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Biomacromolecules 2010, 11, 1118–1124

Autocatalytic Equation Describing the Change in Molecular Weight during Hydrolytic Degradation of Aliphatic Polyesters Harro Antheunis,† Jan-Cees van der Meer,‡ Matthijs de Geus,§ Andreas Heise,§ and Cor E. Koning*,§ Product Development Department, N.V. Organon, P.O. Box 20, 5340 BH Oss, The Netherlands, and Centre for Analysis Scientific Computing and Applications and Laboratory of Polymer Chemistry, Eindhoven University of Technology, Den Dolech 2, 5600 MB Eindhoven, The Netherlands Received February 1, 2010; Revised Manuscript Received February 18, 2010

The autocatalytic equation derived in this study describes and even predicts the evolution of the number average molecular weight of aliphatic polyesters upon hydrolytic degradation. The main reaction in the degradation of aliphatic polyesters is autocatalytic hydrolysis of ester bonds, which causes the molecular weight to decrease. During hydrolysis of the ester bonds in the main chain of the polyester, the chains are cleaved and the end group concentrations will rise. The fundamentals of this equation are based on that principle. To validate the derived equation, the hydrolytic degradation of poly(4-methylcaprolactone), poly(ε-caprolactone), poly(D,L-lactide), and two different poly(D,L-lactide-co-glycolide) copolymers was monitored after immersion in a PBS buffer (pH ) 7.4) at 37 °C. The number average molecular weight, mass loss, and crystallinity were determined after different time intervals. The experimental results confirm that hydrolytic degradation of aliphatic polyesters is a bulk erosion process. When comparing the Mn, calculated with the new autocatalytic equation, with the experimental results, it was found that the new model can predict the decrease of the Mn upon hydrolytic degradation for semicrystalline and amorphous polymers, as well as for copolymers, without the need for complicated mathematics and excessive input parameters. This is a major improvement with respect to earlier proposed models in literature.

Introduction Nowadays, for ecological and environmental reasons there is much interest in replacing synthetic polymers with a long lifetime by biodegradable polymers which have a much shorter lifespan. An additional benefit of these biodegradable polymers, which have a predictable and controllable degradation process, is that they can be used for biomedical and pharmaceutical applications. Therefore, lots of studies have been performed by industrial and academic research groups on applications of such polymers and their degradation mechanisms.1–4 For the design of a degradable product like surgical implants, drug delivery devices, and even for disposable materials, it is of major importance that the degradation speed is controlled. Aliphatic polyesters form one of the main groups of degradable polymers, and within this work we focus on aliphatic polyesters in an aqueous environment. When an aliphatic polyester is submerged in an aqueous environment, the water will only penetrate into the amorphous domains of the polymer because it cannot enter the crystalline regions, if present.5 The diffusion of water into the polymer is faster than the hydrolysis rate of the ester bonds,6 and therefore, it is considered that the degradation of an aliphatic polyester is a bulk erosion process.7–9 With the presence of water and a catalyst, a random hydrolysis of the ester bonds of the polyester backbone in the amorphous domains occurs, making the chains shorter and forming hydroxyl and carboxylic acid end groups. Hydrolysis of an ester bond under mild conditions can only * To whom correspondence should be addressed. Phone: +31-402475353. Fax: +31-40-2463966. E-mail: [email protected]. † N.V. Organon (part of Schering-Plough). ‡ Centre for Analysis Scientific Computing and Applications, Eindhoven University of Technology. § Laboratory of Polymer Chemistry, Eindhoven University of Technology.

