Spinodal Curve for the System Water + 1-Pentanol + 1-Propanol at 25 °C

Abstract

The liquid–liquid equilibrium data at 25 °C of the system water–1-pentanol–1-propanol, that present a I type solubility gap, have been used to calculate the spinodal curve of this system through the use of the Wheeler–Widom model. As in the case of the water–chloroform–acetic acid system, analyzed with the same model in the past, a local fitting method has been necessary. A procedure based on the experimental position of the plait point was used to obtain the single spinodal points. The relative positions of binodal and spinodal curves indicate the presence of a large metastable area in the alcoholic-rich region.

Appendix Brief Presentation of the Generalized Wheeler–Widom Model

The Wheeler–Widom model considers three types of diatomic molecules AA , BB and AB, in which the species AB acts as an amphiphile that increases the mutual solubility of the AA and BB molecules. The molecules are assumed to cover the bonds of a lattice with only one molecule per bond.

The generalized Wheeler–Widom model, used in this article, considers a 2-dimensional honeycomb lattice (denoted as G) with N vertices; on each bond of this lattice is placed a rod-like molecule having end-points of the type AA , BB or AB type; a particular configuration of the molecules is illustrated in Fig. 6. There are interactions only between the 3 nearest neighbor molecular ends, so that at each vertex of the honeycomb lattice there are 3 neighboring molecular ends that have the interaction energies $\varepsilon_{AAA} , \varepsilon_{BBB} , \varepsilon_{ABA}$, and $\varepsilon_{BAB}$. This type of interaction defines the 3–12 lattice (associated to the honeycomb lattice) that is represented in Fig. 7, and it is denoted as $\Uplambda$; this lattice has 2 elements: the triangles denoted as C ₃ and the sides denoted as C ₂.

Fig. 6

The honeycomb lattice having on each side a rod-like molecule; the molecular ends of type A are represented by white discs and the molecular ends of type B are black discs. Each vertex of the lattice has 3 molecular ends, and these molecular ends define a triangle

Fig. 7

The 3–12 lattice (denoted as $\Uplambda$ ) where the triangles formed by the neighboring molecular ends are C ₃ graphs and the sides defined by molecules are C ₂ graphs

We consider that each side of the honeycomb lattice is occupied with one molecule (i.e. there are no vacancies) and a possible configuration of molecules is denoted as $\xi$; then the grand-canonical partition function of this system is:

$$ \Upxi_{\Uplambda} = \sum_{\xi} {\rm e}^{- \beta {\mathcal{H}}_{ \Uplambda}(\xi)} \; , $$

(14)

where $\beta = 1 / (k_B T) ( k_B$ is the Boltzmann constant, and T is the absolute thermodynamic temperature), the summation is performed over all possible configurations and $\mathcal{H}(\xi)$ is the grand-canonical Hamiltonian associated to the configuration $\xi$:

$$ {\mathcal{H}}_{\Uplambda}(\xi) = E(\xi) - \sum_{X,Y \subset \{ A,B \} } \mu_{XY} N_{XY}(\xi) \; . $$

(15)

Here, $E(\xi)$ and $\left\{ N_{XY}(\xi) \right\}_{XY}$ are the energy and, respectively, the number of molecule of XY type in the configuration $\xi$; since the lattice is full, the chemical potentials $\left\{ \mu_{XY} \right\}_{XY}$ tend to infinity, but the differences in them remain as finite quantities. The energy $E(\xi)$ and the numbers of molecules of each species $\left\{ N_{XY} (\xi) \right\}$ for a specific configuration are conveniently represented using the site occupation numbers $\left\{ P^{X}_i \right\}_{i \in \Uplambda}$:

$$ P^X_i = \left\{ \begin{array}{ll}1 \; , & {\rm if }\, i \in \Uplambda \, {\rm is\, occupied\, by\, an}\, X{\rm -type\,end\, in\,} \xi \\ 0 \;, & {\rm otherwise } \end{array}\right. $$

(16)

in this case we have:

$$ {\mathcal{H}}_\Uplambda(\xi) = \!\! \sum_{\langle i, j, k \rangle \subset C_3} \sum_{ X, Y, Z \in \{ A, B \} } \!\! \varepsilon_{XYZ} P_i^X P_j^Y P_k^Z - \!\! \sum_{ \langle i, j \rangle \subset C_2} \sum_{ X, Y \in \{ A, B \} } \!\! \mu_{XY} P^X_i P^Y_j. $$

(17)

The occupation numbers can be represented by the spin variables:

$$ \left\{ \begin{array}{l} P^A_i = \frac{1}{2} \left( 1 + S_i \right) \; , \\ P^B_i = \frac{1}{2} \left( 1 - S_i \right) \; , \end{array} \right. $$

