A Linear Regression Model for Determining the Pre-Exponential Factor and Interfacial Energy Based on the Metastable Zone Width Data

Abstract

A linear regression model is presented in this study to determine the pre-exponential factor and interfacial energy of the crystallized substance based on classical nucleation theory using the metastable zone width data. The nucleation event is assumed corresponding to a point at which the total number density of the nuclei has reached a fixed (but unknown) value. One equation is derived for any temperature-dependent functional form of the solubility. Another equation is derived for the van’t Hoff solubility expression. The pre-exponential factor and interfacial energy obtained from these two equations are found consistent for the studied systems, including glutamic acid, glycine, and 3-nito-1,2,4-triazol-5-one. The results obtained from these two equations are also compared with those obtained from the integral method and classical 3D nucleation theory approach.

1. Introduction

The nucleation behavior in a supersaturated solution is closely related to the induction time or metastable zone width (MSZW) measurements [1,2,3]. As opposed to a lag time during the temperature-decreasing process for the prepared supersaturated solution at a higher temperature lowered to the desired constant temperature in the induction time measurements, there is no lag time for the supersaturated solution cooled at a constant cooling rate in the MSZW measurements. Thus, the MSZW data should be more reliable than the induction time data in determination of the nucleation rate for a system.

In classical nucleation theory (CNT) [1,2,3], the nucleation rate of a crystallization system depends on both the pre-exponential factor and interfacial energy, which are usually determined using induction time data by assuming $J\propto t_i{}^{-1}$, where $J$ is the nucleation rate and $t_i$ is the induction time [4,5,6,7,8,9,10]. Due to the complicated data interpretation method, determination of the pre-exponential factor and interfacial energy using MSZW data has long been a challenging task.

Based on the Nyvlt’s approach [11,12], Sangwal [13,14,15,16] proposed a self-consistent Nyvlt-like equation using the power-law nucleation rate, leading to a linear relationship between $\ln \left(\frac{\Delta T_m}{T_0}\right)$ and $lnb$, where $T_0$ is the initial saturated temperature, $\Delta T_m$ is the MSZW, and $b$ is the cooling rate. Similarly, using the nucleation rate based on CNT, Sangwal [13,14,15,16] developed a linear relationship of ${\left(\frac{T_0}{\Delta T_m}\right)}^2$ versus $lnb$. Kashchiev et al. [17] presented a general expression for the total volume and number of crystallites as functions of the cooling rate for progressive nucleation based on CNT. They verified that the linear dependence of $\ln \left(\frac{\Delta T_m}{T_0}\right)$ on $lnb$ is the linear approximation to their presented equation. For instantaneous nucleation, a more simplified model is derived by Kashchiev et al. [18] which also yields a linear relationship between $\ln \left(\frac{\Delta T_m}{T_0}\right)$ and $lnb$. Kubota [19] proposed a model based on progressive nucleation to account for the MSZW limit corresponding to a point at which the number density of accumulated crystals has reached a fixed (but unknown) value during the cooling process. The simple power-law form of nucleation rate is adopted in the Kubota’s model, leading to a linear relationship of $\ln \Delta T_m$ versus $lnb$ for the MSZW data.

The above-mentioned models have been widely applied in the literature to correlate MSZW with cooling rate in various crystallization systems [20,21,22,23,24,25,26,27]. Although the pre-exponential factor and interfacial energy can provide important information in understanding the nucleation behavior, the pre-exponential factor and interfacial energy of the crystallized substance are usually not determined in these studies. Recently, some researchers [28,29,30] investigated the relationship between MSZW and process parameters in terms of the pre-exponential factor and interfacial energy using classical 3D nucleation theory approach based on the Sangwal’s theory [13,14,15,16]. Shiau and Lu [31,32,33] developed an integral model to determine the pre-exponential factor and interfacial energy based on CNT using the MSZW data. As the nonlinear regression along with numerical integration involved in the integral model is complicated, the objective of this work is to present a linear regression model to determine the pre-exponential factor and interfacial energy of the crystallized substance based on CNT using the MSZW data.