occur when it is acid- or base-catalyzed and not by water alone.10 The formed carboxylic acid end groups in the polymer matrix catalyze further hydrolysis, and when a polymer initially has a high amount of carboxylic acid end groups, the hydrolysis is accelerated from the very beginning of the degradation process.11 With time, the chain fragments will be short enough to dissolve into the aqueous medium and the polymer matrix starts losing weight.12 Before the polymer system loses weight, there is accumulation in the amorphous phase of carboxylic acid end groups, which autocatalytically accelerate the hydrolysis. Furthermore, at higher temperatures, the degradation rate will be faster.13 It should further be kept in mind that the chemical composition of the copolymers14–17 and the crystallinity18,19 may change during degradation, thereby also influencing the degradation rate of the polymer. In our previous work20 a detailed description was given of the hydrolytic degradation of aliphatic polyesters and a detailed mathematical model was developed that describes the development of the total molecular weight distribution (MWD) and the mass loss during the hydrolytic degradation of aliphatic polyesters, including semicrystalline- and copolyesters. The objective of the present study was to develop a very simple model, an equation without complex mathematics, that describes and even predicts the Mn decrease of aliphatic polyesters in an aqueous environment. Additionally, the derived equation is validated by experimental data and is evaluated by comparison to other equations known in literature. To enable a direct and scientifically correct comparison of the current, simplified model with our previous, more complicated model,20 the same experimental data is used for validation. Therefore, some experimental data used in our previous paper20 are repeated in this work.

10.1021/bm100125b  2010 American Chemical Society Published on Web 02/26/2010

Autocatalytic Equation of the Degradation of Polyesters

Existing Degradation Models When in this work the degradation of a polymer is mentioned, all the effects during the hydrolysis of linear aliphatic polyesters are considered, referring to the chain scission of the ester bond within the polymer backbone. Several theories have been developed to describe the degradation of aliphatic polyesters. Both the fundamentals and the assumptions of each approach are explained below, and the weak and strong parts are identified and discussed. Finally, the new autocatalytic model is derived, which gives a straightforward analytical solution to describe the development of the number average molecular weight (Mn) during hydrolytic degradation. Finally, a comparison is made with the other models. Empirical Model/Monte Carlo Model. One of the approaches to describe the degradation of a polymer is to use an empirical model.21 Another approach is based on the Monte Carlo sampling technique. The polymer is divided into a grid and the lifetime of each compartment is determined with random numbers. This approach is mainly used to describe the erosion of the polymer.3,22,23 The advantage of the model is that an estimation of the degradation can be made without knowing the mechanism behind it. The disadvantage of this approach is that the models are not based on the real mechanism of degradation, which makes it difficult to predict the degradation of the polymer based on the initial polymer properties. Pseudo-First-Order Kinetics. In a study toward the reaction kinetics of chain scission of poly(ε-caprolactone) (PCL) and poly(D,L-lactide-co-glycolide) (PLGA) during degradation, Pitt and co-workers24 proposed that normal hydrolysis follows first order kinetics in absence of catalytic effects, whereas autocatalytic hydrolysis follows second order kinetics. In the derivation of this model it is assumed that the degradation follows pseudofirst order kinetics. Only the carboxylic acid concentration is taken into account, whereas the concentration of ester bonds is ignored, while the reaction rate of hydrolysis depends on both concentrations. The advantage of the model is that with simple calculations the decrease of the molecular weight can be calculated. The disadvantage of this pseudofirst order kinetical model is that the autocatalytic behavior of the degradation is not incorporated into the mathematics of the model. Second Order Kinetics. In the work of Lyu et al.25 the authors assumed that the rate of degradation depends on both the concentration of breaking bonds and water. For that a second order kinetic equation is derived. However, the autocatalytic effect of the formation of degradation products is neglected and it is also assumed that the water concentration within the polymer stays constant. The obtained rate equation is a first order rate equation, which is used as a basis for a model describing both the bulk degradation of a polymer and the mass loss during surface degradation or degradation with a moving erosion front in the polymer. The advantage of the second order kinetics model is that the mass loss is predicted for a polymer, which degrades according to a surface process or with a moving erosion front. The disadvantage of model is that the autocatalytic behavior of the degradation is not incorporated into the mathematics of the model. Degradation Theory on a Statistical Basis. To describe the degradation of polymer chains, Montroll and Simha used a statistical approach.26 The model is based on the probability that bonds in a polymer chain can break. To calculate the molecular weight distribution and the degree of degradation the model assumes that all initial molecules have the same molecular weight, that the accessibility to reaction of a bond in a given chain is independent of its position in the chain and independent

