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Crystallization Process and Simulation Calculation, Third Edition
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Crystallization Process and Simulation Calculation, Third Edition

A special issue of Crystals (ISSN 2073-4352). This special issue belongs to the section "Industrial Crystallization".

Deadline for manuscript submissions: 30 May 2025 | Viewed by 2257

Special Issue Editors

Dr. Mingyang Chen

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Guest Editor
State Key Laboratory of Chemical Engineering, Tianjin University, School of Chemical Engineering and Technology, Tianjin 300072, China
Interests: crystallization process; spherical crystallization; nucleation; crystal growth; crystal agglomeration; simulation; particle engineering
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Jinbo Ouyang

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State Key Laboratory of Nuclear Resources and Environment, East China Institute of Technology, Fuzhou, China
Interests: drug crystal design; isolation adsorbent material design; wastewater treatment
Special Issues, Collections and Topics in MDPI journals
Dr. Kangli Li

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Guest Editor
Institute of Shaoxing, Tianjin University, Shaoxing 312300, China
Interests: polymorphism; nucleation; crystal growth; industrial crystallization; crystal engineering
Special Issues, Collections and Topics in MDPI journals
Dr. Mengmeng Sun

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Guest Editor
School of Chemical Engineering and Technology, Hebei University of Technology, Tianjin 300401, China
Interests: particle engineering; oiling-out crystallization; spherical crystallization; nucleation; crystal growth; molecular dynamic simulation
Dr. Yu Liu

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Guest Editor
School of Chemical Engineering, Shenyang University of Chemical Technology, Shenyang 110142, China
Interests: nucleation mechanism; polymorph nucleation; crystal growth; crystal engineering

Special Issue Information

Dear Colleagues,

Following the remarkable success of the first edition and the second edition of this topic, entitled “Crystallization Process and Simulation Calculation” (https://www.mdpi.com/journal/crystals/special_issues/crystallization_process2; https://www.mdpi.com/journal/crystals/special_issues/9LVS3K8K6Q ), we are pleased to announce this third edition.

Crystallization is a crucial unit operation where nucleation, growth, agglomeration, and breakage are regulated to produce high-quality crystals and achieve efficient separation and purification. In recent years, there have been notable advancements in crystallization processes. Process intensification techniques such as ultrasound and wet grinding have been employed to improve the nucleation and breakage processes, thereby preparing ultrafine powders and nanoparticles with different morphologies. Co-crystallization, as a means of crystal engineering, is also widely used to modify crystal structure and morphology, aiming to enhance the physicochemical properties and powder performance of crystal products. Spherical crystallization technology is utilized to generate spherical crystalline particles through crystal growth or agglomeration processes. Continuous crystallization has also gained increasing interest due to its high productivity and consistency in product quality. These studies offer innovative strategies and methods to design processes and control crystallization, ensuring the desired quality attributes and predictable performance of the product. Given the complex nature of crystallization processes, characterized by strong coupling, nonlinearity and large lagging, the challenges that remain include the design of a robust and well-characterized process for efficient crystallization and the production of high-quality crystalline products. The development of process analytical technology that can provide rapid and precise inline or online measurements is critical for designing and controlling crystallization processes. Simulation technology, such as molecular dynamics simulation and hydrodynamics simulation technology, provide insights into the process at multiple scales over time or location. These experimental and simulation tools can greatly enhance the understanding and optimization of crystallization processes.

This Special Issue, “Crystallization Processes and Simulation Calculations, Third Edition”, serves to provide a platform for researchers to report results and findings regarding crystallization process technologies, simulation and process analytical technologies, and relevant crystallization studies.

Dr. Mingyang Chen
Prof. Dr. Jinbo Ouyang
Dr. Kangli Li
Dr. Mengmeng Sun
Dr. Yu Liu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Crystals is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2100 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • nucleation and growth
  • agglomeration and breakage
  • process analytical technology
  • process intensification
  • continuous crystallization
  • spherical crystallization
  • co-crystallization
  • molecular dynamics simulation
  • hydrodynamics simulation

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Published Papers (4 papers)

