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Processing of Ultra-High Molecular Weight Polyethylene
0263–8762/02/$23.50+0.00 # Institution of Chemical Engineers Trans IChemE, Vol 80, Part A, July 2002
www.catchword.com=titles=02638762.htm
PROCESSING OF ULTRA-HIGH MOLECULAR WEIGHT POLYETHYLENE Modelling the Decay of Fusion Defects J. J. WU, C. P. BUCKLEY and J. J. O’CONNOR Department of Engineering Science, University of Oxford, Oxford, UK
A
problem in applications of ultra-high molecular weight polyethylene (UHMWPE) is the tendency for components to contain fusion defects, arising during processing of the as-polymerized powder. These defects have been implicated previously in failures of UHMWPE load-bearing surfaces, in knee and hip prostheses. Recent work of the authors has recognized two forms of defect: voids (Type 1) and particle boundaries de cient in diffusion by reptation (Type 2). To assist process and product design, a method has now been developed for predicting the decay of severity of Type 2 defects during processing, for a component of given shape and process history. A new quanti er was introduced for characterizing the progress of diffusion at Type 2 defects in UHMWPE—the maximum ^ . This was computed using results from reptation theory, embedded reptated molecular mass M within a Finite Element thermal model of the process. The method was illustrated by simulating compression moulding trials already carried out experimentally by the same authors. It was ^ never reached the viscosity average molecular mass of the polymer, discovered that M indicating incomplete boundary diffusion, and explaining the previous observation of Type 2 defects even in fully-compacted, apparently perfect mouldings. The method described has potential as a design tool, especially for optimizing manufacture of UHMWPE prosthesis components. Keywords: UHMWPE; polyethylene; reptation; nite element analysis; thermal model; fusion defects; compression moulding; powder processing.
INTRODUCTION
compliant inter-particle boundaries arising from incomplete inter-particle diffusion6. These are labelled as Type 1 and Type 2 fusion defects respectively. The distinction is important. Type 1 defects may be prevented by proper choice of process parameters, whereas Type 2 defects are probably always present for realistic process conditions, but their severity is sensitive to product and process design. Full details of the experimental study are given elsewhere7. The purpose of this paper is to propose a method of predicting the severity of Type 2 defects in compression-moulded UHMWPE, for given component design and process parameters, to assist manufacturers in optimizing the performance of their products. The engineering problem is complicated by the range of prosthesis designs in use, and the need for each design to be produced in a variety of sizes, to suit the needs of different patients. Therefore, the need is for a exible process/product design tool, that can be used by the material processor to ensure that each product has at least a speci ed minimum level of mechanical integrity. A numerical simulation procedure is proposed that, when embedded within a nite element heat transfer model for calculation of temperature, allows the user to map a numerical quanti er of the state of Type 2 defects at particle interfaces, throughout a threedimensional UHMWPE product.
Ultra-high molecular weight polyethylene (UHMWPE) has been the favoured material for the concave bearing surfaces of total knee and hip joint replacement prostheses since the 1970s, because of its desirable properties of biocompatibility, high impact strength, low friction and high wear resistance1. Although generally these perform well, there is a small but signi cant failure rate, involving painful and expensive revision surgery. This has motivated much research to identify the causes of failure. An important outcome has been the realization that incomplete consolidation of the UHMWPE powder during processing into solid, resulting in ‘fusion defects’, is implicated in the failure of the material due to fatigue2–5. Speci cally, cracking and delamination at fusion defects appear to be associated with fatigue failures found in retrieved knee components5. To solve this problem, there is an urgent need for better understanding of the nature and origin of fusion defects, and for reliable means of controlling them during processing. Recent research in this laboratory has been aimed at achieving these goals. A key recent discovery was that asmoulded UHMWPE is prone to fusion defects of two distinct types: (a) sites of incomplete compaction (voids), and (b) sites where compaction is complete but there remain 423
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The approach adopted here is based on the proposal8 that, establishment of a full bond at a polymer–polymer interface, requires self-diffusion to occur for a period of at least the ‘reptation’ time td . This is the time required for a macromolecule to refresh its conformation completely, by diffusion along the curvilinear tube de ned by the topological constraint of its neighbours9. At an interface, it corresponds to the time taken, from the moment the two surfaces are brought into perfect contact, for molecules to lose by diffusion their conformational memory of the interface. This process knits together the two surfaces and creates a weld that is indistinguishable from the bulk polymer. There is now a substantial body of support for this picture of events, from experimental evidence10,11 and from numerical simulations12. In particular, Wool et al. demonstrated that this approach explained the dependence of the strength of polystyrene on time and molecular weight, when moulded from pellets above the glass transition temperature10,13. Pecorini showed that results from reptation theory can be extended to non-isothermal moulding processes of practical importance. He deduced the effective crack depth associated with weld-lines in injection-moulded cellulose acetate propionate, by calculating the interdiffusion distance achieved at weld-lines during cooling near the mould surface14. Since this distance was less than half the radius of gyration for molecules at the surface, cooled most quickly, the weld line acted as a surface crack. A similar approach to that of Pecorini is applied here. Results from reptation theory are combined with a heat transfer simulation of the moulding process, to deduce the extent to which inter-particle diffusion has progressed by the end of the moulding cycle. To quantify the state of the particle interfaces in the nal product, a new parameter is introduced: the maximum reptated molecular weight, that can be computed throughout a moulding of given shape, for a given process history. THEORY Description of the Problem As-polymerized particles of UHMWPE powder have irregular shapes, with widths of order 100 mm. They appear to be agglomerations of several smaller particles, which are themselves assemblies of spherical sub-particles with diameters of order 1 mm. The literature provides many examples, for powder from all the major industrial suppliers6,15–17. The result is a powder whose particles have a surface of complex topography, with many re-entrant regions. A schematic two-dimensional sketch is given in Figure 1. During processing of the powder into solid, for example by compression moulding or ram extrusion, pressure is applied to compact the powder, the temperature is raised above the crystal melting point, and after a suitable interval of time the polymer is cooled to room temperature. The objective is to convert the powder into a homogeneous solid. This requires a two stage process to be completed. In Stage 1, as shown in Figure 1, the powder surfaces are pressed together until they are in intimate contact at the molecular level. If this Stage is not completed, the nal solid contains intra-particle and inter-particle voids. A typical example is shown in Figure 2. The time taken for
Figure 1. Schematic two-dimensional sketch of UHMWPE powder particles and compaction during Stage 1 of processing the powder into solid. Typical sizes of particle and sub-particle are shown. Stage 2 corresponds to molecular diffusion across the interfaces.
completion of Stage 1 depends on the temperature, pressure and time allowed, and the aspect ratio of surface features18. This stage alone, however, would be insuf cient for creating a solid with acceptable mechanical integrity. The particles would be held together by inter-chain van der Waals bonding only, with weak, compliant interfaces; and would be extremely susceptible to initiation and propagation of interparticle cracks when the eventual solid component is loaded in use. There is thus a further essential stage to be completed— Stage 2, where molecules diffuse across the interfaces and endow the particle boundaries with intra-chain covalent bonding. At the temperatures concerned, there is continuous molecular diffusion by the process of reptation. As soon as particles make molecular contact, boundary molecules will begin to explore the space of their adjacent particles. Eventually, all ‘memory’ of the interfaces will have been lost when there ceases to be any correlation between molecular conformations before and after the surfaces were brought together. The particles will be then fully knitted together.
Figure 2. Typical SEM micrograph showing residual intra-particle and inter-particle voids resulting from incomplete compaction (Stage 1) during processing of UHMWPE. The dotted scale bar represents 120 mm.
