DFT and TST Study of the Calcium Cyanamide Process for Synthesizing Cyanamide and Dicyandiamide

Abstract

Exploring the microscopic reaction mechanism of dicyandiamide (DCD) synthesis using calcium cyanamide (CaCN₂) is highly desirable because of the low conversion of reactants and selectivity of DCD products. DCD synthesis consists of a two-step sequential hydrolysis of CaCN₂, followed by dimerization of cyanamide to DCD in an alkaline environment. Density functional theory (DFT) results revealed that the rate-limiting step (RLS) was the formation of a C-N bond between the cyanamide and cyanamide anion in the dimerization of the DCD reaction. Secondary reactions of cyanamide with water, hydrogen sulfide, and DCD were also analyzed. The effects of solvation on the principal and secondary reactions were systematically explored. A single explicit water molecule can significantly lower the free energy barrier of the RLS. Water molecules facilitate the C-N bonding of the reactants in DCD reactions, resulting in a reduction in the free energy barrier of the RLS. The facilitation of double explicit water for the reaction is weaker than that of single explicit water and even yields negative catalysis. The effect of the [OH(H₂O)₃]⁻ cluster lowering the reaction barrier with the hydrogen-bonding network is the most remarkable, which can alter the reaction path by the direct and indirect involvement of OH⁻ ions. Furthermore, the reaction rate constants were computed by canonical variational theory with the Eckart tunneling correction (CVT/Eckart) and fitted to the Arrhenius expression. The reaction mechanism and kinetics revealed at the microscopic level provide efficient and clean production of DCD with certain theoretical guidance.

1. Introduction

Dicyandiamide (DCD, C₂H₄N₄) is the dimerization product of cyanamide (H₂N-C≡N) under alkaline conditions, which is an attractive raw material for the pharmaceutical, pesticide, plastic, and dyeing industries [1,2]. Dicyandiamide is derived from calcium cyanamide (CaCN₂) hydrolysis, decalcification, polymerization, and crystallization, and the formed dicyandiamide is stable under dry conditions at atmospheric temperature [3]. Dicyandiamide is an amphoteric compound exhibiting high reactivity because it possesses -C≡N, -C=N, -NH₂, and -C-N and also facilitates cyclization reactions, making it widely applicable. Cyanamide, an influential intermediate product in the preparation of dicyandiamide [4], possesses amino (-NH₂) and cyanide (-CN) reactive groups, which are prone to enable cyanamide to be readily polymerized into relatively stable dicyandiamide [5]. Cyanamide is readily hydrolyzed to urea under excessively acidic, alkaline, or heated conditions [6].

The hydrolysis polymerization reaction for the formation of dicyandiamide is not complete, while the secondary reaction is liable to occur when the reaction conditions are not properly controlled [7]. It is essential to reduce the incidence of secondary reactions during the preparation of dicyandiamide. In addition, the current dicyandiamide industry suffers from prominent issues in production, with high material consumption (single-step reaction yield below 90%) and high emissions (1 ton of cyanamide generates 5 tons of offscum and severe dust pollution). Chemical separation is the main challenge of the high emissions in the dicyandiamide industry. In this work, we focus on the high material consumption based on the low yield of cyanamide and dicyandiamide products. This is mainly because the reaction is significantly affected by reaction conditions and impurities [8,9]. Therefore, exploring the reaction mechanism and key kinetics of dicyandiamide is valuable [10,11].

The researches on the conversion of cyanamide and dicyandiamide are mainly concerned with the exploration of the hydrolysis reaction and the tautomerism. In the study of the origin of life, Zhang et al. investigated the formation mechanisms of cyanamide and its isomer carbodiimide, establishing a comprehensive chemical reaction network [12]. For the conformational properties of cyanamide tautomeric reactions, Puzzarini et al. calculated the difference in conformational transition energy between cyanamide and carbodiimide as ~3 kcal/mol, but these two forms are partitioned with a free energy barrier of ~80 kcal/mol by the transition state [13]. Dicyandiamide is a dimer of cyanamide, and theoretical studies on dicyandiamide are concentrated on the molecule structure [2,14], metal complexes [15,16], and measurement technology [17,18]. Dicyandiamide is commonly employed to synthesize target molecules as a nitrogen source because of its economical cost and high nitrogen content (61 wt %) [19,20]. However, there is no systematic theoretical study on the production of dicyandiamide by the calcium cyanamide process. Simultaneously, DFT has already become a remarkably prevalent method in theoretical calculations due to its strong robustness and decent accuracy [21]. The principal reactions for the two-step continuous hydrolysis of CaCN₂ (R1, R2) and the stepwise dimerization of cyanamide (R3, R4, R5), along with the secondary reactions involving cyanamide reacting with water, hydrogen sulfide, and dicyandiamide (R6, R7, R8, respectively), were systematically calculated based on a DFT calculation (Scheme 1). For the tautomerism of the cyanamide and dicyandiamide molecules, a more stable configuration was chosen for the DFT calculation. More detailed information on the tautomer can be seen in the Supplementary Section S1. Ionic equations were utilized to describe the entire reaction system, given the effortless dissociation state of the molecules and material solubility in water. The calcium ion of CaCN₂ in water is not involved in the formation of cyanamide and dicyandiamide. Therefore, the omission of calcium ions in ionic equations has a negligible effect on the study. It interacts with other ions in water mainly through electrostatic interactions. Cyanamide reactions (R1, R2) and dicyandiamide principal/secondary reactions (R3-R8) were carried out in separate reactors under the corresponding reaction conditions in the practical industry. Thus, R2 and R3 reactions proceed in opposite directions. In the DFT calculation, the transition state search of the reaction path and the verification of the RLS is the focus. In reaction R8, the pathway for the synthesis of melamine by-products was examined, referencing the previously proposed synthetic route by Kiyull Yang et al. [22]. The route involves cyanamide as the initial species, which cyclizes with dicyandiamide following hydrogen transfer to ultimately form melamine.

