An entropy spring model for the Young’s modulus change of biodegradable polymers during biodegradation

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Research paper

An entropy spring model for the Young’s modulus change of biodegradable polymers during biodegradation Ying Wang, Xiaoxiao Han, Jingzhe Pan ∗ , Csaba Sinka Department of Engineering, University of Leicester, Leicester, LE2 7HR, UK

A R T I C L E

I N F O

A B S T R A C T

Article history:

This paper presents a model for the change in Young’s modulus of biodegradable polymers

Received 3 October 2008

due to hydrolysis cleavage of the polymer chains. The model is based on the entropy

Received in revised form

spring theory for amorphous polymers. It is assumed that isolated polymer chain cleavage

19 February 2009

and very short polymer chains do not affect the entropy change in a linear biodegradable

Accepted 24 February 2009

polymer during its deformation. It is then possible to relate the Young’s modulus to the

Published online 5 March 2009

average molecular weight in a computer simulated hydrolysis process of polymer chain sessions. The experimental data obtained by Tsuji [Tsuji, H., 2002. Autocatalytic hydrolysis of amorphous-made polylactides: Effects of L-lactide content, tacticity, and enantiomeric polymer blending. Polymers 43, 1789–1796] for poly(L-lactic acid) and poly(D-lactic acid) are examined using the model. It is shown that the model can provide a common thread through Tsuji’s experimental data. A further numerical case study demonstrates that the Young’s modulus obtained using very thin samples, such as those obtained by Tsuji, cannot be directly used to calculate the load carried by a device made of the same polymer but of various thicknesses. This is because the Young’s modulus varies significantly in a biodegradable device due to the heterogeneous nature of the hydrolysis reaction. The governing equations for biodegradation and the relation between the Young’s modulus and average molecular weight can be combined to calculate the load transfer from a degrading device to a healing bone. c 2009 Elsevier Ltd. All rights reserved.

1.

Introduction

Biodegradable fixation devices such as screws, plates and pins made of biodegradable polymers are being increasingly used in orthopaedic surgeries to provide temporary support for broken bones (Rokkanen et al., 2000; Bell and Kindsfater, 2006). A critical issue of using the biodegradable fixation devices is to understand the load transfer from a device to the protected bone as the bone heals and the device degrades. The polymer-chain cleavages in a biodegradable polymer due ∗ Corresponding author. Tel.: +44 116 223 1092; fax: +44 116 252 2525. E-mail address: [email protected] (J. Pan). c 2009 Elsevier Ltd. All rights reserved. 1751-6161/$ - see front matter doi:10.1016/j.jmbbm.2009.02.003

to hydrolysis reaction ultimately lead to the reduction in the Young’s modulus. A simple rule in mechanics tells us that the device shares less load if its stiffness is reduced. If the Young’s modulus reduces too quickly, then the healing bone would be put in danger. On the other hand if the Young’s modulus reduces too slowly, the bone would be weakened by the well known stress-shielding effect. Therefore, there exists an optimised degradation rate which ensures both the complete healing and healthy growth of a broken bone. The current generation of biodegradable devices has not fully exploited