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of the length of its parent chain and that all chains in the mixture are equally accessible to reaction. Second, the model is based on the probability that an ester bond in a polymer chain can be broken. A consequence of this approach is the lack of a time parameter in the model. Therefore, the rate of degradation as function of time cannot be calculated. Only the molecular weight distribution can be calculated as function of the number of chain scissions that has taken place. The advantage of the statistical model is that the complete molecular weight distribution of a polymer can be calculated after a certain degree of degradation. The disadvantage of the model is that the evolution of the degradation in time can not be calculated. Random Chain Scission. In the early 1940s, Simha developed the random chain scission theory to describe the kinetics of degradation and the size distribution of long chain polymers.27 The theory is based on the principle that every connection between two monomer units in the chain can break and that chain scission is a random process. It is assumed that the cleavage of each connection has the same reaction rate constant and that the chain scission reaction follows first order kinetics. For each length of a polymer chain a rate equation exists, which includes the decrease of the original polymer chain length and the formation of a new polymer chain length, being degradation products of longer polymer chains. This approach leads to a set of linear differential equations that can be solved. The big advantage of this model is that all polymer chain lengths are taken into account and can be calculated as a function of time. The disadvantage of this random chain scission model is that it can only describe first order degradation kinetics and no autocatalytic processes can be incorporated. End Chain Scission. Besides the random chain scission theory, Simha also developed a model in which the degradation of the polymer only takes place at the end of the chains.27 In this model it is assumed that the rate of “end scission” is independent of the length of the chain, as well as that the rate of chain scission of all other bonds is assumed to be zero. Those assumptions are applied to a set of linear differential equations describing first order degradation kinetics. The disadvantage of this approach is the limitation that the model only describes first order degradation kinetics and that it is impossible to incorporate autocatalytic processes. Random Chain Scission and End Scission. To describe the drug release out of a spherical bulk-eroding microsphere, Batycky et al. developed a theoretical model.28 Part of that model describes the degradation behavior of the polymer. This model is a combination of the previous two models (“random chain scission” and “end chain scission” models). They considered that random chain scission and end chain scission takes place simultaneously. The rate of the hydrolysis of the ester bonds in the polymer chain constantly increases inward starting from the ends. The same set of coupled linear differential equations has been applied as in the random chain scission model. However, in this model not all reaction rate constants of all the linkages between the monomeric units are considered to be identical. Bonds at the end of the chains are assumed to have a higher reaction rate constant than those in the backbone. The disadvantage is that the model only describes first order degradation kinetics and no autocatalytic processes are incorporated. Improved Mathematical Model. This model is based on an autocatalytic chain scission to describe the kinetics of hydrolytic degradation and the development of the molecular weight distribution of long chain polymers. The model, developed by Antheunis at al.,20 is based on the principle that every connection between

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two monomer residues in the chain can break, that chain scission is a random process, and that after each chain scission the formed carboxylic acid chain ends can catalyze the next chain scission. This model includes that the cleavage of each connection can have a different reaction rate constant, and also a correction for crystalline domains present in semicrystalline polymers can be incorporated. For each length of a polymer chain a rate equation exists, which includes the decrease of the original polymer chain length and the formation of new polymer chain lengths, being the degradation products of longer polymer chains. This approach leads to a set of nonlinear differential equations that can be solved numerically. The advantage of the model is that the change in molecular weight distribution, the decrease of the average molecular weight and the mass loss of the polymer as function of the degradation time can be predicted for semicrystalline as well as amorphous polymers and for copolymers. The disadvantage of this model is that many polymer properties have to be known as input parameter and the calculation has to be performed using computers. Therefore, a simplified model would be most welcome.

Novel, Simplified Calculation of Autocatalytic Degradation

(1)

[A]0 [E]0

c2 )

(5)

Now, let us consider the assumptions. Assumption 1: The mass of the polymer is supposed to be constant during the chain scission process, because the mass of water bound to the polymer during the hydrolysis is negligible as compared to the total mass of the polymer chain. Consequently, the model can only describe the degradation as long as there is no detectable weight loss. Assumption 2: The volume of the swollen polymer is assumed constant during the chain scission process and no oligomers or monomers are supposed to dissolve during the degradation. When these assumptions are used, and taking into account that the ester concentration is not constant during the process, eq 3 can be transformed into eq 6 (see Supporting Information).