Trans IChemE, Vol 80, Part A, July 2002
MODELLING THE DECAY OF FUSION DEFECTS Application of Reptation Theory The authors’ aim is to construct a model that will allow a measure of particle-interface integrity to be evaluated, throughout a polymer component of given shape and process history. There are numerous complicating features of the engineering problem. (1) Stages 1 and 2 proceed concurrently at different locations: there is progressive creation of interfaces during compaction, and hence a range of start times for the ensuing inter-particle diffusion. (2) It is not possible to be sure of the state of molecular segregation or of conformations at the start of Stage 2: (a) because of the need for a nite time for randomization following melting of crystals; and (b) because of the perturbing in uence of surfaces of the small sub-particles (whose dimensions are of the same order of magnitude as typical molecular radius of gyration. (3) There are large temperature gradients within typical mouldings during processing, arising from the low thermal conductivity of the material. (4) There are large changes of temperature during processing—from room temperature to beyond the melting region and back again. To make the problem tractable rst it is idealized with the aid of the following assumptions: (1) stage 1 is complete, and therefore, all intra-particle and inter-particle interfaces are in intimate contact at the molecular level (i.e. wetted), when inter-particle diffusion commences; (2) by the time inter-particle diffusion commences, the polymer molecules (of all lengths) can be treated as random coils, randomly dispersed; (3) temperature differences are negligible on the length scale of an individual molecule. With these assumptions, Stage 2 is modelled by taking results of the ‘tube’ model of polymer dynamics in the molten state, proposed for a material with temperature that is both uniform and constant, generalizing them to the case of a nonuniform, varying temperature, and nally embedding them within a numerical simulation of temperature evolution in a moulding of the required shape. A melt of linear polymer molecules is considered, with monomer molecular mass Mo and j backbone covalent bonds per monomer. Although commercially produced polymers typically contain a distribution of molecular lengths, initially chains of a particular length are focused on, i.e. with a particular molecular mass M, chosen arbitrarily except it must exceed the entanglement value Me . Each of these chains consists of n backbone bonds each of length l: thus n ˆ jM =Mo . The entanglement constraint offered to it by adjacent chains at any instant, is represented by a tube surrounding the molecule, of contour length L. The tube theory given by Doi and Edwards19 provides measures of the motion of such chains-within-tubes, and their escape from the tube ends. For present purposes, the pertinent feature is the starting point of the theory: diffusive motion along the tube, ‘reptation’, is represented by evolution of a probability density function c…n; t† for the centre of mass having diffused a distance n in a time t: this obeys the diffusion equation: @c @2 c ˆ Dc 2 @t @n Trans IChemE, Vol 80, Part A, July 2002
…1†
425
where Dc is the curvilinear diffusion coef cient for onedimensional motion along the tube. Solution of equation (1), together with the appropriate boundary conditions, leads to the important conclusion that correlation between current and initial molecular conformations (expressed as the timecorrelation function of the end-to-end vectors) decays exponentially with time, with the longest time constant given by the ‘reptation time’19: td ˆ
L2 p2 Dc
…2†
This is the time taken for the chains to disengage from their initial tubes and refresh their conformations: that is, to lose all memory of their initial conformations. For molecules at an interface this means: to lose all memory of the interface. To apply the above theory to the present problem, the tube diffusion may be represented by the Rouse bead-spring mechanism, in which ow of molecular segments past their neighbours is thermally activated. Thus, the tube diffusion coef cient at time t is determined by the current temperature T, and may be written in terms of this and its value at some reference temperature (indicated by asterisks): µ ³ ´¶ ED 1 1 ¤ Dc ˆ Dc exp ¡ ¡ …3† R T T¤ where ED is the activation energy for self-diffusion of the molten polymer. These arguments apply, however, only to the wholly amorphous state. In the presence of any signi cant crystallinity, the authors expect self-diffusion to be highly retarded, as a result of pinning by those molecular segments that have crystallized. In the present model, this feature was approximated by assuming Dc to vanish at temperatures below a two-valued critical temperature Tc: the end-of-melting temperature Tem or start-of-crystallization temperature Tsc , during heating and cooling respectively. Results of the theory can be extended to the case of a time-varying temperature T …x; t† by introducing a reptationequivalent time-scale x. If equation (3) is substituted in equation (1), the latter may be re-written » µ ³ ´¶ ¼ E 1 @c @x 1 @2 c ¡ ¤ ˆ D¤c 2 …4† exp D @x @t R T T @n Thus, the solution is the same as that for isothermal diffusion at the reference temperature T ¤ , provided, at any position x, the real time t is replaced by the equivalent time x given by: …t dt0 x…x; t† ˆ …5† 0 0 aT …x; t † where:
0 a¡1 T …x; t †
and: Tc ˆ Tem
µ ³ ´¶ ¡ED 1 1 ˆ exp ¡ T …x; t 0 † T ¤ R 0 a¡1 T …x; t † ˆ 0
if T < Tc ;
for T_ ¶ 0;
Tc ˆ Tsc
if T ¶ Tc ; …6† for T_ µ 0
…7†
thereby ensuring the curly bracket in equation (4) remains unity at all times, and incorporating arrest of self-diffusion
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in the presence of crystallinity. The implied ambiguity in equation (7) in the case T_ ˆ 0 is easily resolved by considering the most recent non-zero value of T_ . The signi cance of time x is that it would produce the same extent of reptative diffusion at the reference temperature, as the real time t does for the time-varying temperature. The variable aT is a time-temperature reduction factor for diffusion. Invoking equation (2) for a time-varying temperature, it is clear that chains will have disengaged from their initial tubes when: x ¶ t¤d ˆ
L2 p2 D¤c
…8†
To apply this result to a particular polymer system, D¤c must be related to the measurable (molecular mass dependent) diffusion coef cient D¤ governing three-dimensional centreof-mass diffusion at the reference temperature. This may be achieved by matching the mean-square distance travelled by the centre of mass along the tube in a given time (by onedimensional diffusion), to the corresponding mean-square distance travelled in three dimensions in the same time (by three-dimensional diffusion). Suppose the rms distance travelled by the centre of mass along the tube is L, then the rms distance travelled in three dimensions must be the rms end-to-end length of the chain hR2 i1=2 . Equating the times for one- and three-dimensional diffusion: hR2 i L2 ˆ 6D¤ 2D¤c
…9†
and, hence, the condition for disengagement may be written: hR2 i …10† 3p2 D¤ The original theory of reptation, as exploited above, is now known to require correction. The diffusing chains are not inextensible strings within their tubes, as originally conceived, but they will be subject to contour length uctuations (CLF)20. The result is that a correction is needed to equation (2) and others derived from it. However, the fractional error in the original expression varies as M ¡0:5 and for the chain lengths of interest in this paper may be neglected. From the de nition of the characteristic ratio C1 , characterizing exibility of the chains, hR2 i may be found in terms of M: x¶
