ARTICLE pubs.acs.org/IECR
Kinetics of Ring-Opening Polymerization of L,L-Lactide Yingchuan Yu, Giuseppe Storti, and Massimo Morbidelli* Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, Swiss Federal Institute of Technology (ETH), 8093 Zurich, Switzerland ABSTRACT: The ring-opening polymerization of L,L-lactide with 2-ethylhexanoic acid tin(II) salt as catalyst, and 1-dodecanol as cocatalyst, is investigated at temperatures ranging from 140 to 180 C and various operating conditions such as different ratios of monomer to catalyst as well as catalyst to cocatalyst amounts. Average molecular weights and byproduct formation have been measured as a function of monomer conversion. A complete kinetic model including intra- and intermolecular transesterifications as well as random chain scission has been developed. All experimental data have been compared with the predictions of the model, which is coherent with the one developed for lower temperature values in a previous work [Yu, Y. C.; Storti, G.; Morbidelli, M. Macromolecules 2009, 42, 8187]. This results in a reliable estimation of the rate coefficients of all involved reactions in a relatively large range of temperatures (130180 C).
’ INTRODUCTION Poly(lactic acid) (PLA) is typically produced either by direct polycondensation1,2 or by ring-opening polymerization (ROP)35 which is usually preferred for large-scale applications in various commercial processes to reach high molecular weights.6,7 During the past half-century, many different catalysts have been studied to increase the reaction productivity. Among them, 2-ethylhexanoic acid tin(II) salt (Sn(Oct)2), approved by the U.S. Food and Drug Administration,8 is the most widely used. The mechanism of ROP of L,L-lactide catalyzed by Sn(Oct)2 has been investigated by many researchers, and different chain initiation mechanisms have been proposed, such as alkoxide initiation,9 monomer activation,10 and cationic initiation.11 A comprehensive discussion of all proposed mechanisms is available in the review by Duda and Penczek.12 The most widely accepted one is the alkoxide mechanism, where stannous octoate reacts with OH-bearing species to form an alkoxide which is the species initiating the polymerization. On the basis of such an initiation step, the same authors proposed a comprehensive kinetic scheme of the polymerization, involving also reversible chain transfer and polymer interchange reactions, so-called “transesterifications”. In our previous modeling study at low temperature,13 i.e., 130 C, it was shown that such reactions are in fact present and are responsible for the fast interchange of active end groups among the polymer chains and affect directly the molecular weight distribution (MWD) of the final polymer. The corresponding experimental data have been described quantitatively with a suitable kinetic model which included catalyst activation, propagation, reversible deactivation, and intermolecular transesterification. On the other hand, the industrial production of PLA is usually run at higher temperatures (at least 180 C) to achieve faster reaction rates and to avoid polymer crystallization and too high viscosity. Under such conditions, the role of other degradation reactions becomes important and, in some cases, dominant. With reference to ROP of L,L-lactide at high temperature, polymer degradation is a severe problem causing low molecular weight values.14 At temperatures as high as 400 C, Mcneill and r 2011 American Chemical Society
Leiper15,16 proposed a comprehensive kinetic scheme for PLA pyrolytic elimination based on thermal volatile analysis (TVA) studies. Kopinke et al.17 reduced the thermal pyrolysis kinetic scheme of PLA to five lumped reactions and concluded that polymer degradation follows two main mechanisms, radical and nonradical. At lower temperatures (180230 C, i.e., close to the temperature values of interest in this study), Wachsen et al.18 proposed two possible degradation reactions: intramolecular transesterifications (back-biting) and nonradical random chain scission, which produce macrocycles and acrylate-ended PLA, respectively. However, since the reactions producing the acrylate-ended PLA chains require higher temperature values, one would expect that intramolecular transesterifications are the main mechanisms responsible for molecular weight decreasing with conversion in these conditions. As discussed in the following this is in fact not the case. According to the theory of Jacobson and Stockmayer (JS theory),19 the concentration of the cycles is a strongly decreasing function of molecular size. Thermodynamics of the ring chain equilibria quantitatively describes the conformational probability of two units along a polymer chain to meet and form a ring. In particular, the cycle formation reaction can be represented as Pn a Pnx þ Cx and the corresponding equilibrium constant is given by19 Keq;c ¼
½Pnx ½Cx = ½Cx ¼ Bx5=2 ½Pn
ð1Þ
where Pn and Pnx are linear polymers with chain length n and n x, respectively, Cx is a cyclic polymer containing x repeating units with a concentration of [Cx], and B is a constant which is characteristic for a given system. This result has been confirmed by Montaudo20 among others in the case of aliphatic polyester, and especially for Received: January 18, 2011 Accepted: April 21, 2011 Revised: March 28, 2011 Published: April 21, 2011 7927
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Figure 1. (a) Y vs time and (b) Mn conversion, at 180 C, [M0]/[C0] = 10 000, and different [OH0]/[C0] values as follows: ), 9.5; 0, 17; O, 29; Δ, 55; r, 95.
Figure 2. Dispersity vs conversion at 180 C at [M0]/[C0] = 10 000 and different [OH0]/[C0] values as follows: 0, 17; O, 29; Δ, 55; r, 95.
PLA, using direct pyrolysis mass spectrometry. On the basis of these results, we can conclude that the reactions leading to macrocycles cannot be regarded as the ones responsible for the decrease of the molecular weight during ROP of L,L-lactide at high temperature. In this work, we study the kinetics of the ROP of L,L-lactide in a wide range of temperatures, namely, from 140 to 180 C. In particular, experimental data of bulk melt ROP of L,L-lactide using Sn(Oct)2 as catalyst and 1-dodecanol as cocatalyst have been collected. Several polymerization reactions have been conducted at various values of the molar ratios between monomer and catalyst and between cocatalyst and catalyst. Conversion and average molecular weights have been measured in all cases as a function of time. In specific cases, i.e., at 180 C, a characterization of the byproducts in the final polymer has also been carried out. The obtained data are compared with the predictions of a suitable kinetic model based on the alkoxide initiation mechanism and accounting for reversible activation, reversible propagation, reversible deactivation, intermolecular transesterifications, and nonradical random chain scission. The values of all rate coefficients involved in the kinetic scheme have been estimated by comparison with the experimental data.
