase region. |
6490738ba2c387fa9a92d863 | 48 | In order to separate the synergistic effects of monomer bulkiness and charge density, in Figure ) we fixed the size of monomeric units to σ mon = 1.00σ ≈ 0.35 nm and we varied only the bond length, yielding the same range of charge densities as in Figure ) while keeping the steric repulsion unchanged. Our results in Figure ) show that a decrease in the charge density shrinks the phase separation region, as predicted by theories and observed in experiments. |
6490738ba2c387fa9a92d863 | 49 | Because of limited availability of suitable inputs, choosing the permittivity of the interpolyelectrolyte complex phase is much more tricky than choosing the sizes of monomeric units or small ions. The supernatant phase is a solution of small ions which is sufficiently dilute to assume that its permittivity is close to that of pure water, ε r ≈ 78. |
6490738ba2c387fa9a92d863 | 50 | However, the polyelectrolyte complex phase should have a lower permittivity than pure water because of high polymer volume fraction, typically about ∼ 10%. This idea is further supported by Ref., where the authors used permittivity of ε r ≈ 30 to match their simulations with experiments. In contrast, the simulations in Ref. just used the permitivitty of water, however, their approach was aimed at the strong coupling regime, where the variation of perimitivitty does not have a strong effect on the phase behavior. |
6490738ba2c387fa9a92d863 | 51 | In Figure ), we show phase diagrams obtained by varying the permititvity of the complex phase, while keeping the permittivity of the supernatant at ε r ≈ 78. A decrease in permitivitty amplifies electrostatic interactions between all charged particles. The cohesive effect of electrostatics not only extends the two phase region, but also causes preferential partitionig of NaCl into the phase of the complex at ε r ≲ 52. This is in line with recent experiments and simulations which acknowledged that the partition coefficients of NaCl can indeed exceed unity if they use short-range attractions between ions and polymers. 131 This, however, unveils a problem -our simulations and those in Ref. both observe qualitatively the same trend in partition coefficients of NaCl, but for different reasons. Even though our model has just a few tunable parameters, it brings about a lot of flexibility and one has to be careful when setting the model parameters to identify the correct source of phenomena with respect to the specific experiment, and to avoid over-fitting. |
6490738ba2c387fa9a92d863 | 52 | Similar to permittivity, another phenomenological parameter to set is the quality of solvent, collectively describing effective interactions between the monomeric units as compared to the polymer-solvent interactions. We characterize these interaction using the second virial coefficient, B 2 , where as explained in Sec. S1.1 in the ESI. The value of B 2 is the highest for purely repulsive interaction (athermal solvent) and it gradually decreases towards the θ-state, which is defined by B 2 ≈ 0. Clearly, aditional attraction between the monomeric units further stabilizes the coacervate phase, extending the coexistence region, in agreement with Ref. Moreover, we observe significant decrease in partition coefficient of NaCl as quality of solvent drops, since the phase of the complex is getting more crowded. |
6490738ba2c387fa9a92d863 | 53 | The finite size of the simulated system may have a significant impact on the location of phase boundaries determined from a simulation with periodic boundary conditions. Only in the limit of infinite system size, such simulations converge to the properties of bulk phase-separated systems. To demonstrate that our results were not significantly affected by the size of the simulated system, in Sec. S2.3 in the ESI we present structural analysis of systems with various sizes, determined by the numbers of polymer chains in the polyelectrolyte complex, N A = N C , ranging from 32 up to 96. In a typical simulation, we observe that L ≈ 8R g , where R g is the mean radius of gyration of the polyions. Therefore the box is large enough, so that chains do not interact strongly with their own periodic images. Accordingly, we observe in Sec. S2.3 that the positions of binodals and partition coefficients of the ions are virtually the same for all simulated sizes of the system. Therefore, we used N A = N C = 32 in all further simulations, which had the most favourable computational costs. |
6490738ba2c387fa9a92d863 | 54 | The harmonic bonds in our model are soft enough to allow crossing of bonds, unlike the standard bead-spring model of Kremer and Grest. The bond crossing has little effect on the static structure and thermodynamic properties but it significantly affects the relaxation dynamics of polymer chains. The Kremer-Grest model was designed to study reptation dynamics of polymer chains in a melt which required that the crossing of bonds must be avoided. In contrast, our model is designed to study only thermodynamic and structural properties of the systems. Therefore, we allow bond crossing, which accelerates the relaxation dynamics, enabling faster equilibration and more efficient sampling of the simulated systems. |
6490738ba2c387fa9a92d863 | 55 | In Figure in the ESI we show the mean-square displacement of centers of mass of the polymer chains as a function of lag-time. We first observe the sub-diffusive regime with scaling exponent ∼ 1/2 at short lag-times and eventually diffusive Brownian regime at large lag-times. Interestingly, these results agree very well with recent simu-lations of Liang et al., 101 who used the polymer model derived from the Kremer-Grest model mentioned above. We surmise, that the bond crossing of our model is rather rare event and it does not destroy the equilibrium dynamics, but one would have to be extremely cautious using this model to measure properties such as viscosity, shear modulus or probing reptation and structure of entanglements in interpolyelectrolyte complexes out of equilibrium or at high densities. |
6490738ba2c387fa9a92d863 | 56 | We have shown that a coarse-grained simulation model, combined with the grandreaction method for simulating acid-base equilibria in two-phase systems, can predict the phase stability of polyelectrolyte complexes (PEC) and partitioning of ionic solutes between the PEC phase and the supernatant. We used the pressure-composition protocol to determine the conditions under which the PEC phase coexists in equilibrium with the supernatant of a given composition. Furthermore, we have shown that the pressure-composition protocol should be supplemented by structural analysis of the PEC phase in order to ensure that phase separation does not occur within the simulation box, resulting in polymer-rich and polymer-poor domains. |