hR2 i ˆ C1 jl2
M Mo
…11†
For many years D was thought to scale with molecular length as M ¡2 , as predicted by reptation theory9,19 and as apparently observed experimentally21,22. However, more recent experimental data and careful re-appraisal of earlier data have a revealed a stronger dependence M ¡a , where a lies between 2 and 2.5, when a wide range of molecular lengths is considered23,24. A recent extension to reptation theory has accounted for this discrepancy in terms of CLF20. Although, as noted above, the effects of CLF on the equations used here may be neglected at molecular lengths of interest for UHMWPE …M > 106 †, the contribution of CLF must be recognized when extrapolating the diffusion coef cient D from low to high molecular masses, as needed
in the current work (see below). Writing D¤¤ as the selfdiffusion coef cient at the reference temperature T ¤ and a reference molecular mass M ¤ …> Me †, equations (8) and (9) may be combined to write the disengagement condition:
x¶
³ ´a‡1 1 C1 jl2 M ¤ M 3p2 D¤¤ Mo M ¤
…12†
The remaining question is how this result may be used to provide a measure of particle interface integrity. The answer is to invert equation (12). Given that the polymer contains a range of molecular masses, inversion of the equation provides the molecular mass of the longest molecules that have had suf cient time to disengage from the tubes they occupied when the surfaces were brought together—the ^ . Those chains will maximum reptated molecular mass M have disengaged for which: ³ ´1=…a‡1† ¤¤ ¤ 2 x…x; t†D Mo ^ M µ M …x; t† ˆ M 3p C1 jl2 M ¤
…13†
With passage of time in the melt state, the nature of ^ increases with x, an increasing interfaces evolves. As M fraction of the molecular mass distribution disengages from ^ passed right through the its original tubes. Eventually, if M distribution, no memory of the interfaces would remain, and the melt would have become homogeneous. Completion of the model requires calculation of temperature evolution throughout the process, for the component and process history of interest, to enable the evolution of x ^ to be computed. During the course of the and hence M process, the temperature at any point x in the polymer rises from room temperature. When it reaches Tem , self-diffusion ^ rises according to equation (13). The commences and M power 1=…a ‡ 1† ensures a rate of increase that is initially very rapid but later decays. When the temperature begins to ^ , and fall, there is a further reduction in rate of increase in M T when the temperature falls below sc the rate of increase falls to zero. When the whole component has cooled below this critical temperature, there remains a frozen-in distribution of maximum reptated molecular mass that persists ^ …x; 1†. The computainde nitely in the solid polymer – M tion of this distribution was the task of the numerical model described below. The set of polymer-speci c parameters required for ^ …x; 1†, for a given temperature history, are computing M collected in Table 1 for polyethylene. The reference temperature was taken to be 176¯ C for consistency with the diffusion data of Klein and Briscoe21, and the reference molecular mass and corresponding diffusion coef cient were taken to be 3000 g mole¡1 and 4:4 £ 10¡12 m2 s¡1, respectively, in line with that data. The value of activation energy ED was obtained from the work of Bartels, Crist and Graessley, who studied the temperature-dependence of selfdiffusion in a series of samples of monodisperse, linear, hydrogenated polybutadiene (hPB)22. The power a has been much debated in the recent literature20,23,24. In the present work it was assigned the value 2.4, giving the best t to a wide range of data for monodisperse hPB, collected by Tao et al.24. Trans IChemE, Vol 80, Part A, July 2002
MODELLING THE DECAY OF FUSION DEFECTS
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Table 1. List of parameters used in the model. Parameter
Value
**
4.4 £ 10¡12m2s¡1 [Ref. 21] 176¯ C [Ref. 21] 3000gmol¡ 1 [Ref. 21] 26kJmol¡ 1 [Ref. 22] 6.7 [Ref. 36] 2 28 154pm 2.4 [Ref. 24]
D T* M* ED C1 j Mo l a
NUMERICAL MODEL AND IMPLEMENTATION Thermal Model The thermal model employed two key simpli cations in order to reduce computing time. Firstly, the compressed UHMWPE powder was assumed to act as a continuous, homogeneous, isotropic medium. Secondly, the mass fraction degree of crystallinity w was assumed to be a unique function of temperature during either heating or cooling— thus the inherent rate-dependence of crystallization of UHMWPE was neglected. Rate-dependence was investigated by heating/cooling differential scanning calorimetry (DSC) measurements on the powder under different cooling rates, but allowance for this in the numerical simulations was found to have a negligible effect on results (see below). In particular, the nal degree of crystallinity varied by only 0.9%, when the cooling rate was varied between 2.5 and 20 K min¡1 . Under these conditions, it is convenient to write the equation of heat conduction as: H:…kHT † ˆ rc0p
@T @t
where c0p ² cp ¡ Dh¤f
dw dT
…14†
where cp and c0p are the speci c heat and an effective speci c heat respectively, k is the thermal conductivity, Dh¤f is the speci c enthalpy of fusion of 100% crystalline polyethylene (ˆ 293 kJ kg¡1 , as shown by Wunderlich and Czornyj25 and r is the density. The three material properties r, cp and k are functions of T and w, whereas for the degree of crystallinity itself w ˆ w…T ; sgn T_ †, because of the melting/crystallization asymmetry. The DSC technique allows measurement of c0p for heating and cooling, and this was used (via integration of the second of equations (14) through the melting exotherm and crystallization endotherm) to determine the temperature-dependence of w for heating and cooling. Results used in the present work were those for a cooling rate of 5 K min¡1 : they are shown in Figure 3. The initial crystallinity of the as-received powder was 76%, with a peak melting temperature of 141¯ C: after heating to 180¯ C and cooling in the DSC at 5 K min¡1 the peak crystallization temperature was 118¯ C and the nal crystallinity was 43%26. Remote from the phase transition, measured values of cp agreed closely with those of Wunderlich and Gaur27 for semicrystalline UHMWPE and molten polyethylene, and the reported values were used for cp in the solid and molten states respectively. The density was calculated as a function of temperature, for heating and cooling, using additivity of Trans IChemE, Vol 80, Part A, July 2002
Figure 3. Temperature dependence of the effective speci c heat c0p for the UHMWPE powder during heating and cooling. Symbols refer to data reported by Wunderlich and Gaur (Ref. 27).
¡1 speci c volumes r¡1 c and ra of crystalline and amorphous polyethylene: ¡1 r¡1 ˆ wr¡1 c ‡ …1 ¡ w†ra
…15†
where temperature-dependent speci c volumes and r¡1 a were obtained from the published data of Chiang and Flory28 and Gubler and Kovacs29 respectively. The resulting variations of density are shown in Figure 4 for heating and cooling. Thermal conductivity k has been reported for polyethylene by several authors30–33. In the solid (semicrystalline) state it increases with decreasing temperature and increasing crystallinity (and hence density). There is also some evidence for separability of the dependences on temperature and density33: k ˆ f …r†g…T †
r¡1 c
…16†
In the melt (amorphous) state k appears constant to within the usual experimental scatter, at least to 200¯ C30,32,33.
Figure 4. Temperature dependence of calculated density for the UHMWPE powder during heating and cooling: full line and dash-dotted line respectively.