2. Theory

2.1. Integral Method

The nucleation rate based on CNT is expressed as [1,2,3]

$$J=A_{J}exp\left[-\frac{16\pi v^2{\gamma }^3}{3k_B{}^3{T}^{3}l{n}^{2}S}\right]$$

(1)

where $A_J$ is the nucleation pre-exponential factor, $\gamma$ is the interfacial energy, $k_B$ is the Boltzmann constant, and $v=\frac{M_w}{{\rho }_c{N}_A}$ is the molecular volume.

To interpret the onset of nucleation based on the progressive nucleation observed in the induction time and MSZW measurements, the nucleation event is assumed corresponding to a point at which the total number density of the nuclei has reached a fixed (but unknown) value, $f_N$ [19,30,31,32]. Thus, one can derive $f_N=J t_i$ at the induction time, which is consistent with $J \propto t_i{}^{-1}$ reported in the literature [1]. As nuclei are continuously born during the cooling process, Shiau and Lu [31,32,33] derived at the MSZW limit:

$$f_N=\underset{0}{\overset{t_m}{{\displaystyle \int }}}J dt$$

(2)

where $f_N$ depends on the measurement device and on the substance.

If the solution is cooled at a constant rate defined as:

$$b=-dT/dt(b>0)$$

(3)

one obtains $T\left(t\right)=T_0-bt$ during the cooling process. Substituting Equations (1) and (3) into Equation (2) yields [31,32,33]:

$$\frac{f_N}{A_J}=-\underset{T_0}{\overset{T_m}{{\displaystyle \int }}}exp\left[-\frac{16\pi v_m{}^2{\gamma }^3}{3k_B{}^3{T}^{3}l{n}^{2}S}\right] \frac{dT}{b}$$

(4)

where, as shown in Figure 1, $T_0$ is the initial saturated temperature at $t=0$, $T_m$ is the maximum supercooling temperature at $t_m$, $\Delta T_m=T_0-T_m$ is the MSZW, $C_0$ is the saturated concentration at $T_0$, $S\left(T\right)=\frac{C_0}{C_{eq}\left(T\right)}$ is the temperature-dependent supersaturation during the cooling process, and $C_{eq}\left(T\right)$ is the temperature-dependent solubility. Note that $C_0$ remains unchanged before the onset of nucleation. As $C_{eq}\left(T\right)$ usually decreases with decreasing temperature, $S\left(T\right)$ increases with decreasing temperature; and subsequently $J$ defined in Equation (1) increases during the cooling process.

Shiau and Lu [31,32,33] proposed the following nonlinear regression procedure to determine two parameters, $\gamma$ and $\frac{A_J}{f_N}$, in Equation (4) from the experimental data of $T_m$ versus $b$: (a) Guess $\gamma$; (b) determine ${(\frac{A_J}{f_N})}_j$ for each pair of $T_m$ versus $R$ data by numerical integration; (c) calculate ${(\frac{A_J}{f_N})}_{av}$; (d) calculate the coefficient of variation $\left(CV\right)$ among all ${(\frac{A_J}{f_N})}_j$. For the guessed $\gamma$, ${(\frac{A_J}{f_N})}_{av}$ is defined as:

$${(\frac{A_J}{f_N})}_{av}=\frac{1}{H}{\displaystyle \sum }_{j=1}^H{(\frac{A_J}{f_N})}_j$$

(5)

where $H$ is the number of data points $\left(H\ge 2\right)$. For the guessed $\gamma$, $CV$ among all ${(\frac{A_J}{f_N})}_j$ is defined as:

$$CV=\frac{\sqrt{\frac{1}{H-1}{\sum }_{j=1}^H{\left[{(\frac{A_J}{f_N})}_j-{(\frac{A_J}{f_N})}_{av}\right]}^2}}{{\left(\frac{A_J}{f_N}\right)}_{av}}$$

(6)

The same procedure from (a) to (d) is repeated by guessing various values of $\gamma$ until the optimal $\gamma$ with the minimum $CV$ is found. Then, the corresponding ${(\frac{A_J}{f_N})}_{av}$ calculated from the optimal $\gamma$ is taken as the optimal $\frac{A_J}{f_N}$. If $f_N$ is known, $A_J$ can be determined.