Mn(t) )

Our previous model accurately predicts the evolution of the molecular weight distribution and the mass loss during the hydrolytic degradation of aliphatic polyesters. However, sometimes a straightforward calculation of the degradation can provide sufficient information. Some analytical equations like the ones of Pitt24 and Lyu25 or the empirical equation are not based on autocatalytic hydrolysis of the ester bonds of the polymer chain. In the following, an equation will be derived that only describes the number average molecular weight development during hydrolytic degradation and is based on the autocatalytic hydrolysis of the ester bonds. Generally, the hydrolysis rate equation of an ester bond, catalyzed by an acid, can be written as

d[E] ) -k[E][A] dt

and

(

[A]0 ec1t - 1 1 + F 1 + c ec1t Mn(0) 2

)

-1

(6)

This is the autocatalytic equation for amorphous homopolyesters, where c1 and c2 are expressed as given in eqs 4 and 5. In eq 6, Mn(0) is the number average molecular weight at time zero (g/mol), Mn(t) is the number average molecular weight at time t (g/mol), and F is the density of the polymer (g/L). Equation 6 is valid for amorphous polymers. To expand eq 6 to semicrystalline polymers, as well as copolymers, additions must be considered. For copolymers, the total amount of different types of ester bonds can be determined. For a copolymer that consists of two different monomeric residues, every specific type of ester bond has its own specific hydrolysis rate constant (kEi).20 The overall hydrolysis rate constant can be expressed by eq 7, wherein xEi is the molar fraction of each corresponding type of ester bond (Ei). m

where [E] is the concentration of ester bonds, [A] is the acid concentration, k is the reaction rate constant of hydrolysis of an ester bond, and t is the time of the hydrolysis process. All the concentrations are molar concentrations and not mass concentrations. The hydrolysis rate law can be rewritten as

du ) k([E]0 - u)([A]0 + u) dt

(2)

Herein, u is the number of carboxylic acid groups generated by hydrolysis after time t, which is equal to the concentration of hydrolyzed ester bonds. This rate equation can be integrated into the following equation (see Supporting Information):

u(t) ) [A]0

ec1t - 1 1 + c2ec1t

∑ xE kE i)1

i

i

(7)

Assuming that the ratio between the ester bonds remains constant during the hydrolysis, the k in eq 1 can be replaced by eq 7. In semicrystalline polymers, parts of the polymer chains are incorporated into crystals and are therefore (in first instance) excluded from hydrolysis, because water cannot, or can only partially, penetrate into the crystals. However, water can enter the amorphous domains,5,9,19 and the ester bonds present in this phase can be hydrolyzed. Therefore, a correction must be made for the hydrolysis rate such that only the concentration of the ester bonds in the amorphous fraction (φA) is considered.20 By incorporating this into eq 1, one gets eq 8. m

(3)

with

c1 ) ([E]0 + [A]0)k

k)

(4)

∑

d[E] xEikEi)[E][A] ) -φA( dt i)1

(8)

When assuming that the crystallinity and the copolymer composition remains constant during degradation, eq 8 can be integrated similarly as before (eq 1). The assumption seems to be valid as long as we limit the application of the model to the

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time interval without detectable weight loss. The validity of constant crystallinity has further been confirmed by Hakkarainen9 and Vert,19 who found that the crystalline regions start to degrade when most or all of the amorphous regions have disappeared. The outcome of the integration is similar as obtained before (eq 6). The only difference is that the expression for c1 (eq 4), which was used to derive eq 6, becomes m

c1 ) ([E]0 + [A]0)φA(

∑ xE kE ) i)1

i

i

(9)

When this expression for c1 is substituted into eq 6, the general version of the autocatalytic equation for semicrystalline homo- and copolyesters is obtained.

