# A Thermodynamic Approach for the Prediction of Oiling Out Boundaries from Solubility Data

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Theory and Model Development

#### 2.1. Fluid Phase Thermodynamics

^{ideal}) is the ideal gas Helmholtz free energy, the second term (A

^{hs}) is the excess free energy of the hard-sphere (reference) fluid. Thus, the sum of the first two terms on the RHS gives the free energy of the reference fluid. The remaining two terms, A₁ and A₂, express the first order and second order perturbation corrections to the reference fluid to yield the Helmholtz free energy of the fluid of interest.

#### 2.1.1. Free Energy of the Reference Fluid

^{ideal}, is given by:

^{hs}), the Carnahan–Starling expression is used [40,41]:

#### 2.1.2. First Perturbation Term

^{hs}(x;η), to approximate the structure of the square-well fluid as [18]:

#### 2.1.3. Second Perturbation Term

#### 2.1.4. Chemical Potential and Pressure

_{l}, and pressure, p

_{l}, of the liquid phases. These quantities can be obtained from the Helmholtz free energy using the standard thermodynamic relations:

#### 2.2. Solid Phase Thermodynamics

_{s}, we use the simple expression due to Asherie et al. [33] and write:

_{s}is the number of molecules whose centers lie within the attractive well of a given molecule. This number is determined by the structure of the crystalline lattice. As a simplification, we follow the approach of Asherie et al., and assume the solid to be of face-centered cubic (fcc) structure and hence consider n

_{s}= 12, i.e., the attractions in the solid are dominated by the molecules in the first coordination shell. This model for the chemical potential of a square-well solid has been found to represent the solid–liquid equilibria of several small molecules and colloidal solutions reasonably well [11,33,44,45].

#### 2.3. Gelation Boundary

_{g}, at which gelation occurs at a given temperature, T. This model relates η

_{g}to the intermolecular interactions through:

_{g}at various temperatures.

#### 2.4. Model Implementation

^{hs}, of the reference hard-sphere fluid and solving Equation (5) numerically. These simplifications make the following approach attractive and readily implementable with minimal computational time.

#### 2.4.1. Obtaining Molecular Interaction Parameters

_{l}= μ

_{s}. Note that the other condition of phase equilibrium, the equality of pressures, is neglected because the solid phase is considered to be incompressible.

_{s}through Equation (17) for each T in the experimental data set. The equation of μ

_{l}= μ

_{s}needs to be solved iteratively by providing an appropriate starting value for η

_{m}, the volume fraction, η, that represents the predicted solubility (solution of the equation) at the chosen T. This initial value of η

_{m}enables one to compute the Helmholtz free energy of the reference fluid by evaluating Equations (3) and (4) for A

^{ideal}and A

^{hs}, respectively. The De Broglie wavelength, Λ, in Equation (3) is neglected, as this term effectively cancels out as an ideal contribution on both sides of the equality of μ

_{l}= μ

_{s}at constant T. One then has a choice of evaluating η

_{eff}using either Equation (11) or (12). From η

_{eff}, A₁ is obtained using Equation (10), and A₂ is obtained by numerically differentiating A₁ and using that derivative in Equation (13). Once the total Helmholtz free energy, A, is obtained for the guessed η

_{m}, this expression is numerically differentiated to evaluate μ

_{l}(Equation (16)). Thus, the relation of μ

_{l}= μ

_{s}is solved to yield the value of η

_{m}for each temperature in the experimental data set. The best-fit model parameters, α₀, α₁, and λ, are those that minimize the objective function:

_{m}and experimental η for the n

_{exp}data points considered.