2.2. Linearized Integral Method

As the nonlinear regression along with numerical integration involved in Equation (4) is complicated, a simplified linear regression model is proposed in this study to extend the applicability of integral model as follows. Based on the trapezoidal rule for the numerical integration, Equation (2) is approximated as:

$$\frac{\Delta T_m}{2b}\left(J_0+J_m\right)=f_N$$

(7)

Here, $\frac{\Delta T_m}{b}$ represents the total time required during the cooling process from $T_0$ to $T_m$ at a constant cooling rate $b$. $J_0$ and $J_m$ represent the nucleation rate at $t=0$ and $t=t_m$, respectively. Note that $J_0=0$ due to $S(T_0)=\frac{C_0}{C_{eq}\left(T_0\right)}=1$.

As shown in Figure 1, $S$ increases gradually from 1 at $C_{eq}\left(T_0\right)=C_0$ during the cooling process. As defined in Equation (1), $J$ starts from $J_0=0$ at $t=0$ and increases gradually as temperature decreases from $T_0$ to $T_m$. The nucleation rate $J_m$ at $t_m$ can be expressed as:

$$J_m=A_{J}exp\left[-\frac{16\pi v^2{\gamma }^3}{3k_B{}^3{T}_m{}^{3}l{n}^2{S}_m}\right]$$

(8)

with

$$S_m=\frac{C_0}{C_m}$$

(9)

where $C_m=C_{eq}\left(T_m\right)$ is the saturated concentration at $T_m$ and $S_m=S_m\left(T_m\right)$ is the supersaturation at $T_m$.

Substituting Equation (8) into Equation (7) yields:

$$exp\left[-\frac{16\pi v^2{\gamma }^3}{3k_B{}^3{T}_m{}^{3}l{n}^2{S}_m}\right]=\frac{2b{f}_N}{\Delta T_m{A}_J}$$

(10)

Taking logarithm on both sides of Equation (10) gives:

$$-\frac{16\pi v^2{\gamma }^3}{3k_B{}^3{T}_m{}^{3}l{n}^2{S}_m}=ln\left[\frac{2b{f}_N}{\Delta T_m{A}_J}\right]$$

(11)

By rearranging Equation (11), the linearized integral method I is expressed as:

$$\frac{1}{T_m{}^{3}l{n}^2{S}_m}=\frac{3}{16\pi }\left(\frac{k_B{}^3}{v^2{\gamma }^3}\right)\left[\ln \left(\frac{\Delta T_m}{b}\right)+\ln \left(\frac{A_J}{2f_N}\right)\right]$$

(12)

A plot of $\frac{1}{T_m{}^{3}l{n}^2{S}_m}$ versus $\ln \left(\frac{\Delta T_m}{b}\right)$ at a given $T_0$ should give a straight line, the slope and intercept of which permit determination of $\gamma$ and $\frac{A_J}{f_N}$, respectively, without the knowledge of $f_N$.

If the temperature-dependent solubility is described in terms of the van’t Hoff equation [1], one obtains:

$$\ln S_m=\ln \left(\frac{C_0}{C_m}\right)=\frac{-\Delta H_d}{R_G}(\frac{1}{T_0}-\frac{1}{T_m})=(\frac{\Delta H_d}{R_G{T}_0})(\frac{\Delta T_m}{T_m})$$

(13)

where $\Delta H_d$ is the heat of dissolution and $R_G$ is the gas constant. By substituting Equation (13) into Equation (12), linearized integral method II is expressed as:

$$\frac{1}{T_m}{(\frac{T_0}{\Delta T_m})}^2=\frac{3}{16\pi }(\frac{k_B{}^3}{v^2{\gamma }^3}){(\frac{\Delta H_d}{R_G})}^2[\ln \left(\frac{\Delta T_m}{b}\right)+\ln \left(\frac{A_J}{2f_N}\right)]$$

(14)

Similarly, a plot of $\frac{1}{T_m}{(\frac{T_0}{\Delta T_m})}^2$ versus $\ln \left(\frac{\Delta T_m}{b}\right)$ at a given $T_0$ should give a straight line, the slope and intercept of which permit determination of $\gamma$ and $\frac{A_J}{f_N}$, respectively, without the knowledge of $f_N$.

