Mathematical Modeling to Predict the Formation of Micrometer-Scale Crystals Using Reverse Anti-Solvent Crystallization
Abstract
:1. Introduction
2. The Establishment of Mathematical Models
2.1. Prelude to the Mathematical Model
2.2. Mechanism Analysis and Mathematical Model Establishment
- (1)
- Each dripping droplet is spherical. When the spherical droplet completely enters the anti-solvent, the solvent could diffuse. At this time, crystallization begins to occur.
- (2)
- The solvent diffusion time in each droplet is the same as the time used for crystal growth, and the crystal nucleus growth time is ignored.
- (3)
- No secondary nucleation occurred in each drop of solution.
- (4)
- The supersaturation as the driving force for crystal growth keeps a constant value when crystal will occur in each solvent droplet.
- (5)
- The crystal growth rate is only related to the temperature and supersaturation.
2.3. Experiment
2.3.1. Materials
- (1)
- The main materials include a φ3 mm silicone tube, a peristaltic pump, a beaker, a Buchner funnel and vacuum filtration device, a small reactor, a set of stirring devices, and two sets of heating devices (ovens);
- (2)
- The test instruments include a scanning electron microscope (Nano SEM 450, Nova) and a laser particle size analyzer (MS2000, Malvern);
- (3)
- The experiment reagents include deionized water, anhydrous ethanol (analytically pure 99.9%), and NaCl (analytically pure).
2.3.2. Experiment Methods
3. Result
4. Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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NO. | Concentration g/g (Water) | Dripping Droplet Velocity μL/min | Agitation Rate rpm/min | Temperature °C |
---|---|---|---|---|
1 | 0.36 | 100 | 500 | 10 |
2 | 0.36 | 100 | 500 | 20 |
3 | 0.36 | 100 | 500 | 30 |
4 | 0.36 | 100 | 500 | 40 |
5 | 0.36 | 100 | 500 | 50 |
1 | 1.257 | 7.279 | 14.012 | 3.738 | 1.752 |
2 | 1.651 | 8.745 | 20.601 | 4.220 | 2.161 |
3 | 1.687 | 11.517 | 24.690 | 4.745 | 1.997 |
4 | 4.822 | 15.637 | 32.965 | 6.814 | 1.799 |
5 | 7.495 | 19.480 | 39.669 | 9.131 | 1.653 |
Parameter [units] | Description | Value |
---|---|---|
[m3/mol] | The molar volume of liquid | 1.8 × 10−5 |
[mol/m3] | Droplet concentration | 55.6 × 103 |
[m2/s] | Diffusion coefficient | 2 × 10−9 |
[m/s] | Growth rate constant | 7.1150 × 10−5 |
[-] | Growth rate order | 1.079 |
[kJ/mol] | Activation energy for growth | 1.9023 |
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Wang, J.; Wang, F.; Wen, X.; Zhang, Y.; Wang, J.; Liu, Y. Mathematical Modeling to Predict the Formation of Micrometer-Scale Crystals Using Reverse Anti-Solvent Crystallization. Crystals 2025, 15, 145. https://doi.org/10.3390/cryst15020145
Wang J, Wang F, Wen X, Zhang Y, Wang J, Liu Y. Mathematical Modeling to Predict the Formation of Micrometer-Scale Crystals Using Reverse Anti-Solvent Crystallization. Crystals. 2025; 15(2):145. https://doi.org/10.3390/cryst15020145
Chicago/Turabian StyleWang, Jianhua, Fawei Wang, Xu Wen, Yankang Zhang, Jiapeng Wang, and Yucun Liu. 2025. "Mathematical Modeling to Predict the Formation of Micrometer-Scale Crystals Using Reverse Anti-Solvent Crystallization" Crystals 15, no. 2: 145. https://doi.org/10.3390/cryst15020145
APA StyleWang, J., Wang, F., Wen, X., Zhang, Y., Wang, J., & Liu, Y. (2025). Mathematical Modeling to Predict the Formation of Micrometer-Scale Crystals Using Reverse Anti-Solvent Crystallization. Crystals, 15(2), 145. https://doi.org/10.3390/cryst15020145