Читать книгу Poly(lactic acid) - Группа авторов - Страница 54

The equilibrium and size distributions can be readily calculated using infinite series when the equilibrium constant is assumed to be independent of the oligomer length. Oligomer distribution is obtained by rearranging the material balances in terms of the equilibrium constant and subsequently empirically determining the equilibrium constant that represents the total titratable acid. The distribution is then verified against the smaller oligomers that are measurable. The equilibrium constant (Equation 2.8) can be rearranged as:

(2.9)

where p is a lumped variable including the lactic acid monomer, free water, and equilibrium constant.

(2.10)

The variable p represents the probability of bond formation. Although the value of p depends on concentration, the quantity is the same for all oligomers at each concentration. Vu et al. [27] used the variable r, which is equivalent to the variable p used here. Using recursion, we recognize that Equation 2.9 can be written as:

(2.11)

(2.12)

Each oligomer of length j contains j lactic acid molecules, so the apparent moles of lactic acid are given by the balance found by the closed form of the sum:

(2.13)

(2.14)

Equation 2.14 can be inserted into Equation 2.12 to give the Flory‐Schulz distribution:

(2.15)

The water in an equilibrated solution is the sum of the apparent water plus the water from the condensation reaction. Each step during the condensation releases a water molecule, so an oligomer of length j releases (j − 1) moles of water (n _W):

(2.16)

Recognizing Equation 2.13, we insert it into Equation 2.16 to obtain:

(2.17)

Inserting Equations 2.14 and 2.17 into Equation 2.10, we develop a relation between the apparent number of moles and K that can be solved to find p

(2.18)

(2.19)

(2.20)

For a given K and apparent moles and Equation 2.20 provides a value of p. Then Equations 2.12 and 2.14 can be used to find the equilibrium moles of lactic oligomers and monomer, while Equation 2.17 provides the equilibrium moles of water. The titratable acidity is a measure of the true number of oligomers in solution because each oligomer has one free carboxylic acid group. The titratable acidity is

FIGURE 2.4 Left axis—total titratable acidity tabulated by Holten [28] from various workers (♢) and measurements by Vu et al. [27] (▪) compared to the model. Right axis—value of p for the model as a function of apparent wt% using K = 0.2023.

(2.21)

Because water is formed by polymerization and can be removed from the solution by evaporation, the material balance constraint for water is for equilibrium moles, n _W ≥ 0, via Equation 2.17 and not the initial moles, . Thus, negative values of are feasible and the apparent weight fractions of water are negative at high degrees of oligomerization. Vu et al. [27] have regressed K to fit titratable acidity and found a value of K = 0.2023. Using K = 0.2023, solutions of up to an apparent lactic acid concentration of 125 wt% lactic acid are theoretically feasible with an apparent water concentration of −25 wt%. The value of p and the titratable acidity are shown in Figure 2.4.

Several important relations can be developed. The distribution of oligomer lengths is given by the Flory‐Schulz distribution [22]. The %EMLA _j for species j is (note that a typographical error in equation 19 of Vu et al. [27] omits the 100):

(2.22)

Other useful results are:

(2.23)

(2.24)

FIGURE 2.5 The percent equivalent monomer lactic acid for L₁ through L₄. Open symbols are from Montgomery [24] and Ueda and Terajima [25]. Solid symbols are measured by Vu et al. [27].

(2.25)

The %EMLA for the short oligomers are shown in Figure 2.5. The equilibrium constant is only weakly temperature dependent. Esterification reactions are commonly nearly thermoneutral. Recently, Feng et al. [29] performed potentiometric titration at temperatures between 8 and 100°C. They found that differences in the titratable acidity were only a couple percent mostly in the range of 60–80 apparent wt% lactic acid; minor differences were observed at other concentrations.