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In the present work, the initial powder can be deduced from its degree of crystallinity to have a density of 961 kgm¡3 at 20¯ C. In its highly compressed state during heating to the melting range, it was assigned the thermal conductivity function k1 …T † given by Hoechst32, for a linear polyethylene of density 950 kgm¡3 —the nearest available data covering the temperature range required. After crystallization, the measured density was close to 940 kgm¡3 at 20¯ C26. Exploiting equation (16), therefore, the thermal conductivity was obtained from k2 ˆ k1 …k…940; 20†/ k…961; 20††, where the ratio of conductivities at 20¯ C was 0.82, as obtained from the data of Sheldon and Lane31. In the melt state, the thermal conductivity was taken to be constant with value ka . Quoted values for this from several sources30,32,33 lie in the range 0:20¡0:26 Wm ¡1 K¡1 . In the present work, it was treated as the only adjustable parameter, and its value found by optimizing the t between the calculated evolution of temperature at the centre of the moulding and its measured counterpart. The best t was obtained with ka ˆ 0:16 Wm¡1 K ¡1 . At temperatures in the ranges of melting or crystallization, k was obtained by linear interpolation with respect to degree of crystallinity, between ka and k1 or k2 respectively. The resulting dependence of k on temperature is shown in Figure 5, for heating and cooling. Finite Element Implementation The continuum nite element (FE) solver employed in the present work was ABAQUS/Standard 5.7 running on a single processor of the Oxford supercomputer OSCAR. The model outlined above was implemented in ABAQUS by use of a user-de ned material FORTRAN subroutine UMATHT (User Material Thermal model). For each time step, the UMATHT was called by ABAQUS at each iteration, for each integration point, to up-date the speci c internal energy U, heat- ux vector f, effective speci c heat c0p , thermal conductivity k, effective time x and maximum ^ , for a given trial up-date of the reptated molecular mass M temperature T. The latter four variables were carried within
the main program as solution-dependent state variables. They were stored at the end of each time increment and retrieved during the initialization of the UMATHT routine at the next time increment. The main program employed iteration to achieve energy balance, to within the required convergence condition. The Appendix details the steps executed within the UMATHT. To illustrate the application of the model, simulations were carried out of the compression moulding process, for the conditions of the experimental study reported previously from this laboratory7,26. The mould simulated was of hardened tool steel and of cylindrical form, with a close tting steel plunger by which pressure was applied to the polymer. The nished cylindrical mouldings had dimensions: 30 mm diameter and 15 mm length. In use, the mould was placed between the temperature-controlled platens of a manually-operated hydraulic press. The mould was instrumented for measurement of pressure and temperature at the mould wall, and of temperature at the centre of the moulding. The process was simulated using a mesh of 4-node linear, diffusive heat transfer, axisymmetric elements (ABAQUS type DCAX4) for the cylinder of compressed polymer. In view of the relatively high thermal conductivity of steel, and short distance between mould temperature measuring point and the mould wall …2 mm†, the measured mould temperature represented well the outer surface temperature of the polymer during compression moulding, provided the surfaces were in intimate contact. The steel mould was, therefore, treated as providing a uniform boundary temperature, equal to that measured in the mould wall. Slight nonaxisymmetry of the real mould, arising from the pressure transducer port, was neglected. Thus, the three dimensional problem was reduced to a two-dimensional one in radial and axial coordinates r, z. Exploiting symmetry of the problem, an 81-element mesh was used …9 £ 9†, covering one quadrant of a plane axial cross-section of the real moulding. Re nement of the mesh to 162 elements was examined, but found to offer no further convergence improvement in the results of interest, but more computing time was required. Boundary conditions applied to the quadrant were of two types. The axis and transverse mid-plane of the cylinder were treated as insulated, in accordance with the symmetry of the problem. The two outer surfaces received an imposed temperature–time sequence, equal to that measured in the mould wall. The thermal model was validated by simulating the benchmark problem of a long cylinder with constant properties and no internal heat source, subject to a step in boundary temperature. The predicted evolution of temperature versus time and position was indistinguishable from that published by Schneider34. RESULTS AND COMPARISON WITH EXPERIMENTS
Figure 5. Temperature dependent thermal conductivity input data for the thermal model. Circles and triangles refer to the heating and cooling process, respectively. Solid line refers to the published data from Ref. 32 for linear polyethylene with density of 950 kg cm¡3 .
A comparison of simulation results with in-situ temperature measurements during compression moulding is shown in Figure 6, for a maximum mould temperature of 176 § 1¯ C. The calculated temperature shows reasonable agreement with experimental data: for the solid state before melting, in the melt state and during crystallization. Within the melting (plateau) region however, the measured temperature Trans IChemE, Vol 80, Part A, July 2002
MODELLING THE DECAY OF FUSION DEFECTS
Figure 6. Comparison of measured UHMWPE temperature data (symbols) with the thermal model simulations (lines) during compression moulding. Filled and opened symbols show measured temperature in the mould wall near the polymer outer surface, and in the polymer at the centre of the moulding, respectively. The dashed line shows calculated temperature at the centre of the moulding during a simulation of the process, with the same boundary temperature sequence (full line).
was higher than that calculated, with a maximum temperature difference of 6:3 K. This is probably because of the effect of pressure on melting temperature, since the measurements of w used in calculating c0p for the model were obtained by DSC at atmospheric pressure. However, in the compression moulding process, melting takes place at higher pressure e.g. 20 MPa7. The rate of change of melting temperature with respect to pressure P is given by the Clausius-Clapeyron equation: dTf Tf¤ …1†Dn¤ ˆ dP Dh¤f
…17†
where Tf¤ …1† is the melting temperature of equilibrium polyethylene with in nite molecular weight, which is 414:6 K25. Dn¤ is the increase in speci c volume during melting for equilibrium polyethylene, and is equal to ¡1 ¡4 3 r¡1 m kg¡1 , employing the data of a ¡ rc ˆ 2:36 £ 10 28 Chiang and Flory and Gubler and Kovacs29 mentioned previously. For a change of pressure from atmospheric to 20 MPa, equation (17) indicates a rise in the melting temperature Tf of 6:7 K. This is satisfactorily close to the maximum temperature difference 6:3 K between the simulations and the measured data. Furthermore, it is interesting to note that the measured temperature indicates a broader melting region than the simulations (see Figure 6). This may be attributed to the pressure build-up during melting. As shown previously7 the pressure rose during melting, and then remained constant until cooling. Clearly, the simulation of temperature could be improved by coupling the thermal model to an FE analysis of pressure evolution during moulding. However, this lay beyond the scope of the present work. For the solid state after crystallization, the numerical simulations indicated a faster cooling than measured experimentally, see Figure 6. The most likely explanation is an insulating air gap developing between the polymer surface and the mould wall during polymer contraction after crystallization. To investigate the importance of the inherent timedependence of crystallization, a set of simulations was carried out using different data sets for the function c0p , Trans IChemE, Vol 80, Part A, July 2002
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Figure 7. Result from a simulation of compression moulding of UHMWPE with a dwell of 10 minutes at 175¯ C: 3-D plot showing the development of ^ with time and radial position, at the transverse mid-plane of the M cylindrical mould.
obtained by DSC by cooling the powder at different rates. There was no discernible difference between the predicted temperatures. Moreover, the cooling rate of the mould during compression moulding was approximately 7 K min¡1 , close to the DSC cooling rate 5 K min¡1 , chosen for providing the input data c0p during crystallization in the model. This con rmed the adequacy of treating the degree of crystallinity w as a unique function of temperature during crystallization, under the conditions of compression moulding modelled here. ^ , for the cylindrical Sample results from calculations of M moulding described above, are shown in Figures 7–9. Figure 7 is a 3-D plot showing the predicted mid-plane maximum reptated molecular weight as a function of radial position and time, for a particular set of process conditions. It reveals how the nite rate of heat transfer causes reptation to proceed more slowly at the interior of the moulding …r ˆ 0†, compared with its surface …r ˆ 15 mm†. Figure 8 shows the predicted spatial variation of nal maximum reptated molecular weight, at the end of a moulding cycle, for moulding prepared with a particular process history of 175¯ C=15 minutes. Simulations for a range of process histories indicated, as expected, that higher temperature,
Figure 8. Result from a simulation of compression moulding of UHMWPE ^ £ 10¡6 at the end of with a dwell of 15 minutes at 175¯ C: contour plot of M a moulding cycle.