Experimental Section Preparation of the Polymer Samples. The preparations of poly(4methylcaprolactone) (P4MC) and poly(ε-caprolactone) (PCL), both with a carboxylic acid end group at every chain end, have been described and evaluated in our previous work.20 Poly(D,L-lactide) (PLA) and poly(D,L-lactide-co-glycolide) with a monomer ratio of 53:47 (PLGA 53:47) and PLGA with a monomer ratio of 75:25 (PLGA 75:25) were obtained from Purac. These polymers did not contain carboxylic acid end groups after their syntheses and the molar ratio between the Dand L-lactide is 1:1. The polymers PCL, PLA, PLGA 53:47, and PLGA 75:25 were placed individually in a melt-indexer (Tinius Φ Olsen, model MP993). Then, after an equilibrium time to allow complete melting, the material was extruded into a rod through a die with a diameter of 2.0 mm, by pressing it with a piston at elevated temperatures (PCL at 75 °C and PLA, PLGA 53:47, and PLGA 75:25 at 140 °C). The P4MC polymer was a thick viscous liquid and therefore could not be processed into rods using the described equipment. Degradation Experiments. The degradation of the polymers was performed in phosphate buffered saline (PBS buffer). In this buffer, the difference in hydrolytic degradation of the core and the surface is smaller than in a nonbuffered aqueous environment.29 Variation of the ionic strength and the pH in the buffer has no influence on the water uptake and on the hydrolysis of the polymer chains.30 PBS buffer was prepared as follows: to a 20 L glass vessel, 174.165 g Na2HPO4, 37.3602 g NaH2PO4 · 2H2O, 0.7059 g NaCl, 7.6310 g NaN3, and 15002.6 g Milli-Q water were added. A top stirrer was placed in the vessel, and the medium was stirred for 1 h. Subsequently, the medium was degassed by purging it with helium for at least 1 min. The mass of small clean vessels was determined before combining the rods and buffer solution. Subsequently, exact amounts of polymer from each batch were individually added to several vessels to generate sample mixtures to be analyzed at different time intervals. From the polymers PLA, PLGA 53:47, and PLGA 75:25, only one single rod was placed into the vessels. The PCL rods were thin, hence, one or more rods were entered into the vessels. P4MC was a viscous liquid, and a small amount of polymer was put into each vessel. To each polymer sample weighed into the vessels, PBS buffer was added and the test tubes were placed in an oven at 37.0 °C. After different time intervals, the polymers were sampled by removing the water with a syringe. The remaining sample was dried at ambient temperature under vacuum (15 mbar), for at least 12 h. Afterward, the mass of the vessels was determined including remaining dried polymer, and the mass loss of the polymer was calculated by subtracting the total mass of remaining polymer and vessel from the initial mass of polymer and vessel. Initially, 40 samples of each polymer were submitted to degradation. Then, for each residual polymer sample the molecular weight distribution and crystallinity were determined with SEC and DSC, respectively. Measurements. SEC analysis was carried out using a Waters GPC equipped with a Waters 712 WISP automatic injector (50 µL), Waters

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Model 510 Pump system, a column oven (40 °C), and a Waters 410 Differential Refractive Index detection. Two MIXED-C 5 µm 300 × 7.5 mm columns (Polymer Laboratories) were used. Tetrahydrofuran (Biosolve, stabilized with BHT) was used as eluent at a flow rate of 1.0 mL/min. Calibration curves were obtained using polystyrene (easy cal.; Polymer Laboratories, M ) 580 to M ) 7.1 × 106 g/mol). Data acquisition was performed using the Waters Millennium32, version 4.0. To calculate a universal calibration curve, the following Mark-Houwink parameters (K and a) were used: PCL (K ) 1.3 × 10-4 dL/g, a ) 0.810), P4MC (K ) 3.2 × 10-4 dL/g, a ) 0.680), PLA (K ) 1.76 × 10-4 dL/g, a ) 0.737), PLGA 53:43 (K ) 1.76 × 10-4 dL/g, a ) 0.726), and PLGA 73:25 (K ) 5.1 × 10-4 dL/g, a ) 0.610).20 The melting points and the crystallinity of the polymers were determined with differential scanning calorimetry (DSC) on a TA Instrument DSC Q100 V9.4 Build 287. The samples were cooled down to -50.0 °C and heated at a rate of 10.0 °C/min to 150.0 °C. From the first heating curve the crystallinity of the PCL, the only semicrystalline polymer used in this study, was determined, using the published melt enthalpy (∆Hm) for 100% crystalline PCL of 139.4 J/g.31