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Research

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8 pages, 2174 KiB  
Article
Effect of Pyrolysis Temperature on Microwave Heating Properties of Oxidation-Cured Polycarbosilane Powder
by Chang-Hun Hwang, Jong-Ha Beak, Sang-In Kim and Se-Yun Kim
Crystals 2024, 14(12), 1080; https://doi.org/10.3390/cryst14121080 - 14 Dec 2024
Viewed by 416
Abstract
Silicon carbide (SiC) has excellent mechanical and chemical properties and is used in a wide range of applications. It has the characteristic of rapidly heating up to several hundred degrees within one minute when irradiated with microwave radiation at 2.45 GHz. In this [...] Read more.
Silicon carbide (SiC) has excellent mechanical and chemical properties and is used in a wide range of applications. It has the characteristic of rapidly heating up to several hundred degrees within one minute when irradiated with microwave radiation at 2.45 GHz. In this study, we investigated the oxidation curing process and microwave heating properties of polycarbosilane (PCS). A PCS disk-shaped green body was fabricated via uniaxial pressure molding. Silicon carbide was prepared by varying the pyrolysis temperature, and the heating characteristics of the microwaves were evaluated. The results showed that the samples pyrolyzed at 1300 °C after oxidation curing for 2 h at 180 °C rapidly heated up to 802 °C within 1 min, and the temperature remained constant for 120 min. The maximum temperature of the samples pyrolyzed at 1500 °C was relatively low, but the rate of heating was the highest. The microstructures and crystal structures of the microwaves as a function of the pyrolysis temperature were investigated. Full article
(This article belongs to the Special Issue Crystallization Process and Simulation Calculation, Third Edition)
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Figure 1

Figure 1
<p>Sample shape after pyrolysis at 1300 °C under different oxidation curing conditions: (<b>a</b>) 180 °C and 30 min, (<b>b</b>) 180 °C and 60 min, (<b>c</b>) 180 °C and 120 min.</p> Full article ">Figure 2
<p>Microstructures according to the pyrolysis temperature: (<b>a</b>) 1200 °C, (<b>b</b>) 1300 °C, (<b>c</b>) 1400 °C, (<b>d</b>) 1500 °C (yellow points: pores size &gt; 0.5 µm).</p> Full article ">Figure 3
<p>Crystal structure analysis of the samples fabricated at various pyrolysis temperatures using XRD: (<b>a</b>) XRD pattern, (<b>b</b>) β-silicon carbide crystal size.</p> Full article ">Figure 4
<p>Microwave irradiation heating test of samples manufactured at different pyrolysis temperatures: (<b>a</b>) 60 s short time. Heating rate according to the thermal decomposition temperature conditions of PCS, (<b>b</b>) 60 min long time heating behavior of silicon carbide samples under microwave irradiation, (<b>c</b>) heating phenomenon according to microwave irradiation time for samples thermally decomposed at 1300 °C for 3 h.</p> Full article ">
12 pages, 4142 KiB  
Article
Batch Cooling Crystallization of a Model System Using Direct Nucleation Control and High-Performance In Situ Microscopy
by Josip Budimir Sacher, Nenad Bolf and Marko Sejdić
Crystals 2024, 14(12), 1079; https://doi.org/10.3390/cryst14121079 - 13 Dec 2024
Viewed by 622
Abstract
The aim of this study was to investigate the use of automated high performance in situ microscopy (HPM) for monitoring and direct nucleation control (DNC) during cooling crystallization. Compared to other techniques, HPM enables the detection of small crystals in the range of [...] Read more.
The aim of this study was to investigate the use of automated high performance in situ microscopy (HPM) for monitoring and direct nucleation control (DNC) during cooling crystallization. Compared to other techniques, HPM enables the detection of small crystals in the range of 1 to 10 μm. Therefore, a novel DNC-controlled variable was investigated to determine the potential improvement of the method. The laboratory system and process control software were developed in-house. A well-studied crystallization model system, the seeded batch cooling crystallization of α-glycine from water, was investigated under normal conditions and temperatures below 60 °C. It was postulated that length-weighted edge-to-edge counts in the range of 1 to 10 μm would be most sensitive to the onset of secondary nucleation and are therefore, used as a control variable. Linear cooling experiments were conducted to determine the initial setpoint for the DNC experiments. Three DNC experiments were then performed with different setpoints and an upper and lower counts limit. It was found that the DNC method can be destabilized with a low setpoint and narrow counts limits. In addition, the new controlled variable is highly sensitive to the formation of bubbles at the microscope window and requires careful evaluation. To address the advantages of the DNC method, an additional linear cooling experiment of the same duration was performed, and it was found that the DNC method resulted in a larger average crystal size. Overall, it can be concluded that the HPM method is suitable for DNC control and could be improved by modifying the image processing algorithm. Full article
(This article belongs to the Special Issue Crystallization Process and Simulation Calculation, Third Edition)
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Figure 1

