Article On Modeling of Synthesis Process of Boron Carbide Based Nanocomposites Levan Chkhartishvili 1,2,*, Levan Antashvili 1 , Lasha Dalakishvili 1, Roin Chedia 3, Otar Tsagareishvili 1 and Archil Mikeladze 1 1 Ferdinand Tavadze Metallurgy and Materials Science Institute, 10 E. Mindeli Str., Tbilisi 0186, Georgia; [email protected] (L.A.); [email protected] (L.D.); [email protected] (O.T.); [email protected] (A.M.) 2 Department of Engineering Physics, Georgian Technical University, 77 M. Kostava Ave., Tbilisi 0160, Georgia 3 Peter Melikishvili Institute of Physical and Organic Chemistry, 31 A. Politkovskaya Str., Tbilisi 0186, Georgia; [email protected] * Correspondence: [email protected] Abstract: Nanocomposites based on boron carbide B4C are hard materials with wide field of applica- tions in modern technologies. A system of first-order ordinary differential equations that simulates the process of chemical synthesis of nanopowders of B4C–TiB2 compositions containing titanium diboride (TiB2) as an additional phase is suggested and resolved numerically for a typical ratio of reaction constants. Reagents and products concentrations are found as time-functions. In this way, the optimal route of production technology of boron carbide-based nanomaterials can be identified. Keywords: modeling; chemical synthesis; liquid charge; nanocomposite; boron compounds Citation: Chkhartishvili, L.; 1. Introduction Antashvili, L.; Dalakishvili, L.; Chedia, R.; Tsagareishvili, O.; Because of complex properties useful for diverse industrial applications, hard nanocom- Mikeladze, A. On Modeling of posites based on boron carbide (with the approximate chemical formula B4C) serve for an Synthesis Process of Boron Carbide important class of advanced materials. In previous works [1–6], we developed an effective Based Nanocomposites. Condens. method of their chemical synthesizing in nanopowdered form from corresponding liquid Matter 2021, 6, 3. https://doi.org/ charges. As is known (e.g., see [7]), optimization of the parameters of chemical technologies 10.3390/condmat6010003 is possible through their mathematical modeling. Here, we develop a set of ordinary differ- ential equations describing evolution of concentrations of reactants (reagents, intermediate Received: 6 January 2021 substances, and products) of chemical reactions, in which the boron carbide/titanium Accepted: 15 January 2021 diboride B4C–TiB2 nanocomposites are formed. Published: 18 January 2021 In the original technology developed, for boron (B), sources include amorphous boron (a-B), boron oxide (B2O3), or boric acid (H3BO3). A carbon source is some liquid organic Publisher’s Note: MDPI stays neutral compound CxHyOz containing carbon (C), hydrogen (H), and oxygen (O). As for the with regard to jurisdictional claims in titanium (Ti) source, usually a titanium dioxide (TiO2) powder is used for this purpose. published maps and institutional affil- The reagents are obtained in the process of a multi-stage heat treatment of the aqueous iations. suspension of these precursors. The B-containing reagent is always a powder of amorphous boron. Thus, the boron source practically does not change when a-B serves as the precursor material. Organic solvent pyrolysis reduces the most part of carbon into carbon black with turbostratic Copyright: © 2021 by the authors. graphite structure. Besides, carbon is emitted with carbon dioxide (CO2). At temperatures Licensee MDPI, Basel, Switzerland. ◦ higher than ~700 C, C reacts with CO2 forming carbon monoxide (CO): CO2 + C ! 2 This article is an open access article CO. Note that the production of a carbide-based composite requires free carbon in excess. distributed under the terms and Under corresponding conditions, C-containing reagents are not CO2 and CO, but C and conditions of the Creative Commons CO. As for carbon, during the heat pretreatment, TiO2 is retained in powdered form almost Attribution (CC BY) license (https:// without changes. creativecommons.org/licenses/by/ 4.0/). Condens. Matter 2021, 6, 3. https://doi.org/10.3390/condmat6010003 https://www.mdpi.com/journal/condensedmatter Condens. Matter 2021, 6, 3 2 of 13 2. Formally Simple Chemical Reactions Representing Technology According to the standard mathematical modeling method [8] of the technological process, the corresponding physical-chemical transformation has to be imagined as a set of (formally) simple, irreversible, homogeneous, and isothermal chemical reactions conducted in a closed volume. Transformation of reagents B, C, CO, and TiO2 into products B4C and TiB2 seems to occur through the following reactions: C + TiO2 ! TiO + CO", (1) 3 CO + TiO ! TiC + 2 CO2", (2) 2 B + TiC ! TiB2 + C, (3) 4 B + C ! B4C. (4) Hence, in addition to boron carbide (B4C) and titanium diboride (TiB2) the product is carbon dioxide (CO2) as well. Free carbon (C) being a reagent in Reactions (1) and (4) is an intermediate substance in Reaction (3). Similarly, carbon monoxide (CO) is a reagent in Reaction (2) and an intermediate substance in Reaction (1). Finally, titanium oxide (TiO)—according to Reactions (1) and (2)—andtitanium