Kinetics of PTSA-Catalysed Polycondensation of Citric Acid with 1,3-Propanediol

Abstract

Recent years have seen the intensive development of novel therapies based on stem cells and advanced materials. Among the latter, polymers, especially polyesters, occupy a special place and are being investigated for use as substrates for cell differentiation and culture. Increasing application requirements are driving demand for such materials. This study aims to obtain a new material with potential biomedical applications, poly(1,3-propanediol citrate). A spectral analysis of the obtained product was carried out. The reaction kinetics of the polycondensation of citric acid and 1,3-propanediol in the presence of a catalyst, p-toluenesulphonic acid (PTSA), are described. The basis for determining the polycondensation kinetics was the assumption of non-equivalence of the carboxyl groups in citric acid. Elementary reaction rate constants and activation energy values were determined. Based on the data obtained, the course of the gelation points of the reaction mixture was estimated in its dependence on the temperature and reaction time. Kinetic research will facilitate the scale-up of the process.

1. Introduction

The main idea of regenerative medicine is to reconstruct, repair, and restore the functionality of tissues and organs damaged by mechanical trauma, disease lesions, or the natural ageing processes. Regenerative medicine is based on a combination of cell therapies and advanced materials that support natural biological processes. The essence is the harnessing of the regenerative potential of isolated cells, which are often further modified or stimulated. This fundamental feature distinguishes these methods from transplantology, which involves whole-organ transplants. Regenerative medicine is based on the aim of personalising the treatment of patients [1,2,3].

In parallel with the advancement in the field of stem cell isolation and applications, there has been the development of tissue engineering, which combines biological sciences with material engineering methods. According to the approach of Robert Langer and Joseph P. Vacanti, neither transplantation of purely biological material nor completely synthetic implants are sufficient. The optimal strategy is hybrid therapy. The cells’ regenerative capacity should be supported by a suitable biomaterial, which forms a scaffold for reconstructing tissue. The third element is a variety of signalling factors that support cell proliferation and direct cell differentiation [4,5,6,7,8,9,10,11,12].

The requirements for biomaterials in tissue engineering, i.e., bioresorbability, bioactivity, porosity, and various mechanical properties, have led to a great deal of interest in polymers in recent years [13]. Polymeric materials are more similar to the extracellular matrix compared to other types of materials (metallic, ceramic) [14]. The use of polymer–inorganic composites improves their mechanical strength and biological activity [15]. The development of polymeric biomaterial engineering consists of three main directions: the search for new materials, the development of methods for their processing into cellular scaffolds, and methods for modifying the biological activity of these scaffolds [16,17,18,19].

Both natural and synthetic biopolymers are used in tissue engineering. Materials from natural sources have the advantage of being non-toxic and minimising the host’s immune response. However, increasing application requirements are leading to an increasing emphasis on the ability to functionalise materials and the ease of processing them into a product with specific properties that are tailored to a particular application. As a result, synthetic polymers are becoming more critical. The solution to their biocompatibility problem is the use of substrates that are naturally present in the body [20,21,22,23,24,25].

Citric acid is of growing interest, although it is more widely used in biomaterial chemistry as a cross-linking agent. This is due to the high functionality of the molecule (three carboxyl groups and one hydroxyl group). The concept of producing cross-linked polymers represents a significant advance in biomaterial engineering, as the cross-linking process provides an opportunity to modify the mechanical properties and degradation rate of the biomaterial. Thus, biodegradable elastomers can be tailored to the requirements of a specific tissue application [26,27,28,29,30]. Citric acid is widely used in the food and pharmaceutical industries. As an intermediate in the Krebs cycle, citrate is present in every cell of every anaerobic organism. Citrate-based pharmaceuticals and supplements are an important part of modern medicine [31,32,33]. Citric acid is often used as a cross-linking agent, e.g., in the preparation of hydrogels or in systems for the extended release of active substances [34,35,36,37].