6490738ba2c387fa9a92d863 | 57 | Our model is suitable for predicting the phase coexistence far from the critical point. However, it cannot correctly describe the critical region because it assumes that no polymer is present in the supernatant phase. In accordance with the literature, our simulations predict that the stability window of the PEC phase shrinks as the salt concentration is increased. Furthermore, they predict that the partition coefficients of monovalent ions are slightly below unity, indicating a slight preference for the supernatant phase. Unlike monovalent ions, divalent ions strongly prefer the PEC phase, resulting in partition coefficients on the order of ten. These results semi-quantitatively agree with our own experiments and also with other experiments from the literature. |
6490738ba2c387fa9a92d863 | 58 | Next, we have shown that the ionization of a weak diprotic acid is significantly enhanced in the PEC phase, as compared to the supernatant. Therefore, the partition-ing of such a weak acid between the two phases can be tuned by varying the pH. At low pH, the weak diprotic acid is uncharged, therefore, steric repulsion dominates its interactions, causing that it prefers the supernatant phase. As the pH is increased, the preference for the PEC phase increases concomitantly. Divalent form of the diprotic acid starts to dominate in the PEC phase already at pH values when only its neutral and signle-ionized forms would be found in the supernatant. Consequently, the diprotic acid significantly accumulates in the PEC phase already at much lower pH values than could be expected from its ionization in the aqueous solution of the supernatant. This observation demonstrates that the partitioning of weak multiprotic acids can be easily tuned by varying the pH, providing interesting opportunities for engineering applications, such as separation of small acids based on their pK A, . |
6490738ba2c387fa9a92d863 | 59 | We determined how the model parameters affect the partition coefficients and shape of the phase diagram, particularly the region of phase coexistence. We showed that an increase in monomer size or a decrease in charge density both shrink the coexistence region whereas a decrease in permittivity of the PEC phase or a decrease in solvent quality, quantified by the second virial coefficient B 2 , both expand the coexistence region. Simultaneously, charge density and solvent quality have only negligible impact on the partition coefficients of small ions whereas the monomer size and permittivity of the PEC phase significantly affect the partition coefficients to such an extent that variation of these parameters may even switch the preferential accumulation of the ions from supernatant to the PEC phase. Thus, it should be always possible to tune the simulation model in order to match a specific set of experimental results. Additionally, various combinations of these parameters can lead to nearly identical phase diagrams, producing a good match with the experiment. Therefore, even a quantitative agreement between simulations and experiments does not automatically imply that the model parameters have been chosen correctly. Nonetheless, the generic patterns in the partitioning of multivalent ions, described above, may be quantitatively but not qualitatively affected by small changes in parameters of the model. where we see |
6490738ba2c387fa9a92d863 | 60 | Similarly for the succinic acid, the choice of pK A,1 and pK A,2 together with the equilibrium constant, defined in Eq. 16, uniquely defines all other equilibrium constants, Eq. 17 dominant form, without loss of generality, in the limit of high pH is SuA 2 -. Accordingly, the mean counts of forms HSuA -an H 2 SuA are low and the concentrations can not be determined accurately due to the finite-size effects on the fluctuations. |
6759b06df9980725cfbc8cef | 0 | The average global temperature has been steadily rising since the late 19 th century and was c.a. 1.2 Celsius degree warmer in 2023 than the preindustrial average. This warming is largely driven by greenhouse gas emissions, with CO₂ accounting for over 60 % of the effect. Amongst various strategies to manage CO2 emission levels, direct air capture (DAC) offers a solution by capturing CO2 directly from ambient air. This process is challenging due to trace amounts of CO2 concentration (around 400 ppm) and low selective capturing of CO2 over atmospheric water vapor. |
6759b06df9980725cfbc8cef | 1 | To date, DAC technologies have predominantly relied on chemisorption of CO2 using amines and solid alkali hydroxides, due to their strong affinity for CO2. However, the regeneration of adsorbents is the dominant source of the total operational costs. To mitigate these issues, physisorption-based materials with weaker CO2 interactions have been explored to reduce regeneration costs with repeated usage. Among these, metal-organic frameworks (MOFs) stand out due to their high surface area and remarkable tunable properties. However, the chemical space for forming MOFs is vast due to the large number of possible chemical building blocks and topologies. |
6759b06df9980725cfbc8cef | 2 | To expedite the identification and design of promising MOFs for CO2 capture, large-scale screening has been widely applied. These are typically based on classical molecular simulations with force fields, such as universal force fields (UFFs) with point charges (e.g. Qeq, EQeq, and DDEC ). For CO2 capture, Boyd et al., designed new MOFs suitable for flue gas CO2 capture by introducing CO2 responsive pockets acquired from screening results. Findley et al. screened the CoRE MOF database to evaluate whether MOFs are suitable for DAC application. While these methods are time-efficient, they struggle to accurately describe the interaction between the MOF frameworks and the gas molecules such as CO2 and H2O, due to the diversity of chemical environments and the complexity of the short-range potential energy landscape. While ab initio calculations offer a more accurate alternative by explicitly accounting for electron density distributions, the computational cost increases substantially and are impractical for large-scale screening or extensive data accumulation for porous solids. |