(18)

so if the site i of the $\Uplambda$ lattice is occupied with an A molecular end, then S _i  =  +1, and if this site is occupied with an molecular end of B type, then S _i  =  −1. Using the spin representation the grand-canonical Hamiltonian becomes:

$$ {\mathcal{H}}_\Uplambda(\xi) = K_I + J_3 \sum_{(i,j,k) \subset C_3} S_i S_j S_k + J_1 \sum_{(i,j) \subset C_3} S_i S_j + \mu_I \sum_{(i,j) \subset C_2} S_i S_j + h_I \sum_{i \in \Uplambda} S_i \; , $$

(19)

where K _I is a constant independent of the molecular energies and chemical potentials, and $J_3 , J_1 , \mu_I , h_I$ are constant quantities that depend linearly on the molecular energies and chemical potentials. Then the grand-canonical partition function can be written in the following form:

$$ \Upxi_{\Uplambda} = {\rm e}^{- \beta K_I} \; Z_{\Uplambda} (R_3, R, L, h) \; , $$

(20)

where:

$$ \begin{aligned} &Z_{\Uplambda}(R_3, {\bf R}, L, h) \\ &\quad = \sum_{ \{ S \} } \hbox{exp} \left\{ R_3 \! \sum_{(i,j,k) \subset C_3} \!\! S_i S_j S_k + R \! \sum_{(i,j) \subset C_3} \!\! S_i S_j+ L \! \sum_{(i,j) \subset C_2} \!\! S_i S_j + h \sum_{i \in \Uplambda} S_i \right\} , \end{aligned} $$

(21)

and the reduced parameters have the expressions:

$$ R_3 = - \beta J_3= - \frac{\beta}{8} \left( \varepsilon_{AAA} - \varepsilon_{BBB} + 3 \varepsilon_{ABA} - 3 \varepsilon_{BAB} \right) \; , $$

(22a)

$$ R = - \beta J_1 = - \frac{\beta}{8} \left( \varepsilon_{AAA} + \varepsilon_{BBB} - \varepsilon_{ABA} + \varepsilon_{BAB}\right) \; , $$

(22b)

$$ L = - \beta \mu_I = - \frac{\beta}{4} \left( 2 \mu_{AB} - \mu_{AA} - \mu_{BB} \right) \; , $$

(22c)

$$ h = \beta h_I = \frac{\beta}{8} \left( \varepsilon_{BBB} - \varepsilon_{AAA}- \varepsilon_{ABA} + \varepsilon_{BAB} \right)+ \frac{\beta}{4} \left( \mu_{AA} - \mu_{BB} \right) \; . $$

(22d)

From Eq. 21 $Z_{\Uplambda}(R_3, R, L, h)$ is the canonical partition function of an Ising model having the spins located on the $\Uplambda$ lattice; there are three-body interactions between the spins on the triangle with the coupling constant J ₃, binary interactions between the spins on neighboring vertices on the same triangle with the coupling constant J ₁, binary interactions between the neighboring spins on different triangles with the coupling constant $\mu_I$, and each spin interacts with the external magnetic field h _I . Thus, from Eq. 20 it follows that the molecular model on the honeycomb lattice is equivalent to an Ising model on the 3-12 lattice.

Then, using two spin transformations, the star-triangle transformation and the double decoration iteration transformation [15–19], the partition function $Z_{\Uplambda} (R_3, R, L, h)$ becomes:

$$ Z_{\Uplambda}(R_3, R, L, h) = \frac{B^{\frac{3}{2} N} }{A^N} Z_G(K, H) \; , $$

(23)

where the coefficients A and B are functions of the reduced parameters (i.e. R ₃ , R , L , h), and Z _G (K, H) is the partition function of an Ising model defined on the honeycomb lattice having binary interactions between the nearest neighboring spins with the coupling constant K and each spin interacts with the external field H:

$$ Z_G(K, H) = \sum_{ \{ S \} } \hbox{exp} \left\{ K \sum_{ \langle i,j \rangle \subset G} S_i S_j + H \sum_{i \in G} S_i \right\} \; . $$

(24)

Figure 8 shows the honeycomb lattice having one spin in each vertex. Therefore, the Ising model on $\Uplambda$ with an external field and both two-spin and three-spin interactions is equivalent to an Ising model on G with an external field and only two-spin interactions.

Fig. 8

The honeycomb lattice where the spins are located on all of the vertices; the spins with positive values (S _i  =  +1) are represented by white discs, and the spins with negative values (S _i  =  −1) are represented by black discs

Since there are functional dependences between the 4 reduced parameters of the Ising model on the $\Uplambda$ lattice and the 2 reduced parameters of the Ising model on the G lattice, it is convenient to choose as independent parameters R , R ₃ , K and H. Because $R = {\frac{- J_1}{k_B T}}$, it is possible to define the reduced temperature $\tau = {\frac{1} {R}}$ (for ferromagnetic interactions R > 0); also, instead of R ₃ it is convenient to use the asymmetry parameter $\Updelta = {\frac{J_3}{J_1}} = {\frac{R_3}{R}}$ (the name is due to the fact that the three-spin interactions produce asymmetric phase diagrams). Therefore, the set of effective parameters to describe the Ising model on $\Uplambda \,{\rm lattice are}: \tau , \Updelta , K , H$.