The investigation of the competition mechanism between primary and secondary reactions plays a crucial role in enhancing the yield of the target product and minimizing the occurrence of secondary reactions [23]. To this point, reports of kinetic data on the reaction of dicyandiamide by the calcium cyanamide process are quite rare, and the relevant literature is excessively outdated. Transition state theory (TST) is currently the most prevalent method for calculating the rate constants of reactions with explicit energy barriers. In particular, the variational transition state theory (VTST) [24] combined with rational correction for the tunneling effect can yield more satisfactory kinetic results. A more detailed discussion of the kinetic theory and tunneling correction can be found in the Supplementary Section S2.

In this paper, the reaction pathway of the corresponding system was initially computed by DFT calculation, whereby the RLS and its energy barrier values were obtained. The solvation effect on the reaction mechanism was systematically investigated using a hybrid solvent model [25,26]. With the participation of single/double explicit water molecules and [OH(H₂O)₃]⁻ clusters, the reaction mechanism of water molecules and clusters that lower the hydrogen transfer energy barrier was revealed systematically. Furthermore, the canonical variational theory (CVT) [22,23] with the unsymmetrical Eckart tunneling correction [27] was employed to calculate the rate constants and transmission coefficients at the temperature of the operating range (R1-R2: 283.15–323.15 K, R3-R8: 323.15–363.15 K). The CVT/Eckart approach for computing the rate constants contributes to the reference value for kinetic studies of cyanamide and dicyandiamide production.

2. Computational Methods

2.1. Mechanism Method

All molecular geometry optimization and vibrational frequencies were performed in Gaussian 09 D.01 software [28], where the calculation of the stationary points under the implicit solvent model and single/double explicit waters was conducted at the B2PLYP [29]-D3 [30,31]/def2-TZVP [32] level. After adding the [OH(H₂O)₃]⁻ cluster into the reaction system, optimization and vibration analysis were performed at the B3LYP [33]/def2-SVP [32] level combined with DFT-D3 correction [30] considering the computation cost and time. Here, the ZPE correction factors [34] were considered (Supplementary Section S3). The subsequent electronic energy calculation was all performed at the B2PLYP-D3/ma-QZVPP [35,36] level with higher accuracy. All the transition structures are searched by the opt = TS method, while the QST3 method is also valid (Supplementary Section S4). The intrinsic reaction coordinate (IRC) was employed to verify whether the transition states correctly connected the reactants and products of the corresponding reactions. To depict the water solvent environment of the reaction process more realistically, all calculations were performed using the implicit solvent model. Furthermore, the effect of water molecules was explicitly described in the investigation of the solvation effect via a hybrid solvent model (hybrid method) [25,26]. The integral equation formalism of the polarizable continuum model (IEFPCM) [37] was used for geometry optimization and vibration analysis. The solvation model, the density (SMD) [38] model, was used for single-point calculations of the solvation free energy (ΔG_solv). Water was selected as the solvent for this reaction. The default criteria for convergence of Gaussian 09 software were adopted for all calculations.

The solution-phase free energy (G_soln) of the whole reaction system is the electron energy (E), thermal correction to the Gibbs free energy (G_corr) adding the solvation free energy (ΔG_solv) [39,40], and standard state transformation free energy difference (1.89 kcal/mol). The calculation process can be described as follows in Equation (1):

$$G_{\mathrm{soln}}=E+G_{\mathrm{corr}}+\Delta G_{\mathrm{solv}}+RT\ln \left({\displaystyle \frac{RT}{P}}\right)$$

(1)

where E is the electron energy under vacuum conditions based on the optimized structure. For better calculation results, the special solvent model parameterized calculation level M05-2X/6-31G* is applied to calculate the solvation free energy (ΔG_solv) [38]. ΔG_solv is the difference between free energy of the solute in the solution (M05-2X/6-31G*//SMD level) and the gas phase (M05-2X/6-31G* level) based on the optimized structure with IEFPCM, refer to the practice of Truhlar et al. (the proposer of SMD model) [39,40]. The last term (1.89 kcal/mol) is the transition from the gas phase standard state (298.15 K, 1atm) to the solution phase standard state of 1 M, namely 1.89 kcal/mol. In the calculation of the solution-phase free energy (G_soln), the electron energy was calculated using the Gaussian 09 software. The free energy calculation, after considering ZPE corrections and the solvent environment, was calculated using the Shermo 2.3.4 program [41]. Concretely, the output file after geometry optimization and vibration analysis by Gaussian 09 software was taken as an input file to the Shermo 2.3.4 program. The higher accuracy electronic energy values and ZPE correction factor were then entered into the settings.ini file (setup operation parameters) of the Shermo 2.3.4 program, and finally, the sum of E and G_corr was obtained. All computed data and structural details for the optimized reaction molecule can be found in the Supplementary Sections S5, S6, and S17, respectively. All values (G_soln) in the energy profile considered the ZPE correction and implicit solvent model.