6759b06df9980725cfbc8cef | 3 | Machine learning force fields (MLFFs) have come to the forefront to tackle this challenge (Figure and (b)), combining the low computational cost of classical force fields with the high accuracy of ab initio methods. While various materials including alloys, metal oxides, and perovskites have been widely studied, the applications of MLFF for MOFs have still been limited, especially for heterogeneous systems that contain the MOF material itself and the guest molecules. Although a few studies have explored the development of MLFFs for the combined system of MOF with gas molecules such as CO2 or H2, the transferability across different MOF systems remains significantly limited. This limitation is primarily due to the lack of comprehensive databases that contain density functional theory (DFT) simulation data involving the combined system of MOFs with gas molecules due to their expensive computational cost. |
6759b06df9980725cfbc8cef | 4 | In this work, we introduce a MLFF that is both accurate for DAC applications and transferable across a wide range of MOFs, as shown in Figure . By including configurations of MOFs interacting with CO2 and H2O gas molecules contained in our curated GoldDAC dataset, we finetune a foundation model that was pretrained on diverse crystalline compounds. To deploy this model, we have developed the DAC-SIM package, which combines our MLFF with the statistical Widom insertion method , molecular dynamics, and structure optimization. We use DAC-SIM to conduct high-throughput screening of 8,131 experimental MOFs in the CoRE MOF database . Comparison of the distributions in CO2 adsorption and selectivity highlights several limitations in the traditional screening approach and we identify several novel candidates and design principles. |
6759b06df9980725cfbc8cef | 5 | A key challenge for DAC lies in identifying materials that exhibit strong CO₂ affinity and high selectivity over H₂O. To address this, Monte Carlo (MC) simulations such as Widom insertion and Grand Canonical Monte Carlo (GCMC) are used to determine gas adsorption properties of materials by calculating the heat of adsorption (Qst) and Henry's law coefficient (KH). These ensemble-averaged properties enable a more comprehensive evaluation than relying solely on single configuration properties such as adsorption energies at the most likely stable binding sites. |
6759b06df9980725cfbc8cef | 6 | The Widom insertion method, as outlined in Figure (c), is well-suited for DAC under the condition of low CO2 concentration, where modeling a dilute system is applicable. The simulation involves numerous individual calculations to build the ensemble, where single gas molecules are randomly inserted into MOF frameworks to compute the ensemble-averaged interaction energy, 𝑈 !"# . The interaction energy 𝑈 !"# is determined from: |
6759b06df9980725cfbc8cef | 7 | , where 𝑘 % is the Boltzmann constant and 𝑇 is the temperature. Without the contribution from the thermal term 𝑘 % 𝑇 in 𝑄 $# , the value represents the ensemble-averaged interaction energy, denoted as ∆ℎ ! . For convenience within this study, the magnitude of Qst is represented with an inverse sign. |
6759b06df9980725cfbc8cef | 8 | that accurately reproduce the DFT results for a diverse set of crystalline materials. These models have undergone extensive pretraining on datasets with total DFT energies (E) and atomic forces (F), mainly sampling from geometry optimization trajectories such as MPtrj 41 . Among these, MACE-MP-0 model (referred to hereafter as MACE-MP for simplicity) stands out with its combination of the atomic cluster expansion for structure representation with equivariant and high body-order message passing graph neural networks for regression. The pretraining datasets for these models are dominated by inorganic compositions with fewer organic and organic-inorganic solids; however, MACE-MP model has shown reasonable preliminary results on MOF systems. Therefore, we have built on this to finetune a transferable MLFF, termed MACE-DAC, tailored to MOFs and their interaction with relevant gases. This analysis encompasses both the foundational MACE-MP model and its variant, MACE-MP-0b that incorporates the analytic Ziegler-Biersack-Littmark (ZBL) repulsive potential to remove pathologies in the PES at short distances. The mean absolute error (MAE) for the average of total PES data, denoted as 'All' in Figure (c) and Table , exhibits a notable reduction compared to the original foundation models with respect to interaction energy and force predictions. This improvement is mainly attributed to enhanced predictive accuracy in the repulsive region of the finetuning model, assisted by configurations sampled by GoldDAC dataset. |
6759b06df9980725cfbc8cef | 9 | The finetuning model does decline in performance within the equilibrium and weak-attraction regions compared to the pretraining MACE-MP models. This degradation can be attributed to the complexity in the interaction energy which is not determined solely by the combined system's total energy but by its energy relative to the isolated framework and gas molecule. The finetuning process can disrupt the delicate balance originally established with biases toward the data represented in the finetuning dataset. |
6759b06df9980725cfbc8cef | 10 | This issue aligns with the phenomenon of "catastrophic forgetting", wherein knowledge acquired during pretraining is disrupted or overwritten during the subsequent finetuning process. To remedy this issue, we implemented a refined finetuning strategy with continual learning loss. This additional loss term is designed to preserve the pretraining model's knowledge during the finetuning. |
6759b06df9980725cfbc8cef | 11 | Specifically, the continual learning loss is implemented by freezing the weights of the pretraining model and minimizing discrepancies in energy and force predictions between the pretraining and the finetuning model during the finetuning process, as illustrated in . This approach ensures that the finetuning process does not introduce significant deviations from the pretraining model, thereby ensuring the original energy balance between components in the interaction energy calculation. |