#### 2.4.2. Practical Considerations

_{l}= μ

_{s}may have up to three solutions (i.e, one, two, or three solutions) for η

_{m}due to the cubic nature of the μ

_{l}vs. η curve. The volume fraction, η, is obviously constrained between 0 and 1, but volume fractions greater than 0.5 are unphysical (for the liquid phase). Hence, it is desirable to constrain the numerical solution of μ

_{l}= μ

_{s}to 0 ≤ η

_{m}≤ 0.5. Finding the η

_{m}that corresponds to the minimum of all the possible solutions in this range (which is a guaranteed solution) ensures that the solution obtained is physically meaningful and the iterative procedure for parameter estimation converges. Also, one needs to be mindful of the range of λ over which the chosen model for η

_{eff}(Equation (11) or (12)) is applicable and constrain the search space for λ appropriately during the regression. The strength of interaction, ε, between the molecules is expected to be in a range of 1 to 5 kT. Hence, one may have to limit the search space for α₀ and α₁ accordingly during parameter estimation, so that the values of the resulting ε/k (= α₀ + α₁ T) fall in the range of 1 to 5 T, where T is the absolute temperature. The bounds on the possible parameter space and the resulting constrained error minimization, while help keep the result of the calculation physically relevant, make it hard to estimate the uncertainties in the parameters.

#### 2.4.3. Calculation of Other Phase Boundaries

_{c}, and critical volume fraction, η

_{c}, at which the binodal and spinodal meet. At this upper consolute temperature (UCST) and composition, the derivatives of the chemical potential with respect to the volume fraction must vanish. From this condition, we obtain T

_{c}and η

_{c}by solving ${\left(\partial \text{}{\mu}_{\mathrm{L}}\u2215\partial \text{\hspace{0.05em}}\eta \right)}_{T}=0\text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}}{\left({\partial}^{2}{\mu}_{\mathrm{L}}\u2215\partial \text{\hspace{0.17em}}{\eta}^{2}\right)}_{T}=0$ simultaneously using the values of α₀, α₁, and λ regressed from the solubility data. Once T

_{c}and η

_{c}are obtained, we reduce the temperature slightly and calculate ${\eta}^{\mathrm{I}}\text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}\hspace{0.05em}}{\eta}^{\mathrm{II}}$ at that temperature by using ${\eta}_{\mathrm{c}}+\delta \eta \text{\hspace{0.17em}\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}\hspace{0.17em}}{\eta}_{\mathrm{c}}-\delta \eta $ as initial guesses for ${\eta}^{\mathrm{I}}\text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}\hspace{0.05em}}{\eta}^{\mathrm{II}}$, respectively, toward solving ${p}_{\mathrm{L}}^{\mathrm{I}}={p}_{\mathrm{L}}^{\mathrm{II}}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}\hspace{0.17em}}{\mu}_{\mathrm{L}}^{\mathrm{I}}={\mu}_{\mathrm{L}}^{\mathrm{II}}$. Here, δη is a small value, typically of the order of 0.001. The freshly obtained ${\eta}^{\mathrm{I}}\text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}\hspace{0.05em}}{\eta}^{\mathrm{II}}$ are then passed on as initial guesses for the next temperature, which will be slightly less than the previous one (about 0.5 °C). This procedure results in quick convergence of the iterative numerical computation and will not result in trivial solutions. The spinodal curve can also be calculated using this procedure. In this case, we need to solve only ${\left(\partial \text{}{\mu}_{\mathrm{L}}\u2215\partial \text{\hspace{0.05em}}\eta \right)}_{T}=0$ for the two solutions of η at a given temperature.

## 3. Experimental Section — Methods

#### 3.1. Pyraclostrobin – Isopropanol/Cyclohexane

#### 3.2. Compound **Z** – Methanol/Water

**Z**) was crystallized in 2% (w/w) methanol – 98% (w/w) water. The solubility data were determined in the temperature range of 5 to 65 °C using the Crystal 16 apparatus at a 0.5 °C/min cooling rate. The oiling-out data were also determined using this apparatus separately at a 1.0 °C/min cooling rate. These relatively fast cooling rates in this case were necessary due to the possible decomposition of the compound in extended contact with water at elevated temperatures. An additional data point on the LLPS curve was determined separately using a cold stage microscopy technique, which agreed with the LLPS data trend obtained with Crystal 16.

#### 3.3. Idebenone – Hexane/Methylene Chloride

#### 3.4. C₃₅H₄₁Cl₂N₃O₂ – Ethanol/Water

#### 3.5. Vanillin – 1-Propanol/Water

#### 3.6. Data Extraction and Processing

_{s}is the density of the crystal, MW is the molecular weight of the solute, and N

_{av}is the Avogadro number. We implemented all the phase diagram calculations in Wolfram Mathematica (v11.3, Wolfram Research, Champaign, IL, U.S.A.). While the curve fits were obtained by considering the errors in solubility in terms of volume fractions, the modeling results are shown below with concentrations expressed in wt% for convenience.