Results and Discussion Preparation of the Polymer Samples and the Hydrolytic Degradation. In our previous paper, the many aspects that play a role in the hydrolytic degradation process of aliphatic polyesters were discussed extensively.20 Within the present study the focus lies on the validation and evaluation of a newly developed, simple autocatalytic model, and therefore the experimental focus of the degradation study is only on the Mn and mass loss. Despite that the number average molecular weight (Mn) compared to the weight average molecular weight (Mw) is very sensitive toward small variations in the integration of the chromatograms, it needs to be used, as it represents the number of polymer chains present in the sample, from which also the number of carboxylic acid end groups can be derived (see further). The ram-extrusion of the different polymers (PCL, PLA, and PLGA) was performed as described before.20 The Mn values of PLA and PLGA, without detectable initial carboxylic acid end groups present in the synthesized polymer samples, were determined with SEC before and after ram-extrusion and proved to have reduced by 6 and 20%, respectively. Due to this decrease, new polymer chains were formed, comprising newly generated carboxylic acid end groups. The exact amount of carboxylic acid end groups present in the sample, right before starting the hydrolysis experiments, must be taken into account when calculating the decrease of the molecular weight during hydrolysis. The Mn of PCL was only determined after ramextrusion. P4MC was a viscous liquid and therefore not processed by ram-extrusion. Both polymers had initial carboxylic acid end groups present in the synthesized polymer samples. In the Validation of the Model section (see further), it will be explained how the amount of COOH groups was determined. The time that the polymers were exposed to elevated temperatures during ram-extrusion was not measured, and hence, the extent of degradation can only be used indicatively. In the SEC chromatograms of the samples, no unreacted ε-caprolactone, 4-methylcaprolactone, lactide, or glycolide was detected and, therefore, no effects of unreacted monomers on hydrolysis rate are expected.32 The mass loss of the polymers during hydrolysis was monitored.20 The polymers PCL and P4MC did not show any mass loss during the studied period, while the polymers PLA, PLGA 53:47, and PLGA 75:25 started to lose weight after, respectively, 80, 20, and 40 days. For validation of the new

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Figure 1. Experimental evolution of number average molecular weight as a function of the degradation time for PLA (diamonds), PLGA 75: 25 (squares), and PLGA 53:47 (triangles), which all are amorphous polymers.

Figure 2. Experimental evolution of number average molecular weight as a function of the degradation degradation time for semicrystalline PCL (diamonds) and amorphous P4MC (squares).

model, only the Mn values obtained before mass loss occurred can be used. Upon hydrolytic degradation of PLA and both PLGAs (as synthesized all without carboxylic acid end groups, but the ramextrusion has introduced just a few COOH end groups, see further), the decrease of Mn has a typical S-shaped curve (Figure 1). At the start of the degradation, the molecular weight decreases slowly, and over time, the decrease of Mn accelerates due to the newly formed carboxylic acid groups, which autocatalyze the hydrolytic degradation. In the final stage, most of the ester bonds have reacted, and the Mn decrease slows down. The polymers PLC and P4MC as synthesized have a significant number of carboxylic acid end groups, and this relatively high initial carboxylic acid concentration accelerates the hydrolytic degradation directly from the start of the degradation study. Because no “initiation period” is required during which the COOH concentration is built up, the degradation profile does not exhibit an S-shape (like for the almost COOH-free PLA and PLGA (co)polymers, shown in Figure 1) but, in fact, only corresponds to the most right part of the degradation curve, describing the decrease of Mn (see Figure 2). During the degradation period before mass loss of the polymers was observed, the molecular weight is diminished, which is an indication that the degradation follows bulk erosion. During the hydrolytic degradation of PLA a skin layer was observed in samples after day 28, hence, from that moment on the degradation can be considered to follow partially surface erosion and partially bulk erosion. Therefore, data points taken after 28 days can not be used for the evaluation of the model. The skin layer became white and brittle while the core was a

Antheunis et al.