6759b06df9980725cfbc8cef | 12 | By introducing this regularization loss, the finetuning process is guided to adapt more gradually to the new MOF data while retaining the knowledge learned from pretraining on inorganic materials. As shown in Figure (c), this method prevents performance degradation in the equilibrium and weak-attraction regions, observed in the finetuning model without the continual learning loss (termed as MACE-DAC-0), while maintaining the trend that significantly reduces loss in the repulsion region. Therefore, the continual learning model (termed as MACE-DAC-1) achieved the lowest MAE across all PES data. |
6759b06df9980725cfbc8cef | 13 | Subsequently, geometry optimizations were carried out using MACE-DAC-1, where the details of geometry optimization are provided in the Method section. Following these optimizations, Qst and KH values were calculated by the Widom insertion simulations for 6,461 MOFs, after excluding MOFs encountering convergence issues and consistency problems (where the standard deviation from two repeated folds of averaged interaction energy calculations exceeded 0.05). promising for DAC application within physisorption mechanism. In line with this criterion, it is observed that 364 MOFs (5.63 %) exhibit CO2 Qst values exceeding 50 kJ/mol, and 167 of them even surpassing those for H2O. presents the relationship between Qst for CO2 and the selectivity derived from the ratio of KH for CO2 to H2O. Although KH generally follows the trend of Qst as shown in |
6759b06df9980725cfbc8cef | 14 | In contrast to the MC method which computes ensemble average properties for gas adsorption, an alternative approach can utilize single configuration properties such as the lowest interaction energy at the most favorable binding site within a given MOF framework. This approach, used in the ODAC23, involved calculating the lowest interaction energy at the most favorable binding site within a given MOF framework using DFT. To identify the most stable adsorption site, it begins with generating potential sites through classical force fields or random selection, yielding multiple candidate configurations per structure. MOFs with less than four configurations were excluded in this work. Each configuration then undergoes geometry optimization via DFT to identify the most energetically favorable binding site. |
6759b06df9980725cfbc8cef | 15 | Although this approach is effective for identifying the optimal interaction site, it may not comprehensively represent the overall gas adsorption performance of MOFs. Indeed, the lowest CO2 interaction energy of 2,794 MOFs derived from the ODAC23 database and the corresponding Qst values for CO2 calculated by DAC-SIM do not exhibit a strong correlation, as shown in the scatter plot in Figure (c). The same trend was observed for the H2O molecule in Supplementary Figure . |
6759b06df9980725cfbc8cef | 16 | The Widom insertion simulation results derived from our developed MLFF (MACE-DAC-1) using DAC-SIM were systematically compared with those obtained from a classical force field approach. In the classical method, the UFF 18 combined with density derived electrostatic and chemical (DDEC) charges (hereinafter referred to as UFF+DDEC) were used to simulate the CO2 and the H2O adsorption properties in the MOFs. UFF is interatomic potential consisting of empirical parameters and DDEC charges are derived from DFT calculations. A total of 2,654 MOFs, having the calculated DDEC charges provided by Kancharlapalli et al. were used for this analysis. Figures 4(a) and present the Qst distributions for CO₂ and H₂O, respectively, as computed using the DAC-SIM package with the developed MLFF and the RASPA software with the UFF+DDEC method. The overall distribution of CO2 Qst values from DAC-SIM closely resembles that obtained with UFF+DDEC; however, a notable difference is a minor peak at the lower range near 0 kJ/mol. As shown in Supplementary Figure , the LCD distribution of these MOFs exhibiting CO2 Qst values below 2.5 kJ/mol shifts to a lower range compared to the structures before undergoing geometry optimization. That is, the shrinkage in pore size occurred during the prior geometry optimization stage before Widom insertion simulation, thereby preventing them from accommodating CO₂ molecules. This highlights the advantage of DAC-SIM, enabling computationally efficient geometry optimization with accuracy near that of DFT prior to Widom insertion simulation, by improving force predictions in the finetuning stage. |
6759b06df9980725cfbc8cef | 17 | While most MOFs have H2O Qst values below 100 kJ/mol in the DAC-SIM calculations, over 17.8 % of MOFs exceed 100 kJ/mol when using UFF+DDEC. These higher Qst values in the UFF+DDEC calculations are primarily associated with MOFs containing lanthanide metals. Supplementary Figure shows the most frequently observed metal types in MOFs with H2O Qst values exceeding 100 kJ/mol in UFF+DDEC calculations, largely comprising lanthanide elements such as Gd, Nd, Tb, La, Er, Pr, Dy, Ce, Sm, Ho, and Tm. |
6759b06df9980725cfbc8cef | 18 | Supplementary Figure further indicates that there is no significant deviation in distribution between the overall and lanthanide MOFs regarding the lowest interaction energies from the ODAC23 database based on DFT calculations, while a pronounced increase in Qst values is observed for lanthanide MOFs in UFF+DDEC calculations. It suggests that the UFF+DDEC method tends to overestimate Qst values for H2O in lanthanide MOFs. Supplementary Figure illustrates that UFF+DDEC calculations produce significantly higher H₂O Qst values compared to CO2 for MOFs containing lanthanide metals. In contrast, the DAC-SIM results, similar to the DFT results of interaction from the ODAC23 database, do not exhibit such a pronounced difference between lanthanide-containing MOFs and those with other metals. |
6759b06df9980725cfbc8cef | 19 | This inconsistency between classical force field and DFT calculations arises from the chemically complex inorganic-organic hybrid compositions of the MOFs. Simulating interactions involving metals and neighboring organic atoms (e.g. oxygen or nitrogen atoms bonded to metal) within these hybrid environments requires careful parameter fitting regarding each individual chemical context. However, UFF employs fixed parameters, which limit its ability to accurately represent varying chemical environments. While point charges offer an additional degree of freedom for simulating interactions, UFF was originally developed without incorporating point charges. Thus, it remains uncertain whether point charges effectively compensate for or exacerbate simulation errors. |