Taking into account the relation between the partition functions of the two corresponding Ising models, Eq. 23, the magnetization $m_{\Uplambda}$ and the spin–spin correlation function $\sigma_{\Uplambda}$ of the Ising model on the 3-12 lattice can be expressed in terms of the magnetization M _G ≡ M and the spin–spin correlation function $\Upsigma_G \equiv \Upsigma$ of the Ising model on the honeycomb lattice:

$$ m_{\Uplambda} \equiv \left\langle S_i \right\rangle_{i \in \Uplambda} = \frac{\partial }{\partial h} \lim_{N \rightarrow \infty} \left( \frac{ \ln Z_{\Uplambda} }{3 N} \right) = c_0 + c_s \Upsigma + c_m M \; , $$

(25a)

$$ \sigma_{\Uplambda} \equiv \left\langle S_i S_j \right\rangle_{(i,j) \subset C_2} = \frac{\partial }{\partial L} \lim_{N \rightarrow \infty} \left( \frac{ \ln Z_{\Uplambda} }{ \frac{3}{2} N} \right) = g_0 + g_s \Upsigma + g_m M \;, $$

(25b)

where the coefficients c ₀ , c _s  , c _m  , g ₀ , g _s  , g _m are definite (but very complicated) functions of $\tau , \Updelta ,\, K\, {\rm and } \,H$.

The ferromagnetic Ising model on the honeycomb lattice (K > 0) exhibits a phase transition in the absence of the external magnetic field (H = 0) and for strong enough coupling constant K > K _c . In this case it is possible to obtain exact solutions for the magnetization M and for the spin–spin correlation function $\Upsigma$, at a given reduced temperature ($\tau$), for a well defined asymmetry parameter ($\Updelta$), and for a given coupling constant (K); the phase equilibrium diagram is obtained by changing the values of the coupling constant.

However, the Mean Field Approximation allows solutions even for a non-vanishing external magnetic field, so in this case it is possible to find the spinodal as the limit of the metastable states. In this case the magnetization satisfies the transcendental equation:

$$ M = \tanh \left( 3 K M + H \right) \; , $$

(26)

and the spin–spin correlation function is:

$$ \Upsigma = M^2 \; . $$

(27)

Since the coupling constant will be changed to obtain the phase diagram, it is convenient to solve Eq. 26 with respect to K , so the variable will be M (instead of K). The phase equilibrium curve corresponds to vanishing magnetic field (H = 0) and for the spinodal there are the following equations:

$$ M_s(K) = \sqrt{1 - \frac{1}{3 K} } \; , $$

(28a)

$$ H_s = \frac{1}{2} \ln \left( \frac{1 + M_s}{1 - M_s} \right) + \frac{3}{2} K M_s \; . $$

(28b)

The expressions for the coefficients for the magnetization and the spin–spin correlation function for the Ising model on the 3-12 lattice (i.e. c ₀ , c _s  , c _m  , g ₀ , g _s  , g _m which are used in Eqs. 25), both for the binodal and for spinodal are presented in the Appendix of our previous article [29].

For the ternary molecular system the mole fractions of the components satisfy the conservation relation:

$$ X_{AA} + X_{BB} + X_{AB} = 1 \; ; $$

(29a)

in addition, using Eq. 20, it results the following relations:

$$ X_{AA} - X_{BB} = M_{\Uplambda} \; , $$

(29b)

$$ X_{AA} + X_{BB} - X_{AB} = \Upsigma_{\Uplambda} \; . $$

(29c)

Then, taking into account the correspondence between the molecular model and the associated Ising model, it is possible to obtain the binodal and the spinodal of the ternary molecular system by knowing the magnetization $m_{\Uplambda} (M; \tau, \Updelta)$ and the spin–spin correlation function $\sigma_{\Uplambda} (M; \tau, \Updelta)$ of the associated Ising model.

In conclusion, we considered that the framework needed to describe the molecular generalized Wheeler–Widom model and the ternary system water + 1-pentanol + 1-propanol are very similar, so we can obtain the properties of the liquid–liquid equilibria of the ternary system using the correspondence between the generalized Wheeler–Widom model with the associated Ising model. In this case the magnetization of the Ising model is related to the difference of the mole fractions of the substances which are partially soluble in the corresponding binary system, and the spin–spin correlation function is related to the mole fraction of the amphiphile substance, as results from Eqs. 29.