’ EXPERIMENTAL PART Materials. (S,S)-3,6-Dimethyl-1,4-dioxane-2,5-dione (L,L-lactide; PURAC, g99.5% GC; water < 0.02%) was further recrystallized in toluene (Sigma-Aldrich, puriss. p.a., g99.7% GC; H2O
< 0.001%). The acidic residues of purified monomer have been previously estimated13 as about 0.25 mequiv kg1. 2-Ethylhexanoic acid tin(II) salt (Sn(Oct)2; Sigma Aldrich, 95% purity), 1-dodecanol (Fluka, 99.5% purity), toluene anhydrous used to facilitate catalyst transfer (99.8%, packaged under argon), and methanol (Sigma Aldrich, puriss) were used as received. For size exclusion chromatography (SEC) analyses, polystyrene standards from 500 to 2 000 000 Da (Sigma Aldrich) were used for calibration, and chloroform (J. T. Baker) was utilized as eluent. Reaction Procedure. L,L-Lactide (LA) was melted at temperature below 100 C in a stirred flask in a glovebox, and, then, Sn(Oct)2 (represented by C in the following text) and 1-dodecanol were prepared in the glovebox at a given molar ratio with respect to monomer in toluene (10 wt %). Anhydrous toluene was used to prevent contamination during the transfer of catalyst and cocatalyst to the reaction vessel. In particular, both mixtures were transferred to glass vials and sealed with T-type poly(tetrafluoroethylene) caps in order to prevent the loss of LA during the reaction by vaporization and recrystallization. All vials were finally transferred into a controlled heating block set at different reaction temperatures, ranging from 145 to 180 C. Note that all reaction times have been subtracted by 30 s, which is the estimated average time required to melt the solid reactant mixture in the heating block at 180 C. PLA products in the different reaction vials were finally quenched in an ice bath at different times and kept for further characterizations. Reactions at constant amount of catalyst and different cocatalyst/catalyst ratios have been carried out. The polymerization byproducts were isolated by polymer precipitation. The crude polymer was first dissolved in chloroform and dropwise added in excess amount of cold methanol. The polymer was precipitated subsequently, and the liquid phase was dried in a rotating evaporator. SEC Analysis. Conversion and MWD of all samples were characterized by SEC (Agilent, 1100 series) equipped with two detectors, ultraviolet (UV) and differential refractive index (RI). Depending upon the molecular weight of the specific sample, a precolumn with an oligopore column (Polymer Laboratories; length, 300 mm; diameter, 7.5 mm; measuring range, 04500 Da) or a precolumn with two PLgel 5 μm MIXED-C columns (Polymer Laboratories; length, 300 mm; diameter, 7.5 mm; measuring range, 20002 000 000 Da) have been used. Chloroform was used as eluent at a flow rate of 1 mL min1 and temperature of 30 C. Universal calibration was applied, based 7928
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Figure 3. (a) Conversion vs time and (b) Y vs time at [M0]/[C0] = 10 000, [OH0]/[C0] = 17, and the following temperatures: r, 130; Δ, 145; O, 160; 0, 180 C.
calibration standard I (500017 000 Da), and protein calibration standard II (20 00050 000 Da) from Care (Bruker).
’ EXPERIMENTAL RESULTS The living behavior of ring-opening polymerization of L,Llactide is well-known.12,24,25 Accordingly, the reaction proceeds at a constant number of growing chains and the following relationships for monomer concentration, [M], as a function of time, t, as well as number average molecular weight, Mn, as a function of monomer conversion, X, defined as ([M0] [M])/ [M0], can be applied:24,25 " # ð½M0 ½Meq Þ Y ¼ ln ð3Þ ¼ kp ½R t ð½M ½Meq Þ Figure 4. Mn (solid dots) and Mw (empty dots) vs time at [M0]/[C0] = 10 000, [OH0]/[C0] = 17, and the folllowing temperatures: 1, r, 130; 2, Δ, 145; b, O, 160; 9, 0, 180 C.
on poly(styrene) standards and the following equation:21 1 þ a1 1 K1 ln ðM2 Þ ¼ ln ðM1 Þ þ ln ð2Þ 1 þ a2 1 þ a2 K2 where a and K are the MarkHouwink constants for PLA (index 2) and the reference polymer, poly(styrene) (index 1). The values of MarkHouwink constants for poly(L,L-lactide), 22 K2 = 0.0171 mL g1 and a2 = 0.806, are used in this work along with the following values for poly(styrene): K1 = 0.0049 mL g1 and a1 = 0.794.23 MALDI-TOF Mass Spectrometry. MALDI TOF mass spectra were measured by an Ultraflex II TOF Bruker spectrometer (Bremen, Germany) using 2-[(2E)-3-(4-tert-butylphenyl)-2methylprop-2-enylidene]malononitrile (DCTB) as matrix material. Samples cocrystallized with the matrix on the probe were ionized by Smart Bean laser pulse (337 nm) and accelerated under 25 kV with time-delayed extraction before entering the time-of-flight mass spectrometer. Matrix (DCTB) and sample were separately dissolved in dichloromethane and mixed in a ratio of 10:1 matrix/sample. To produce some special adducts, sodium ions are added (1% sodium acetate in methanol). A 1 μL mixture of matrix and sample was applied to a MALDI-TOF MS probe and air-dried. All spectra were performed in positive reflection mode. External calibration was performed by using peptide calibration standard II (7003200 Da), protein
Mn ¼
½M0 X mmon ½R þ ½D
ð4Þ
where [R*] is the concentration of living chains, [D] is that of dormant (reversibly terminated) chains, [M0] and [Meq] are the initial and equilibrium monomer concentrations, respectively, mmon is the molar mass of the monomer repeating unit, and Y is the logarithm of the normalized conversion defined in eq 3. The experimental data of Y vs time and Mn vs conversion are shown for various cocatalyst to catalyst ratios in Figure 1a,b, respectively, at temperature of 180 C and up to conversion values of about 90%. The linearity of Y vs time is not so clear in the experiments at large values of the [OH0]/[C0] ratio; however, it is quite convincing in terms of Mn vs conversion, thus proving the living nature of the reaction. The lack of linearity in the first case is believed to be imputed to experimental error. As a matter of fact, the best linearity in Figure 1a is exhibited by the data at the lowest value of [OH0]/[C0], i.e., when the reaction is slow and reliable sampling is facilitated. The dispersity (Pd = Mw/Mn) values shown in Figure 2 as a function of conversion indicate that the MWD is narrow only at low conversion, while it broadens as conversion increases, most probably because of intermolecular transesterifications. Note that not all reactions presented in Figure 1 have been considered for better visualization of the dispersity profile. As expected, the conversion vs time plots shown in Figure 3 reveal that the polymerization rate increases with temperature. Even though the linearity of Y vs time is not always fully satisfactory, the living behavior is well-accessed in all cases. 7929
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Figure 5. RI signal of SEC chromatograms of PLA samples taken during polymerization at 180 C, [M0]/[C0] = 10 000, [OH0]/[C0] = 17, and the following reaction times: (a) 1, 2, 3, and 5 min; (b) 7.5 and 10 min and 3.3, 24, and 120 h. Arrows: direction of increasing reaction time.