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this optimisation potential due to the lack of a predicting tool for the stress transfer. To be on the “safe” side, the polymers are often over-designed using very high molecular weight and degree of crystallinity. The degradation time of these devices are very long, sometime over 4 years (Barber et al., 2000). Extensive experimental studies have been carried out on the degradation behaviour of biodegradable polymers. For example, Tsuji and his co-workers (Tsuji, 2000; Tsuji and Ikada, 2000; Tsuji and Muramatsu, 2001; Tsuji, 2002; Tsuji and Saha, 2006) published a series of experimental data on the degradation of polylactic acids (PLAs) and their copolymers. These are long term data, some over a period of 36 months, showing average molecular weight, degree of crystallinity, weight loss, Young’s modulus, ultimate strength and elongation to failure as functions of the degradation time. One observation of the data is that the property reduction significantly lags behind the reduction in the average molecular weight for most of the biodegradable polymers. In fact the Young’s modulus often increases, instead of decreasing, at the early stage of the degradation. A reasonable understanding has also been obtained on the “degradation pathways”. It has been concluded that the water molecules diffuse into the amorphous region of the polymers relatively quickly and attack the backbones of the amorphous polymer chains. There is however some confusion about whether the chain cleavage occurs randomly along the polymer chains (van Nostrum et al., 2004; Shih, 1995; de Jong et al., 2001; Ratner et al., 2003), referred to as random scission, or dominantly at the end of the polymer chains like unzipping, referred to as end scission. It has been reported that random scission dominates the early stage of the degradation while end scissions dominate the later stage (Shih, 1995). As degradation proceeds further, the water molecules enter the narrow amorphous gaps between the crystalline lamellae causing chain scissions there (Zong et al., 1999). At the final stage of the degradation, water attacks the crystalline phase which takes a much longer time to degrade. One complication is that the degree of crystallinity can increase significantly during the degradation process (Zong et al., 1999; Tsuji and Ikada, 2000). This has been explained as that the chain cleavage provides the amorphous polymer chains with extra mobility allowing small crystals to form first in the large amorphous region and later between the gaps of the lamellae (Zong et al., 1999). The cleavageinduced crystallisation may partially explain the increase in Young’s modulus at the early stage of the degradation. Recently Pan and his co-workers developed a mathematical model for the biodegradation firstly for amorphous biodegradable polymers (Wang et al., 2008) and later for semicrystalline biodegradable polymers (Han and Pan, 2009). The model simplified and extended the classical reaction diffusion equation, and can predict the average molecular weight and degree of crystallinity as functions of location and time in a device of any sophisticated shape. Such a model, when connected with a model of predicting the change in elastic properties, would be a powerful tool to assist the device manufacturers and orthopaedic surgeons to optimise the device design and applications. However existing theories for the degradation of elastic properties of polymers do not capture the observed behaviour of biodegradable polymers. The

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modern constitutive laws for elastomers have taken the effect of chain cleavage into account (Wineman, 2005; Shaw et al., 2005), which were built on the experimental and theoretical work by Tobolsky (1960). However these constitutive laws were developed for rubbers undergoing oxidation. The fundamental trend in these materials is that the Young’s modulus reduces exponentially with time, which is well captured by Tobolsky theory (Tobolsky, 1960). The same theory cannot be applied to biodegradable polymers which show an incubational behaviour (Tsuji, 2002; Tsuji and Saha, 2006). The purpose of this paper is to present a model relating the Young’s modulus to the average molecular weight for biodegradable polymers. The model is based on the entropy spring theory for amorphous polymers, which predicts that the Young’s modulus depends linearly on the number of polymer chains in a unit volume of the material (Ward and Hadley, 1993). A central issue in using the entropy theory is how to count the number of molecular chains as chain scissions occur. Two concepts, a molecular weight threshold and a so-called “no-rise rule”, are proposed. The model is then used to explain the experimental data for poly(L-lactic acid) (PLLA) and poly(D-lactic acid) (PDLA) by Tsuji (2002). A demonstration case is provided to show how to connect the biodegradation model developed by Pan and his co-workers (Wang et al., 2008; Han and Pan, 2009) with the current model for a three-dimensional device. The focus of this paper is on amorphous polymers because (a) some biodegradable polymers remain amorphous throughout the biodegradation process and (b) the amorphous region is the weak link in the degradation of semicrystalline biodegradable polymers; and a composite theory can be used to predict the Young’s modulus of a semicrystalline polymer from its degree of crystallinity and Young’s modulus of the amorphous region.

2.

Entropy spring model of polymer elasticity

A force applied on a sample of material, f , is related to its stretched length, l, through its internal energy, U, and entropy, S, such that     ∂U ∂S f= −T (1) ∂l T ∂l T in which T represents the temperature. For most engineering materials, the entropy term can be ignored. For amorphous polymers, the entropy spring model assumes that the internal energy term can be ignored and it is the entropy increase of the polymer chains from a disordered state to a more ordered state during deformation that provides the elasticity of the material. This is schematically shown in Fig. 1 (a). The entropy spring theory reflects the fact that very little force is carried by the polymer backbone during deformation; hence the total internal energy change is small relative to the entropy change. The entropy theory leads to the following prediction of Young’s modulus E (see the textbook by Ward and Hadley (1993)): E = 3NkT

(2)

in which N represents the number of polymer chains per unit volume, k is the Boltzmann constant and T is the

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Fig. 1 – Schematic demonstration of the entropy spring model showing an initially intact polymer chain embedded in many other polymer chains which are represented by the shaded background. (a) The entropy of the polymer chains is reduced during deformation as the long molecules become more ordered, giving an elastic resistance to the deformation. (b) Neither an isolated cleavage nor very short chains affects the entropy resistance to deformation. (c) Sufficient chain cleavages break down the polymer chain which no longer contributes to the entropy resistance to deformation.