To elucidate the reaction mechanism at the microscopic level, the charge transfer amount of the rate-determining step was quantitatively calculated using the atomic dipole moment-corrected Hirshfeld population (ADCH) [42]. ADCH charge analysis was performed using the Multiwfn 3.8 dev program [43]. Electron density difference (EDD) analyses were utilized to further reveal the change in the charge density of bond formation and breakage in the reaction using the Multiwfn 3.8 dev program. The visual molecular dynamics (VMD 1.9.3) program [44] was applied to obtain isosurface maps involving the density of the reaction transition state system minus the density of the fragment species. In the isosurface map below, the red region denotes an increase in electron density and the blue region denotes a decrease in electron density. The IR spectroscopy, molecular electrostatic potential map (MEP), and NBO analysis were also conducted to identify the stability and rationality of the cyanamide and dicyandiamide molecule.

2.2. Kinetics Calculation

In the kinetic calculation, rate constants have been calculated, employing the RLS of the principal and secondary reactions. Given that the reaction process involves the hydrogen transfer, the tunneling effect cannot be ignored for more rational calculation. In this research work, the transmission coefficient was calculated with Eckart correction [27]. The rate constant (k) and transmission coefficient (κ) were calculated by Equations (2) and (3), respectively. All reactions are calculated using the TST/CVT combined with Eckart tunneling correction by the KiSThelP 2021 package [45].

$$k=\sigma {\displaystyle \frac{\kappa k_{B}T}{h}}{e}^{-{\displaystyle \frac{\Delta G^{0,\ne }}{RT}}}$$

(2)

$$\kappa ={\displaystyle \frac{\exp (\Delta H_f^{\ne ,0K}/k_{B}T)}{k_{B}T}}{\displaystyle {\int }_0^{\infty }p(E)\exp (-E/k_{B}T)}dE$$

(3)

where σ represents the reaction path degeneracy; and k_B and h represent the Boltzmann constant and Planck’s constant. ΔG^0,≠ is the standard Gibbs free energy barrier. ${\mathsf{\Delta }H}_f^{\ne ,0\mathrm{K}}$ is the zero-point corrected activation enthalpy in the forward direction. p(E) is the transmission probability with the corresponding 1D barrier at energy E.

3. Results and Discussion

3.1. The Structure of Cyanamide and Dicyandiamide Structure

To study the stability and rationality of cyanamide and dicyandiamide structure in this work, we compared the main peak positions of cyanamide and dicyandiamide in the calculated and experimental IR spectroscopy (Supplementary Section S7). The similar position of the main peak indicates that the cyanamide and dicyandiamide configurations used in this work are reasonable. In addition, NBO analysis is applied to study the stability of cyanamide and dicyandiamide. According to the second-order perturbation theory analysis, there is a strong interaction between NBO orbital 10 (lone pair electrons orbital of N1) and orbital 104 (σ-bonding of -C4≡N5) in cyanamide, resulting in the decrease in the energy of the whole system by 39.81 kcal/mol to maintain the molecule stability. A similar situation occurs in the dicyandiamide, as shown in the Supplementary Section S8. In the molecular electrostatic potential map (MEP), the blue region near the N atoms shows the negative electrostatic potential, which is consistent with the high electronegativity of lone pair electrons of the N atoms (Supplementary Section S9). On the contrary, the H atoms are in the red region for the positive electricity. N atoms with high negative electrostatic potential could be the nucleophilic site during the reaction, which could be studied in the mechanism analysis.

3.2. Reaction Mechanism Under the Implicit Solvent Model

Here, the reaction mechanism of the calcium cyanamide process for synthesizing cyanamide and dicyandiamide was explored exhaustively via DFT calculation. And the possible reaction pathways in aqueous media were proposed. Furthermore, ADCH charge and EDD analysis collectively reveal the electron transfer mechanism of the reaction, elucidating the essential cause of bond formation and breaking. The reaction path under the implicit solvent model described in this section refers to the absence of explicit water molecule participation.