Figure 6. MALDI-TOF spectrum of low molecular weight fraction extracted from the polymer produced at 180 C.
Figure 7. MALDI-TOF spectrum of crude PLA produced at 180 C: peaks with 0, linear PLA; those with O, cyclic PLA.
The corresponding Mn and Mw are shown in Figure 4 as a function of time. While they continuously grow in time at the lowest temperature (130 C), at higher temperatures they reach a maximum value and then decrease, and this happens sooner for higher temperature. For example, this decrease appears within about 0.15 h at 180 C, thus indicating that degradation reactions are effective at high conversion, well above 80% (cf. Figure 3). To obtain high molecular weight PLA, it is then clear that such degradation reactions have to be investigated in detail, particularly at high temperatures. The refractive index signals of SEC traces are shown in Figure 5 for polymer samples taken at various reaction times. It is seen that, at low conversion values, i.e., Figure 5a, the second peak
(at elution time equal to 19.5 min) corresponding to unreacted monomer decreases in time due to the polymerization reaction. The first peak corresponds to the polymer chains and moves to shorter elution times, thus indicating increasing molecular weight at increasing conversion. On the other hand, at higher conversion (Figure 5b), the monomer peak remains practically constant, because the polymerization reaction has achieved equilibrium conditions. Nevertheless, the peak corresponding to polymer chains moves now to longer elution times, indicating decreasing molecular weight and increasing dispersity. It is worth noticing that if such a decrease would correspond to the formation of cyclic PLA, a corresponding peak at high elution time (i.e., about 1820 min corresponding to cyclic oligomers) should appear. The fact that this does not happen confirms that cycle formation by intramolecular transesterification plays a minor role. In addition, the presence of such reactions would eventually lead to the ringchain equilibrium given by eq 1, while instead the SEC chromatograms show that the polymer molecular weight decreases continuously without reaching any equilibrium state. This behavior is coherent with the random et al.26 chain scission degradation process proposed by S€odergard To further support the preceding conclusion, fractions at low molecular weight have been isolated from the crude polymer by precipitation in excess methanol and both crude polymer and extracted polymer have been analyzed by MALDI-TOF. Once subtracted, the molecular weight contribution of sodium ions, a series of peaks which correspond to multiples of 72 Da, the molecular weight of the lactoyl repeating unit, is clearly identified in the MALDI-TOF spectra such as the one shown in Figure 6. From these data, it is found that PLA cycles involving from 6 to 50 repeating units (i.e., from 455 to 3523 Da) are indeed present. This is in agreement with Duda and Penczek12 and Kowalski et al.,27 who observed the existence of macrocyclic PLA as one of the side products of ROP of L,L-lactide at 80 C. The corresponding MALDI-TOF spectrum of crude PLA is shown in Figure 7, where the peaks with empty squares represent the fraction of linear PLA and those with open circles represent the fraction of cyclic PLA. It can be seen that the ratio of abundances of the cyclic to linear PLA, RA, decreases quickly at increasing number of lactoyl repeating units according to a power law with exponent 2.41, as shown in the insert. The experimental value of the power exponent in this equation is in close agreement with the Jacobson and Stockmayer theory expressed by eq 1, whose exponent is 2.5, thus confirming that 7930
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Scheme 1. Nonradical PLA Degradation during LA Polymerization at 180 C
Figure 8. Kinetic scheme for ROP of PLA with Sn(Oct)2 as catalyst (C) and an alcohol as cocatalyst (D0) at temperatures ranging from 130 to 180 C.
Table 1. Average Natural Impurities Bearing OH Groups ([M0]/[C0] = 10 000; 180 C) [OH0,nom]/[C0] 9.5
[OH0,exp]/[C0]
[IM]/[C0]
[IM] (mmol L1)
12.8
3.3
2.6
17
20
3.0
2.3
29
34
5.0
3.9
55
60
5.0
3.9
95
100
5.0
3.9
Table 2. kp Values at Different Temperatures of ROP of LA with Sn(Oct)2 as Catalyst temperature
temperature kp (L mol1 h1)
(C)
kp (L mol1 h1)
130
4 500
160
17 500
145
10 000
180
37 000
(C)
only small cycles are formed. It is worth noticing that the amount of cycles measured is by far insufficient to explain the molecular weight decrease in Figure 3. Having excluded intramolecular transesterification, the best candidate as responsible for the decrease of molecular weight with conversion is the random chain scission reaction illustrated in Scheme 1. This corresponds to the pyrolytic elimination, also known as nonradical random chain scission reaction.1517 This mechanism has been investigated by Fan et al.28 and Nishida et al.29 through dynamic pyrolysis of PLA in the presence of stannous
Figure 9. Arrhenius plot of the propagation rate constants for the ROP of LA.
octoate. In particular, the presence of this reaction at the conditions used in this work in terms of catalyst amount and temperature has been proved. More recently, Wang et al.30 reported a strong correlation between increasing UV absorption of the PLA melt and molar mass reduction at 180 C. Since the red shift of the UV spectra can be attributed to the formation of conjugated species, rather than to PLA cyclic oligomers or hydrolysis products, this again is evidence of chain scission. In addition, evidence exists28,29,31 that the PLA degradation is catalyzed by stannous octoate.