It should be noted in the application of Equations (12) and (14) that $\gamma$ is not influenced by the chosen value of $f_N$ although $A_J$ needs to be determined based on $f_N$. Based on the study of 28 inorganic systems, Mersmann and Bartosch [34] concluded that the minimum detectable volume fraction of nuclei in solution corresponds to $f_V={10}^{-4}-{10}^{-3}$ with the minimum detectable size of $10-100 \mathsf{\mu }\mathrm{m}$. The intermediate value, $f_V=4\times {10}^{-4}$, was adopted at the detection of the nucleation point for the Lasentec focus beam reflectance measurements reported by Lindenberg and Mazzotti [35] and for the turbidity measurements reported by Shiau et al. [31,32,33,36]. If the uniform-sized spherical nuclei of $10 \mathsf{\mu }\mathrm{m}$ with $k_V=\frac{\pi }{6}$ are assumed for simplicity, $f_V=4\times {10}^{-4}$ corresponds to $f_N=7.64\times {10}^{11} {\mathrm{m}}^{-3}$ [32].

2.3. Classical 3D Nucleation Theory Approach

Sangwal [13,14,15,16] related $J_m$ with the rate of change of solution supersaturation as:

$$J_m=f\frac{\Delta C}{C_m\Delta t}$$

(15)

where $\Delta C=C_0-C_m$, $\Delta t=t_m$, and $f$ is a constant defined as the number of clusters per unit volume. The value of $f$ is governed by the aggregation and diffusion processes in the solution.

If $S_m$ is slightly greater than 1, one obtains:

$$\ln S_m\cong S_m-1$$

(16)

Combining Equations (13) and (16) yields:

$$S_m-1=(\frac{\Delta H_d}{R_G{T}_0})(\frac{\Delta T_m}{T_m})$$

(17)

Substituting Equation (9) into Equation (17) gives:

$$\frac{\Delta C_m}{\Delta T_m}=(\frac{\Delta H_d}{R_G{T}_0})(\frac{C_m}{T_m})$$

(18)

By combining Equations (8), (15) and (18), Sangwal [13,14,15,16] derived the classical 3D nucleation theory approach as:

$${\left(\frac{T_0}{\Delta T_m}\right)}^2=F\left(1-Zlnb\right)$$

(19)

where:

$$Z=-1/ln(\frac{f\Delta H_d}{A_J{R}_G{T}_0{T}_m})$$

(20)

$$F=\frac{1}{Z}[\frac{3}{16\pi }{\left(\frac{k_B{T}_0}{v^{2/3}\gamma }\right)}^3{\left(\frac{\Delta H_d}{R_G{T}_0}\right)}^2]$$

(21)

A plot of ${\left(\frac{T_0}{\Delta T_m}\right)}^2$ versus $lnb$ at a given $T_0$ should give a straight line, the slope and intercept of which permit determination of $Z$ and $F$, respectively. Subsequently, the values of $\gamma$ and $\frac{A_J}{f}$ can be calculated without the knowledge of $f$.

It should be noted in the application of Equation (19) that $\gamma$ is not influenced by the chosen value of $f$ although $A_J$ needs to be determined based on $f$. Sangwal [14] proposed that the upper limit of $f$ may be estimated from solute concentration in the saturated solution. For example, $C_0=230 \mathrm{kg} /{\mathrm{m}}^3$, the saturated concentration for aqueous glycine solutions at 308.15 K ($M_w=0.075 \mathrm{kg}/\mathrm{mole}$), corresponds to $C_0=1.84\times {10}^{27} {\mathrm{m}}^{-3}$ based on the number of solute molecules per unit solution volume, which is close to $f={10}^{27} {\mathrm{m}}^{-3}$ proposed by Sangwal [14].