The preliminary hydrolysis of CaCN₂ (R1) is a simple hydrogen transfer reaction in which the reactant molecules first form the R1 reactant complex (RC). The energy profiles of the cyanamide reactions are shown in Figure 1a. The O-H bond length of the water molecule in R1 RC was extended compared to of that the isolated water molecules, indicating that the H atom of water was attracted by the fairly electronegative N atom of CN₂²⁻. When the R1 transition state (TS) is formed, the O-H bond length of water is further elongated, and the H atom of water is closer to the N atom of CN₂²⁻. Eventually, the H atom in the water molecule is bonded to CN₂²⁻, at which stage HCN₂⁻ and OH⁻ are derived. Unfortunately, R2 underwent numerous attempts and no search for a transition state was obtained, which can be utilized to validate the rationality of R2 TS. As illustrated by the relaxed scan curve in the Supplementary Section S10, the reaction system exhibits a monotonically increasing trend in energy as the H2 atom of the water molecule approaches the N6 atom of HCN₂⁻, and no maximum value point emerges. Thus, the existence of R2 TS was verified based on the relaxed scan for R2.

The RLS in the principal reaction comprising R1 and R2 is the formation of the R1 RC with a free energy barrier of 14 kJ/mol. The presence of a low barrier normally indicates that the process of R1 occurs easily and is dominated by diffusion control. It is noteworthy that the Gibbs free energy of R1 TS is lower than that of R1 RC, while the electronic energy of R1 TS at the geometry optimization level (B2PLYP-D3/def2-TZVP) is higher than that of R1 RC. This discrepancy can be attributed to the free energy correction. Only in the case where the calculation levels of single-point energy and configuration optimization are identical will the electron energy of the TS be higher than its connected stationary point in IRC. The electron energies of the R1 RC, R1 TS, and R1 PC at the B2PLYP-D3/def2-TZVP level can be seen in the Supplementary Section S11. Similar samples can also be seen in these Refs. [46,47,48,49,50]. Hence, it is plausible that this circumstance has emerged.

Subsequently, the ADCH charge analysis was performed on the RLS of preliminary hydrolysis of CaCN₂ (R1). The charge distribution on the atoms is shown in Figure 1b. Here, the charge of the isolated molecule is treated as the initial charge value. In the formation of R1 PC from isolated reactant molecules in the preliminary hydrolysis of CaCN₂, the H2 atom of the water gradually approaches the N4 atom of CN₂²⁻, along with the H2 atom in the water, which obtains more electrons (0.416 → 0.367 → 0.322 → −0.183), and the N4 atom of CN₂²⁻ loses electrons (−1.116 → −1.058 → −0.971 → −0.540), indicating that electrons are transferred from the N4 atom of CN₂²⁻ to the H2 atom of the water. Simultaneously, the dissociation of the H2 atom in the water causes the charge of the O1 atom in the water to be redistributed, which displays an increase toward negative values (−0.832 → −0.906 → −0.998 → −1.016). EDD analysis was performed to further reveal the charge transfer in R1, as shown in Figure 1c. The region of the N4 atom in R1 TS presents a blue area, indicating that the charge density in this part is decreasing. Also, the red region of the O1 atom of the water shows an increase in the negative charge due to the departure of the H2 atom, which is consistent with the results of the ADCH analysis.

The reaction mechanism for dicyandiamide principal and secondary reactions (R3-R8) under the implicit solvent model can be seen in the Supplementary Section S12. It can be found that the RLS energy barrier of dicyandiamide reactions is generally high and does not occur easily at room temperature without the assistance of explicit water and another catalytic molecule (or high energy input.)

3.3. Solvation Effect on the Reaction Mechanism

In the previous section of the reaction pathway with the implicit solvent model, the aqueous solvent is treated as the homogeneous bulk, which can eliminate the conformation and spatial configuration of the solvent molecules. Generally, it can reliably depict the contribution of solvation effects to the system when a strong interaction between solvent and solute is not involved. Since the reaction path involves the proton transfer process, the role of water in these reactions is worth further discussion. Therefore, the effect of water molecules is described explicitly in this section. In detail, the direct addition of one and two water molecules at the periphery of reactant molecules explicitly depicts the direct solvent-solute interaction by the explicit solvent model [51,52]. The bulk of solvent outside of the explicit water molecule still utilizes the implicit solvent model just like the previous section, which is the hybrid solvent model [25,26]. To consider the effect of the alkaline environment during cyanamide/dicyandiamide production, a hydrated hydroxide anion cluster [OH(H₂O)₃]⁻ saturated in the first coordination layer of the OH⁻ anion is added to the reaction system [53,54,55]. The influence of the localized microenvironment constructed by the [OH(H₂O)₃]⁻ cluster on the reaction mechanism was analyzed systematically. In the following sections, _w, _2w, and _c represent stationary points in reaction paths under the single explicit water, double explicit waters, and [OH(H₂O)₃]⁻ cluster, respectively.