6759b06df9980725cfbc8cef | 20 | Given that the unique chemical environments of MOFs are primarily attributed to the presence of metals, the role of metals was investigated. Supplementary Figure illustrates the distribution of point charges across various metals, revealing that metals with higher point charges tend to exhibit greater discrepancies between UFF+DDEC and DAC-SIM results in Supplementary Figure ). Metals with elevated point charges can function as hard Lewis acids and polarize neighboring atoms nearby metals (typically oxygen or nitrogen atoms within MOFs). This polarization effect decreases the polarizability of adjacent atoms relative to similar atoms in different environments, while the metal's polarizability remains largely unchanged. However, since UFF utilizes fixed parameters, it is unable to accurately account for this variation in polarizability. Although point charges such as DDEC are generally applied to enhance interaction modeling by providing additional electrostatic potential, they are only partially effective in specific cases and often exacerbate discrepancies. Therefore, the overestimation of Qst in UFF+DDEC calculations is particularly pronounced for H2O molecule, which can induce stronger electrostatic potential compared to CO2 molecule. |
6759b06df9980725cfbc8cef | 21 | Due to overestimation of H2O Qst, only few candidates were rarely identified as suitable for DAC application from the screening from CoRE MOF database using classical force field. Likewise, promising MOFs identified in DAC-SIM simulations similarly fail to exhibit high performance using the UFF+DDEC model classical as shown in . We conducted an in-depth investigation into the origin of this difference using three MOFs, whose REFCODEs are GUJVOX (Mn2(ttmb)2(SO4)2), YUFLOC (AEMOF-5), and FECWOB01 (Zn-TBC), as case studies (filled square points in Figure ). |
6759b06df9980725cfbc8cef | 22 | Firstly, in the case of GUJVOX, while the CO2 Qst remained above 50 kJ/mol, the H2O Qst exhibits a substantial increase in the UFF+DDEC calculation. The presence of the electronegative functional group (SO4 2-) within the framework, which acts as a strong binding site for H2O, appears to be a major factor in this discrepancy. The point charges for S in SO4 2-are +1.5, while for O in SO4 2-, they are -0.7, indicating significant polarization within this moiety (see Figure ). As discussed with Figure (b), UFF+DDEC encounters challenges in accurately simulating highly polarized moieties, leading to a general overestimation of interactions with H2O compared to CO2. |
6759b06df9980725cfbc8cef | 23 | However, the sigma values for some metal atoms tend to be overestimated, preventing guest molecules from approaching the OMS closely enough. Consequently, the interaction between the MOF and the guest molecule is underestimated. Unlike H2O, CO2 cannot generate sufficient electrostatic potential to counterbalance this underestimation in interaction energy. For FECWOB01, the Qst values for both CO2 and H2O are underestimated, though the trend of CO2 Qst being higher than H2O Qst remained consistent. FECWOB01 contains an azole linker, and since the electronegativity of azoles varies with the nitrogen content, the force field parameters of nitrogen require careful consideration. For example, a study on ZIF-8, which contains imidazole as an organic linker, derived a tailored force field by using nitrogen parameters from Generalized Amber Force Field (GAFF). In this case, the epsilon value was set at 85.5 K, which is 2.5 times larger than that in UFF, while the sigma value remained nearly identical. Similarly, applying a larger epsilon for nitrogen in azoles would likely increase the interaction strength for both CO2 and H2O. |
6759b06df9980725cfbc8cef | 24 | To conduct a systematic analysis, interaction energies between MOFs and gas molecules were compared by examining the PES using UFF+DDEC and MACE-DAC-1 for the GoldDAC test set as provided in Supplementary Figure . Overall, the MLFF model exhibits more accurately capturing the interactions between MOFs and CO2 or H2O molecules, aligned with the DFT calculations. While extensive parameter fitting within classical force fields can yield high accuracy, such methods are typically confined to a limited set of similar MOFs and lack broad transferability. Therefore, using our DAC-SIM process to screen adsorption properties offers a more precise and time-efficient alternative to traditional calculations based on classical force fields. |
6759b06df9980725cfbc8cef | 25 | The screening results obtained using MACE-DAC-1 with DAC-SIM allowed for a systematic analysis of chemical features within promising MOFs. All 161 promising candidates in the CoRE MOF database were further investigated. 42 chemically incorrect MOFs (e.g., missing hydrogen atoms or overlapping atoms due to symmetry issues, see Supplementary Figure ) were excluded. |
6759b06df9980725cfbc8cef | 26 | Before proceeding with the analysis of the remaining MOFs, we evaluated the necessity of finetuning by comparing MACE-DAC-1 to MACE-MP. As shown in Supplementary Figure , S4 and S15, MACE-MP tended to overestimate interaction energies, leading to an overestimation of H2O Qst compared to MACE-DAC-1. In this context, similar to the observation that not all promising MOFs demonstrate desirable performance in UFF+DDEC calculations, 47 out of 119 MOFs (approximately 40 %) fail to be identified as promising when using MACE-MP instead of MACE-DAC-1. |
6759b06df9980725cfbc8cef | 27 | As shown in Figure (a), seven chemical features, including OMS, parallel benzene ring (PAR), metalelectronegative atom-metal (M-X-M), uncoordinated nitrogen, azole linker, methyl functional group (-CH3), and electronegative functional group (-EN), were identified in the promising MOFs. 4 of them (OMS, PAR, M-X-M, and uncoordinated nitrogen) were suggested in previous screening works, and others (azole linker, -CH3, and -EN) were intermittently described in experimental works. Among these, OMS and the azole linker appeared in nearly half, or more, out of the MOF candidates. The OMS, a well-known chemical moiety, facilitates strong interactions between positively charged metal atoms and negatively charged oxygen atoms in gas molecules. However, the stronger interaction between the more negatively charged O atom in H2O, compared to that in CO2, leads to H2O typically interacting more strongly with OMS than CO2. This suggests that OMS utilization in DAC application should be approached with caution. |