Of the citric-acid-based biopolyesters, poly(1,8-octanediol citrate) (POC) is the best known. It shows application potential in soft-tissue engineering [20]. POC biocomposites with mesoporous bioactive glass are being investigated for use in wound healing due to their antibacterial properties [38]. An appealing direction for development is the production of similar materials by using other dihydroxy alcohols. The shorter carbon chain separating the hydroxyl groups should increase the hydrophilicity of the final product. This, in turn, should promote cellular adhesion to the scaffold made of such polyester. It is assumed that the ability of cells to adhere increases on hydrophilic surfaces and decreases on hydrophobic surfaces, even though hydrophobic surfaces are generally considered to promote protein adsorption [14,39]. The type of protein that is most strongly bound is crucial. Initial interactions are usually dominated by albumin, which prefers hydrophobic substrates. Albumin does not promote cellular adhesion, and its strong and prolonged adsorption to the surface reduces the likelihood of substitution by proadhesive proteins, such as fibronectin and vitronectin. The affinity of a protein for a material may be based on the recognition of specific functional groups, which is similar in the case of cells [40,41].

There is a lack of information in the literature on poly(1,3-propanediol citrate), particularly in biomedical applications. The research to date has mainly focused on 1,2-propanediol and has explored copolymers of citric acid polyesters with other acids/diols, e.g., sebacic acid. However, before the application possibilities of this polyester are explored, its preparation must be studied and described. The reaction kinetics of the polycondensation of citric acid and 1,3-propanediol are a good starting point for more extensive studies, as they allow the gel point of the reaction mixture to be determined. This allows a safer scale-up of the process. For example, similar studies have been carried out for poly(glycerol citrate) [42].

The aim of this work is to carry out the synthesis of a new material, poly(1,3-propanediol citrate), perform a spectral analysis of the products obtained, describe the kinetics of the reaction, and determine the gelation limit of the reaction mixture depending on the time and temperature at which the synthesis is carried out. The results will enable the establishment of the synthesis conditions in order to maximise acid group conversion. This, in turn, will minimise the potential cytotoxicity of the polymer resulting from rapid acidification of the culture medium, which would lead to cell death. By terminating the reaction close to the gel point, the time required for controlled cross-linking of the prepolymer in the next step will be reduced. A high degree of cross-linking should extend the degradation time of the material.

2. Materials and Methods

2.1. Synthesis Procedure

Poly(1,3-propanediol citrate) syntheses were carried out by using the IKA RCT basic magnetic stirrer with a heating function and equipped with the IKA ETS-D5 temperature sensor and four IKA 28.5 heating mantles. Anhydrous citric acid (5.75 g; ≥99.5%, ACS reagents, ACROS ORGANICS) and 1,3-propanediol (3.43 g; ≥99.7%, CHEMSOLUTE) were used as reactants in solvent-free polycondensation in the presence of p-toluenesulfonic acid (PTSA) catalyst (0.06 g; ≥98.5%, ACS reagents, SIGMA-ALDRICH). Each time, the molar ratio of the functional groups (COOH:OH) was 1:1, not including the hydroxyl group of citric acid. Mantles were heated and thermostated to set the temperatures (120, 135, 150 °C) before the syntheses. Syntheses were carried out in flat-bottom test tubes with magnetic stirring and water removal. Scheme 1 presents a simplified version of the reaction.

For this work, the following nomenclature is established: (1) A carbonyl group bonded with an α carbon is abbreviated as α-C(O)O-, and (2) a carbonyl group bonded with a β carbon is abbreviated as β-C(O)O-; Scheme 1 shows citric acid with the α and β carbons marked.

2.2. Fourier-Transform Infrared Spectroscopy

A Fourier-transform infrared spectroscopy (FTIR) (Alpha II BRUKER) analysis was performed to confirm the polyester structure of the products. Measurements were carried out by using the attenuated total reflectance (ATR) technique. A total of 32 scans in the range of 400–4000 cm⁻¹ were performed and averaged for each sample.

2.3. Nuclear Magnetic Resonance Spectroscopy

Nuclear magnetic resonance (NMR) spectroscopy was used to confirm the ester structure and to determine the conversion of the carboxyl groups of citric acid. Spectra were obtained by using an Agilent 400 MHz spectrometer. The ¹³C spectra without NOE were assessed and used to determine the conversion. Each time, poly(1,3-propanediol citrate) was dissolved in deuterated acetone, and the solution was transferred to an NMR tube.