3. Results and Discussion

The experimental MSZW data for three crystallization systems, including glutamic acid, glycine and, 3-nito-1,2,4-triazol-5-one (NTO), reported in the literature are analyzed as follows. Figure 2a shows that the MSZW data of aqueous glutamic acid solutions fitted to linearized integral method I Equation (12) at various $T_0$, where the original experimental MSZW data listed in

Table 1. The experimental MSZW data of aqueous glutamic acid solutions taken from Shiau and Lu [32], where $T_0$ is the initial saturated temperature at the corresponding initial saturated concentration $C_0$.

${\mathit{T}}_0\left(\mathbf{K}\right)$	${\mathit{C}}_0\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$	$\mathbf{\Delta }{\mathit{T}}_{\mathit{m}}\left(\mathbf{K}\right)$
		$\mathit{b}=0.00083 \mathbf{K}/\mathbf{s}$	$\mathit{b}=0.0017 \mathbf{K}/\mathbf{s}$	$\mathit{b}=0.0025 \mathbf{K}/\mathbf{s}$	$\mathit{b}=0.0042 \mathbf{K}/\mathbf{s}$	$\mathit{b}=0.005 \mathbf{K}/\mathbf{s}$
308.15	12.2	12	17	19.2	20.9	24
313.15	15.1	11	14	17	19	20.5
323.15	22.3	7.5	9.5	12	12.5	15
328.15	26.6	6.0	7.5	8.5	10	11

are taken from Shiau and Lu [32] in a 200 mL vessel. The solubility of glutamic acid in water is taken as $C_{eq}\left(T\right)=9.58654\times {10}^{-3}{T}^2-5.3778T+759.067$; $C_{eq} in \mathrm{kg} /{\mathrm{m}}^3 , T in \mathrm{K}$ [37]. Note that $v=1.674\times {10}^{-28} {\mathrm{m}}^3$ for glutamic acid. Figure 2b shows the same MSZW data of aqueous glutamic acid solutions fitted to linearized integral method II Equation (14) at various $T_0$, where $\Delta H_d=31.8 \mathrm{kJ}/\mathrm{mol}$ is used for the van’t Hoff solubility equation. For comparison, Figure 2c,d show the same MSZW data of aqueous glutamic acid solutions fitted to the classical 3D approach Equation (19) and integral method Equation (4), respectively, at various $T_0$.

Figure 3a shows that the MSZW data of aqueous glycine solutions fitted to linearized integral method I Equation (12) at various $T_0$, where the original experimental MSZW data listed in

Table 2. The experimental MSZW data of aqueous glycine solutions taken from Shiau [38], where $T_0$ is the initial saturated temperature at the corresponding initial saturated concentration $C_0$.

${\mathit{T}}_0\left(\mathbf{K}\right)$	${\mathit{C}}_0\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$	$\mathbf{\Delta }{\mathit{T}}_{\mathit{m}}\left(\mathbf{K}\right)$
		$\mathit{b}=0.000417 \mathbf{K}/\mathbf{s}$	$\mathit{b}=0.00833 \mathbf{K}/\mathbf{s}$	$\mathit{b}=0.01111 \mathbf{K}/\mathbf{s}$	$\mathit{b}=0.01389 \mathbf{K}/\mathbf{s}$
308.15	230	8.2	10.2	11	11.7
318.15	261	6.9	8.5	9.1	9.9
323.15	276	6.0	7.5	8.0	8.6
328.15	292	4.9	6.0	6.5	7.1

are taken from Shiau [38] in a 200 mL vessel. The solubility of glycine in water is taken as $C_{eq}\left(T\right)=5.4397\times {10}^{-3}{T}^2-3.2022\times {10}^{-1}T-188.2$; $C_{eq} in \mathrm{kg} /{\mathrm{m}}^3 , T in \mathrm{K}$ [39]. Note that $v=7.76\times {10}^{-29}{\mathrm{m}}^3$ for glycine. Figure 3b shows the same MSZW data of aqueous glycine solutions fitted to linearized integral method II Equation (14) at various $T_0$, where $\Delta H_d=10.2 \mathrm{kJ}/\mathrm{mol}$ is used for the van’t Hoff solubility equation. For comparison, Figure 3c,d show the same MSZW data of aqueous glycine solutions fitted to classical 3D approach Equation (19) and integral method Equation (4), respectively, at various $T_0$.