430
WU et al. and users of their products that the severity of the fusion defects lies within acceptable bounds? The dilemma is particularly acute for load-bearing prosthesis parts, where the consequences of an in-use fatigue failure are extreme for the patient, and where a given product is produced in a range of sizes to suit different patients. The model presented here provides a means of overcoming this dif culty, as it allows a quanti ed measure of the severity of Type 2 fusion defects to be mapped in advance for a given component, to be subjected to a given process history. CONCLUSIONS
Figure 9. Sample results from a series of simulations of compression moulding, showing the predicted dependence on temperature–time history ^ , for the centre in the melt of the nal maximum reptated molecular mass M of the moulding. Circle and triangle symbols represent different values of melt temperature during the dwell time: 165¯ C and 175¯ C, respectively. The dashed lines indicate the speci ed upper and lower limits on viscosity average molecular mass, for the medical grade UHMWPE used in the current work.
longer dwell time, processes produce mouldings with higher ^ …x; 1†26. The worst case (lowest and more uniform M ^ M …x; 1†) clearly occurs at the centre of the cylindrical mouldings simulated here. Figure 9 summarizes the predicted dependence on dwell time and temperature in the melt state, as obtained from a series of simulations. The effects of temperature and time are clearly apparent. However, the most remarkable feature of this graph is that, ^ …x; 1† come for none of the conditions simulated does M close to the range of viscosity average molecular mass MZ speci ed for the grade of UHMWPE used in the actual mouldings being simulated—indicated by dashed lines in Figure 9. The sample results presented here have demonstrated that, for the process conditions simulated, there is incomplete reptation in UHMWPE. The maximum reptated molecular mass is predicted not to reach the range of the viscosity average molecular mass. This means that the nal solid mouldings retained memory of the original molecular conformations in the powder particles. Since this includes molecules located at particle boundaries, it follows that mouldings also retained a memory of the particle interfaces, in terms of incomplete molecular diffusion across them. Thus, simulations have predicted the existence of Type 2 fusion defects. It is interesting to note, therefore, that these were discovered by the authors experimentally, in the same mouldings whose processing has been simulated here6. As anticipated by the simulations, they were observed even for the longest times and highest temperatures employed during processing, in fully-compacted mouldings that appeared defect-free until tested using the special procedure described before6,7. It is clear that Type 2 fusion defects cannot be avoided in UHMWPE, for reasonable process conditions. However, another prediction of the present work is that the severity ^ , is a strong function of of them, as expressed through M process history, and typically varies signi cantly within a moulding. In this case, how do processors satisfy themselves
A process-modelling tool has been devised, for predicting particle interface integrity in compression-moulded UHMWPE, processed from as-polymerized powder. This information is central to optimizing performance of UHMWPE bearing surfaces in hip and knee prostheses, since it is known that fatigue failure in-use is associated with fusion defects. Earlier work from the authors’ laboratory identi ed two distinct types of fusion defect: voids (Type 1 defects) and diffusion–de cient interfaces (Type 2 defects). The new quanti er of the extent of self-diffusion introduced ^ —charachere—maximum reptated molecular weight M terizes the degree of severity of Type 2 fusion defects, since it represents the size of the longest molecules that have had time to nd a totally new conformation since particle surfaces were brought together. For a given size and shape of component, it depends on process conditions, speci cally the time-temperature history in the melt state, and hence, on location within the mould. The present model, combining results from reptation theory, extended to the non-isothermal case, with a Finite Element heat transfer ^ solver, allows computation of the spatial variation of M within the nal solid component. The usefulness of the model has been clearly demonstrated by the simulation of compression moulding trials described in the authors’ earlier publication, for one of the two instrumented cylindrical moulds used. Results showed how interface ^ , varied throughout mouldintegrity, quanti ed in terms of M ings, for typical values of temperature and time spent in the ^ melt state during the moulding cycle. In all cases, however, M everywhere remained signi cantly below the range of viscosity average molecular weight for the grade of UHMWPE studied, thus predicting the presence of Type 2 fusion defects. The modelling results explain, therefore, the microscopic evidence for Type 2 fusion defects observed previously in these mouldings6. Given the apparent impossibility of reaching times long enough for full reptation of the entire sample during compression moulding, it must be concluded that UHMWPE components always contain Type 2 fusion defects. In this situation, a model of the form proposed here becomes a useful tool for computer-aided evaluation of the distribution of the degree of severity of the defects, either as a process/product design tool, or as a means of certifying the state of Type 2 defects in existing products35. Although promising, the approach described here needs further development to achieve its potential. In particular, there is a need to include erasure of Type 1 defects, to achieve a full model of the consolidation of UHMWPE during processing. Furthermore, to achieve maximum relevance to in-use performance, the state of Type 1 and Type 2 Trans IChemE, Vol 80, Part A, July 2002
MODELLING THE DECAY OF FUSION DEFECTS defects, as predicted, needs to be correlated with the resulting in-vivo fatigue and wear performance of components. These issues remain as objectives for further work. APPENDIX Steps Executed by the User-De ned Material Subroutine On a call to the subroutine UMATHT at discrete time t ‡ Dt, it was provided with a trial new temperature t‡Dt T ˆ t T ‡ DT and temperature gradient vector t‡Dt HT, together with previous values of the speci c internal energy t U and the solution-dependent state variables. The steps executed were as follows. (1) Update the effective speci c heat to t‡Dt c0p using the new temperature and the sign of DT , and, hence, compute the new speci c internal energy: t‡Dt
U ˆ t U ‡t‡Dt c0p DT
…A1†
The neglect of the distinction between cp and cv is justi ed on the grounds that, at the pressures involved (maximum 20 MPa) the relative difference is only of order 1%. (2) Update the thermal conductivity t‡Dt k using t‡Dt T and the sign of DT , and hence, compute the new heat ux vector: t‡Dt
f ˆ ¡t‡Dt k t‡Dt
t‡Dt t‡Dt
HT
…A2†
t‡Dt 0 t‡Dt cp , k.