carbide (TiC)—according to Reactions (2) and (3)—serve only as intermediate substances. By definition, the order of a chemical reaction is the total number of particles— “molecules”—participating in an elementary collision act. Therefore, for Reactions (1)–(4), the orders are 2, 4, 3, and 5, respectively. As practice shows, the order of a simple chemical reaction cannot exceed 3 and, therefore, Reactions (2) and (4) seem to be complex. However, the establishment of real, i.e., effective, orders for these reactions requires more detailed consideration of their chemical mechanisms. First, note that the icosahedral cluster B12 is the main structural motif for any con- densed phase of elemental boron (as well as boron-rich compounds), including amorphous boron [9]. It remains stable only in the material’s bulk and, upon (in our case, high- temperature) ablation from the powder particle surface, takes a quasiplanar form with broken bonds and, accordingly, greatly increased chemical activity [10]. For this reason, it is natural to think that, not four individual B atoms, but a B12 quasiplanar cluster en- ter into the reaction. Clusters formation is also characteristic of carbon ablation process. One of the highest mass spectrometric peaks corresponds to C3 clusters [11]. Therefore, the mechanism of Reaction (4) can be represented as the interaction of only two particles: clusters B12 and C3. Similarly, the mechanism of Reaction (3), in which elemental boron also acts as a reagent, seems to be the interaction of B12 cluster with the (TiC)6 fragment of the titanium carbide structure. At high temperatures, in a medium containing both carbon and oxygen, various oxycarbon structures are formed. Among the small oxycarbonic clusters [12], only (CO)3 is metastable and, for this reason, hyperactive chemically. Therefore, the mechanism of Reaction (2) can be imaged as the interaction of a metastable cluster (CO)3 with a fragment of the titanium oxide structure in the form of a diatomic TiO molecule. In its original form, the order of Reaction (1) is already 2. However, it is most likely carried out not by the interaction between C atom and TiO2 molecule [13], but some small carbon cluster, for example the same C3 and the corresponding fragment (TiO2)3 of the titanium dioxide structure. Thus, the production of B4C–TiB2 nanocomposite powder is (formally) represented by the following combination of simple chemical reactions: C3 + (TiO2)3 ! 3 TiO + (CO)3", (5) (CO)3 + TiO ! TiC + 2 CO2", (6) B12 + (TiC)6 ! 6 TiB2 + 2 C3, (7) Condens. Matter 2021, 6, 3 3 of 13 B12 + C3 ! 3 B4C. (8) Note that all of them are of secondorder, since reagents are represented by pairs of clusters. Because all reactants (reagents, intermediates, and products) are in dispersive (nanopowdered or even free cluster) or gaseous forms, their mixing is assumed to be good enough to consider distribution of reactants concentrations in the reaction space, i.e., chamber volume, as almost homogeneous. Now, we have to discuss another key assumption of the model used: chemical close- ness of the reaction chamber or lack of exchange of matter with environment. Most reactants are in the solid phase and, consequently, remain in the chamber during whole technology process. There are only two reactants, CO and CO2, in the gas phase. In Reaction (2), gaseous CO produced from the precursor mixture serves for a reagent together with inter- mediate CO produced in Reaction (1). Thus, at the properly chosen charge composition, no presence of free (not reacted) CO is expected in the system. CO2, the final gaseous product, is carried away from the chamber by an inert gas (e.g., argon) flow. As mentioned above, in principle, carbon dioxide can react with carbon. However, at the properly chosen charge composition, no presence of free (not reacted) C is expected. It means that CO2 left in the system would behave as an inert gas and then dispute the exchange of CO2 gas with environment, the reaction chamber effectively remaining closed. 3. Reactions Rates From the acting masses law, the rates V1, V2, V3, and V4 of second-order chemical Reactions (5)–(8) are square functions of corresponding reagents or intermediates concen- trations (number of formula unit “molecules” per unit volume) [B], [C], [CO], [TiO2], [TiO], and [TiC]: V1 = k1[C][TiO2], (9) V2 = k2[CO][TiO], (10) V3 = k3[B][TiC] (11) V4 = k4[B][C]. (12) By definition, the reaction constants k1, k2, k3, and k4 are positive and not dependent on the reagent concentrations. However, their temperature-dependence (Arrhenius law) is quite significant, and, when the thermal treatment is not isothermal, reaction constants are also complex time-functions. In the technology under consideration, at the heatingstage, reagents are formed from precursor charge, while the basic chemical Reactions (5)–(8) proceed too slowly. As for the coolingstage, it starts after all the reagents and products are completely consumed and formed, respectively. Therefore, the assumption that the technology is almost isothermal and the corresponding reaction rates are time-constants is acceptable.