Figure 1
<p>Experimental setup and communication protocol.</p> Full article ">Figure 2
<p>Laboratory system set-up.</p> Full article ">Figure 3
<p>DNC algorithm logic flow diagram.</p> Full article ">Figure 4
<p>Counts and temperature trends during linear cooling experiments.</p> Full article ">Figure 5
<p>Counts and temperature trends during DNC experiments.</p> Full article ">Figure 6
<p>Bubble formation on microscope window toward the end of the experiment D3.</p> Full article ">Figure 7
<p>Control variable comparison for experiment D3.</p> Full article ">Figure 8
<p>Sieving particle size distribution comparison between DNC and linear cooling.</p> Full article ">Figure 9
<p>Agglomerated crystals at the end of experiment D2.</p> Full article ">
14 pages, 2173 KiB  
Article
Crystallization Kinetics of Tacrolimus Monohydrate in an Ethanol–Water System
by Suoqing Zhang, Jixiang Zhao, Ming Kong, Jiahui Li, Mingxuan Li, Miao Ma, Li Tong, Tao Li and Mingyang Chen
Crystals 2024, 14(10), 849; https://doi.org/10.3390/cryst14100849 - 28 Sep 2024
Viewed by 655
Abstract
Nucleation and growth during the crystallization process are crucial steps that determine the crystal structure, size, morphology, and purity. A thorough understanding of these mechanisms is essential for producing crystalline products with consistent properties. This study investigates the solubility of tacrolimus (FK506) in [...] Read more.
Nucleation and growth during the crystallization process are crucial steps that determine the crystal structure, size, morphology, and purity. A thorough understanding of these mechanisms is essential for producing crystalline products with consistent properties. This study investigates the solubility of tacrolimus (FK506) in an ethanol–water system (1:1, v/v) and examines its crystallization kinetics using batch crystallization experiments. Initially, the solubility of FK506 was measured, and classical nucleation theory was employed to analyze the induction period to determine interfacial free energy (γ) and other nucleation parameters, including the critical nucleus radius (r*), critical free energy (∆G*), and the molecular count of the critical nucleus (i*). Crystallization kinetics under seeded conditions were also measured, and the parameters of the kinetic model were analyzed to understand the effects of process states such as temperature on the crystallization process. The results suggested that increasing temperature and supersaturation promotes nucleation. The surface entropy factor (f) indicates that the tacrolimus crystal growth mechanism is a two-dimensional nucleation growth. The growth process follows the particle size-independent growth law proposed by McCabe. The estimated kinetic parameters reveal the effects of supersaturation, temperature, and suspension density on the nucleation and growth rates. Full article
(This article belongs to the Special Issue Crystallization Process and Simulation Calculation, Third Edition)
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Figure 1

Figure 1
<p>Solubility of tacrolimus monohydrate in an ethanol–water system.</p> Full article ">Figure 2
<p>The variation in transmittance over time.</p> Full article ">Figure 3
<p>Relationship between induction time and supersaturation.</p> Full article ">Figure 4
<p>Relationship between and <math display="inline"><semantics> <mrow> <mi>l</mi> <mi>n</mi> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mi>i</mi> <mi>n</mi> <mi>d</mi> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msup> <mrow> <mn>1</mn> <mo>/</mo> <mfenced separators="|"> <mrow> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">n</mi> <mi>S</mi> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Relationships between nucleation parameters and supersaturation: (<b>a</b>) the radius of the critical nucleus, (<b>b</b>) the critical free energy of the nucleus, (<b>c</b>) primary nucleation rate, and (<b>d</b>) the molecular number of the critical nucleus.</p> Full article ">Figure 6
<p>Relationship between particle density and particle size.</p> Full article ">Figure 7
<p>Relationship between suspension density and nucleation rate.</p> Full article ">Figure 8
<p>Relationships between supersaturation and nucleation (growth).</p> Full article ">Figure 9
<p>Comparison between experimental and theoretical kinetics rate for FK506 crystallization: (<b>a</b>) nucleation rate and (<b>b</b>) growth rate.</p> Full article ">