## 4. Results

_{eff}are close to each other and either of those curves can be used as a guideline in crystallization process development.

**Z**– methanol/water system. For this case also, we note a agreement between the LLPS line (the binodal calculated using the square-well parameters regressed from the solubility data) and the experimental data. Table 2 gives the best-fit parameters for this case. From Table 2, we note that for this system, even though the estimated parameters are nearly equal in both the cases of using Equations (11) and (12) for η

_{eff}, the critical temperatures predicted by the models are significantly different. The critical concentration, however, is predicted to be about the same.

## 5. Discussion

#### 5.1. Relevance of the Spinodal and Gelation Lines to Process Development

#### 5.2. Limitations of the Current Modeling Approach

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Experimental data and model predictions for the pyraclostrobin – isopropanol/cyclohexane system. The open circles and diamonds indicate the experimental data points for solubility and oiling out (binodal), respectively, adapted from [4]. Curve 1 represents the model (μ

_{l}= μ

_{s}) fit to solubility data using Equations (16) and (17). The square-well interaction parameters (listed in Table 1) obtained from this fit were used to further calculate the binodal (curve 2), spinodal (curve 3), and the gelation (curve 4) boundaries. The dotted lines are the curves obtained using Equation (11) and the solid lines are those obtained with Equation (12), respectively, for calculating η

_{eff}. The dotted and solid lines for curve 4 are almost identical.

**Figure 2.**Experimental data and model predictions for the compound

**Z**– methanol/water system showing (

**a**) the complete phase diagram, and (

**b**) magnified view of the experimental data region. The open circles and diamonds indicate the experimental data points for solubility and oiling out (binodal), respectively, adapted from [11]. Curve 1 represents the model (μ

_{l}= μ

_{s}) fit to solubility data using Equations (16) and (17). The square-well interaction parameters (listed in Table 2) obtained from this fit were used to further calculate the binodal (curve 2), spinodal (curve 3), and the gelation (curve 4) boundaries. The dotted lines are the curves obtained using Equation (11) and the solid lines are those obtained with Equation (12), respectively, for calculating η

_{eff}. The dotted and solid lines for curve 4 are almost identical.

**Figure 3.**Experimental data and model predictions for the idebenone – hexane/methylene chloride (7:1 by volume) system showing (

**a**) the complete phase diagram, and (

**b**) magnified view of the experimental data region. The open circles and diamonds indicate the experimental data points for solubility and oiling out (binodal), respectively, adapted from [5]. Curve 1 represents the model (μ

_{l}= μ

_{s}) fit to solubility data using Equations (16) and (17). The square-well interaction parameters (listed in Table 3) obtained from this fit were used to further calculate the binodal (curve 2), spinodal (curve 3), and gelation (curve 4) boundaries. The dotted lines are the curves obtained using Equation (11) and the solid lines are those obtained with Equation (12), respectively, for calculating η

_{eff}. The dotted and solid lines for curve 4 are almost identical.

**Figure 4.**Experimental data and model predictions for (

**a**) C₃₅H₄₁Cl₂N₃O₂ (form I) – ethanol/water system [48] and (

**b**) for vanillin in 1-propanol/water [52]. In both (

**a**) and (

**b**), the open circles and diamonds indicate the experimental data points for solubility and oiling out (binodal), respectively. In (

**a**), the triangles are the spinodal points reported in [48] obtained through extrapolation of the light scattering intensity data [50]. Curve 1 represents the model (μ

_{l}= μ

_{s}) fit to solubility data using Equations (16) and (17). The square-well interaction parameters (listed in Table 4) obtained from these fits were used to further calculate the binodal (curve 2) and spinodal (curve 3) boundaries. The dotted lines are the curves obtained using Equation (11) and the solid lines are those obtained with Equation (12), respectively, for calculating η

_{eff}.