Figure 4a shows that the MSZW data of aqueous NTO solutions fitted to linearized integral method I Equation (12) at various $T_0$, where the original experimental MSZW data listed in

Table 3. The experimental MSZW data of aqueous NTO solutions taken from Kim et al. [40], where $T_0$ is the initial saturated temperature at the corresponding initial saturated concentration $C_0$.

${\mathit{T}}_0\left(\mathbf{K}\right)$	${\mathit{C}}_0\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$	$\mathbf{\Delta }{\mathit{T}}_{\mathit{m}}\left(\mathbf{K}\right)$
		$\mathit{b}=0.000167 \mathbf{K}/\mathbf{s}$	$\mathit{b}=0.000833 \mathbf{K}/\mathbf{s}$	$\mathit{b}=0.00167 \mathbf{K}/\mathbf{s}$	$\mathit{b}=0.00833 \mathbf{K}/\mathbf{s}$	$\mathit{b}=0.0167 \mathbf{K}/\mathbf{s}$
338.15	54.7	6.2	7.3	7.8	9.2	9.5
348.15	72.7	5.6	6.8	7.1	8.3	8.8
358.15	93.8	5.2	6.2	6.7	7.8	8.4
367.15	115.4	4.8	5.7	6.2	7.2	7.6

are taken from Kim et al. [40] in a 300 mL vessel. The solubility of NTO in water is taken as $C_{eq}\left(T\right)=0.0153T^2-8.7002T-1247.2$; $C_{eq} in \mathrm{kg} /{\mathrm{m}}^3 , T in \mathrm{K}$ [40]. Note that $v=8.86\times {10}^{-29}{\mathrm{m}}^3$ for NTO. Figure 4b shows the same MSZW data of aqueous NTO solutions fitted to linearized integral method II Equation (14) at various $T_0$, where $\Delta H_d=26.1 \mathrm{kJ}/\mathrm{mol}$ is used for the van’t Hoff solubility equation. For comparison, Figure 4c,d show the same MSZW data of aqueous NTO solutions fitted to the classical 3D approach Equation (19) and integral method Equation (4), respectively, at various $T_0$.

The fitted results for glutamic acid, glycine, and NTO are listed in

Table 4. The fitted results of $\gamma$ and $A_J$ at various $T_0$ for aqueous glutamic acid solution based on $f_N=7.64\times {10}^{11} {\mathrm{m}}^{-3}$ and $f={10}^{27} {\mathrm{m}}^{-3}$.

${\mathit{T}}_0\left(K\right)$	Linearized Integral Method I Equation (12)				Linearized Integral Method II Equation (14)
	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{{\mathit{f}}_{\mathit{N}}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	${\mathit{R}}^2(-)$	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{{\mathit{f}}_{\mathit{N}}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	${\mathit{R}}^2(-)$
308.15	4.4	8.84 × 10−4	6.76 × 108	0.797	4.0	6.30 × 10−4	4.81 × 108	0.829
313.15	4.0	8.05 × 10−4	6.15 × 108	0.924	3.8	7.51 × 10−4	5.73 × 108	0.930
323.15	2.9	1.01 × 10−3	7.75 × 108	0.886	2.9	1.04 × 10−3	7.96 × 108	0.887
328.15	2.5	1.47 × 10−3	1.12 × 109	0.982	2.5	1.52 × 10−3	1.16 × 109	0.983
${\mathit{T}}_0\left(K\right)$	Integral Method Equation (4)				Classical 3D Approach Equation (19)
	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{{\mathit{f}}_{\mathit{N}}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	$\mathit{C}\mathit{V}(-)$	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{\mathit{f}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	${\mathit{R}}^2(-)$
308.15	3.2	5.06 × 10−4	3.87 × 108	0.175	4.6	3.62 × 10−4	3.62 × 1023	0.893
313.15	3.1	6.27 × 10−4	4.79 × 108	0.106	4.3	3.80 × 10−4	3.80 × 1023	0.949
323.15	2.2	7.91 × 10−4	6.04 × 108	0.149	3.3	3.34 × 10−4	3.34 × 1023	0.939
328.15	2.0	1.35 × 10−3	1.03 × 109	0.047	2.9	2.87 × 10−4	2.87 × 1023	0.979