’ KINETIC SCHEME AND PRELIMINARY ANALYSIS Based on the previous kinetic study13 at 130 C and the experimental results at higher temperature discussed in the section above, the kinetic scheme shown in Figure 8 is considered in this work. In Figure 8, A is octanoic acid which is produced by Sn(Oct)2 (represented by C) activation, Rn and Dn the active (all species containing tin alkoxide) and dormant (all [OH]-bearing species) chains of length n (n ∈ [0, ¥)). Note that the value of n represents the number of lactoyl units and that each L,L-lactide monomer unit contributes two repeating units during the chain growth step. When n = 0, R0 and D0 represent the initiator (tin alkoxide) and alcoholic cocatalyst (1-dodecanol), respectively. Moreover, M is the monomer L,L-lactide, whereas Gn is terminated PLA chains produced by nonradical chain scission reaction.17,29,31 Thus, summarizing, the kinetic scheme in Figure 8 includes the following reactions: reversible activation (reaction a; ka1 and ka2), reversible propagation (reaction b; kp and kd), reversible chain transfer (reaction c; ks1 = ks2 = ks), intermolecular transesterification (reactions df; kte1 = kte2 = 7931
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Figure 10. Comparison between experimental (symbols) and calculated (eq 11) values of [R*]/[C0] at [M0]/[C0] = 10 000 and the following temperatures: (a) 130, (b) 145, (c) 160, and (d) 180 C.
kte), and, finally, nonradical random chain scission (reactions gi; kde). According to this last reaction, active, dormant, and dead polymer chains can degrade in the presence of catalyst producing two shorter chains, one of which is the dead polymer chain, Gn. Note that the reactivity of reaction steps cf is assumed independent upon the direction; i.e., the corresponding equilibrium constant is equal to 1. While activation and reversible chain transfer reactions are at equilibrium,13 the remaining reactions (propagation, intermolecular transesterification, and nonradical random chain scission) define the process kinetics. This means that all polymer chains are either active or dormant, i.e., bearing an OH terminal group. Moreover, since the living behavior is maintained for most of the reaction duration (cf. Figure 1b), it is clear that reactions cf are much more relevant than the termination reactions gi. Accordingly, a preliminary kinetic analysis can be carried out neglecting the latter degradation reactions and focusing on the system behavior in the first part of the reaction where the system is still living. Note that the same analysis has been already reported with detail in our previous paper:13 therefore, it is only summarized here to show that the final result is not affected and to correct a misprint present in the corresponding final equation (eq 24 in ref 13). We evaluate the total concentrations of active, [R*], and OHbearing species, [OH] (previously indicated by [D] in eq 4) as follows: ¥
½R ¼
∑ ½R n n¼0
½OH ¼
∑ ½Dn n¼0
¥
ð5Þ
ð6Þ
Table 3. Equilibrium Constant Values Keq,a at Different Temperatures for the ROP of LA temperature (C)
Keq,a
temperature (C)
Keq,a
130
0.045
160
0.126
145
0.101
180
0.256
so that the material balance of the active chain is simply expressed as d½R ¼ ka1 ½C½OH ka2 ½R ½A dt
ð7Þ
The material balances of catalyst, OH-bearing species, and acid are given by very similar expressions and in particular the following differential constraints apply: d½R ¼ d½OH ¼ d½C ¼ d½A
ð8Þ
By integrating eq 8, the amounts of the different species at any time are related to the initial values ([R*] = 0, [OH] = [OH0], [C] = [C0], and [A] = [A0] at t = 0) as follows: ½R ¼ ½OH0 ½OH ¼ ½C0 ½C ¼ ½A ½A 0
ð9Þ
It is worth noticing that these relationships apply all along the process, from the beginning to the end. Therefore, given the time evolution of a reference variable, e.g., the catalyst concentration [C], the total concentrations of all the other species (active ([R*]), OH-bearing ([OH]), and acid ([A])) are readily evaluated. As indicated above, the linearity of Y vs time and Mn vs conversion plots (Figure 1) demonstrates that all living and dormant chains are formed immediately after the reaction start, and their number does not change later on during the living stage 7932
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of the polymerization. This means that equilibrium conditions are fully established for the activation reaction a, so that the material balances (eq 7) can be replaced by the following equation: ½R ½A ka1 ¼ Keq, a ½C½OH ka2
ð10Þ
where Keq,a is the equilibrium constant of the catalyst activation reaction. Combining this equation with eq 9, [C] can be obtained as a function of the initial concentrations and of the activation equilibrium constant by solving the resulting quadratic algebraic equation as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B þ B2 þ 4½C0 ðKeq, a 1Þð½C0 þ ½A 0 Þ ð11Þ ½C ¼ 2ðKeq, a 1Þ where B = [A0] þ 2[C0] þ Keq,a ([OH0] [C0]). As expected, the equilibrium concentration of the catalyst is equal to its initial value for zero alcohol concentration while it approaches zero for [OH0] . [C0], [A0], respectively. To evaluate the actual amount of residual catalyst from eq 11, the initial amounts of cocatalyst and acid are needed; such values are indeed difficult to estimate because of the contribution of impurities entering the system with the reagents and from the environment. According to eq 9, the overall concentration of polymer chains, NC = [R*] þ [OH], is equal to the initial concentration of OH-bearing species. On the other hand, the actual value of this same quantity is readily estimated through eq 4: therefore, any discrepancy with respect to the initial alcohol concentration is indicative of the contribution of OH-bearing species associated with environmental impurities. The results of such evaluations are summarized in Table 1 for a specific set of
Figure 11. Semilogarithmic plot of the reversible activation equilibrium constant at different temperatures.