Review

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38 pages, 23114 KiB  
Review
Mathematical Modeling of Properties and Structures of Crystals: From Quantum Approach to Machine Learning
by Grzegorz Matyszczak, Christopher Jasiak, Gabriela Rusinkiewicz, Kinga Domian, Michał Brzozowski and Krzysztof Krawczyk
Crystals 2025, 15(1), 61; https://doi.org/10.3390/cryst15010061 - 9 Jan 2025
Viewed by 98
Abstract
The crystalline state of matter serves as a reference point in the context of studies of properties of a variety of chemical compounds. This is due to the fact that prepared crystalline solids of practically useful materials (inorganic or organic) may be utilized [...] Read more.
The crystalline state of matter serves as a reference point in the context of studies of properties of a variety of chemical compounds. This is due to the fact that prepared crystalline solids of practically useful materials (inorganic or organic) may be utilized for the thorough characterization of important properties such as (among others) energy bandgap, light absorption, thermal and electric conductivity, and magnetic properties. For that reason it is important to develop mathematical descriptions (models) of properties and structures of crystals. They may be used for the interpretation of experimental data and, as well, for predictions of properties of novel, unknown compounds (i.e., the design of novel compounds for practical applications such as photovoltaics, catalysis, electronic devices, etc.). The aim of this article is to review the most important mathematical models of crystal structures and properties that vary, among others, from quantum models (e.g., density functional theory, DFT), through models of discrete mathematics (e.g., cellular automata, CA), to machine learning (e.g., artificial neural networks, ANNs). Full article
(This article belongs to the Special Issue Crystallization Process and Simulation Calculation, Third Edition)
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Figure 1