**Table 1.**Best fit parameters for the pyraclostrobin – isopropanol/cyclohexanone system estimated through fitting the experimental data of Li et al. [4]. Parameters obtained using both the models for the first perturbative term of the Helmholtz free energy (i.e., η

_{eff}through Equations (11) and (12)), are listed. Also given are the estimated critical temperature (T

_{c}) and critical concentration (W

_{c}). The calculations were performed with σ = 0.896 nm.

α₀ | α₁ | λ | T_{c}, °C | W_{c}, wt% |

Equation (11) | ||||

1399.4 | −3.3992 | 1.2866 | 36.63 | 37.45 |

Equation (12) | ||||

1399.4 | −3.3985 | 1.2861 | 37.72 | 37.63 |

**Table 2.**Best fit parameters for the compound

**Z**– methanol/water system estimated through fitting the experimental data of Bhamidi et al. [11]. Parameters obtained using both the models for the first perturbative term of the Helmholtz free energy (i.e., η

_{eff}through Equations (11) and (12)) are listed. Also given are the estimated critical temperature (T

_{c}) and critical concentration (W

_{c}). The calculations were performed with σ = 0.664 nm.

α₀ | α₁ | λ | T_{c}, °C | W_{c}, wt% |

Equation (11) | ||||

713.83 | −0.7412 | 1.3006 | 117.57 | 32.24 |

Equation (12) | ||||

713.86 | −0.7413 | 1.3005 | 121.74 | 32.66 |

**Table 3.**Best fit parameters for the idebenone – hexane/ methylene chloride (7:1 by volume) system estimated through fitting the experimental data of Lu et al. [5]. Parameters obtained using both the models for the first perturbative term of the Helmholtz free energy (i.e., η

_{eff}through Equations (11) and (12)), are listed. Also given are the estimated critical temperature (T

_{c}) and critical concentration (W

_{c}). The calculations were performed with σ = 0.872 nm.

α₀ | α₁ | λ | T_{c}, °C | W_{c}, wt% |

Equation (11) | ||||

512.93 | −0.4744 | 1.3704 | 88.16 | 34.59 |

Equation (12) | ||||

522.23 | −0.4830 | 1.3521 | 89.81 | 36.37 |

**Table 4.**Best fit parameters for the C₃₅H₄₁Cl₂N₃O₂ – ethanol/water system estimated through fitting the experimental data of Veesler et al. [48], and for the vanillin – 1-propanol/water system estimated through fitting the experimental data of Zhao et al. [52]. Parameters obtained using both the models for the first perturbative term of the Helmholtz free energy (i.e., η

_{eff}through Equations (11) and (12)), are listed. Also given are the estimated critical temperature (T

_{c}) and critical concentration (W

_{c}). The calculations were performed with σ = 1.044 nm for C₃₅H₄₁Cl₂N₃O₂ and 0.697 nm for vanillin.

α₀ | α₁ | λ | T_{c}, °C | W_{c}, wt% |

C₃₅H₄₁Cl₂N₃O₂ – ethanol/water | ||||

Equation (11) | ||||

995.26 | −1.9216 | 1.3002 | 57.68 | 33.44 |

Equation (12) | ||||

1001.3 | −1.9373 | 1.2976 | 59.40 | 33.97 |

Vanillin – 1-propanol/water | ||||

Equation (11) | ||||

1056.0 | −2.3813 | 1.2284 | 16.15 | 32.49 |

Equation (12) | ||||

1073.0 | −2.4226 | 1.2188 | 16.36 | 31.62 |

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**MDPI and ACS Style**

Bhamidi, V.; Abolins, B.P.
A Thermodynamic Approach for the Prediction of Oiling Out Boundaries from Solubility Data. *Processes* **2019**, *7*, 577.
https://doi.org/10.3390/pr7090577

**AMA Style**

Bhamidi V, Abolins BP.
A Thermodynamic Approach for the Prediction of Oiling Out Boundaries from Solubility Data. *Processes*. 2019; 7(9):577.
https://doi.org/10.3390/pr7090577

**Chicago/Turabian Style**

Bhamidi, Venkateswarlu, and Brendan P. Abolins.
2019. "A Thermodynamic Approach for the Prediction of Oiling Out Boundaries from Solubility Data" *Processes* 7, no. 9: 577.
https://doi.org/10.3390/pr7090577