reactions ([M0]/[C0] = 10 000; different values of [OH0]/[C0], 180 C). In the table, the [OH0]/[C0] values from the recipe are indicated as “nominal” (subscript nom), while those estimated from the molecular weight, as “experimental” (subscript exp). In all cases, the experimental values are larger than the nominal ones, thus confirming the presence of OH-bearing impurities; moreover, the estimated values of the concentration of such impurities, [IM], are quite constant, ranging from 3 to 5 in terms of molar ratio with respect to the initial amount of catalyst. This same analysis has been applied to all reactions carried out in this work, and the actual values of [OH0]/[C0] (=[OH0,exp]/[C0]) estimated this way have been always used in model simulations. The range of values of estimated impurities concentration in monomer bulk was 25 mmol L1. A complete collection of the specific values at all temperatures and reaction conditions can be found elsewhere.32 On the other hand, the initial concentration of acidic impurities is also needed. As mentioned above, the acidic impurities in the fresh monomer bulk are around 0.25 mequiv kg1; when the monomercatalyst ratio is equal to 10 000, this value corresponds to [A0]/[C0] = 0.36. Exactly this value will be used in all of the simulations presented in the following, thus assuming that such impurities come from the monomer and merging them with the “natural” acid in the system, i.e., octanoic acid. According to the analysis above, when an excess amount of cocatalyst is used, the catalyst is fully activated and [R*] = [C0]. On the other hand, from eq 3, the slopes of the lines in Figure 1a are equal to the product of kp and [R*] so that, with [R*] = [C0], kp is readily evaluated. At the conditions under examination (180 C), a value of 37 000 L mol1 h1 is obtained. Applying similar preliminary analysis to experimental data at other temperature values, the corresponding kp values have been estimated and collected in Table 2. Note that, from our previous study,13 although [OH0]/[C0] = 100, the catalyst was still not fully activated, thus resulting in an underestimated kp value. The new experiments at 130 C have been implemented in order to cover [OH0]/[C0] ranging from 0.5 to 600, and the new propagation rate coefficient has been estimated as 4500 L mol1 h1. The corresponding activation energy and preexponential factor have been estimated from the Arrhenius plot in Figure 9. The values are Ea,p = 63.3 kJ mol1 and kp0 = 7.4 1011 L mol1 h1, respectively, and the average error of these estimations is (3.6%. Having estimated kp, we can now estimate the values of [R*]/[C0] as a function of [OH0]/[C0] from the slopes in Figure 1a, as shown in Figure 10. On the other hand, for the same values of the initial concentrations, we can compute the ratio [R*]/[C0] using eqs 9 and 11 for any given value of the equilibrium constant, Keq,a. The curve shown in Figure 10d corresponds to the value Keq,a = 0.256, which has been established as the best fit of the experimental data.
Table 4. Numerical Values of Selected Model Parameters at 180 C parameter
symbol
value
source
activation equilibrium constant
Keq,a
0.256
this work
activation rate coefficients
ka1
106 L mol1 h1
13
propagation rate coefficient
kp
3.7 104 L mol1 h1
this work
monomer equilibrium concentration
[Meq]
0.225 mol L1
33
chain-transfer rate coefficients intermolecular transesterification rate coefficient
ks1, ks2 kte
106 L mol1 h1 90 L mol1 h1
13 model fitting
nonradical chain scission reaction rate coefficient
kde
2.8 104 h1
model fitting
7933
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Industrial & Engineering Chemistry Research Similar analysis can be applied to the data at 130, 145, and 160 C,32 and the corresponding fittings have been shown in Figure 10ac, respectively. As mentioned above, the kp at 130 C has been underestimated, thus resulting in a wrong Keq,a as previously estimated.13 In this work, a set of new experimental data is introduced and refitted by our model and the estimated Keq,a have been all collected in Table 3 and plotted in Figure 11 in
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Figure 12. Conversion vs time for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C, [M0]/[C0] = 10 000, and different [OH0]/[C0] values (experimental data): 0, 20; O, 34; Δ, 60; r, 100. Simulation results: dashed lines (reaction rate increases with increasing values of [OH0]/[C0]).
logarithmic scale vs the reciprocal of the corresponding temperatures. These values indicate a slightly endothermal reaction, positively affected by the increasing temperature. Model Development. We now consider the population balance equations corresponding to the kinetic scheme in Figure 8. Here, in contrast to the simplified approach discussed in the previous section, all of the reactions are accounted for in order to calculate correctly the polymer molecular weight. The model equations accounting for reactions ae in Figure 8 are the same considered in our previous work.13 Therefore, the equations reported as follows consider in addition the nonradical random chain scission reactions gi, along with the balances for the new polymer species Gn, the so-called dead polymer chains. Moreover, transesterification between living and dead chains, i.e,. reaction f in Figure 8, has been included in the new kinetic scheme. Before discussing the new model equations, we note that intermolecular transesterification reactions are very typical in polyester production and correspond to chain reshuffling through breakage of a generic ester bond.9,12,33 Therefore, to keep track of this kinetic event, we need to consider an independent variable in the molecular weight distribution calculation, the number of lactoyl groups in the chain (-OCH(CH3)CO-) rather than the number of lactoyl lactate units (-OCH(CH3)COOCH(CH3)CO-, also known as dilactide), as done in our previous model.13 Note that this also affects the value of the reaction rate coefficient of intermolecular transesterifica-
Figure 13. Mn vs conversion for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C, [M0]/[C0] = 10 000, and different [OH0]/[C0] values (experimental data): 0, 20; O, 34; Δ, 60; r, 100. Simulation results: dashed lines.
Figure 15. Mw vs conversion for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C. [M0]/[C0] = 10 000, and different [OH0]/[C0] values (experimental data): 0, 20; O, 34; Δ, 60; r, 100. Simulation results: dashed lines.