Figure 1
<p>(<b>A</b>) Crystal structures of some of the phases in the Co-Al system: (<b>a</b>) Al<sub>9</sub>Co<sub>2</sub> (space group: <span class="html-italic">P2<sub>1</sub>/c</span>), (<b>b</b>) Al<sub>13</sub>Co<sub>4</sub> (space group: <span class="html-italic">Pmn2<sub>1</sub></span>), (<b>c</b>) Al<sub>3</sub>Co (space group: <span class="html-italic">C2/m</span>), and (<b>d</b>) Al<sub>5</sub>Co<sub>2</sub> (space group: <span class="html-italic">P6<sub>3</sub>/mmc</span>). (<b>B</b>) A comparison of the experimental Co-Al phase diagram (dashed black lines) with the one calculated using DFT (red lines). Reproduced from ref. [<a href="#B15-crystals-15-00061" class="html-bibr">15</a>] under the Creative Commons license.</p> Full article ">Figure 2
<p>(<b>A</b>) A phase diagram of platinum obtained using the Z methodology: fcc–Pt melting curve (green line), liquid Pt solidified into solid fcc (green bullets), 9R-Pt melting curve (blue line), liquid Pt solidified into solid 9R (blue bullets), and the fcc–9R solid–solid phase boundary (violet). (<b>B</b>) A phase diagram of tantalum obtained from the Z methodology: bcc–Ta melting curve (green), Pnma–Ta melting curve (blue), and the bcc–Pnma solid–solid phase boundary (red). Reproduced from ref. [<a href="#B20-crystals-15-00061" class="html-bibr">20</a>] under the Creative Commons license.</p> Full article ">Figure 3
<p>(<b>a</b>) The calculated crystal structure of K<sub>2</sub>SiF<sub>6</sub> (violet—K, blue—Si, and grey—F). (<b>b</b>) The dependence of the bandgap value of K<sub>2</sub>SiF<sub>6</sub> on the external pressure. Reproduced from ref. [<a href="#B22-crystals-15-00061" class="html-bibr">22</a>] under the Creative Commons license.</p> Full article ">Figure 4
<p>Structures with CHA framework: (<b>a</b>) chabazite, (<b>b</b>) CHA, and (<b>c</b>) AlPO<sub>4</sub>-34. Red balls indicate O atoms, blue indicate N atoms, cyan indicate Al atoms, and purple indicate P atoms. Reproduced from ref. [<a href="#B24-crystals-15-00061" class="html-bibr">24</a>] under the Creative Commons license.</p> Full article ">Figure 5
<p>E–V fitting for all-silica chabazite based on following sets of pseudopotentials: (<b>a</b>) PBE_mGGA, (<b>b</b>) PBE_GW and PAW_PBE, (<b>c</b>) PAW_GGA, (<b>d</b>) LDA_PP, (<b>e</b>) USPP_GGA, and (<b>f</b>) USPP_LDA. Reproduced from ref. [<a href="#B24-crystals-15-00061" class="html-bibr">24</a>] under the Creative Commons license.</p> Full article ">Figure 6
<p>Comparison of calculated and experimental: (<b>a</b>) XRD pattern and (<b>b</b>) IR spectra. Reproduced from ref. [<a href="#B34-crystals-15-00061" class="html-bibr">34</a>] under the Creative Commons license.</p> Full article ">Figure 7
<p>A collection of views (along the b-axis) of crystal structures of three polymorphs of Cu<sub>3</sub>VSe<sub>4</sub> predicted by the USPEX algorithm. Reproduced from ref. [<a href="#B36-crystals-15-00061" class="html-bibr">36</a>] under the Creative Commons license.</p> Full article ">Figure 8
<p>A collection of views (along the c-axis) of predicted (by the USPEX algorithm) Cu<sub>4</sub>TiSe<sub>4</sub> crystal structures and the experimental disordered structure. Reproduced from ref. [<a href="#B38-crystals-15-00061" class="html-bibr">38</a>] under the Creative Commons license.</p> Full article ">Figure 9
<p>The initial state of the model. (<b>a</b>) Molecular simulation model of nanoindentation for three types of SiC single crystals: (<b>b</b>) 3C-SiC, (<b>c</b>) 4H-SiC, and (<b>d</b>) 6H-SiC. Reproduced from ref. [<a href="#B48-crystals-15-00061" class="html-bibr">48</a>] under the Creative Commons license.</p> Full article ">Figure 10
<p>A cross-sectional view of nanoindentation for three types of SiC single crystals at completed loading and completed unloading: (<b>a</b>) (100) plane in 3C-SiC, (<b>b</b>) (1-210) plane in 4H-SiC, and (<b>c</b>) (1-210) plane in 6H-SiC. Reproduced from ref. [<a href="#B48-crystals-15-00061" class="html-bibr">48</a>] under the Creative Commons license.</p> Full article ">Figure 11
<p>Growing crystalline particles (<b>a</b>) of BCC Fe at 900 K (blue) and FCC Cu (green) at 850 K (<b>b</b>). HCP stacking faults are shown in red. Reproduced from ref. [<a href="#B49-crystals-15-00061" class="html-bibr">49</a>] under the Creative Commons license.</p> Full article ">Figure 12
<p>(<b>a</b>): Atomic fluctuations for each amino acid residue, emphasizing the larger fluctuations in the disordered regions (D1 and D2) and the GATase domain compared to the ATPPase and dimerization domains. (<b>b</b>–<b>f</b>): The domain-specific fluctuations when various domains or subunits were fixed, confirming that the GATase domain moves more independently than the others. Reproduced from ref. [<a href="#B52-crystals-15-00061" class="html-bibr">52</a>] under the Creative Commons license.</p> Full article ">Figure 13
<p>(<b>a</b>) A visualization of PSII water channels (O1, O4, and Cl1) with marked crystallographic water positions (blue) and nearby amino acids. (<b>b</b>) Histogram showing the distance between crystallographic waters and their nearest MD-predicted water positions, highlighting high agreement within ~1 Å. (<b>c</b>) Correlation between MD-predicted electron density heights and experimental values, demonstrating consistency with higher density in structured regions. (<b>d</b>,<b>e</b>) Electron density maps for selected water clusters near the catalytic center, with higher density peaks indicating ordered water structures in channel regions. Reproduced from ref. [<a href="#B53-crystals-15-00061" class="html-bibr">53</a>] under the Creative Commons license.</p> Full article ">Figure 14
