# Insight into the Nucleation Mechanism of p-Methoxybenzoic Acid in Ethanol-Water System from Metastable Zone Width

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## Abstract

**:**

## 1. Introduction

## 2. Experiments and Methods

#### 2.1. Materials

#### 2.2. Measurement of Solubility

#### 2.3. Metastable Zone Measurement Experiment for PMBA

#### 2.4. X-ray Powder Diffraction

## 3. Theory

#### 3.1. Nývlt’s Metastable Zone Model

#### 3.2. Sangwal Metastable Zone Model

#### 3.3. Modified Sangwal Metastable Zone Model

## 4. Results and Discussion

#### 4.1. Solubility of PMBA in Ethanol Solution

^{2}of the fitted line is 0.9598, 0.9692, and 0.9853, respectively, indicating that the solubility has a good linear relationship with the temperature, and the temperature coefficient ${\left(dc/dT\right)}_{T}$ is a constant. Therefore, the Nývlt’s metastable zone model can be used to fit metastable zone data. At the same time, the solubility was fitted with the Van’t Hoff equation ($\mathrm{ln}x=-\frac{\Delta {H}_{s}}{{R}_{g}T}+\frac{\Delta S}{{R}_{g}}$) (as shown in Figure 4) to obtain the corresponding dissolution enthalpy $\Delta {H}_{s}$ and dissolution entropy $\Delta S$. The results are shown in Table 1. $\Delta {H}_{s}$ can reflect the difference in the solvation intensity of solutes in different solution compositions to a certain extent. Table 1 shows that $\Delta {H}_{s}$ values in different ethanol mass fractions ranked as 0.6 > 0.8 > 1.0 in sequence. The results show that with an increase in the ethanol mass fraction, the enthalpy of dissolution decreases, and the solubility increases.

#### 4.2. Solid-State Characterization of PMBA

#### 4.3. Effects of Saturation Temperature, Cooling Rate, and Ethanol Mass Fractions on MSZW

#### 4.4. Nucleation Kinetic Parameters and Crystal Habit

^{27}m

^{−3}[20], the value of $A$ can be obtained. The calculation results show that the low saturation temperature and fast cooling rate mean that the formation rate of supersaturation is fast, resulting in a large kinetic factor $A$.

#### 4.5. Critical Nucleation Parameters and Nucleation Kinetics

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## References

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**Figure 1.**Metastable zone determination experimental flow chart (

**1**: cooling circulating water machine,

**2**: digital magnetic stirrer,

**3**: digital thermometer,

**4**: crystallizer,

**5**: FBRM probe,

**6**: FBRM workstation,

**7**: computer).

**Figure 2.**Schematic diagram of measuring the metastable zone by the Focused Beam Reflectance Meter (FBRM) (taking the saturation temperature of 293.15 K, stirring rate of 300 rpm, ethanol mass fraction of 0.6, and cooling rate of 20 K/h as an example).

**Figure 3.**Relationship between solubility and temperature of PMBA under different ethanol mass fractions.

**Figure 5.**X-ray powder diffraction patterns of the PMBA solid samples. (

**a**) is the raw material; (

**b**–

**d**) stands for the PXRD patterns of the samples obtained from solvent with ethanol mass fractions of 0.6, 0.8, and 1.0, respectively.

**Figure 6.**Relationship between MSZW and cooling rate at different saturation temperatures fitted by Nývlt’s model: (

**a**) ethanol mass fraction of 0.6, (

**b**) ethanol mass fraction of 0.8, and (

**c**) ethanol mass fraction of 1.0.

**Figure 7.**Relationship between MSZW and cooling rate at different saturation temperatures fitted by the Sangwal model: (