Figure 14. Mn vs time for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C. [M0]/[C0] = 10 000, and different [OH0]/[C0] values (experimental data): 0, 20; O, 34; Δ, 60; r, 100. Simulation results: dashed lines. Panels a and b show results at short and long time ranges, respectively. 7934
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Figure 16. Mw versus time for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C, [M0]/[C0] = 10 000, and different [OH0]/[C0] values (experimental data): 0, 20; O, 34; Δ, 60; r, 100. Simulation results: dashed lines. Panels a and b show results at short and long time ranges, respectively.
d½A ¼ ka1 ½Cμ0 ka2 ½Aλ0 dt
ð13Þ
dM ¼ kp Mλ0 þ kd ðλ0 R 1 R 0 Þ dt
ð14Þ
d½R n ¼ ka1 ½Dn ½C ka2 ½R n ½A þ 1 δn, 0 1 δn, 1 kp ½R n2 ½M dt kd ½R n Þ þ kd ½R nþ2 kp ½R n ½M ks ½R n μ0 þ ks ½Dn λ0 ¥ kte f½R n ðλ1 λ0 þ R 0 Þ 1 δn, 0 λ0 ½R j
þ 1 δn, 0
1 δn, 1 ðn 1Þ½R n λ0
∑
j ¼ nþ1
Figure 17. Dispersity vs conversion for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C, [M0]/[C0] = 10 000, and different [OH0]/[C0] values (experimental data): 0, 20; O, 34; Δ, 60; r, 100. Simulation results: dashed lines.
n1 1 δn , 0 ½R i 1 δn , 0 λ 0
∑ ½Di þ ½R n γ1 γ0 þ G0 i¼n þ 1
Table 5. Numerical Values of All Model Parameters at 145 and 160 C
1 δn, 0 λ0
∑ ½Gi g kde ðn 1Þ½R n i¼n þ 1
∑ i¼0
value parameter
symbol
intermolecular
130 C
145 C
þ kde 160 C
kte (L mol1 h1)
6a
kde (h1)
1 105 4 105 1.2 104
12
chain scission rate coefficient
ref 13.
¥
∑ ½R i i¼n þ 1
ð15Þ
kde ðn 1Þ½Dn þ kde
tion, kte, since double amounts of repeating units have to be considered to properly account for all available ester bonds. Therefore, the previously estimated value of kte (12 L mol1 h1) is twice the one consistent with choosing the number of lactoyl groups into the chains as the independent variable.13 We now consider the set of material and population balances, with reference to a well-stirred, homogeneous batch reactor, for the kinetic scheme in Figure 8: d½C ¼ ka1 ½Cμ0 þ ka2 ½Aλ0 dt
¥
∑
rate coefficient
a
¥
d½Dn ¼ ka1 ½Dn ½C þ ka2 ½R n ½A þ ks ½R n μ0 ks ½Dn λ0 dt ¥ n1 þ kte 1 δn, 0 ½R i ½Dk i¼0 k¼n þ 1 i kte 1 δn, 0 1 δn, 1 ðn 1Þ½Dn λ0
30
transesterification nonradical random
¥
∑ ½R k þ ½R n μ1 μ0 þ D0 k¼n þ 1 i
∑
¥
∑ ½Di i¼n þ 1
¥ d½Gk ¼ kte 1 δn, 0 λ0 ½Gi dt i ¼ nþ1 kte 1 δn, 0 1 δn, 1 ðn 1Þ½Gn λ0 2kde 1 δn, 0 1 δn, 1 ðn 1Þ½Gn ¥ þ2kde 1 δn, 0 ½Gm þ kde 1 δn, 0
ð16Þ
∑
∑ m ¼ nþ1
þkde 1 δn, 0
ð12Þ 7935
¥
∑ ½Dk k ¼ nþ1
¥
∑ ½R k k ¼ nþ1 ð17Þ
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Figure 18. Mn vs time at different time scale for ROP of LA at 130 C, [M0]/[C0] = 6000, and [OH0]/[C0] = 9. Simulation results: dashed curves. Panels a and b show results at short and long time ranges, respectively.
Table 6. Activation Energies and Preexponential Factors for Intermolecular Transesterification and Nonradical Random Chain Scission Reaction reaction intermolecular transesterification nonradical random chain scission
parameter 1
value
Ea,te (kJ mol )
83.3
kte0 (L mol1 h1)
3.38 1011
1
Ea,de (kJ mol )
101.5
kde0 (h1)
1.69 108
Figure 19. Arrhenius plot of intermolecular transesterification rate coefficient in ROP of LA. Solid line: linear fit.
Figure 21. Conversion vs time for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C, [M0]/[C0] = 20 000, and various [OH0]/[C0] values (experimental data): 0, 40; O, 60; Δ, 120. Simulation results: dashed lines.
Figure 20. Arrhenius plot of the nonradical random chain scission rate coefficient in ROP of LA. Solid line: linear fit.
Equations 1214 are the material balances of the monomeric species present in the kinetic model (Figure 8); eqs 1517 are population balances of all active and dormant as well as dead chains, respectively, considering reversible activation, propagation, chain transfer, intermolecular transesterification, and the newly developed irreversible termination, random chain scission. All these equations have been numerically solved using the method of moments. The resulting moment equations involve specific chain lengths (such as [R0] and (R1]) and such terms have been omitted in the final equations. Therefore, they are not
fully general but indeed applicable to high molecular weights, like in the cases under examination in this work. The corresponding simplified moment equations are reported in the Appendix. Parameter Evaluation. Since the reacting system under examination is relatively complex, the proposed kinetic model involves many parameters, such as reaction rates and equilibrium constants. To maximize the model reliability, it is advisible to minimize the number of parameters estimated by direct fitting to the experimental data while proceeding whenever possible with independent parameter estimation. Accordingly, some of them have been evaluated through the preliminary analysis presented before, i.e., the activation equilibrium constant, Keq,a, and the propagation rate coefficient, kp. Moreover, the value of the propagation equilibrium constant kp/kd, in terms of monomer 7936
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equilibrium concentration [Meq], is taken from the literature33 and the value of the chain-transfer equilibrium constant was set equal to 1.13 Finally, under the assumption of fast activation and deactivation reactions, the corresponding rate coefficients, ka1 and ks1, respectively, have been set to a very large value (106 L mol1 h1), large enough to ensure equilibrium conditions.13 All of these values are summarized in Table 4 along with the corresponding sources. Such values apply to a reaction temperature of 180 C. This way, we are left with only two parameters: kte and kde. They have been estimated by direct fitting of the model predictions to the experimental data, and the obtained values are also reported in Table 4. The comparison between experimental data and simulation results is shown in Figures 1217. The prediction of conversion in Figure 12 is quite satisfactory: it is seen that even when small catalyst concentrations are used, the polymerization rate of LA is very fast and propagation equilibrium is established in a few minutes. Some discrepancies in the equilibrium conversion values appear and have been discussed in our previous study due to the limitation of characterization which caused the underestimation of the monomer equilibrium concentration.13 However, the literature value of [Meq] was used without any parameter adjustment. In Figures 13 and 14, we see that Mn is controlled by the cocatalyst concentration. At longer reaction times, Mn exhibits a clear decrease (see Figure 14b) which is less significant for increasing cocatalyst amounts, as correctly predicted by the model. The decreasing molecular weight is well-predicted also