3.3.1. Reaction Mechanism Under the Single Explicit Water

Cyanamide reactions (R1 and R2) do not take into account the reaction path under the single explicit water due to the diffusion control domination. The reaction path for dicyandiamide principal reactions (R3-R5) under the implicit solvent model without considering the auxiliary effect of water is shown in the Supplementary Section S12. The R3 is a simple H-abstraction process for the H₂CN₂ molecule by the OH⁻ ion, which is the barrierless reaction. Subsequently, the binding of the N atom of HCN₂⁻ to the C atom of H₂CN₂ obtains the H₃C₂N₄⁻ ion. The R5 is the H-transfer reaction for hydrogen migrating from the amino group of H₂CN₂ to the imino group of H₃C₂N₄⁻ ion. The C-N bonding process between the H₂CN₂ and HCN₂⁻ ions in R4 is the RLS of dicyandiamide principal reactions. So, in R3-R5, one explicit water molecule was considered in the reaction process from R4 reactants to R4 IM1 (including RLS). As can be seen in Figure 2, water molecules can form a six-membered ring structure with a reaction intermediate. The electronegative nitrogen and oxygen atoms contribute to the formation of hydrogen bonds, leading to a six-membered ring structure. A water molecule was engaged in the formation of R4 IM1_w, spanning a free energy barrier of RLS (ΔG_RLS) with 98 kJ/mol. ΔG_RLS decreased from 182 kJ/mol (pure implicit solvent model) to 98 kJ/mol, and it can be assumed that the water molecule plays a catalytic role in the reaction. In the RLS of dicyandiamide reactions, the formation of R4 TS requires a high free energy barrier. A water molecule can lower the potential barrier by facilitating the C-N bonding process of R4 IM1_w. Then, the water molecule acts as the bridge that hydrogen is indirectly transferred: from R4 IM1_w to R4 IM2_w.

In the secondary reactions, the RLS of the thiourea secondary reaction (R7) was merely explored due to the lowest free energy value of RLS among secondary reactions under the implicit solvent model. The detailed discussion on R7 under the implicit solvent model is shown in the Supplementary Section S12. The formation of thiourea undergoes the C-S bonding between the H₂CN₂ and H₂S and the migration movement of hydrogen. The RLS of the R7 reaction path is the first stage of the C-S atom bonding process. As shown in Figure 3, a six-membered ring structure also was obtained under the hydrogen-bond interaction of water molecules with the intermediate. In contrast to dicyandiamide reactions (R3, R4, and R5), the sulfur atom was included in the formation of a six-membered ring in addition to the C-H-O-N atoms. The RLS is the formation of R7 TS1_w, which requires a free energy barrier of 150 kJ/mol. ΔG_RLS of dicyandiamide reactions decreased by 8 kJ/mol (from 158 to 150 kJ/mol) after considering the proton transfer of water. It is assumed that a water molecule catalyzes the thiourea reaction with the help of the proton transfer process. The overall reduction in ΔG_RLS was slight compared to the dicyandiamide reactions (R3, R4, and R5). It might be attributed to the fact that high energy was required for the bonding process in C-S atoms in R7 TS1 and the reduction in the N-C-N bond angle in H₂CN₂ forming R7 IM1. And water molecule has a minor effect on these processes.

Similarly, water molecules can act as the catalyst for the urea reaction (R6) and melamine reaction (R8) (Supplementary Section S13). Under the single explicit water, free energy barriers of the RLS are lowered in R6 and R8. A hydrosolvent can considerably reduce the free energy barrier of the reaction path. It is mostly ascribed to the formation of a hydrogen-bonded ring exposing the electronegativity of oxygen and nitrogen atoms. The formation of hydrogen-bonded complexes depends on the conformation property of water molecules, which comprises both electron-deficient hydrogen atoms and oxygen atoms with lone electron pairs. Thus, water molecules can act as either a hydrogen-bond donor or a hydrogen-bond acceptor and are capable of forming hydrogen-bonded ring complexes with plentiful polar molecules in aqueous solution [56]. Similar hydrogen-bonded complex studies also can be found in these works [56,57,58].

Furthermore, the reaction mechanism under the double explicit waters was explored systematically. A more detailed discussion can be found in the Supplementary Section S14. The results show that double explicit water molecules can also play a catalytic role in lowering the reaction barrier in most reactions. Whereas, the promotion of double explicit waters for the reaction is slightly weaker than single explicit water.

3.3.2. Reaction Mechanism and Kinetics Under the [OH(H₂O)₃]⁻ Cluster

Reaction Mechanism

In the alkaline aqueous solution, the hydroxyl ion exists mainly in the form of a hydrated hydroxide anion cluster [OH(H₂O)_n]⁻ [53,55,59]. Some researches indicate the first coordination shell of the OH⁻ anion, consisting of three water molecules [54,55,60,61]. After n = 3 in the [OH(H₂O)_n]⁻ cluster, the water molecules start to fill up the second solvation shell [53]. Therefore, the general cluster form [OH(H₂O)₃]⁻ for n = 3 is selected to explicitly describe the first coordination layer of OH⁻ in this section. The structural details of the [OH(H₂O)₃]⁻ cluster are shown in the Supplementary Section S6. The effect of the localized microenvironment established by the [OH(H₂O)₃]⁻ cluster on the reaction path was discussed so as to rationally describe the formation process of dicyandiamide in real industrial manufacture.

With regard to the cyanamide reactions (R1, R2), the RLS is the H-abstraction process of the water molecule in [OH(H₂O)₃]⁻ cluster by the HCN₂⁻ anion with the energy barrier of 60 kJ/mol. The hydrogen-bonding network constructed by the OH⁻ anion can transform the reaction path compared with the pathway under the implicit solvent model. Even though the OH⁻ anion of the cluster is not directly involved in the hydrolysis reaction. A more detailed discussion of the cyanamide reactions can be found in the Supplementary Section S15.