The conversion of citric acid was calculated according to the following formulas:

$$\begin{gathered} X_{\mathsf{\alpha }}^{\mathrm{NMR}}=\frac{\int E_{\mathsf{\alpha }}}{\int A_{\alpha }+\int E_{\alpha }}\times 100\% \\ {X}_{\mathsf{\beta }}^{\mathrm{NMR}}=\frac{\int E_{\mathsf{\beta }}}{\int A_{\beta }+\int E_{\beta }}\times 100\% \\ {X}^{NMR}=\frac{\int E_{\mathsf{\alpha }}+\int E_{\mathsf{\beta }}}{\int A_{\alpha }+\int E_{\alpha }+\int A_{\beta }+\int E_{\beta }}\times 100\% \end{gathered}$$

where: $X_{\mathsf{\alpha }}^{\mathrm{NMR}}$—conversion of an α-C(O)O-acid group, $X_{\mathsf{\beta }}^{\mathrm{NMR}}$—conversion of a β-C(O)O-acid group, $X^{NMR}$—overall conversion of acid groups, $\int A_{\mathsf{\alpha } \mathrm{or} \mathsf{\beta }}$—integral of the signal from α-C(O)O-acid or β-C(O)O-acid, and $\int E_{\mathsf{\alpha } \mathrm{or} \mathsf{\beta }}$—integral of the signal from an α-C(O)O-ester or β-C(O)O-ester. The conversion of the hydroxyl groups of 1,3-propanediol is equated with the overall conversion of the carboxyl groups of citric acid.

Signals from the α and β carbon atoms of the acid or ester groups are distinguishable on the ¹³C NMR spectrum (Figure 1), and this was used to determine the conversion of citric acid.

3. Results and Discussion

3.1. Spectral Analysis

3.1.1. FTIR

As a result of the syntheses that were performed, products in the form of a colourless or slightly yellowish resin with a high viscosity that increased with the reaction time were obtained. The formation of polyester was confirmed based on the FTIR spectra of the selected samples (Figure 2).

The spectra of the substrates included:

• For citric acid:
  • around 3500 cm⁻¹ (A), a narrow band of stretching vibrations of O-H groups that did not participate in hydrogen bonding;
  • a broad band near 3250 cm⁻¹ (B) that extended down to about 2600 cm⁻¹ (C) and originated from O-H groups bound by hydrogen bonding; in this area, bands originating from the free-standing hydroxyl group and the O-H bond in the carboxyl group, as well as a stretching vibration band of aliphatic C-H bonds, most likely overlapped;
  • a characteristic narrow band of high intensity around 1715 cm⁻¹ (D) of the stretching vibrations of the C=O bond in a carboxylic group; a double band indicates non-equivalence of carboxylic groups (α-C(O)O-H and β-C(O)O-H), which could also have resulted from the presence of hydrogen bonds; then, the band of the carbonyl group of the associated molecules appeared at higher frequencies than with non-associated ones;
  • bands in the range of 1430–1100 cm⁻¹ (E,F) corresponding to deformation vibrations of the C-O-H group and stretching vibrations of C-O bonds.
• For 1,3-propanediol:
  • a characteristic broad band of high intensity in the range of 3600–3000 cm⁻¹ (G) originating from O-H bond stretching vibrations; the presence of hydrogen bonds significantly influenced the shape of the band;
  • band of about 2900 cm⁻¹ (H) of stretching vibrations of aliphatic C-H bonds;
  • deformation vibration band of an O-H bond near 1430 cm⁻¹ (I);
  • a strong C-O stretching vibration band around 1050 cm⁻¹ (J);
  • a broad band corresponding to the deformation vibrations of the hydrogen-bonded O-H group in the range of 550–750 cm⁻¹ (K).