,

Table 5. The fitted results of $\gamma$ and $A_J$ at various $T_0$ for aqueous glycine solutions based on $f_N=7.64\times {10}^{11}{ \mathrm{m}}^{-3}$ and $f={10}^{27}{ \mathrm{m}}^{-3}$.

${\mathit{T}}_0\left(K\right)$	Linearized Integral Method I Equation (12)				Linearized Integral Method II Equation (14)
	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{{\mathit{f}}_{\mathit{N}}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	${\mathit{R}}^2(-)$	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{{\mathit{f}}_{\mathit{N}}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	${\mathit{R}}^2(-)$
308.15	2.5	5.21 × 10−3	3.98 × 109	0.979	2.4	5.16 × 10−3	3.95 × 109	0.980
318.15	2.2	6.46 × 10−3	4.93 × 109	0.990	2.1	6.26 × 10−3	4.78 × 109	0.989
323.15	1.9	6.97 × 10−3	5.32 × 109	0.976	1.9	7.00 × 10−3	5.34 × 109	0.976
328.15	1.6	8.38 × 10−3	6.40 × 109	0.996	1.7	8.43 × 10−3	6.44 × 109	0.996
${\mathit{T}}_0\left(K\right)$	Integral Method Equation (4)				Classical 3D Approach Equation (19)
	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{{\mathit{f}}_{\mathit{N}}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	$\mathit{C}\mathit{V}(-)$	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{\mathit{f}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	${\mathit{R}}^2(-)$
308.15	2.0	4.95 × 10−3	3.78 × 109	0.044	2.7	5.47 × 10−4	5.47 × 1023	0.983
318.15	1.8	6.25 × 10−3	4.78 × 109	0.032	2.4	5.23 × 10−4	5.23 × 1023	0.990
323.15	1.6	6.73 × 10−3	5.14 × 109	0.048	2.2	4.91 × 10−4	4.91 × 1023	0.981
328.15	1.4	8.22 × 10−3	6.28 × 109	0.018	1.9	4.71 × 10−4	4.71 × 1023	0.995

and

Table 6. The fitted results of $\gamma$ and $A_J$ at various $T_0$ for aqueous NTO solutions based on $f_N=7.64\times {10}^{11}{ \mathrm{m}}^{-3}$ and $f={10}^{27}{ \mathrm{m}}^{-3}$.

${\mathit{T}}_0\left(K\right)$	Linearized Integral Method I Equation (12)				Linearized Integral Method II Equation (14)
	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{{\mathit{f}}_{\mathit{N}}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	${\mathit{R}}^2(-)$	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{{\mathit{f}}_{\mathit{N}}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	${\mathit{R}}^2(-)$
338.15	5.8	5.34 × 10−2	4.08 × 1010	0.967	5.4	5.89 × 10−2	4.50 × 1010	0.967
348.15	5.2	5.18 × 10−2	3.96 × 1010	0.952	5.0	5.67 × 10−2	4.33 × 1010	0.952
358.15	4.6	4.22 × 10−2	3.22 × 1010	0.968	4.7	4.54 × 10−2	3.47 × 1010	0.968
367.15	4.3	5.18 × 10−2	3.96 × 1010	0.962	4.4	5.52 × 10−2	4.22 × 1010	0.962
${\mathit{T}}_0\left(K\right)$	Integral Method Equation (4)				Classical 3D Approach Equation (19)
	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{{\mathit{f}}_{\mathit{N}}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	$\mathit{C}\mathit{V}(-)$	$\mathit{\gamma }\left(mJ/m^2\right)$	$\frac{{\mathit{A}}_{\mathit{J}}}{\mathit{f}}$($s^{-1}$)	${\mathit{A}}_{\mathit{J}}\left(m^{-3}{s}^{-1}\right)$	${\mathit{R}}^2(-)$
338.15	5.5	1.17 × 10−1	8.94 × 1010	0.284	5.6	1.01 × 10−2	1.01 × 1025	0.969
348.15	4.8	9.43 × 10−2	7.2 × 1010	0.309	5.2	8.37 × 10−3	8.37 × 1024	0.955
358.15	4.3	8.04 × 10−2	6.14 × 1010	0.250	4.8	6.03 × 10−3	6.03 × 1024	0.969
367.15	4.0	1.04 × 10−1	7.95 × 1010	0.286	4.6	6.37 × 10−3	6.37 × 1024	0.964