absolute temperature. The theory seems to predict an everincreasing Young’s modulus for a degrading polymer became the chain cleavage always increases N. However, the entropy theory was developed for fresh polymers and a fundamental assumption leading to Eq. (2) is that the end to end distance of a single polymer chain is much smaller than the extended chain length (Ward and Hadley, 1993). Under this assumption, the end to end distance follows the Gaussian distribution. This assumption is no longer valid if random chain cleavage occurs. It is complicated to calculate the entropy taking the random scission into account. Instead an intuitive argument is suggested here to modify the entropy spring theory. As shown in Fig. 1(b), an isolated chain scission of a very long chain should not affect the entropy change during the deformation of the polymer because the long polymer chain is constrained by its surrounding chains (represented by the shaded background in the figure). It is then reasonable to assume that N does not increase after a chain cleavage. This is referred to as the “no-rise rule” in the following discussion. Furthermore, a very short chain does not contribute to the entropy change during deformation. It has long been recognised that polymers with very small degree of polymerisation have little strength and stiffness

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(Kaufman and Falcetta, 1976). Therefore chains shorter than that of a critical degree of polymerisation should not be counted when using Eq. (2). Consequently, a polymer chain should be removed from the entropy calculation if enough cleavages have occurred so that its molecular weight (or degree of polymerisation) is smaller than a threshold. This is schematically shown in Fig. 1(c). From a known initial distribution of polymer molecular weight, the random and end scissions are simulated numerically. Using the no-rise rule and the molecular weight threshold, the value of N, and hence the Young’s modulus, and the average molecular weight are calculated. A relationship between the Young’s modulus and the average molecular weight is then determined. A computer code is developed to obtain the Young’s modulus–average molecular weight relation. Fig. 2 shows the flowchart of the numerical scheme. The simulation starts from numerically generating polymer chains of random degrees of polymerisation according to the initial molecular weight distribution. The sample size, i.e. the initial number of polymer chains in the simulation, has to be chosen arbitrarily. The effect of the sample size will be further discussed in the following section. Chain scissions are then simulated according to the ratio of the random scission rate over the end scission rate. At each scission step, a random value R less than unity is generated to represent the probability of random scission. A random scission occurs if R/(1 − R) is less than the ratio of random scission rate over the end scission rate timing the ratio of the number of total ester bounds over the total number of chain ends. The value of N is then calculated according to the molecular weight threshold and the no-rise rule. Simultaneously, the average molecular weight of the polymer chains is calculated.

3. Comparison with the experimental data by Tsuji (2002) The experimental data on poly(L-lactide) (PLLA) and poly(Dlactide) (PDLA) by Tsuji (2002) are examined here using the entropy spring model. These particular data are selected because (a) PLLA and PDLA are commonly used biodegradable polymers and (b) Tsuji reported that the polymers remain amorphous throughout the hydrolysis experiment so that the complication of crystallinity is excluded. The samples were kept under 37 ◦ C in a phosphate-buffered solution of pH 7.4 and tested every 4 months using gel permeation chromatography (GPC) and tensile testing for their molecular weight distributions and mechanical properties. Very thin samples were used to eliminate the self-catalytic effect so that the samples degraded uniformly. Fig. 3 shows the reproduced PLLA chain distributions at different times of the degradation test. The y-axis of the curves has been transformed into the number of polymer chains from the standard mass fraction of polymer chains (GPC curves) as in Tsuji’s data for convenience in our calculation. Tsuji reported that the Young’s modulus of the PLLA samples at the t = 0 and t = 24 months are 1813 and 950.6 MPa, respectively (Tsuji, 2002). The total number of polymer chains larger than a molecular weight threshold can be calculated by integrating

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Fig. 2 – Flowchart showing the numerical scheme for simulating the polymer chain cleavages and calculating the corresponding Young’s modulus and average molecular weight.

the distribution curve in Fig. 3 on the right-hand side of the threshold. It is found that a threshold of Mth = 1.3 × 105 g/mol leads to the ratio of Nt=0 E = 1.8 ≈ t=0 . Nt=24 Et=24

(3)