The energy profile of dicyandiamide principal reactions (R3-R5) is shown in Figure 4. The first generation step of HCN₂⁻ (R3) is a spontaneous reaction without an energy barrier with free energy of −10 kJ/mol. In R4, the two water molecules of the [OH(H₂O)₃]⁻ cluster interact with H₂CN₂ and HCN₂⁻ by hydrogen bonding interaction, respectively. The cyclic reactant complex R4 RC_c was formed. Thereafter, the N-C-N skeleton of H₂CN₂ bent, and the N atom of HCN₂⁻ was bonded to the C atom of H₂CN₂ with a free energy span of 72 kJ/mol. The water molecule of the cluster formed a six-membered hydrogen-bond ring with the intermediate, and the indirect transfer of hydrogen was accomplished with a dihydrogen transfer mechanism to produce R4 IM2_c. The energy barrier of indirect hydrogen transfer was nearly a barrierless process. R4 IM2_c underwent the space isomerism with the free energy of 36 kJ/mol obtaining the R4 product. In the formation of dicyandiamide (R5), it is a simple hydrogen transfer process. Hydrogen is transferred from the H₂CN₂ molecule to the H₃C₂N₄⁻ ion.

The formation of R4 TS_c is the RLS in the dicyandiamide consecutive reactions, corresponding to the C-N bonding process between H₂CN₂ and HCN₂⁻ ion. The energy barrier for the RLS is 72 kJ/mol, which is considerably lower than the RLS energy barrier under the implicit solvent model. It indicated that the hydrogen-bonding network constructed by the [OH(H₂O)₃]⁻ cluster can enhance the C-N bonding process. Even though the hydroxide ion was not directly involved in the reaction, the dihydrogen transfer process assisted by the water molecule of the cluster lowered the energy barrier at this step.

The energy profile of urea secondary reaction (R6) under the [OH(H₂O)₃]⁻ cluster is shown in Figure 5. After the reactant molecules form R6 RC_c with the cluster, the OH⁻ ion can interact with the C atom of the H₂CN₂ molecule. The N-C-N skeleton of H₂CN₂ bent while the hydrogen of OH⁻ dissociated and moved closer to the cyano group of H₂CN₂, forming R6 TS1_c. When R6 IM2_c formed, three water molecules of the cluster were attracted by the imino group of H₂CN₂ and gradually migrated. One of the water molecules of the cluster interacted with the imino group by hydrogen bond obtaining R6 IM3_c. With the H dissociation of the water molecule, R6 TS2_c formed. After the H-transfer process, the imino group of the intermediate was converted to the amino group producing the urea molecule.

The RLS of the whole process was the formation of R6 TS1_c. It is the C-O bonding process of H₂CN₂ and OH⁻ and the dissociation of the hydrogen of OH⁻. The reaction energy barrier of RLS was lower than the reaction path under the implicit solvent model and single/double explicit waters. Uniquely, the OH⁻ of the cluster was directly involved in the formation of the carbonyl group in the urea, while the reactant water molecule simply acted as a bystander. The OH⁻ was not consumed, indicating that the OH⁻ played a catalytic role in the alkaline environment and changed the reaction mechanism. And the facilitation of OH⁻ was notably larger than the single/double explicit waters, enabling the formation of urea to occur more readily.

The energy profile of the thiourea secondary reaction (R7) is displayed in Figure 6. After the formation of the reactant complex, the H atom of the amino group in H₂CN₂ was dissociated by the attraction of the O atom in the water molecule. The skeleton of H₂CN₂ was deformed, and the S atom of H₂S interacted with the C atom of H₂CN₂, indicating a tendency toward bonding. Subsequently, the N atom of the cyano group in H₂CN₂ attacked the H atom of H₂S to form R7 TS1_c, which crossed the energy barrier of 48 kJ/mol. Next, the imino group of H₂CN₂ captured the H atom of the cluster to produce R7 TS2_c. In this process, the C-S bond was continuously shortened to facilitate the formation of the thiourea molecule. The final step in the formation of thiourea was the H-transfer process from the S atom to the imino group. The thiourea molecule was finally obtained.

The final process of the H-transfer was the RLS of R7, and R7 TS3_c was the rate-determining state. The energy barrier span of the RLS was 102 kJ/mol, and the RLS was no longer the previous C-S bonding between H₂CN₂ and H₂S, which spanned only 52 kJ/mol with the [OH(H₂O)₃]⁻ cluster participation. The energy barrier for C-S bonding was considerably lower than that of the reaction process under the implicit solvent model or single/double explicit waters. The results indicated that the cluster could stabilize the formation of C-S bonding and counteract the steric effect caused by N-C-N skeleton deformation of H₂CN₂. Then, the RLS for the generation of thiourea was altered.

The energy profile of the melamine secondary reaction (R8) is displayed in Figure 7. The reaction path after the participation of the cluster can be divided into two stages, before and after the introduction of the H₂CN₂ molecule into the system. In the first stage, the hydroxide ion extracted hydrogen from the amino group of the dicyandiamide molecule forming R8 IM2_c. The H of the imino group of dicyandiamide then underwent rotational isomerization to obtain the final state intermediate R8 IM3_c of the first stage.