in terms of Mw (Figures 15 and 16), as well as the corresponding increase of dispersity (Figure 17). However, model prediction of Mw is slightly overestimated according to the experimental results of Mw vs t. Since the Mn is mainly determined by the cocatalyst and the natural impurities such as water, Mw will be mainly affected by intermolecular transesterification. When the reaction system has a small amount of cocatalyst, the molecular weight of the resulting polymer is higher than 100 000 Da with very high viscosity. Therefore, the chain mobility has been reduced so that the transesterification reaction rate might be reduced. Thus, the higher weight average molecular weight is predicted from the model since the ideal batch reactor is assumed to be used. Compared to the data at 130 C1, the increase of the polymer dispersity at higher conversion values is much more pronounced at high [OH0]/[C0] ratios, thus confirming that temperature is the most important variable in determining the relevance of intermolecular transesterification reaction. Finally, it is worth noting that the chain scission reactions are much slower than all of the other reactions operating in the system. This clearly appears from the data shown in Figures 12 and 14. When the molecular weights start decreasing, chain propagation is practically over and therefore high molecular weight can be preserved by quenching the reaction at a proper time. The estimated values of the rate coefficients for intermolecular transesterification and nonradical random chain scission degradation are reported in the last two rows in Table 4. The ratio between kp and kte is 411, which is slightly smaller than the
Figure 22. Mn vs conversion for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C, [M0]/[C0] = 20 000, and various [OH0]/[C0] values (experimental data): 0, 40; O, 60; Δ, 120. Simulation results: dashed lines.
Figure 24. Mw vs conversion for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C. [M0]/[C0] = 20 000, and various [OH0]/[C0] values (experimental data): 0, 40; O, 60; Δ, 120. Simulation results: dashed lines.
Figure 23. Mn vs time for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C, [M0]/[C0] = 20 000, and various [OH0]/[C0] values (experimental data): 0, 40; O, 60; Δ, 120. Simulation results: dashed lines. Panels a and b show results at short and long time ranges, respectively. 7937
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Figure 25. Mw vs time for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C, [M0]/[C0] = 20 000, and various [OH0]/[C0] values (experimental data): 0, 40; O, 60; Δ, 120. Simulation results: dashed lines. Panels a and b show results at short and long time ranges, respectively.
Figure 26. Dispersity vs conversion for LA polymerization with Sn(Oct)2 and 1-dodecanol at 180 C, [M0]/[C0] = 20 000, and various [OH0]/[C0] values (experimental data): 0, 40; O, 60; Δ, 120. Simulation results: dashed lines.
corresponding value 750 at 130 C. This means that reversible propagation has lower activation energy than intermolecular transesterification. For the nonradical chain scission, the estimated value is in the same order of the literature value obtained at 180 C,31 and falls in the usual range of values for polyester degradation reactions.16,34 Following the same approach, experimental data at 145 and 160 C have been fitted32 in order to estimate the corresponding kte and kde values at those temperatures. The obtained values are reported in Table 5. Note that in order to estimate the experimental value of kde at 130 C reported in the table, an additional experimental run has been performed at very long times to highlight the effect of the scission reaction. The corresponding results are shown in Figure 18 in terms of Mn as a function of time. Combining the rate coefficients determined in this work with the corresponding values available in the literature at 130 C1, the Arrhenius plots of kte and kde shown in Figures 19 and 20 are obtained. The linearity of the data is quite satisfactory, and the corresponding values of activation energy and preexponential factor are summarized in Table 6. The activation energy of this reaction is in good agreement with literature values of the activation energy of random chain scission for PLA16,31 estimated at temperatures ranging from 180 to 400 C. The uncertainties in estimating the activation energies and preexponential factors of transesterification and random chain scission are (6.9 and (17%, respectively.
Model Validation. To test the model reliability, its results are compared in a fully predictive way to a set of experimental data different from the one used in the previous section for parameter estimation. In particular, we checked the capability of the model to predict the effect of the catalyst concentration and cocatalyst amount. The value of the ratio [M0]/[C0] was changed from 10 000 to 20 000 and [OH0]/[C0] from 40 to 120 while keeping the temperature at 180 C. All model parameter values have not been changed and have been kept as in Table 4, while the model predictions are compared with the experimental data in Figures 2126 in terms of conversion as a function of time (Figure 21), Mn and Mw as a function of conversion and time (Figures 2225) and dispersity as a function of conversion (Figure 26). The agreement is generally good, thus confirming the validity of the selected kinetic scheme as well as of the corresponding parameters values.