However, this calculation of N has counted all the polymer chains larger than the threshold while the no-rise rule dictates that cleaved polymer chains should not be counted twice. Therefore the threshold value obtained in this way can only be used as an initial guess. Using this value of the threshold and the no-rise rule, the normalised Young’s modulus versus the number average molecular weight is obtained for PLLA, which is shown in Fig. 4. In the figure the solid line is the model prediction while the discrete dots are the average experimental data with error bars

obtained by Tsuji (2002). It is surprising to observe that the initial guess of the threshold value gives a very good fit between the experimental data and the model prediction. Our further numerical tests showed that the shape of the Young’s modulus–average molecular weight curve is sensitive to the threshold value. The dashed line in Fig. 5 shows the same Young’s modulus–average molecular weight curve obtained using Mth = 1.1 × 105 g/mol. The good fit shown in Fig. 4 is therefore not a coincidence. The results indicate that the hydrolysis reaction in Tsuji’s degradation experiment was controlled by end scissions, because only in such a case can the threshold obtained according to Eq. (3) be accurate. Indeed several authors, including Shih (1995), have suggested that the hydrolysis reaction in PLAs is controlled by end scissions. In our simulation, the ratio between the random scission and

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Fig. 3 – Distributions of number of polymer chains of different molecular weight reproduced from the GPC curves obtained by Tsuji (2002) for PLLA at 0, 8, 16, and 24 months, respectively (from right to left) of degradation.

end scission was set as 1:12 following Shih (1995). To study the effect of this ratio on the model prediction, the simulation was repeated using the random to end scission ratios of 3:1, 1:1, and 1:50. The results are presented in Fig. 5 together with the data shown in Fig. 4. It can be seen that the model prediction is insensitive to the choice of the ratio. In other words, the numerical model predicts that the relationship between the Young’s modulus and the average molecular weight does not depend on whether the hydrolysis reaction is controlled by end scissions or random scissions. This is a significant proposition which requires further experimental validation and should not be confused with the fact that the threshold value, Mth , cannot be determined using Eq. (3) if the hydrolysis is controlled by random scissions. In the numerical simulation, 3000 polymer chains were used at the beginning of the hydrolysis reaction. To check if the initial sample size of 3000 polymer chains was large enough, the simulation was repeated using sample sizes of 10 000 and 30 000 respectively. Again these results are plotted in Fig. 5 together with the curve in Fig. 4. It can be observed from Fig. 5 that all these curves are reasonably close to each other, showing that the sample size of 3000 polymer chains is large enough. The above procedure is repeated using the data for PDLA obtained by Tsuji (2002). Fig. 6 shows the reproduced polymer chain distributions at four different times of degradation. The molecular weight threshold was determined as Mth = 9.36 × 104 g/mol shown in the figure using the shaded area. Fig. 7 compares the predicted (solid line) and experimental (discrete dots with error bars) relation between the Young’s modulus and average molecular weight. In the simulation, 3000 initial polymer chains were used and the random to end scission ratio was set as 1:12. It can be observed from Figs. 4 and 7 that the simple model proposed in this paper can capture the trend in the Young’s modulus reduction very well. The input to the model is the initial polymer chain distribution. The model prediction is insensitive to the ratio between random and end scissions. Therefore the only parameter in the model is the molecular weight threshold. The two thresholds determined for PLLA and PDLA

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Fig. 4 – Normalised Young’s modulus versus normalised average molecular weight for PLLA used by Tsuji (2002). Solid line: model prediction using Mth = 1.3 × 105 g/mol, initial sample size of 3000 polymer chains and the random to end scission ratio of 1:12; discrete dots with error bars: experimental data obtained by Tsuji (2002).

Fig. 5 – Normalised Young’s modulus versus normalised average molecular weight for PLLA used by Tsuji (2002). Solid lines: model prediction using Mth = 1.3 × 105 g/mol, initial sample size of 3000, 10 000 and 30 000 polymer chains and the random to end scission ratio of 3:1, 1:1, 1:12 and 1:50, respectively; discrete dots: experimental data obtained by Tsuji (2002). The dashed line was obtained using Mth = 1.1 × 105 g/mol.

Fig. 6 – Distributions of number of polymer chains of different molecular weight reproduced from the GPC curves obtained by Tsuji (2002) for PDLA at 0, 8, 16, and 24 months, respectively (from right to left), of the degradation test.

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Fig. 7 – Normalised Young’s modulus versus normalised average molecular weight for the PDLA used by Tsuji (2002). Solid line: model prediction using Mth = 9.36 × 104 g/mol, initial sample size of 3000 polymer chains and the random to end scission ratio of 1:12; discrete dots: experimental data obtained by Tsuji (2002). Fig. 8 – A biodegradable rod and its finite element model. correspond to degrees of polymerisation of about 600–1000. These are consistent with the well known threshold of degree of polymerisation for polymers to possess any useful mechanical properties (Kaufman and Falcetta, 1976).