6759b06df9980725cfbc8cef | 28 | Additionally, the azole linker, commonly used in MOF synthesis due to its robust bonding with transition metals, contains nitrogen and other non-carbon atoms (e.g., nitrogen, sulfur, or oxygen) that actively interact with CO2 molecules. Given the frequent application of nitrogen-rich MOFs and porous materials in DAC application, azole linkers may serve a similar function within MOFs for facilitating CO2 capture in humid condition. |
6759b06df9980725cfbc8cef | 29 | Apart from the two frequently observed features, the performance of promising MOFs containing one of the remaining five chemical features is shown in to examine the role of those features in DAC application. Among them, PAR, identified through screening study using classical force fields as one of effective features for CO2 capture in humid conditions, appeared frequently in the results from the DAC-SIM as well. The strong dispersion force created between parallel benzene rings form selective pockets conducive to capturing CO2. Due to its larger molecular size and unique pi bonds between C and O atoms, CO2 is more responsive to the chemical environment dominated by dispersion forces, unlike H2O. Furthermore, PAR is less affected by the stronger electrostatic interactions induced by H2O in comparison to CO2. As a result, 9 out of 11 promising MOFs containing PAR show high selectivity, with KH ratio for CO2 to H2O exceeding 100. |
6759b06df9980725cfbc8cef | 30 | The M-X-M, a generalized form of metal-oxygen-metal (M-O-M) bridges, was proposed as a feature for strong CO2 adsorption from screening study with the classical force field approach. However, this feature was not considered essential for constructing MOFs specifically for selective CO2 capture, as many MOFs with M-O-M bridges interact strongly with both H2O and CO2. However, given that the overestimation of H2O interaction, which was observed in calculations using UFF and point charges, was corrected in this study, M-X-M still appeared in over 20 % of the promising MOFs often associated with high CO2 selectivity over H2O. |
6759b06df9980725cfbc8cef | 31 | Uncoordinated nitrogen, as defined in this study, refers to nitrogen atoms that are not part of metal coordination complexes or azole linkers, such as those in amine groups or azines. They have been experimentally recognized as promising functional groups for DAC applications. The lone pair on nitrogen atom create localized pockets within the framework that act as nucleophilic site, facilitating strong interactions with electrophilic CO2 molecules. Consequently, 87.5 % of MOFs containing uncoordinated nitrogen exhibit CO2 Qst values exceeding 58.9 kJ/mol (the top 25 % CO2 Qst value of the candidates) and selectivity greater than 100. Notably, selective CO2 capture associated with amine groups occurs generally via chemical reaction, highlighting the need for further research to fully understand this selective behavior. |
6759b06df9980725cfbc8cef | 32 | For the -CH3 group, known for its hydrophobic properties, it is commonly employed in MOFs to impart hydrophobic characteristics. As the smallest hydrocarbon unit, -CH3 becomes increasingly hydrophobic with greater carbon chain length. Previous study has demonstrated that organic ligands with extended hydrocarbon chains are particularly suitable for DAC application when integrated into MOFs. Likewise, promising MOFs functionalized with -CH3 groups exhibit selectivity values consistently above 7.5, with 59 % of them exceeding 100. |
6759b06df9980725cfbc8cef | 33 | Lastly, the -EN group, capable of readily accumulating electrons, can function as a nucleophilic site, selectively interacting with the electrophilic C atom within CO2 molecule. Due to the high electronegativity of fluorine, fluoride functional groups are frequently employed to design optimal pockets for selective CO2 capture within MOFs. A few MOFs incorporating fluoride functional groups have been experimentally validated to maintain their CO2 capture efficiency under humid conditions. |
6759b06df9980725cfbc8cef | 34 | Additionally, five representative MOFs (RIPPEN, GAVYAG, ARAHIM, GELVID01, and PEGBUA), which have been computationally or experimentally studied for CO2 capture purpose, are illustrated in Figure (b). Among these MOFs, ARAHIM, also known as NbOFFIVE-1-Ni or KAUST-7, 67 has even been proven to maintain trace CO2 capture performance in humid conditions. Moreover, the common names of promising MOFs were examined to enhance accessibility. As shown in Table , 12 MOFs were presented with the corresponding common name. Given the extensive tunability of MOFs, which allows for a wide range of chemical environments, all structures except ARAHIM exhibit multiple chemical features. Thus, the diverse chemical attributes of these MOFs are anticipated to synergistically contribute to DAC. |
6759b06df9980725cfbc8cef | 35 | In the same vein, combining these features offers an effective strategy for designing MOFs tailored for DAC application. For example, MOF named SIFSIX-18-Ni, which was demonstrated exceptional trace CO2 capture capabilities in humid conditions, incorporates three key features: azole linker, -CH3, and -EN. As shown in Supplementary Figure , SIFSIX-18-Ni consists of a tetramethyl bipyrazole organic linker (a methyl functionalized azole-based linker) and a SiF6 2-pillar (an electronegative functional group). The methyl groups on the organic linker act as water-blocking groups and the F atoms of the SiF6 2-pillar are working as nucleophile sites to interact with the electrophilic C atom of the CO2 molecule. |
6759b06df9980725cfbc8cef | 36 | with MOFs and integrated it with a simulation workflow to calculate Qst and KH with high accuracy and efficiency. This workflow is provided as a Python package, DAC-SIM, for ease of use for various user purposes. We screened the CO2 and H2O adsorption properties of over 8,000 experimentally synthesized frameworks and identified 161 promising candidates that were overlooked by conventional molecular simulation methods based on classical force fields. Additionally, we categorized features that can be introduced to design porous frameworks for DAC applications. Our workflow addresses prior limitations associated with classical force fields and the computational demands of direct ab initio methods, thus providing a solid foundation for accurate and fast virtual materials screening. |