The FTIR spectra of the substrates were compared with those of the products. For this purpose, two samples (from the initial and final phases) from each measurement series were selected. The most important bands were distinguished as follows:

• a broad band in the range of 3700–3100 cm⁻¹ (L) that formed as a result of the overlapping of bands corresponding to stretching vibrations of O-H bonds in molecules of the unreacted substrates, i.e., A, B, and G;
• a band of approximately 2900 cm⁻¹ (M) of stretching vibrations of C-H aliphatic bonds in both a polyester chain and the molecules of substrates;
• a band at about 2600 cm⁻¹ (N) from O-H groups bonded by hydrogen bonding, indicating the presence of unreacted acid (the corresponding band is marked with C);
• a strong narrow band around 1730 cm⁻¹ (O) corresponding to the stretching vibrations of the C=O bond of the ester; the visible broadening of the band on the right-hand side is due to the incompletely reacted acid, where the C=O bond is part of the carboxyl group.

In addition to the noticeable shift in the frequency of the stretching band of the carbonyl group, another confirmation of the polyester being obtained is the bands coming from the two groupings containing the C-O bond:

• at about 1050 cm⁻¹ (R), the C-O bond stretching vibration band in the O-C-C grouping (Figure 3a);
• at about 1240 cm⁻¹ (P), the C-O bond stretching vibration band in the ester grouping C-C(O)-O (Figure 3b).

It should be noted that as the reaction time increased, the intensity of the bands corresponding to the unreacted substrates decreased, which confirmed the progress of the polycondensation reaction towards the predicted products. The change in intensity was least noticeable at the lowest temperature tested, which was due to the slower changes in the conversion.

3.1.2. NMR

Table 1. Conversion of α-C(O)O-H (X_α^NMR), β-C(O)O-H (X_β^NMR), and hydroxyl (X_OH^NMR) groups depending on the reaction time and temperature.

Reaction Time [min]	120 °C			135 °C			150 °C
	XαNMR	XβNMR	XOHNMR	XαNMR	XβNMR	XOHNMR	XαNMR	XβNMR	XOHNMR
5	18%	42%	35%	21%	54%	44%	28%	57%	48%
7	–	–	–	–	–	–	33%	66%	56%
9		–	–	–	–	–	45%	74%	65%
10	22%	55%	44%	30%	62%	52%	–	–	–
11	–	–	–	–	–	–	41%	73%	63%
13	–	–	–	–	–	–	48%	78%	68%
15	25%	60%	50%	36%	70%	59%	50%	81%	71%
17	–	–	–	–	–	–	57%	80%	72%
19	–	–	–	–	–	–	mixture gelled
20	30%	67%	57%	45%	76%	68%	–	–	–
21	–	–	–	–	–	–	mixture gelled
25	34%	68%	58%	50%	78%	69%	–	–	–
30	32%	69%	57%	54%	80%	72%	–	–	–
40	31%	58%	49%	57%	81%	74%	–	–	–
50	44%	73%	63%	mixture gelled			–	–	–
60	48%	74%	65%				–	–	–
70	43%	72%	63%				–	–	–
80	54%	74%	68%				–	–	–
90	59%	74%	69%				–	–	–
100	57%	74%	68%				–	–	–

summarises the conversion of the carboxylic and hydroxyl groups as determined from the ¹³C-NOE NMR spectra of all of the samples before the gelation point. The reason for the higher frequency of measurement points at 150 °C was the assumption that the reaction mixture would gel after about 20 min, as confirmed by the experimental results.

The molar concentrations of α-C(O)O-H, β-C(O)O-H, and OH groups were calculated from the conversion of citric acid. It was assumed that the volume of the reaction mixture was constant over time, while its density changed. The volume was determined for a selected reference sample (135 °C, 20 min). In the calculation of the kinetic model, the outliers, which are marked in grey in

Table 2. Molar concentrations of α-C(O)O-H (C_α), β-C(O)O-H (C_β), and OH groups of 1,3-propanediol (C_OH) depending on the reaction time and temperature.