**a**) ethanol mass fraction of 0.6, (

**b**) ethanol mass fraction of 0.8, and (

**c**) ethanol mass fraction of 1.0.

**Figure 8.**Crystal morphology of PMBA at ethanol mass fraction of (

**a**) 0.6, (

**b**) 0.8, and (

**c**) 1.0 (saturation temperature 293.15 K, cooling rate 20 K/h, the scale in all figures is 20 µm).

**Figure 9.**Relationship between MSZW and cooling rate at different ${T}_{0}$ fitted by the modified Sangwal model: (

**a**) ethanol mass fraction of 0.6, (

**b**) ethanol mass fraction of 0.8, and (

**c**) ethanol mass fraction of 1.0.

**Figure 10.**Relation between ${r}_{crit}$ and $\Delta \mu $ at different ${T}_{0}$: (

**a**) ethanol mass fraction of 0.6, (

**b**) ethanol mass fraction of 0.8, (

**c**) ethanol mass fraction of 1.0.

**Figure 11.**Relationship between $\Delta {G}_{crit}$ and $\Delta \mu $ at different ${T}_{0}$: (

**a**) ethanol mass fraction of 0.6, (

**b**) ethanol mass fraction of 0.8, (

**c**) ethanol mass fraction of 1.0.

**Figure 12.**Relationship between $J$ and $\Delta \mu $ at different ${T}_{0}$: (

**a**) ethanol mass fraction of 0.6, (

**b**) ethanol mass fraction of 0.8, (

**c**) ethanol mass fraction of 1.0.

**Figure 13.**Relationship between $\mathrm{ln}{r}_{crit}$ and $\mathrm{ln}\Delta \mu $ at different ${T}_{0}$ : (

**a**) ethanol mass fraction of 0.6, (

**b**) ethanol mass fraction of 0.8, (

**c**) ethanol mass fraction of 1.0.

**Figure 14.**Relationship between $\mathrm{ln}\Delta {G}_{crit}$ and $\mathrm{ln}\Delta \mu $ at different ${T}_{0}$ : (

**a**) ethanol mass fraction of 0.6, (

**b**) ethanol mass fraction of 0.8, (

**c**) ethanol mass fraction of 1.0.

**Figure 15.**The relationship between interfacial energy $\gamma $ and the change in ethanol mass fraction (saturated temperature 293.15 K, cooling rate 10, 20, 30, and 60 K/h, respectively).

**Figure 16.**The relationship between interfacial energy and the change in saturated temperature (ethanol mass fraction 0.8, cooling rate 10, 20, 30, and 60 K/h, respectively).

**Figure 17.**The relationship between interfacial energy and the change in cooling rate (saturated temperature 303.15 K, ethanol mass fraction 0.6, 0.8, and 1.0, respectively).

$\mathit{\omega}$ | ∆H_{S} (J/mol) | ∆S (J/mol/K) | R^{2} |
---|---|---|---|

0.6 | 35032.33 | 70.75 | 0.9919 |

0.8 | 30061.06 | 60.87 | 0.9966 |

1.0 | 29637.00 | 62.92 | 0.9974 |

$\mathit{\omega}$ | ${\mathit{T}}_{0}/\mathbf{K}$ | $\Delta {\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}/\mathbf{K}$ | |||
---|---|---|---|---|---|

R = 10 K/h | R = 20 K/h | R = 30 K/h | R = 60 K/h | ||

0.6 | 293.15 | 2.900 | 3.350 | 3.650 | 4.075 |

0.6 | 303.15 | 1.875 | 2.350 | 2.776 | 3.275 |

0.6 | 313.15 | 0.800 | 1.125 | 1.500 | 2.075 |

0.8 | 293.15 | 3.575 | 3.875 | 4.150 | 4.525 |

0.8 | 303.15 | 2.075 | 2.525 | 3.000 | 3.575 |

0.8 | 313.15 | 1.550 | 2.025 | 2.550 | 3.075 |

1 | 293.15 | 2.700 | 3.025 | 3.400 | 3.675 |

1 | 303.15 | 0.800 | 1.100 | 1.500 | 1.800 |

1 | 313.15 | 0.450 | 0.775 | 1.150 | 1.500 |

$\mathit{\omega}$ | Nucleation Order m | ||
---|---|---|---|

T_{0} = 293.15 K | T_{0} = 303.15 K | T_{0} = 313.15 K | |

0.6 | 3.18 | 3.16 | 1.85 |

0.8 | 7.50 | 3.23 | 2.55 |

1.0 | 5.63 | 2.14 | 1.46 |

$\mathit{\omega}$ | T_{0}/K | $\mathit{\gamma}$ (mJ/m^{2}) | |||
---|---|---|---|---|---|