6759b06df9980725cfbc8cef | 37 | To encompass the full range of the potential energy surface of interaction energy between the framework and gas molecules (CO2 and H2O), interaction energies were measured by adjusting the positions of the molecules within the structure. Starting from the equilibrium state with the presumably most stable configuration, two repulsive configurations were collected by moving the gas molecule closer to the structure, while three weak-attraction configurations were collected by moving the guest molecule toward the pore center (details on configuration construction are provided in Supplementary Note S1). |
6759b06df9980725cfbc8cef | 38 | MOFs that were identified as promising for CO2 capture or DAC applications. The MLFF model was trained based on optimized configurations collected from the ODAC23 database (training and validation set). First, the lowest CO2 and H2O interaction energies (interaction energies in this case were calculated after geometry optimization of MOF+gas system) were gathered for MOFs using the IS2RE database from the ODAC23 database. MOFs that met the following two conditions were considered: (1) the lowest CO2 and H2O binding energy is smaller than -0.1 eV, and (2) the number of atoms is less than 250. Lastly, given that the top 20 most frequently appearing metal atoms in the ODAC23 database were provided in the original paper (see Supplementary Figure ), 3 MOFs were randomly selected from those containing each metal type. Therefore, 60 MOFs were considered, and 120 equilibrium configurations were collected from the ODAC23 database, considering both CO2 and H2O molecules. For the evaluation of the trained MLFF model, an additional database was constructed based on the equilibrium states obtained from the manually selected 26 MOFs. A single guest molecule was introduced into the framework, where it was allowed to relax while the framework remained fixed. Three to five different initial points were sampled, considering active adsorption sites for the guest molecules, and the configuration with the lowest energy was selected as the equilibrium state. In total, 52 equilibrium configurations were collected for both CO2 and H2O molecules. Further details and illustrations of these 26 MOFs are provided in Supplementary |
6759b06df9980725cfbc8cef | 39 | Finetuning was carried out over 100 epochs with a batch size of 4. The Adam optimizer was adopted for parameter updates, featuring a learning rate of 1e-4 and zero weight decay. The loss function (𝐿 *,, ) for finetuning was balanced equally between total energy and force predictions, with a ratio set of 1:1. |
6759b06df9980725cfbc8cef | 40 | The DAC-SIM package is a python package integrated with the developed MLFF through the ASE calculator interface, for molecular simulations including Widom insertion, molecular dynamics, and geometry optimization. The Widom insertion simulations were conducted on the CoRE MOF 2019 dataset to determine Henry's law coefficient (KH) and heat of adsorption (Qst). Before performing the simulation, geometry optimization was implemented for each MOF structure. The optimization process includes a total of 30 steps, each consisting of 50 iterations of cell relaxation followed by 50 iterations of internal relaxation with the cell held fixed. The convergence criterion for the maximum force was set at 0.05 eV/Å to balance computational efficiency. |
6759b06df9980725cfbc8cef | 41 | For the following Widom insertion simulation, A grid with a spacing of 0.15 Å was generated to define potential insertion points across the unit cells. Grid points located within 1.5 Å of the central positions of gas molecules, such as the C atom in CO₂ and the O atom in H₂O, were excluded to avoid inserting molecules into blocked regions. In each iteration, a gas molecule was randomly rotated and placed at grid points, and their interaction energies between frameworks and gas molecules were calculated using MLFFs. To enhance statistical reliability, two independent Widom insertion simulations were performed, with each consisting of 10,000 insertions. The results from these simulations were averaged to calculate the final KH and Qst. |
6759b06df9980725cfbc8cef | 42 | DFT calculations were conducted using Vienna Ab initio Simulation Package (VASP). The exchangecorrelation terms were calculated using generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) functional. The recommended pseudopotentials from VASP manual except Eu atom (Eu_3 was used instead of Eu_2 following pseudopotential selection in QMOF database 86 ) were used to describe the ion-electron interactions. The energy cutoff was selected as 600 eV and the electronic relaxations were iterated until the energy changes below the 10 -5 eV. The gamma points were only used during calculation along with the gaussian smearing method (sigma = 0.05 Å). Dispersion correction using the D3 (Becke-Johnson) scheme were only applied during preparing equilibrium state configurations for test set and ionic relaxations were iterated until the forces less than 0.03 eV/Å. Spin-polarization calculations were only considered for frameworks with open-shell metal atoms, and U corrections were applied for d orbitals of Cu, Mn, Ni, Fe, Co, V, Nb, and Ti (more details on Supplementary Note S2). |
6759b06df9980725cfbc8cef | 43 | The interaction energies using a classical force field of MOFs in the test set were computed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software. Consistent with the approach taken in adsorption property calculations, a cutoff distance of 12.8 Å was employed, and the Ewald summation method was used to account for long-range coulombic interactions with an accuracy threshold of 1 x 10 -5 . |
6759b06df9980725cfbc8cef | 44 | For classical force field calculations, framework atoms were modeled using the universal force field (UFF), with point charges assigned based on the density derived electrostatic and chemical (DDEC) scheme. If the .cif file for a structure contained pre-defined charges, these values were retained without further modification. For guest molecules, specifically CO2 and H2O, the TraPPE 90 and SPC/E 91 force fields were employed, respectively. |