Reaction Time [min]	120 °C			135 °C			150 °C
	${\mathbf{C}}_{\mathsf{\alpha }} \left[\frac{\mathrm{mol}}{\mathrm{d}{\mathrm{m}}^3}\right]$	${\mathbf{C}}_{\mathsf{\beta }} \left[\frac{\mathrm{mol}}{\mathrm{d}{\mathrm{m}}^3}\right]$	${\mathbf{C}}_{\mathbf{OH}} \left[\frac{\mathrm{mol}}{\mathrm{d}{\mathrm{m}}^3}\right]$	${\mathbf{C}}_{\mathsf{\alpha }} \left[\frac{\mathrm{mol}}{\mathrm{d}{\mathrm{m}}^3}\right]$	${\mathbf{C}}_{\mathsf{\beta }} \left[\frac{\mathrm{mol}}{\mathrm{d}{\mathrm{m}}^3}\right]$	${\mathbf{C}}_{\mathbf{OH}} \left[\frac{\mathrm{mol}}{\mathrm{d}{\mathrm{m}}^3}\right]$	${\mathbf{C}}_{\mathsf{\alpha }} \left[\frac{\mathrm{mol}}{\mathrm{d}{\mathrm{m}}^3}\right]$	${\mathbf{C}}_{\mathsf{\beta }} \left[\frac{\mathrm{mol}}{\mathrm{d}{\mathrm{m}}^3}\right]$	${\mathbf{C}}_{\mathbf{OH}} \left[\frac{\mathrm{mol}}{\mathrm{d}{\mathrm{m}}^3}\right]$
0	4.18	8.37	12.56	4.18	8.37	12.56	4.18	8.37	12.56
5	3.43	4.88	8.20	3.29	3.83	7.00	3.03	3.57	6.48
7	–	–	–	–	–	–	2.80	2.85	5.63
9	–	–	–	–	–	–	2.30	2.19	4.40
10	3.27	3.79	6.98	2.92	3.17	6.05	–	–	–
11	–	–	–	–	–	–	2.46	2.23	4.61
13	–	–	–	–	–	–	2.17	1.84	4.05
15	3.12	3.32	6.30	2.67	2.52	5.10	2.08	1.60	3.60
17	–	–	–	–	–	–	1.80	1.65	3.49
19	–	–	–	–	–	–	mixture gelled
20	2.92	2.72	5.48	2.30	1.97	4.22	–	–	–
21	–	–	–	–	–	–	mixture gelled
25	2.76	2.65	5.26	2.10	1.83	3.96	–	–	–
30	2.83	2.63	5.35	1.92	1.69	3.53	–	–	–
40	2.88	3.53	6.36	1.79	1.61	3.30	–	–	–
50	2.34	2.30	4.58	mixture gelled			–	–	–
60	2.16	2.19	4.40				–	–	–
70	2.38	2.32	4.70				–	–	–
80	1.91	2.15	4.05				–	–	–
90	1.72	2.17	3.87				–	–	–
100	1.80	2.19	3.98				–	–	–

, were not considered.

3.2. Kinetic Model

The polycondensation under study followed the mechanism of the Fischer esterification reaction. In the first step, a proton from an acidic catalyst (PTSA) is transferred to carbonyl oxygen in the citric acid molecule. This increases the electrophilicity of the carbonyl carbon and facilitates the nucleophilic attack of the alcohol molecule. Access to the carbonyl oxygen atom is easier for β-C(O)O-H groups than for α-C(O)O-H groups (Figure 4, carbonyl oxygen atoms are indicated by arrows). This is based on distinguishing two types of carboxylic groups in citric acid in the described model.

The starting point for the determination of the polycondensation kinetics was the assumption that the reactivity of the α-C(O)O-H and β-C(O)O-H groups changed due to the esterification of neighbouring carboxylic groups. At the same time, the length of the polyester chain was irrelevant. The following terminology of the reacting carboxyl groups was adopted: X-Y1-Y2, where X is the reacting group and Y1 and Y2 are the neighbouring esterified groups. To simplify the notation of the kinetic equations, α-C(O)O-H and β-C(O)O-H were abbreviated to α and β, respectively. The reaction rate constants are marked numerically (Figure 5).