’ CONCLUSION A model for L,L-lactide polymerization at temperatures ranging from 130 to 180 C in bulk, with Sn(Oct)2 as the catalyst and 1-dodecanol as cocatalyst, has been developed, including interand intramolecular transesterification reactions. Model validation and parameter evaluation were carried out by comparison with experimental data at different values of the ratios of monomer to catalyst and monomer to cocatalyst. The reaction responsible for the significant decrease of molecular weight taking place at sufficiently long reaction time, which actually decreases significantly as the temperature increases, is random chain scission reaction. It is found that a satisfactory prediction of the MWD is achieved when both intermolecular transesterification and nonradical random chain scission are included in the kinetic scheme. The values of the kinetic parameters of all reactions in the considered kinetic scheme have been estimated, and the model predictions have been validated in a wide range of operating conditions, including the temperature range of 130180 C and the monomer to catalyst ratio range of 10 00020 000 as well as the cocatalyst to catalyst ratio range of 0.5100. Overall, the average errors of the model predictions at various conditions with respect to experimental data are well below 5%, which indicates the reliability of this model. ’ APPENDIX j Moment equations of the active chains (λj = Σ¥ n=0 n [Rn]; j = 0, 1, 2): 7938
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along with the corresponding closure formulas:35
dλ1 ¼ ka1 μ1 ½C ka2 λ1 ½A þ 2kp ½Mλ0 2kd λ0 ks λ1 μ0 dt 1 þ ks μ1 λ0 kte λ1 ðμ1 μ0 Þ þ kte λ0 ðμ2 μ1 Þ 2 1 1 kte λ1 ðγ1 γ0 Þ þ kte λ0 ðγ2 γ1 Þ kde ðλ2 λ1 Þ 2 2 dλ2 ¼ ka1 μ2 ½C ka2 λ2 ½A þ 4kp ½Mðλ1 þ λ0 Þ þ 4kd ðλ0 λ1 Þ dt 1 ks λ2 μ0 þ ks μ2 λ0 þ kte λ0 ðλ1 λ3 Þ þ kte λ1 ðλ2 λ1 Þ 3 1 kte λ2 ðμ1 μ0 Þ þ kte λ0 ð2μ3 3μ2 þ μ1 Þ 6 1 kte λ2 ðγ1 γ0 Þ þ kte λ0 ð2γ3 3γ2 þ γ1 Þ 6 1 kde ð4λ3 3λ2 λ1 Þ 6 j Moment equations of the dormant chains (μj = Σ¥ n=0 n [Dn]; j = 0, 1, 2):
dμ0 ¼ ka1 μ0 ½C þ ka2 λ0 ½A dt dμ1 ¼ ka1 μ1 ½C þ ka2 λ1 ½A þ ks λ1 μ0 ks μ1 λ0 dt 1 1 þ kte λ1 ðμ1 μ0 Þ kte λ0 ðμ2 μ1 Þ kde ðμ2 μ1 Þ 2 2 dμ2 ¼ ka1 μ2 ½C þ ka2 λ2 ½A þ ks λ2 μ0 ks μ2 λ0 dt þ kte λ2 ðμ1 μ0 Þ þ kte λ1 ðμ2 μ1 Þ 1 1 þ kte λ0 ð4μ3 þ 3μ2 þ μ1 Þ kde ð4μ3 3μ2 μ1 Þ 6 6 j Moment equations of the terminated chains (γj = Σ¥ n=0 n [Gn]; j = 0, 1, 2):
dγ0 ¼ kde ðλ1 λ0 Þ þ kde ðμ1 μ0 Þ dt dγ1 1 ¼ kte λ1 ðγ1 γ0 Þ kte λ0 ðγ2 γ1 Þ kde ðγ2 γ1 Þ 2 dt 1 1 þ kde ðλ2 λ1 Þ þ kde ðμ2 μ1 Þ 2 2 dγ2 ¼ kte λ2 ðγ1 γ0 Þ þ kte λ1 ðγ2 γ1 Þ dt 1 1 þ kte λ0 ð4γ3 þ 3γ2 þ γ1 Þ kde ð4γ3 3γ2 γ1 Þ 6 3 1 1 þ kde ð2λ3 3λ2 þ λ1 Þ þ kde ð2μ3 3μ2 þ μ1 Þ 6 6
λ3 =
λ2 ð2λ2 λ0 λ1 2 Þ ; λ1 λ0
γ3 =
γ2 ð2γ2 γ0 γ1 2 Þ γ1 γ0
μ3 =
μ2 ð2μ2 μ0 μ1 2 Þ ; μ1 μ0
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT The financial support provided by the Swiss Commission for Technology and Innovation (KTI/CTI; Project No. 8611.2 PFIW-IW) is gratefully acknowledged. ’ NOTATION A = acid [A] = concentration of acid [A0] = initial concentration of acid a, K = MarkHouwink constants B = characteristic constant of PLA ring-chain equilibrium C = catalyst, Sn(Oct)2 [C] = concentration of catalyst [C0] = initial concentration of catalyst CPx = cyclic polymer with x repeating units D, OH = dormant chains, OH-bearing species [D], [OH] = concentration of cocatalyst, 1-dodecanol [D0], [OH0] = initial concentration of cocatalyst, 1-dodecanol Dn = dormant chains with n repeating units Ea,de = activation energy of nonradical random chain scission Ea,p = activation energy of reversible propagation Ea,te = activation energy of intermolecular transesterification Gn = terminated PLA chains IM = OH-bearing impurities [IM] = concentration of OH-bearing impurities ka1, ka2 = reversible catalyst activation rate coefficient kd = depropagation rate coefficient kde = nonradical random chain scission rate coefficient kde0 = preexponential factor of nonradical random chain scission rate coefficient Keq,a = reversible catalyst activation equilibrium constant Keq,c = equilibrium constant of PLA ring-chain equilibrium kp = propagation rate coefficient kp0 = preexponential factor of propagation rate coefficient ks = reversible chain transfer rate coefficient kte = intermolecular transesterification rate coefficient kte0 = preexponential factor of intermolecular transesterification rate coefficient mmon = molar mass of monomer M = monomer M1, M2 = polymer molecular weights, cf. eq 2 [M] = instantaneous monomer concentration [M0] = initial monomer concentration [Meq] = equilibrium monomer concentration Mn = number average molecular weight Mw = weight average molecular weight 7939
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Industrial & Engineering Chemistry Research MWD = molecular weight distribution Nc = overall concentration of polymer chains [OH0,nom] = nominated initial cocatalyst concentration [OH0,exp] = experimental initial cocatalyst concentration Pd = dispersity PLA = poly(lactic acid) Pn = polymer chain with chain length n R* = active chains [R*] = concentration of active chains R0 = activated catalyst, tin alkoxide Rn = active chains with n repeating units RA = ratio of MALDI-TOF abundances of cyclic to linear PLA ROP = ring-opening polymerization SEC = size exclusion chromatography t = time T = temperature X = conversion Y = ln[([M0] [Meq])/([M] [Meq])] λ0, λ1, λ2, λ3 = zero, first, second, and third moments of active chains, respectively μ0, μ1, μ2, μ3 = zero, first, second, and third moments of dormant chains, respectively γ0, γ1, γ2, γ3 = zero, first, second, and third moments of dead chains, respectively δi,j = Kroenecker delta (1 if i = j, 0 otherwise)
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