R = 10 K/h | R = 20 K/h | R = 30 K/h | R = 60 K/h | ||

0.6 | 293.15 | 1.8450 | 1.8437 | 1.8426 | 1.8404 |

0.6 | 303.15 | 1.3658 | 1.3650 | 1.3644 | 1.3636 |

0.6 | 313.15 | 0.7090 | 0.7088 | 0.7085 | 0.7080 |

0.8 | 293.15 | 2.3195 | 2.3187 | 2.3179 | 2.3169 |

0.8 | 303.15 | 1.4552 | 1.4545 | 1.4537 | 1.4528 |

0.8 | 313.15 | 1.0368 | 1.0363 | 1.0357 | 1.0351 |

1.0 | 293.15 | 1.774 | 1.7733 | 1.7726 | 1.7720 |

1.0 | 303.15 | 0.6518 | 0.6516 | 0.6513 | 0.6511 |

1.0 | 313.15 | 0.4246 | 0.4244 | 0.4243 | 0.4241 |

$\mathit{\omega}$ | T_{0}/K | A/f (s^{−1}) | |||
---|---|---|---|---|---|

R = 10 K/h | R = 20 K/h | R = 30 K/h | R = 60 K/h | ||

0.6 | 293.15 | 6.4345 | 6.4478 | 6.4601 | 6.4830 |

0.6 | 303.15 | 5.7808 | 5.7899 | 5.7981 | 5.8078 |

0.6 | 313.15 | 3.1034 | 3.1066 | 3.1104 | 3.1161 |

0.8 | 293.15 | 45.8461 | 45.8936 | 45.9373 | 45.9970 |

0.8 | 303.15 | 26.4174 | 26.4570 | 26.4989 | 26.5497 |

0.8 | 313.15 | 3.5892 | 3.5947 | 3.6008 | 3.6069 |

1.0 | 293.15 | 16.9758 | 16.9949 | 17.0169 | 17.0330 |

1.0 | 303.15 | 2.9710 | 2.9740 | 2.9779 | 2.9809 |

1.0 | 313.15 | 2.0819 | 2.0841 | 2.0866 | 2.0890 |

Ethanol Mass Fraction | ${\mathit{T}}_{0}/\mathbf{K}$ | |||||
---|---|---|---|---|---|---|

293.15 | 303.15 | 313.15 | 293.15 | 303.15 | 313.15 | |

N | M | |||||

0.6 | −13.22 | −32.68 | −234.00 | −85.41 | −213.90 | −1671.07 |

0.8 | −4.90 | −20.13 | −55.03 | −21.57 | −99.58 | −380.02 |

1.0 | −10.66 | −215.71 | −780.63 | −57.51 | −1535.38 | −5809.89 |

$\gamma $ (mJ/m^{2}) | A/f | |||||

0.6 | 1.8498 | 1.3681 | 0.7098 | 6.5891 | 6.0545 | 3.3360 |

0.8 | 2.3254 | 1.4519 | 1.0384 | 44.3009 | 25.6925 | 3.6232 |

1.0 | 1.7777 | 0.6524 | 0.4249 | 16.1821 | 2.8894 | 2.0881 |

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## Share and Cite

**MDPI and ACS Style**

Wang, G.; Shang, Z.; Liu, M.; Dong, W.; Li, H.; Yin, H.; Gong, J.; Wu, S. Insight into the Nucleation Mechanism of *p*-Methoxybenzoic Acid in Ethanol-Water System from Metastable Zone Width. *Molecules* **2022**, *27*, 4085.
https://doi.org/10.3390/molecules27134085

**AMA Style**

Wang G, Shang Z, Liu M, Dong W, Li H, Yin H, Gong J, Wu S. Insight into the Nucleation Mechanism of *p*-Methoxybenzoic Acid in Ethanol-Water System from Metastable Zone Width. *Molecules*. 2022; 27(13):4085.
https://doi.org/10.3390/molecules27134085

**Chicago/Turabian Style**

Wang, Guangle, Zeren Shang, Mingdi Liu, Weibing Dong, Haichao Li, Haiqing Yin, Junbo Gong, and Songgu Wu. 2022. "Insight into the Nucleation Mechanism of *p*-Methoxybenzoic Acid in Ethanol-Water System from Metastable Zone Width" *Molecules* 27, no. 13: 4085.
https://doi.org/10.3390/molecules27134085