viscous liquid. Probably skin degradation was slowed down by penetration of buffer,20 which neutralizes the acidic degradation products which could autocatalyze further degradation. Also the PLA sample showed a mass increase between day 50 to day 90, which shows that the drying procedure before weighing the samples appears to be too short. The polymer PCL is semicrystalline with an initial crystallinity of 60%.20 Upon degradation, the crystallinity slightly increased to a weight fraction of 78% and the average weight fraction during degradation was 66%. The enhanced crystallinity is most probably caused by the increased chain flexibility in the amorphous domains due to the penetration of water into the polymer, which acts as a plasticizer. This type of annealing of polymers has also been observed upon water penetration into nylons.33 The Mn versus time plots for some of the analyzed polyesters show a relatively large scattering for the first 120 days of the monitoring. During this period, the corresponding SEC chromatograms are causing an error in the Mn determination of PCL and P4MC, more specifically in the integration of the raw data of the chromatogram. After the polymer peak, in these chromatograms, the signal did not return to exactly the same level of the baseline as before the peak, not even after the molecular cut off position of the column. As a result, an error occurs in the estimation of the amount of oligomers and other small molecules. Because the calculation of Mn strongly depends on the amount of small molecules taken into account, a small overestimation or underestimation of the amount of low molecular weight products results in a too low, respectively, too high value of the experimentally determined Mn. After 120 days, the GPC baseline had stabilized and less scattering is observed in the Mn versus time plots. The calculation of the weight average molecular weight (Mw) is much less dependent on the amount of small molecules, and therefore, in our previous paper, there was not such a scattering in the Mw versus time degradation curve of PCL and P4MC.20 Validation of the Model. The autocatalytic equation was developed (eq 6) to provide an easy method for predicting the evolution of the average molecular weight (Mn) of aliphatic polyesters submitted to hydrolytic degradation. To validate this equation the initial number average molecular weight (Mn), the crystallinity, the density of the polymer, the type of the end group(s), the concentration of the initial carboxylic acid groups present, and the ratio of the two different monomer residues in the copolymer were used. The initial number of carboxylic acid groups present in a polymer is calculated as follows: For polymers with carboxylic acid end groups present on every polymer molecule after syntheses, the initial Mn is measured (after ram-extrusion), which corresponds with the concentration of polymer chains present (eq 12, Supporting Information). Every chain has one carboxylic acid end group and therefore the concentration of chains is equal to the concentration of carboxylic acid end groups. For polymers without carboxylic acid end groups present after syntheses, the Mn is measured before and after ram-extrusion (initial) and the concentration of polymer chains can be calculated (eq 12, Supporting Information). The low initial concentration of carboxylic acid end groups present in the polymer materials before the degradation is initiated can be calculated by subtracting the concentration of polymer chains present before ram-extrusion from the concentration of polymer chains after ram-extrusion. To be able to fit the calculated Mn with the measured Mn, a bisectional34 method was used to calculate the hydrolysis rate constant (k). The rate constants are presented in Table 1. The

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Table 1. Determined Hydrolysis Rate Constants (k) polymer

ester bond

PCLa P4MC PLA PLGA 75:25 PLGA 53:47

CL-CL 4MC-4MC LA-LA GA-GA GA-GA

autocatalytic equationb model Antheunis et al.c (L mol-1 day-1) (L mol-1 day-1) 0.50 × 10-3 0.33 × 10-3 6.5 × 10-3 20 × 10-3 20 × 10-3

0.42 × 10-3 0.40 × 10-3 6.4 × 10-3 10 × 10-3 13 × 10-3

a The value of the hydrolysis rate constant of PCL only refers to the amorphous phase. φA ) 0.33 is the fraction amorphous material. b The autocatalytic equation is derived in this work. c The values of the hydrolysis rate constants are obtained from literature.20