The calculations were based on the following assumptions: (1) The water concentration was constant and equal to 0, which excluded the hydrolysis of ester bonds, (2) the reactivity of 1,3-propanediol was independent of its conversion, (3) the volume of the reacting phase did not change, (4) the reaction mixture was far enough from the gel point that the effects of steric hindrance and a sudden change in viscosity were negligible, (5) the heating time of the reaction mixture to a given temperature was negligibly short and the mixture was homogeneous from the start, and (6) all reactions were first-order reactions in each reactant, where the functional group was considered to be the reactant.

The kinetic model is based on the elementary reactions shown in Figure 5. Given the assumptions that were made, the process is described by the following equations:

$$\frac{\mathrm{d}\left[\mathsf{\alpha }\right]}{\mathrm{dt}}=-k_1\left[\mathsf{\alpha }\right]\left[\mathrm{OH}\right]-k_2\left[\mathsf{\beta }\right]\left[\mathrm{OH}\right]$$

(1)

$$\frac{\mathrm{d}\left[\mathsf{\beta }\right]}{\mathrm{dt}}=-2k_2\left[\mathsf{\beta }\right]\left[\mathrm{OH}\right]-2k_1\left[\mathsf{\alpha }\right]\left[\mathrm{OH}\right]$$

(2)

$$\frac{\mathrm{d}\left[\mathsf{\alpha }-\mathsf{\beta }\right]}{\mathrm{dt}}=k_2\left[\mathsf{\beta }\right]\left[\mathrm{OH}\right]-k_3\left[\mathsf{\alpha }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]-k_5\left[\mathsf{\beta }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]$$

(3)

$$\frac{\mathrm{d}\left[\mathsf{\alpha }-\mathsf{\beta }-\mathsf{\beta }\right]}{\mathrm{dt}}=k_5\left[\mathsf{\beta }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]-k_6\left[\mathsf{\alpha }-\mathsf{\beta }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]$$

(4)

$$\frac{\mathrm{d}\left[\mathsf{\beta }-\mathsf{\alpha }\right]}{\mathrm{dt}}=2k_1\left[\mathsf{\alpha }\right]\left[\mathrm{OH}\right]-2k_4\left[\mathsf{\beta }-\mathsf{\alpha }\right]\left[\mathrm{OH}\right]$$

(5)

$$\frac{\mathrm{d}\left[\mathsf{\beta }-\mathsf{\beta }\right]}{\mathrm{dt}}=k_2\left[\mathsf{\beta }\right]\left[\mathrm{OH}\right]-k_5\left[\mathsf{\beta }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]-k_3\left[\mathsf{\alpha }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]$$

(6)

$$\frac{\mathrm{d}\left[\mathsf{\beta }-\mathsf{\alpha }-\mathsf{\beta }\right]}{\mathrm{dt}}=k_3\left[\mathsf{\alpha }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]+k_4\left[\mathsf{\beta }-\mathsf{\alpha }\right]\left[\mathrm{OH}\right]-k_7\left[\mathsf{\beta }-\mathsf{\alpha }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]$$

(7)

$$\frac{\mathrm{d}\left[\mathsf{\alpha } \mathrm{total}\right]}{\mathrm{dt}}=-k_1\left[\mathsf{\alpha }\right]\left[\mathrm{OH}\right]-k_3\left[\mathsf{\alpha }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]-k_6\left[\mathsf{\alpha }-\mathsf{\beta }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]$$

(8)

$$\frac{\mathrm{d}\left[\mathsf{\beta } \mathrm{total}\right]}{\mathrm{dt}}=-k_2\left[\mathsf{\beta }\right]\left[\mathrm{OH}\right]-k_4\left[\mathsf{\beta }-\mathsf{\alpha }\right]\left[\mathrm{OH}\right]-k_5\left[\mathsf{\beta }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]-k_7\left[\mathsf{\beta }-\mathsf{\alpha }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]$$