After the introduction of the H₂CN₂ molecule into the reaction system, the H atom of the amino group of H₂CN₂ transferred to the imino group of the dicyandiamide intermediate. During the formation of R8 TS2_c, the C atom of H₂CN₂ bonded to the N atom of the dicyandiamide intermediate. A bending deformation of the N-C-N skeleton of H₂CN₂ can be clearly observed to facilitate the occurrence of C-N bonding with the energy barrier of 103 kJ/mol. Then, the OH⁻ of cluster attacked the H atom of the intermediate, and a hydrogen extraction reaction occurred to obtain R8 IM7_c. The N atom of the intermediate extracted the H atom of water and underwent R8 TS4_c to form R8 IM9_c. The C and N atoms of the intermediate then bonded to form the nitrogen heterocycle via R8 TS5_c. Finally, the trihydrogen transfer mechanism mediated by two water molecules enabled the indirect transfer of hydrogen from the nitrogen heterocycle to the imino group of the intermediate. It is a two-step H-transfer process with a negative energy barrier to gain the melamine molecule.

As for the single-step process, the closure for cyclization of the C-N atom in R8 TS5_c was the step with the highest barrier span in the whole process. It was no longer the C-N bonding process between the H₂CN₂ and C₂H₄N₄ intermediate under the implicit solvent model and single/double explicit waters. This suggested that the addition of the cluster can change where the single-step RLS occurs. Compared with the reaction path under the implicit solvent model and single/double explicit waters, the energy barrier of RLS decreased more notably under the cluster. It showed that OH⁻ and water molecules can directly participate in the reaction and alter the reaction path in the hydrogen-bonding network formed by the cluster. This was relatively consistent with the practical reaction rule that a strong base environment was conducive to the formation of melamine impurities.

Reaction Kinetics

An Eckart tunneling calculation was performed by the KiSThelP package based on the DFT results. The calculated results at operating temperature are summarized in

Table 1. The reaction rate constant k and transmission coefficient κ for each reaction system at operating temperature. (T_R1-R2 = 303.15 K, T_R3-R8 = 343.15 K).

Reaction	Reaction Type	κ	k (cm3 mol−1 s−1)
R1-R2	Principal reaction	1.447	7.117 × 102
R3-R4-R5	Principal reaction	3.234	1.995 × 10−4
R6	Secondary reaction	52.638	3.495 × 10−5
R7	Secondary reaction	9.095	1.278 × 10
R8	Secondary reaction	2.290	5.322 × 10−18

. More detailed data on the temperature of the manufacturing operating range are in the Supplementary Section S16. The reaction rate constants for synthesizing cyanamide and dicyandiamide at operating temperature (R1-R2: 303.15 K, R3-R8: 343.15 K) follow the order k_cyanamide > k_thiourea > k_{dicyandiamide} > k_urea > k_melamine. It can be found that the rate constant of principal reactions (R1-R5) is larger than secondary reactions, except for the thiourea secondary reaction. The results showed that the formation of thiourea was faster than dicyandiamide at 343.15 K. This is consistent with the practical law for industrial production of dicyandiamide. If the feedstock CaCN₂ contains sulfur impurities, it would be readily introduced into the thiourea by-products. In another aspect, transmission coefficients of the total reactions follow the order κ_urea > κ_thiourea > κ_{dicyandiamide} > κ_melamine > κ_cyanamide. The extent of variation in the energy barrier due to the tunneling effect can be directly described by the transmission coefficient. The transmission coefficient is equivalent to reducing the activation energy and will enlarge the value of rate constants [62]. The curve of the transmission coefficient with temperature at the manufacturing operating range can be seen in Figure 8a. It can be observed that there is a negative correlation between the transmission coefficient and temperature. Among all the reactions, the tunneling effect of urea (R6) and thiourea (R7) secondary reaction is the most prominent, whose influence on the kinetic parameter cannot be ignored.

Then, the Arrhenius equation is fitted to the rate constants of cyanamide and dicyandiamide reaction systems at 283.15–323.15 K and 323.15–363.15 K based on the manufacture operating temperature. The fitting data and Arrhenius equations are listed in

Table 2. Arrhenius equations fitting data and Arrhenius equations of calcium cyanamide process for synthesizing cyanamide and dicyandiamide.

Reaction	Slope	Intercept	A	Ea (kJ/mol)	Arrhenius Formula
R1-R2	−6.91	29.36	5.62 × 1012	57.44	k = (5.62 × 1012) exp(−6.91/T)
R3-R4-R5	−10.18	21.15	1.53 × 109	84.64	k = (1.53 × 109) exp(−10.18/T)
R6	−8.20	13.66	8.53 × 105	68.17	k = (8.53 × 105) exp(−8.20/T)
R7	−8.16	26.34	2.75 × 1011	67.87	k = (2.75 × 1011) exp(−8.16/T)
R8	−20.53	20.06	5.17 × 108	170.73	k = (5.17 × 108) exp(−20.53/T)

. As can be seen from the Arrhenius plot (Figure 8b), the rate constants of all reactions are positively correlated with temperature. The reaction rate of cyanamide is faster than other reactions, which is consistent with the practical production. The activation energy (E_a) and pre-exponential factor (A) were derived from the slope and intercept of equation regression, as well as the Arrhenius equations.