651a9bb1ade1178b2484ea17 | 0 | In essence, organic chemistry is all about the interaction between electron-rich atoms or molecules and electron-deficient atoms or molecules. The concept of nucleophiles and electrophiles dates back to 1933, 2 when it was introduced by Ingold based on Lewis's electronic theory of valency and the acid-base theory of Brøndsted and Lowry. Over the years, nucleophilicity and electrophilicity have been quantified to create reactivity scales that explain why some atoms or molecules are more reactive than others. Most noteworthy is the work by Mayr and co-workers, who experimentally studied the reactivity of organic molecules that led to the Mayr-Patz equation; |
651a9bb1ade1178b2484ea17 | 1 | Essential to Equation 1 is that k 20 • C can span from non-observable reactions (k 20 • C < 10 -5 M -1 s -1 ) to diffusion-controlled reactions (k 20 • C > 10 9 M -1 s -1 ), and its applicability has been confirmed for various molecules resulting in Mayr's database that currently holds experimental reactivity parameters for 352 electrophiles and 1,261 nucleophiles. However, measuring reaction rates to extract reactivity parameters is generally time-consuming and often difficult at the extremes of the reactivity scales. Thus, several in silico methods have been proposed to circumvent this problem. This includes estimating the rate constant using the Eyring equation and computing the reactivity parameters from frontier molecular orbital (FMO) energies or chemical affinities . Furthermore, a lot of new ML approaches based on Mayr's database have recently emerged. In this work, we will focus on the studies by Van Vranken and Baldi showing that calculated methyl cation affinities (MCAs) and methyl anion affinities (MAAs) of structurally different molecules correlate with Mayr's N •s N and E, respectively, when considering solvent effects. Based on these findings, they created two QM-derived datasets with reactivity parameters for 1,232 nucleophiles and 1,113 electrophiles (we have excluded 76 duplicates) covering ∼ 50 orders of magnitude in each case. The datasets were used to train different ML models by treating the affinities as either atomic, functional group, or molecular properties. The best-performing architecture was an atom-based graph attention network (GAT) achieving 10-fold cross-validation R 2 coefficients of 0.92 ± 0.02 and 0.94 ± 0.02 for MCAs and MAAs, respectively. However, it is worth noting that the two datasets contain molecules that are not in Mayr's database, and the ML models have not been validated against experimental reactivity parameters. Furthermore, the atom sites are identified by hand, which makes it difficult to apply their method to arbitrary molecules in an automated fashion. This work will address these limitations by introducing a fast and fully automated quantum chemistry-based workflow that detects relevant sites and calculates associated MCAs and MAAs. In addition, we will apply the workflow to different tasks to highlight the applicability of MCAs and MAAs as quantitative measures of chemical reactivity. |
651a9bb1ade1178b2484ea17 | 2 | We have previously presented fully automated quantum chemistry-based workflows for predicting the regioselectivity of EAS reactions and palladium-catalyzed Heck reactions . Following this work, we introduce a workflow that, given a SMILES string as input, identifies possible nucleophilic and electrophilic sites and individually attaches methyl cations or anions to calculate associated MCA or MAA. The input SMILES string is modified using a set of SMIRKS and RDKit , which can easily be adjusted to include/exclude certain functional groups. The nucleophilic sites include double/triple-bonded atoms, singly charged anions, atoms with lone pairs, and specific functional groups such as aldehydes, amides, amines, carbanions, carboxylic acids, cyanoalkyl/nitrile anions, enolates, esters, ethers, imines, isonitriles, ketones, nitranions, nitriles, and nitronates. The electrophilic sites include double/triple-bonded atoms, singly charged cations, and specific functional groups such as acyl halides, aldehydes, amides, anhydrides, carbocations, esters, imines, iminium ions, ketones, Michael acceptors, and oxonium ions. The calculated MCAs and MAAs are defined in Equations 2 and 3 as the negative energy difference (-∆E) of a nucleophile (Nuc) reacting with a methyl cation (CH + 3 ) and an electrophile (Elec) reacting with a methyl anion (CH - 3 ), respectively. |
651a9bb1ade1178b2484ea17 | 3 | For each compound, we embed min(1 + 3 • n rot , 20) conformers using RDKit, where n rot is the number of rotatable bonds. The conformers are prescreened with force-field optimizations in dimethyl sulfoxide (DMSO, dielectric constant = 46.68) using GFNFF-xTB and the analytical linearized Poisson-Boltzmann (ALPB) solvation model as implemented in the open source semiempirical software package xtb. Only unique conformers within a 10 kJ/mol cut-off are carried forward by selecting the centroids of a Butina clustering using a pairwise heavy-atom position root mean square deviation (RMSD) with a threshold of 0.5 Å. The remaining conformers are then reoptimized with GFN1-xTB ALPB(DMSO) before refining the energy of the lowest energy conformer with single-point DFT calculations in DMSO using the r 2 SCAN-3c composite electronic structure method and the SMD solvation model as implemented in the quantum chemistry program ORCA version 5.0.1. |
651a9bb1ade1178b2484ea17 | 4 | In Figure , we present a comparison of the automated quantum chemistry-based workflow against the experimental and higher-level QM-derived data. The reference data are provided for a single atom in each molecule, so the calculated MCAs and MAAs are only evaluated for these labeled atom sites. However, such labels are not provided for the data in Figure . In this case, we use the highest calculated MAAs, although some associated sites have been compared to the molecular coordinates provided by Mood et al. and confirmed as the actual reaction centers. |
651a9bb1ade1178b2484ea17 | 5 | The top panels in Figure show that the workflow can reproduce a strong correlation between experimental reactivity parameters and calculated MCAs and MAAs as previously observed by Van Vranken and Baldi. Specifically, the R 2 coefficients of 0.84 and 0.94 are somewhat similar to 0.89 and 0.96 for MCAs and MAAs, respectively, with the latter based on Gibbs free energies at the PBE0-D3(BJ)/DEF2-TZVP COSMO(∞) level of theory. |
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