U, (3) Return to the FE solver: f, (4) Update the further solution-dependent state variables ^ and return them to the FE solver. From x and M equation (6): µ ³ ´¶ ED 1 1 t‡Dt aT ˆ exp ¡ …A3† R t‡Dt T T ¤ hence: t‡Dt
Dt x ˆ t x ‡ t‡Dt aT
…A4†
431
3. Landy, M. M. and Walker, P. S., 1988, J Arthroplasty (Suppl), 3: S73. 4. Blunn, G. W., Joshi, A. B., Lilley, P. A., Engelbrecht, E., Ryd, L., Lidgren, L., Hardinge, K., Nieder, E. and Walker, P. S., 1992, Acta Orthop Scand, 63: 247. 5. Wrona, M., Mayor, M., Collier, J. and Jensen, R., 1994, Clin Orthop, 299: 9. 6. Wu, J. J., Buckley, C. P. and O’Connor, J. J., 2001, Journal of Materials Science Letters, 20: 473. 7. Wu, J. J., Buckley, C. P. and O’Connor, J. J., Biomaterials, in press. 8. De Gennes, P. G., 1983, Physics Today, June: 33. 9. De Gennes, P. G., 1971, J Chem Phys, 55: 572. 10. Wool, R. P., Yuan, B.-L. and McGarel, O. J., 1989, Polym Eng Sci, 29: 1340. 11. Wool, R. P., 1995, Polymer Interfaces: Structure and Strength (Hanser Publishers, Munich, Germany). 12. Haire, K. R. and Windle, A. H., 2001, Computational and Theoretical Polymer Science, 11: 227. 13. Wool, R. P. and O’Connor, K. M., 1981, J Appl Phys, 52: 5953. 14. Pecorini, T., 1997, Polym Eng Sci, 37: 308. 15. Truss, R. W., Han, K. S., Wallace, J. F. and Geil, P. H., 1980, Polym Eng Sci, 20: 747. 16. Gao, P., Cheung, M. K. and Leung, T. Y., 1996, Polymer, 37: 3265. 17. Farrar, D. F. and Brain, A. A., 1997, Biomaterials, 18: 1677. 18. Gao, P. and Mackley, M. R., 1994, Polymer, 35: 5210. 19. Doi, M. and Edwards, S. F., 1986, The Theory of Polymer Dynamics (Clarendon Press, Oxford, UK). 20. Frischknecht, A. L. and Milner, S. T., 2000, Macromolecules, 33: 5273. 21. Klein, J. and Briscoe, B. J., 1979, Proc Roy Soc A, 365: 53. 22. Bartels, C. R., Crist, B. and Graessley, W. W., 1984, Macromolecules, 17: 2702. 23. Lodge, T. P., 1999, Phys Rev Lett, 83: 3218. 24. Tao, H., Lodge, T. P. and von Meerwall, E. D., 2000, Macromolecules, 33: 1747. 25. Wunderlich, B. and Czornyj, G., 1977, Macromolecules, 10: 906. 26. Wu, J. J., 2000, DPhil Thesis (University of Oxford, UK). 27. Wunderlich, B. and Gaur, U., 1981, J Phys Chem Ref Data, 10: 119. 28. Chiang, R. and Flory, P. J., 1961, J Am Chem Soc, 83: 2857. 29. Gubler, M. G. and Kovacs, A. J., 1959, J Polymer Sci, 34: 551. 30. Hansen, D. and Ho, C. C., 1965, J Polymer Sci, Part A, 3: 659. 31. Sheldon, R. P. and Lane, S. K., 1965, Polymer, 6: 205. 32. Anon, 1980, Hostalen (Hoechst Plastics). 33. Kraybill, R. R., 1981, Polym Eng Sci, 21: 124. 34. Schneider, P. J., 1963, Temperature Response Charts (Wiley, New York, USA). 35. O’Connor, J. J., Wu, J. J. and Buckley, C. P., Compression Moulding, UK Patent, led April 6th, 1999 (Application No. 99 07843.8). 36. Flory, P. J., 1989, Statistical Mechanics of Chain Molecules (Hanser Publishers, Munich, Germany).
ACKNOWLEDGEMENTS
and: t‡Dt
^ ˆ g… M
1=…a‡1†
x†
where the constant g is given by: ³ ´ =…a‡ D¤¤ M0 1 1† g ˆ M ¤ 3p 2 C1 jl2 M ¤
…A5†
The authors thank Biomet Merck Ltd for providing the UHMWPE powder. J. J. Wu also thanks the K. C. Wong Foundation and St. Hugh’s College, Oxford for the award of scholarships.
…A6†
ADDRESS
REFERENCES
Correspondence concerning this paper should be addressed to Dr C. P. Buckley, Department of Engineering Science, University of Oxford, Parks Rd, Oxford OX1 3PJ, UK. E-mail: [email protected]
1. Li, S. and Burstein, A. H., 1994, J Bone and Joint Surg, 76-A: 1080. 2. Rose, R. M., Crugnola, A., Ries, M., Cimino, W. R., Paul, I. and Radin, E. L., 1979, Clin Orthop, 145: 277.
The manuscript was received 30 November 2001 and accepted for publication 22 April 2002.
Trans IChemE, Vol 80, Part A, July 2002
www.catchword.com=titles=02638762.htm
PROCESSING OF ULTRA-HIGH MOLECULAR WEIGHT POLYETHYLENE Modelling the Decay of Fusion Defects J. J. WU, C. P. BUCKLEY and J. J. O’CONNOR Department of Engineering Science, University of Oxford, Oxford, UK
A
problem in applications of ultra-high molecular weight polyethylene (UHMWPE) is the tendency for components to contain fusion defects, arising during processing of the as-polymerized powder. These defects have been implicated previously in failures of UHMWPE load-bearing surfaces, in knee and hip prostheses. Recent work of the authors has recognized two forms of defect: voids (Type 1) and particle boundaries de cient in diffusion by reptation (Type 2). To assist process and product design, a method has now been developed for predicting the decay of severity of Type 2 defects during processing, for a component of given shape and process history. A new quanti er was introduced for characterizing the progress of diffusion at Type 2 defects in UHMWPE—the maximum ^ . This was computed using results from reptation theory, embedded reptated molecular mass M within a Finite Element thermal model of the process. The method was illustrated by simulating compression moulding trials already carried out experimentally by the same authors. It was ^ never reached the viscosity average molecular mass of the polymer, discovered that M indicating incomplete boundary diffusion, and explaining the previous observation of Type 2 defects even in fully-compacted, apparently perfect mouldings. The method described has potential as a design tool, especially for optimizing manufacture of UHMWPE prosthesis components. Keywords: UHMWPE; polyethylene; reptation; nite element analysis; thermal model; fusion defects; compression moulding; powder processing.
INTRODUCTION
compliant inter-particle boundaries arising from incomplete inter-particle diffusion6. These are labelled as Type 1 and Type 2 fusion defects respectively. The distinction is important. Type 1 defects may be prevented by proper choice of process parameters, whereas Type 2 defects are probably always present for realistic process conditions, but their severity is sensitive to product and process design. Full details of the experimental study are given elsewhere7. The purpose of this paper is to propose a method of predicting the severity of Type 2 defects in compression-moulded UHMWPE, for given component design and process parameters, to assist manufacturers in optimizing the performance of their products. The engineering problem is complicated by the range of prosthesis designs in use, and the need for each design to be produced in a variety of sizes, to suit the needs of different patients. Therefore, the need is for a exible process/product design tool, that can be used by the material processor to ensure that each product has at least a speci ed minimum level of mechanical integrity. A numerical simulation procedure is proposed that, when embedded within a nite element heat transfer model for calculation of temperature, allows the user to map a numerical quanti er of the state of Type 2 defects at particle interfaces, throughout a threedimensional UHMWPE product.