. As integral method Equation (4) is numerically integrated without any approximations in this study, $\gamma$ and $\frac{A_J}{f_N}$ obtained from integral method Equation (4) represent the exact solution to Equation (2) based on the nucleation event assumed corresponding to a point at which the total number density of the nuclei has reached $f_N$. For all three studied systems, as opposed to $\gamma$ obtained from classical 3D approach Equation (19) compared with that obtained from integral method Equation (4), $\gamma$ obtained from linearized integral methods Equations (12) and (14) is closer to that obtained from integral method Equation (4) at each condition. Furthermore, $\frac{A_J}{f_N}$ obtained from linearized integral methods Equations (12) and (14) is also consistent with that obtained from integral method Equation (4) at each condition. As $f_N$ and $f$ are two different parameters, $A_J$ determined from Equations (4), (12) and (14) based on $f_N=7.64\times {10}^{11} {\mathrm{m}}^{-3}$ is not strictly comparable with that determined from Equation (19) based on $f={10}^{27} {\mathrm{m}}^{-3}$. It should be noted in Table 4, Table 5 and Table 6 that $\gamma$ and $\frac{A_J}{f_N}$ (or $\frac{A_J}{f}$) are determined first without the knowledge of $f_N$ (or $f$). Consequently, $\gamma$ is not influenced by the chosen value of $f_N$ (or $f$) although $A_J$ needs to be determined based on $f_N$ (or $f$). For example, if the chosen value of $f_N$ (or $f$) is increased by ten times, $A_J$ is also increased by ten times at each condition while $\gamma$ remains unchanged.

As compared in Table 4, Table 5 and Table 6, $\gamma$ and $A_J$ obtained from linearized integral method I Equation (12) are consistent with those obtained from linearized integral method II Equation (14) at each condition for all three studied systems. Thus, both equations can be applied to determine $\gamma$ and $A_J$ of the crystallized substance using the MSZW data. As opposed to the temperature-dependent solubility required for linearized integral method I Equation (12), only the value of $\Delta H_d$ is required in application of linearized integral method II Equation (14).

4. Conclusions

A linear regression method is proposed in this work to determine the pre-exponential factor and interfacial energy based on CNT using the MSZW data. Linearized integral method I Equation (12) is derived for any temperature-dependent functional form of the solubility while linearized integral method II Equation (14) is derived for the van’t Hoff temperature-dependent solubility. Only the value of $\Delta H_d$ is required in application of linearized integral method II Equation (14), as opposed to the temperature-dependent solubility required for linearized integral method I Equation (12). The experimental MSZW data for all three studied systems, including glutamic acid, glycine, and NTO, are fitted well to these two equations. The pre-exponential factor and interfacial energy obtained from linearized integral method I Equation (12) are consistent with those obtained from linearized integral method II Equation (14) for these systems.

As the integral method is numerically integrated without any approximations, the pre-exponential factor and interfacial energy obtained from the integral method represent the exact values based on the nucleation event assumed corresponding to a point at which the total number density of the nuclei has reached a fixed value. As opposed to the interfacial energy obtained from classical 3D nucleation theory approach compared with that from the integral method, the interfacial energy obtained from linearized integral methods Equations (12) and (14) is closer to that from the integral method at each condition. Furthermore, the pre-exponential factor obtained from linearized integral method Equations (12) and (14) is also consistent with that from the integral method at each condition.