The rate constants of these reactions have been calculated by TST and CVT with Eckart tunneling correction to evaluate the deviation of the calculation. As shown in Figure 9, the rate constants of all reaction systems are larger than the original values after Eckart correction. The values of the rate constant are all larger than the values of conventional transition state theory after considering the variational TST. Significantly, whether CVT is considered or not, the rate constant of the urea secondary reaction is equal to the value of TST. This indicated that the variational effect in kinetics was not notable at the current temperature, and the tunneling effect was more pronounced in R6. The reaction proceeds more slowly at a lower temperature range and the rate constants for each reaction are closer, even though the tunnelling effect is pronounced. As the temperature increases, the rate constants calculated by different methods are gradually differentiated.

The calcium cyanamide process for synthesizing cyanamide and dicyandiamide is highly sensitive to the pH range of aqueous solution. The effect of the alkaline microenvironment constructed by the [OH(H₂O)₃]⁻ cluster on the reaction was further investigated. The reaction process mostly involves the H-transfer movement in the hydrogen-bonding network. Normally, the strong alkali environment favors hydrogen abstraction and is not conducive to hydrogenation. The difficulty of H-abstraction and hydrogenation can reflect the effect of the basicity of the cluster on the reaction. Therefore, the hydrogen dissociation energy (HDE) and hydrogen binding energy (HBE) were calculated for the reactions involving intramolecular H-transfer in the RLS. As shown in

Table 3. Hydrogen dissociation energy (HDE) and hydrogen binding energy (HBE) values for the rate-limiting step (kJ/mol). (Values are obtained by the difference in the electron energy between the whole and fragment of the reaction intermediate or transition state of RLS).

Reaction System	Hydrogen Dissociation Energy	Hydrogen Binding Energy
Dicyandiamide principal reactions (R3-R5)	395	−445
Urea secondary reaction (R6)	287	−435
Thiourea secondary reaction (R7)	389	−452

, all the absolute values of HBE are larger than the HDE. It indicated that the energy released by the hydrogen bonding process is larger than the energy required for hydrogen dissociation. The hydrogenation process is more prone to occur than the H-abstraction, and alkaline conditions can facilitate H-abstraction. For the dicyandiamide principal reactions and thiourea secondary reaction, the OH⁻ in the cluster does not directly participate in the reaction. The values of HDE and HBE are similar. The H-abstraction process of these two reaction systems is more difficult than hydrogenation. In actual production, the pH range for the formation of dicyandiamide and thiourea is the same (10 ˂ pH ˂ 10.5), which corresponds to the close H-abstraction and hydrogenation difficulty of two reactions.

Different from the R3-R5 and R7, R6 is an addition reaction in which OH⁻ of the cluster is directly added to the H₂CN₂ molecule to obtain urea. The addition process is also the RLS of R6, which indicates that the direct participation of the OH⁻ ion can promote the formation of urea. The generation of urea by-product is favored when the concentration of OH⁻ ion is higher. This is consistent with the suitable formation of urea impurities in the stronger base environment (pH > 12). In brief, the difficulty of H-abstraction and hydrogenation by HDE and HBE calculation can reflect the effect of alkalinity on the reaction well.

4. Conclusions

The reaction paths for principal and secondary reactions of the calcium cyanamide process were systematically investigated by DFT calculation, for which the RLS were derived. The reaction path of principal reactions is consecutive hydrolysis of CN₂²⁻-yielding cyanamide and dimerization of cyanamide forming dicyandiamide. Three reactions that produce urea, thiourea, and melamine are secondary in the dicyandiamide reaction system. Hydrogen abstraction of CN₂²⁻ from a water molecule forming an initial complex is the RLS for cyanamide synthesis under the pure implicit solvent model. Furthermore, reaction paths under the single/double explicit waters and hydrated hydroxide anion cluster [OH(H₂O)₃]⁻ were exploited employing a hybrid solvent model. The formation of a hydrogen-bonded ring complex is a typical characteristic of structural change in a hybrid solvent model. The research results indicated that single explicit water can dramatically reduce the free energy barrier of the RLS and play a catalytic role for indirect H-transfer by forming the hydrogen-bond cyclic network. The promotion of double explicit waters is weaker than single explicit water and even raises the negative catalysis. [OH(H₂O)₃]⁻ cluster can alter the RLS in cyanamide reactions. The free energy barrier reduction in the RLS for dicyandiamide principal and secondary reactions with a cluster was remarkably larger than single/double explicit waters. An OH⁻ ion in the cluster includes the direct and indirect involvement in reactions. Based on the DFT results, the rate constants were derived by the CVT with the Eckart tunneling correction. The rate constants for all reactions on the temperature of the operating range were fitted to the Arrhenius expression. In summary, this study provides a deeper understanding of the calcium cyanamide process and the role of aqueous solvents in the reaction, offering partial guidance for the industrial production of dicyandiamide.