Ultra-high molecular weight polyethylene (UHMWPE) has been the favoured material for the concave bearing surfaces of total knee and hip joint replacement prostheses since the 1970s, because of its desirable properties of biocompatibility, high impact strength, low friction and high wear resistance1. Although generally these perform well, there is a small but signi cant failure rate, involving painful and expensive revision surgery. This has motivated much research to identify the causes of failure. An important outcome has been the realization that incomplete consolidation of the UHMWPE powder during processing into solid, resulting in ‘fusion defects’, is implicated in the failure of the material due to fatigue2–5. Speci cally, cracking and delamination at fusion defects appear to be associated with fatigue failures found in retrieved knee components5. To solve this problem, there is an urgent need for better understanding of the nature and origin of fusion defects, and for reliable means of controlling them during processing. Recent research in this laboratory has been aimed at achieving these goals. A key recent discovery was that asmoulded UHMWPE is prone to fusion defects of two distinct types: (a) sites of incomplete compaction (voids), and (b) sites where compaction is complete but there remain 423
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The approach adopted here is based on the proposal8 that, establishment of a full bond at a polymer–polymer interface, requires self-diffusion to occur for a period of at least the ‘reptation’ time td . This is the time required for a macromolecule to refresh its conformation completely, by diffusion along the curvilinear tube de ned by the topological constraint of its neighbours9. At an interface, it corresponds to the time taken, from the moment the two surfaces are brought into perfect contact, for molecules to lose by diffusion their conformational memory of the interface. This process knits together the two surfaces and creates a weld that is indistinguishable from the bulk polymer. There is now a substantial body of support for this picture of events, from experimental evidence10,11 and from numerical simulations12. In particular, Wool et al. demonstrated that this approach explained the dependence of the strength of polystyrene on time and molecular weight, when moulded from pellets above the glass transition temperature10,13. Pecorini showed that results from reptation theory can be extended to non-isothermal moulding processes of practical importance. He deduced the effective crack depth associated with weld-lines in injection-moulded cellulose acetate propionate, by calculating the interdiffusion distance achieved at weld-lines during cooling near the mould surface14. Since this distance was less than half the radius of gyration for molecules at the surface, cooled most quickly, the weld line acted as a surface crack. A similar approach to that of Pecorini is applied here. Results from reptation theory are combined with a heat transfer simulation of the moulding process, to deduce the extent to which inter-particle diffusion has progressed by the end of the moulding cycle. To quantify the state of the particle interfaces in the nal product, a new parameter is introduced: the maximum reptated molecular weight, that can be computed throughout a moulding of given shape, for a given process history. THEORY Description of the Problem As-polymerized particles of UHMWPE powder have irregular shapes, with widths of order 100 mm. They appear to be agglomerations of several smaller particles, which are themselves assemblies of spherical sub-particles with diameters of order 1 mm. The literature provides many examples, for powder from all the major industrial suppliers6,15–17. The result is a powder whose particles have a surface of complex topography, with many re-entrant regions. A schematic two-dimensional sketch is given in Figure 1. During processing of the powder into solid, for example by compression moulding or ram extrusion, pressure is applied to compact the powder, the temperature is raised above the crystal melting point, and after a suitable interval of time the polymer is cooled to room temperature. The objective is to convert the powder into a homogeneous solid. This requires a two stage process to be completed. In Stage 1, as shown in Figure 1, the powder surfaces are pressed together until they are in intimate contact at the molecular level. If this Stage is not completed, the nal solid contains intra-particle and inter-particle voids. A typical example is shown in Figure 2. The time taken for
Figure 1. Schematic two-dimensional sketch of UHMWPE powder particles and compaction during Stage 1 of processing the powder into solid. Typical sizes of particle and sub-particle are shown. Stage 2 corresponds to molecular diffusion across the interfaces.
completion of Stage 1 depends on the temperature, pressure and time allowed, and the aspect ratio of surface features18. This stage alone, however, would be insuf cient for creating a solid with acceptable mechanical integrity. The particles would be held together by inter-chain van der Waals bonding only, with weak, compliant interfaces; and would be extremely susceptible to initiation and propagation of interparticle cracks when the eventual solid component is loaded in use. There is thus a further essential stage to be completed— Stage 2, where molecules diffuse across the interfaces and endow the particle boundaries with intra-chain covalent bonding. At the temperatures concerned, there is continuous molecular diffusion by the process of reptation. As soon as particles make molecular contact, boundary molecules will begin to explore the space of their adjacent particles. Eventually, all ‘memory’ of the interfaces will have been lost when there ceases to be any correlation between molecular conformations before and after the surfaces were brought together. The particles will be then fully knitted together.
Figure 2. Typical SEM micrograph showing residual intra-particle and inter-particle voids resulting from incomplete compaction (Stage 1) during processing of UHMWPE. The dotted scale bar represents 120 mm.
Trans IChemE, Vol 80, Part A, July 2002
MODELLING THE DECAY OF FUSION DEFECTS Application of Reptation Theory The authors’ aim is to construct a model that will allow a measure of particle-interface integrity to be evaluated, throughout a polymer component of given shape and process history. There are numerous complicating features of the engineering problem. (1) Stages 1 and 2 proceed concurrently at different locations: there is progressive creation of interfaces during compaction, and hence a range of start times for the ensuing inter-particle diffusion. (2) It is not possible to be sure of the state of molecular segregation or of conformations at the start of Stage 2: (a) because of the need for a nite time for randomization following melting of crystals; and (b) because of the perturbing in uence of surfaces of the small sub-particles (whose dimensions are of the same order of magnitude as typical molecular radius of gyration. (3) There are large temperature gradients within typical mouldings during processing, arising from the low thermal conductivity of the material. (4) There are large changes of temperature during processing—from room temperature to beyond the melting region and back again. To make the problem tractable rst it is idealized with the aid of the following assumptions: (1) stage 1 is complete, and therefore, all intra-particle and inter-particle interfaces are in intimate contact at the molecular level (i.e. wetted), when inter-particle diffusion commences; (2) by the time inter-particle diffusion commences, the polymer molecules (of all lengths) can be treated as random coils, randomly dispersed; (3) temperature differences are negligible on the length scale of an individual molecule. With these assumptions, Stage 2 is modelled by taking results of the ‘tube’ model of polymer dynamics in the molten state, proposed for a material with temperature that is both uniform and constant, generalizing them to the case of a nonuniform, varying temperature, and nally embedding them within a numerical simulation of temperature evolution in a moulding of the required shape. A melt of linear polymer molecules is considered, with monomer molecular mass Mo and j backbone covalent bonds per monomer. Although commercially produced polymers typically contain a distribution of molecular lengths, initially chains of a particular length are focused on, i.e. with a particular molecular mass M, chosen arbitrarily except it must exceed the entanglement value Me . Each of these chains consists of n backbone bonds each of length l: thus n ˆ jM =Mo . The entanglement constraint offered to it by adjacent chains at any instant, is represented by a tube surrounding the molecule, of contour length L. The tube theory given by Doi and Edwards19 provides measures of the motion of such chains-within-tubes, and their escape from the tube ends. For present purposes, the pertinent feature is the starting point of the theory: diffusive motion along the tube, ‘reptation’, is represented by evolution of a probability density function c…n; t† for the centre of mass having diffused a distance n in a time t: this obeys the diffusion equation: @c @2 c ˆ Dc 2 @t @n Trans IChemE, Vol 80, Part A, July 2002
…1†
425