(9)

$$\begin{gathered} \frac{\mathrm{d}\left[\mathrm{OH}\right]}{\mathrm{dt}}=-k_1\left[\mathsf{\alpha }\right]\left[\mathrm{OH}\right]-k_2\left[\mathsf{\beta }\right]\left[\mathrm{OH}\right]-k_3\left[\mathsf{\alpha }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]-k_4\left[\mathsf{\beta }-\mathsf{\alpha }\right]\left[\mathrm{OH}\right]-k_5\left[\mathsf{\beta }-\mathsf{\beta }\right]\left[\mathrm{OH}\right] \\ -k_6\left[\mathsf{\alpha }-\mathsf{\beta }-\mathsf{\beta }\right]\left[\mathrm{OH}\right]-k_7\left[\mathsf{\beta }-\mathsf{\alpha }-\mathsf{\beta }\right]\left[\mathrm{OH}\right] \end{gathered}$$

(10)

Equations (1)–(7) are partial balance equations. They refer to groups that react according to the specific rate constants of elementary reactions, as shown in Figure 5. The symbols α (Equation (1)) and β (2) are to be understood as the respective carboxyl groups in the unreacted acid molecule. The coefficient of 2 appearing in Equations (2) and (5) results from the fact that in a given elementary reaction, the reactivity of two carboxylic groups simultaneously changes into an esterified or adjacent-to-esterified one. Equations (8) and (9) are total balances of α-C(O)O-H and β-C(O)O-H groups, respectively. Equation (10) describes the total balance of hydroxyl groups.

Table 3. Calculated reaction rate constants for the three selected temperatures.

	${\mathbf{k}}_1·{10}^{-4} \left[\frac{\mathrm{d}{\mathrm{m}}^3}{\mathrm{mol} · \mathrm{s}}\right]$	${\mathbf{k}}_2·{10}^{-4}{ }_{ }\left[\frac{\mathrm{d}{\mathrm{m}}^3}{\mathrm{mol} · \mathrm{s}}\right]$	${\mathbf{k}}_3·{10}^{-4}{ }_{ }\left[\frac{\mathrm{d}{\mathrm{m}}^3}{\mathrm{mol} · \mathrm{s}}\right]$	${\mathbf{k}}_4·{10}^{-4}{ }_{ }\left[\frac{\mathrm{d}{\mathrm{m}}^3}{\mathrm{mol} · \mathrm{s}}\right]$	${\mathbf{k}}_5·{10}^{-4}{ }_{ }\left[\frac{\mathrm{d}{\mathrm{m}}^3}{\mathrm{mol} · \mathrm{s}}\right]$	${\mathbf{k}}_6·{10}^{-4}{ }_{ }\left[\frac{\mathrm{d}{\mathrm{m}}^3}{\mathrm{mol} · \mathrm{s}}\right]$	${\mathbf{k}}_7·{10}^{-4}{ }_{ }\left[\frac{\mathrm{d}{\mathrm{m}}^3}{\mathrm{mol} · \mathrm{s}}\right]$
120 °C	0.50	2.00	0.35	0.20	0.55	0.65	0.07
135 °C	1.00	5.00	0.80	0.50	1.35	1.55	0.30
150 °C	2.00	7.50	1.50	1.00	2.50	3.00	0.55

shows the determined reaction rate constants for the polycondensation of citric acid and 1,3-propanediol at temperatures of 120, 135, and 150 °C. The calculations were carried out using the Scilab program.

The changes in the total concentrations of carboxyl and hydroxyl groups over time are shown in Figure 6.

The natural logarithm of each of the seven rate constants as a function of the inverse of the absolute temperature can be described by a linear relationship (Figure 7) according to the Arrhenius equation:

$$\ln k_i=\ln A-\frac{E_a}{RT}$$

where: A—pre-exponential factor, i.e., rate constant at an infinitely high temperature, E_a—activation energy (J/mol), R—universal gas constant (8.314 J/(mol∙K)), and T—temperature (K).

The values of the activation energy and pre-exponential factor were determined based on the results obtained for the developed kinetic model (

Table 4. Values of the activation energy (E_a) and pre-exponential factors (A) for the developed model.

	k1	k2	k3	k4	k5	k6	k7
$E_a\left[\frac{\mathrm{kJ}}{\mathrm{mol}}\right]$	63.89	71.96	67.18	69.94	70.60	74.29	95.48
$A·105^{ }\left[\frac{{\mathrm{dm}}^3}{\mathrm{mol}·\mathrm{s}}\right]$	0.15	7.54	0.30	1.12	1.60	1.53	384.34

).