4. Connecting biodegradation with elasticitydegradation Next we demonstrate how to use the entropy spring model and the biodegradation model developed by Pan and his co-workers (Wang et al., 2008; Han and Pan, 2009) to predict the stiffness change of a biodegradable device. In the biodegradation model, a biodegradable polymer is simplified to four constituent species: (a) water molecules, which are assumed abundant anywhere in the device; (b) amorphous polymer molecules, which can hydrolyse but are too large to diffuse; part of the polymer molecules can also crystallise; (c) monomers and water-soluble oligomers, which are the product of the hydrolysis reaction and can diffuse out of the system; (d) polymer crystals, which will nucleate and grow but do not hydrolyse. The state of a biodegrading polymer is then described using Ce , the mole concentration of the ester bounds of the amorphous polymer, Cm , the mole concentration of monomers remaining in the device, and Xc , the degree of crystallinity. The governing equations for these state variables include (Han and Pan, 2009) !  n dCm Cm = k1 C e + k2 C e + div D grad (Cm ) (4) xi dt 1 − Xc xi and   n  dCe Cm Ce dXc = − k1 Ce + k2 Ce − dt 1 − Xc 1 − Xc dt

(5)

in which k1 and k2 are the reaction constants for the non-autocatalytic and autocatalytic hydrolysis reactions, the power of n accounts for the dissociation of the acid end

groups, and D is a effective diffusion coefficient depending on the porosity and degree of crystallinity. Further equations are required when crystallisation is considered (Han and Pan, 2009), which is not a concern of the current paper. Subject to appropriate boundary conditions, these equations can be solved using the finite element method for devices of any sophisticated geometry, giving the spatial and temporal distributions of Ce , Cm and Xc (Wang et al., 2008). For simplicity the degradation model considers the molecular weight distribution as a bimodal distribution, which is characterised by Ce and Cm . Assuming the monomers are too small to detect using standard experimental techniques and that end scissions dominates over random scissions, the measured average molecular weight M can be related to Ce such that M Ce = M0 Ce0

(6)

in which M0 and Ce0 are the initial values of the average molecular weight and ester bound concentration respectively. The model does not distinguish between number averaged and weight averaged molecular weights, which is a shortcoming of the simplification. One needs to choose one of the averaged molecular weights. The parameters in the model, k1 and k2 , are then defined accordingly. The spatial and temporal distribution of Young’s modulus in a device can be obtained by firstly solving Eqs. (4) and (5), and then using the relation between Young’s modulus and average molecular weight given by, for example, Fig. 4. As a demonstration case, we consider a biodegradable rod made of PLLA, shown in Fig. 8. A small section of the rod is modelled using the finite element method. Despite the obvious axisymmetry of the problem, we still solve it as a general threedimensional problem to illustrate the validity of the method for more general cases. The boundary conditions are that there is no diffusive flux of Cm normal to the top and bottom cross-sections and that Cm = 0 at the outer surface of the rod. The following non-dimensional parameters were used k1 in the analysis (Wang et al., 2008): k¯ = = 0.5 and ¯ = D

1 k2 C n e0 D0 = 0.1. Here D is the diffusion coefficient of 0 Cn k R2 e0 2

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(a) t¯ = 1.

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(b) t¯ = 2.

Fig. 9 – Distribution of normalised average molecular weight obtained by solving Eqs. (4) and (5) using the finite element method at two normalised times of degradation.

Fig. 10 – Effective Young’s modulus for a biodegradable rod in its axial direction as a function of the normalised degradation time.

of PLAs reported by Tsuji (2002) using the entropy spring theory. Assuming that isolated chain cleavages and short polymer molecules do not affect the entropy change of an amorphous polymer during its deformation, the Young’s modulus of the degrading polymer is related to its average molecular weight in a computer simulated process of random and end scissions of the polymer. The only input of the model is the initial molecular weight distribution of the polymer and a threshold for the molecular weight. The numerically obtained relation between Young’s modulus and average molecular weight was then used in combination with a biodegradation model to predict the spatial and temporal distribution of the Young’s modulus in a cylindrical rod. It is shown that the Young’s modulus reduction depends on the diameter of the rod.

REFERENCES

the monomers in the non-degraded polymer and R is the radius of the rod. The diffusion coefficient D of the monomers in the degrading polymer is assumed to linearly depend on
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5.

Concluding remarks

This paper shows that it is possible to provide a common thread through the set of long term data for the degradation

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