monomer residues of the copolymer PLGA are of the lactic acid (LA) and glycolic acid (GA) type. Therefore, four different types of ester bonds are present, namely: LA-LA, LA-GA, GA-LA, and GA-GA. One of the assumptions made in deriving the model is that the ratio of ester bonds does not change during the degradation. In first approximation and for convenience it is also assumed that the GA-LA and LA-GA bonds have a hydrolysis rate constant which is exactly the average of the two hydrolysis rate constants of GA-GA and LA-LA bonds. As can be seen in Table 1, the fitted hydrolysis rate constants of PCL and P4MC are almost identical, both for the autocatalytic equation and for the extensive Antheunis model. This was expected because of the structural similarity of both polymers. The methyl group in the 4-position was expected to be too distant from the ester bond to influence its hydrolysis rate. The values of the hydrolysis rate constants obtained from the simple autocatalytic equation and the detailed Antheunis model are very close (20% off), indicating that the kinetics of both models are very similar. The fitted LA-LA hydrolysis rate constants of PLA, either derived from the autocatalytic equation or obtained with the Antheunis model, are almost identical as well, indicating that both equations can be used both for polymers with significant amounts of carboxylic acid end groups (PCL and P4MC) and for polymers with (initially) hardly any carboxylic acid end groups (PLA and both PLGAs). For the fitting calculation with the experimental results of both PLGA samples, the hydrolysis rate constant for the LA-LA bond obtained from the PLA calculation was used. The values of the fitted GA-GA hydrolysis rate constants of PLGA 53:47 and PLGA 75:25 are for the autocatalytic equation almost identical, as expected. However, if the values are compared with the hydrolysis rate constant of GA-GA obtained from the Antheunis model, there is a factor 1.5 to 2 difference between the two models (50% off). The difference between both models can be explained by realizing that in this paper the hydrolysis rate constant is fitted to the decrease of the number average molecular weight, while for the published, more complicated model of Antheunis et al. this rate constant was fitted to the decrease of the weight average molecular weight. The determined Mn is very sensitive toward small variations in the integration of the chromatograms, which introduces a (systematic) error. Additionally, it was assumed in the newly developed autocatalytic equation that the ratio between the different ester bonds in copolymers does not change during degradation, while the GA-GA ester bonds hydrolyze faster than LA-LA ester bonds. The latter also introduces an error. The measured decrease of the Mn as function of the degradation time of an amorphous polymer carrying a carboxylic acid end group on every molecule (P4MC) is compared with the data calculated with the autocatalytic equation in Figure 3. For an amorphous polymer with a much lower concentration of

Figure 3. Measured and calculated Mn during degradation of amorphous P4MC homopolymester carrying a carboxylic acid end group on every polymer chain.

Figure 4. Measured and calculated Mn during degradation of amorphous PLGA 75:25 copolyester, initially (at t ) 0) with hardly any carboxylic acid end groups.

Figure 5. Measured and calculated Mn during degradation of semicrystalline PCL homopolyester with a carboxylic acid end group on every polymer chain.

carboxylic acid end groups, right before the hydrolysis is started (PLGA), the comparison is made in Figure 4, and for a semicrystalline polymer (PCL) with a carboxylic acid end group on every polymer chain the comparison is made in Figure 5. Due to the earlier mentioned unstable baseline in the SEC measurements on the PCL only the measured data from day 110 and onward are reliable enough to be compared with the calculated values. The calculated line has been extrapolated from day 110 backward to the initial value. It can be concluded that the autocatalytic equation is an excellent tool to calculate the decrease of the Mn during the autocatalytic hydrolysis of amorphous homo- and copolyesters and semicrystalline homopolyesters. However, the autocatalyic

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equation is not able to describe the development of other parameters during degradation, such as the complete molecular weight distribution (MWD) and the mass loss. To calculate the development of those properties during degradation a more complex model has to be used, and from previous work it turned out that the Antheunis model seems to be the most advanced and accurate one.

Conclusion The newly proposed autocatalytic equation that can be used to describe the decrease of the number average molecular weight during the degradation is based on the kinetics of the autocatalytic hydrolysis of aliphatic polyesters. The equation has been validated using experimental results, which were used to determine the values of the hydrolysis rate constants and to compare experimental results with the calculated theoretical degradation profiles. These calculations show that the new equation is able to describe the decrease of the number average molecular weight for aliphatic polyesters before mass loss occurs. The autocatalytic equation can be used for homo(P4MC) and copolyesters (PLA and both PLGAs) and for semicrystalline homopolyesters (PCL), provided that for copolyesters different hydrolysis rates of the different ester bonds are taken into account and that for the semicrystalline polyesters a correction is introduced for the crystalline fraction, which is initially not participating in the hydrolysis. Acknowledgment. We thank Purac B.V. for sending the polymer samples for free and R. Maalcke of the Hoge School van Arhnem en Nijmegen and W. de Graaff and H. Veenstra of N.V. Organon (part of Schering-Plough) for their support. Supporting Information Available. The integration of the hydrolysis rate law (eq 2), by using partial fractions,35 resulting in eq 6, giving the number average molecular weight as a function of time under autocatalytic conditions. This material is available free of charge via the Internet at http://pubs.acs.org.

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