The values of the seven reaction rate constants correspond to the expected course of polycondensation. The fastest reacting carboxyl group is β-C(O)O-H in the unreacted acid molecule, followed by the (β-β) and (β-α) groups, according to the established nomenclature. This results from the β-C(O)O-H group being more accessible than the α-C(O)O-H group, even when one adjacent group is esterified. The steric hindrance created by the oligomer/polymer chain is more negligible when the substitution of the β-C(O)O-H group is involved. As a result, the rate constant k₅ is greater than k₄.

In the case of the esterification of the α-C(O)O-H groups, the unreacted acid molecule is most favoured, followed by the molecule with one esterified group and the molecule with two esterified groups. Access of the hydroxyl group of 1,3-propanediol to the partially reacted acid is hindered by the presence of oligomer/polymer chains. Esterification of the (α-β-β) group contributes significantly to the gelation process.

The similar activation energy values for all of the reactions indicate similar mechanisms. The selectivity of the process does not depend on temperature, which means that the preferred direction of attack of the hydroxyl group on the carboxyl group, which is understood as the choice of a given type of group, does not change. The β-C(O)O-H group is the most rapidly esterified in the unreacted acid molecule.

The available literature lacks information on the polycondensation kinetics of citric acid and 1,3-propanediol. Kinetic studies have referred to reactions without a catalyst, which makes it impossible to compare the results. The most similar is the polyester obtained from sebacic acid and glycerol. The monomer’s functionality is the opposite, i.e., a bifunctional (dicarboxylic) acid and a trifunctional (trihydroxylic) alcohol.

3.3. Gel-Point Determination of the Reaction Mixture

A model was derived based on the calculated values of the activation energies and pre-exponential factors. The gelation point was the limiting time from the start of the reaction depending on the temperature. The gelation was made dependent upon the degree of conversion of the carboxyl groups of citric acid: α-C(O)O-H ≥ 58%, β-C(O)O-H ≥ 82%, based on the experimental data. It was assumed that both conditions must be met simultaneously. The assumed conversion limits for the α-C(O)O-H and β-C(O)O-H groups gave a total conversion of carboxyl groups of 73%. The results are presented in Figure 8.

The model presented here is a good approximation of reality. For the reaction that was carried out at 135 °C, the predicted gelation time was 49 min (experimental: 50 min). At 150 °C, the times were 22 and 19 min, respectively. The determined gel points may be the basis for planning subsequent experiments with the reservation that they describe this particular reaction system of citric acid and 1,3-propanediol with the stoichiometric equilibrium of functional groups and with the addition of the PTSA catalyst. The gelation limit was not determined for temperatures outside the range of 100–170 °C because they are not of practical importance.

4. Conclusions

A new polyester was obtained from citric acid and 1,3-propanediol by using a catalyst, p-toluenesulfonic acid. A spectral analysis of the products obtained was carried out. A series of experiments were carried out, and the reaction’s kinetics at 120, 135, and 150 °C were described. The values of the kinetic constants and the activation energies of seven elementary reactions included in our kinetic model were determined. Based on the data obtained, the course of the gel points of the reaction mixture was estimated as a function of the temperature and reaction time. This will facilitate the scaling up of the process.

The presented research is a prelude to further analysis of poly(1,3-propanediol citrate) as a material for fabricating cell culture scaffolds. In order to reduce the potential cytotoxicity, a material with the highest possible degree of carboxyl group conversion should be obtained. This will minimise the risk of significant acidification of the medium and that of accelerated autocatalysed degradation. The developed gelation model of the reaction mixture will facilitate synthesis that will lead to the desired products. In addition, it will allow estimation of the time required to cross-link the prepolymer to obtain the polymer film. Poly(1,3-propanediol citrate) can be used, for example, as a hydrophilising additive to polylactide in the preparation of scaffolds for ex-vivo cell culturing. Increased hydrophilicity will be possible due to the short carbon chain in the diol molecule.