<p>Histogram comparing the distribution of the peak height of simulated waters inside the three channels of interest (O1, Cl, and O4) vs. outside the channels. Reproduced from ref. [<a href="#B53-crystals-15-00061" class="html-bibr">53</a>] under the Creative Commons license.</p> Full article ">Figure 15
<p>Weight-averaged stem length (d) in the crystalline state formed by supercooling at T<sub>q</sub> = 418, 412.5, and 385 K for different volume fractions (ϕ). Reprinted with permission from ref. [<a href="#B43-crystals-15-00061" class="html-bibr">43</a>]. Copyright 2016 American Chemical Society.</p> Full article ">Figure 16
<p>A schematic of the MD cutting model. (<b>a</b>) The modification of the workpiece. (<b>b</b>) The determination of the tool trajectory. (<b>c</b>) The morphology of the modified MD model. Green atoms represent the deformation region in one vibration cycle. Reproduced from ref. [<a href="#B54-crystals-15-00061" class="html-bibr">54</a>] under the Creative Commons license.</p> Full article ">Figure 17
<p>The structure of an AlMg<sub>5</sub> alloy where (<b>a</b>) shows the modeled structure in the final stage of solidification, and (<b>b</b>) shows the actual structure in the die casting. Reproduced from ref. [<a href="#B58-crystals-15-00061" class="html-bibr">58</a>] under the Creative Commons license.</p> Full article ">Figure 18
<p>Solute map at different simulation times using the CA-FE model with preferred growth orientations of 0o with respect to the horizontal direction. Reproduced under permission from ref. [<a href="#B59-crystals-15-00061" class="html-bibr">59</a>].</p> Full article ">Figure 19
<p>(<b>a</b>) Hexagonal grid and (<b>b</b>) transformed hexagonal grid. Reproduced from ref. [<a href="#B55-crystals-15-00061" class="html-bibr">55</a>] under the Creative Commons license. Copyright AIDIC/CET.</p> Full article ">Figure 20
<p>Uranium crystal growth modeled by CA: (<b>a</b>) evolution in time, (<b>b</b>) the influence of initial concentration (C<sub>0</sub>) on crystal growth, and (<b>c</b>) the influence of initial temperature (T<sub>0</sub>) on crystal growth. Reproduced from ref. [<a href="#B55-crystals-15-00061" class="html-bibr">55</a>] under the Creative Commons license. Copyright AIDIC/CET.</p> Full article ">Figure 20 Cont.
<p>Uranium crystal growth modeled by CA: (<b>a</b>) evolution in time, (<b>b</b>) the influence of initial concentration (C<sub>0</sub>) on crystal growth, and (<b>c</b>) the influence of initial temperature (T<sub>0</sub>) on crystal growth. Reproduced from ref. [<a href="#B55-crystals-15-00061" class="html-bibr">55</a>] under the Creative Commons license. Copyright AIDIC/CET.</p> Full article ">Figure 21
<p>Snowflake growth based on a simulation where N = the number of iterations used for the simulation, and each row has a different set of α from the top, with α set to 0.2, 0.4, and 0.5, to the last row, where α is set to 0.6, and the β parameter is set to zero. Reproduced under permission from ref. [<a href="#B56-crystals-15-00061" class="html-bibr">56</a>].</p> Full article ">Figure 22
<p>The experimental (<b>a</b>,<b>b</b>) and CA (<b>c</b>,<b>d</b>) results for microstructures formed at deformation temperatures of 350 °C (<b>a</b>,<b>c</b>) and 450 °C (<b>b</b>,<b>d</b>); (<b>e</b>) the variation in the recrystallized volume fraction and the mean grain size at temperatures of 350 °C and 450 °C. Reproduced from ref. [<a href="#B61-crystals-15-00061" class="html-bibr">61</a>] under the Creative Commons license.</p> Full article ">Figure 22 Cont.
<p>The experimental (<b>a</b>,<b>b</b>) and CA (<b>c</b>,<b>d</b>) results for microstructures formed at deformation temperatures of 350 °C (<b>a</b>,<b>c</b>) and 450 °C (<b>b</b>,<b>d</b>); (<b>e</b>) the variation in the recrystallized volume fraction and the mean grain size at temperatures of 350 °C and 450 °C. Reproduced from ref. [<a href="#B61-crystals-15-00061" class="html-bibr">61</a>] under the Creative Commons license.</p> Full article ">Figure 23
<p>The overall scheme of the described CSP method. Reproduced from ref. [<a href="#B71-crystals-15-00061" class="html-bibr">71</a>] under the Creative Commons license.</p> Full article ">Figure 24
<p>A schematic of MolXtalNet. ngc is the number of graph convolution layers in the model. Reproduced from ref. [<a href="#B72-crystals-15-00061" class="html-bibr">72</a>] under the Creative Commons license.</p> Full article ">Figure 25
<p>A schematic of the simulation procedure in LAQA, which is automated in CrySPY. Reproduced from ref. [<a href="#B73-crystals-15-00061" class="html-bibr">73</a>] under the Creative Commons license.</p> Full article ">Figure 26
<p>A schematic of the PIRNN-based process. Reproduced from ref. [<a href="#B75-crystals-15-00061" class="html-bibr">75</a>] under the Creative Commons license.</p> Full article ">Figure 27
<p>A schematic representation of the workflow for computing ab initio, quantum anharmonic Gibbs free energies for candidate crystal structures. Upper section shows the main steps: (1) generating ab initio reference data on which to (2) train a combined MLP, which can then be used to (3) compute MLP Gibbs free energies, which one can finally (4) promote to ab initio Gibbs free energies. Lower section (shaded in blue) details the key aspects of how each of these steps is performed in practice. Reproduced from ref. [<a href="#B77-crystals-15-00061" class="html-bibr">77</a>] under the Creative Commons license.</p> Full article ">
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