where Dc is the curvilinear diffusion coef cient for onedimensional motion along the tube. Solution of equation (1), together with the appropriate boundary conditions, leads to the important conclusion that correlation between current and initial molecular conformations (expressed as the timecorrelation function of the end-to-end vectors) decays exponentially with time, with the longest time constant given by the ‘reptation time’19: td ˆ
L2 p2 Dc
…2†
This is the time taken for the chains to disengage from their initial tubes and refresh their conformations: that is, to lose all memory of their initial conformations. For molecules at an interface this means: to lose all memory of the interface. To apply the above theory to the present problem, the tube diffusion may be represented by the Rouse bead-spring mechanism, in which ow of molecular segments past their neighbours is thermally activated. Thus, the tube diffusion coef cient at time t is determined by the current temperature T, and may be written in terms of this and its value at some reference temperature (indicated by asterisks): µ ³ ´¶ ED 1 1 ¤ Dc ˆ Dc exp ¡ ¡ …3† R T T¤ where ED is the activation energy for self-diffusion of the molten polymer. These arguments apply, however, only to the wholly amorphous state. In the presence of any signi cant crystallinity, the authors expect self-diffusion to be highly retarded, as a result of pinning by those molecular segments that have crystallized. In the present model, this feature was approximated by assuming Dc to vanish at temperatures below a two-valued critical temperature Tc: the end-of-melting temperature Tem or start-of-crystallization temperature Tsc , during heating and cooling respectively. Results of the theory can be extended to the case of a time-varying temperature T …x; t† by introducing a reptationequivalent time-scale x. If equation (3) is substituted in equation (1), the latter may be re-written » µ ³ ´¶ ¼ E 1 @c @x 1 @2 c ¡ ¤ ˆ D¤c 2 …4† exp D @x @t R T T @n Thus, the solution is the same as that for isothermal diffusion at the reference temperature T ¤ , provided, at any position x, the real time t is replaced by the equivalent time x given by: …t dt0 x…x; t† ˆ …5† 0 0 aT …x; t † where:
0 a¡1 T …x; t †
and: Tc ˆ Tem
µ ³ ´¶ ¡ED 1 1 ˆ exp ¡ T …x; t 0 † T ¤ R 0 a¡1 T …x; t † ˆ 0
if T < Tc ;
for T_ ¶ 0;
Tc ˆ Tsc
if T ¶ Tc ; …6† for T_ µ 0
…7†
thereby ensuring the curly bracket in equation (4) remains unity at all times, and incorporating arrest of self-diffusion
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in the presence of crystallinity. The implied ambiguity in equation (7) in the case T_ ˆ 0 is easily resolved by considering the most recent non-zero value of T_ . The signi cance of time x is that it would produce the same extent of reptative diffusion at the reference temperature, as the real time t does for the time-varying temperature. The variable aT is a time-temperature reduction factor for diffusion. Invoking equation (2) for a time-varying temperature, it is clear that chains will have disengaged from their initial tubes when: x ¶ t¤d ˆ
L2 p2 D¤c
…8†
To apply this result to a particular polymer system, D¤c must be related to the measurable (molecular mass dependent) diffusion coef cient D¤ governing three-dimensional centreof-mass diffusion at the reference temperature. This may be achieved by matching the mean-square distance travelled by the centre of mass along the tube in a given time (by onedimensional diffusion), to the corresponding mean-square distance travelled in three dimensions in the same time (by three-dimensional diffusion). Suppose the rms distance travelled by the centre of mass along the tube is L, then the rms distance travelled in three dimensions must be the rms end-to-end length of the chain hR2 i1=2 . Equating the times for one- and three-dimensional diffusion: hR2 i L2 ˆ 6D¤ 2D¤c
…9†
and, hence, the condition for disengagement may be written: hR2 i …10† 3p2 D¤ The original theory of reptation, as exploited above, is now known to require correction. The diffusing chains are not inextensible strings within their tubes, as originally conceived, but they will be subject to contour length uctuations (CLF)20. The result is that a correction is needed to equation (2) and others derived from it. However, the fractional error in the original expression varies as M ¡0:5 and for the chain lengths of interest in this paper may be neglected. From the de nition of the characteristic ratio C1 , characterizing exibility of the chains, hR2 i may be found in terms of M: x¶
hR2 i ˆ C1 jl2
M Mo
…11†
For many years D was thought to scale with molecular length as M ¡2 , as predicted by reptation theory9,19 and as apparently observed experimentally21,22. However, more recent experimental data and careful re-appraisal of earlier data have a revealed a stronger dependence M ¡a , where a lies between 2 and 2.5, when a wide range of molecular lengths is considered23,24. A recent extension to reptation theory has accounted for this discrepancy in terms of CLF20. Although, as noted above, the effects of CLF on the equations used here may be neglected at molecular lengths of interest for UHMWPE …M > 106 †, the contribution of CLF must be recognized when extrapolating the diffusion coef cient D from low to high molecular masses, as needed
in the current work (see below). Writing D¤¤ as the selfdiffusion coef cient at the reference temperature T ¤ and a reference molecular mass M ¤ …> Me †, equations (8) and (9) may be combined to write the disengagement condition:
x¶
³ ´a‡1 1 C1 jl2 M ¤ M 3p2 D¤¤ Mo M ¤
…12†
The remaining question is how this result may be used to provide a measure of particle interface integrity. The answer is to invert equation (12). Given that the polymer contains a range of molecular masses, inversion of the equation provides the molecular mass of the longest molecules that have had suf cient time to disengage from the tubes they occupied when the surfaces were brought together—the ^ . Those chains will maximum reptated molecular mass M have disengaged for which: ³ ´1=…a‡1† ¤¤ ¤ 2 x…x; t†D Mo ^ M µ M …x; t† ˆ M 3p C1 jl2 M ¤
…13†
With passage of time in the melt state, the nature of ^ increases with x, an increasing interfaces evolves. As M fraction of the molecular mass distribution disengages from ^ passed right through the its original tubes. Eventually, if M distribution, no memory of the interfaces would remain, and the melt would have become homogeneous. Completion of the model requires calculation of temperature evolution throughout the process, for the component and process history of interest, to enable the evolution of x ^ to be computed. During the course of the and hence M process, the temperature at any point x in the polymer rises from room temperature. When it reaches Tem , self-diffusion ^ rises according to equation (13). The commences and M power 1=…a ‡ 1† ensures a rate of increase that is initially very rapid but later decays. When the temperature begins to ^ , and fall, there is a further reduction in rate of increase in M T when the temperature falls below sc the rate of increase falls to zero. When the whole component has cooled below this critical temperature, there remains a frozen-in distribution of maximum reptated molecular mass that persists ^ …x; 1†. The computainde nitely in the solid polymer – M tion of this distribution was the task of the numerical model described below. The set of polymer-speci c parameters required for ^ …x; 1†, for a given temperature history, are computing M collected in Table 1 for polyethylene. The reference temperature was taken to be 176¯ C for consistency with the diffusion data of Klein and Briscoe21, and the reference molecular mass and corresponding diffusion coef cient were taken to be 3000 g mole¡1 and 4:4 £ 10¡12 m2 s¡1, respectively, in line with that data. The value of activation energy ED was obtained from the work of Bartels, Crist and Graessley, who studied the temperature-dependence of selfdiffusion in a series of samples of monodisperse, linear, hydrogenated polybutadiene (hPB)22. The power a has been much debated in the recent literature20,23,24. In the present work it was assigned the value 2.4, giving the best t to a wide range of data for monodisperse hPB, collected by Tao et al.24. Trans IChemE, Vol 80, Part A, July 2002
MODELLING THE DECAY OF FUSION DEFECTS
427
Table 1. List of parameters used in the model. Parameter
Value
**
4.4 £ 10¡12m2s¡1 [Ref. 21] 176¯ C [Ref. 21] 3000gmol¡ 1 [Ref. 21] 26kJmol¡ 1 [Ref. 22] 6.7 [Ref. 36] 2 28 154pm 2.4 [Ref. 24]
D T* M* ED C1 j Mo l a
NUMERICAL MODEL AND IMPLEMENTATION Thermal Model The thermal model employed two key simpli cations in order to reduce computing time. Firstly, the compressed UHMWPE powder was assumed to act as a continuous, homogeneous, isotropic medium. Secondly, the mass fraction degree of crystallinity w was assumed to be a unique function of temperature during either heating or cooling— thus the inherent rate-dependence of crystallization of UHMWPE was neglected. Rate-dependence was investigated by heating/cooling differential scanning calorimetry (DSC) measurements on the powder under different cooling rates, but allowance for this in the numerical simulations was found to have a negligible effect on results (see below). In particular, the nal degree of crystallinity varied by only 0.9%, when the cooling rate was varied between 2.5 and 20 K min¡1 . Under these conditions, it is convenient to write the equation of heat conduction as: H:…kHT † ˆ rc0p
@T @t
where c0p ² cp ¡ Dh¤f
dw dT
…14†
where cp and c0p are the speci c heat and an effective speci c heat respectively, k is the thermal conductivity, Dh¤f is the speci c enthalpy of fusion of 100% crystalline polyethylene (ˆ 293 kJ kg¡1 , as shown by Wunderlich and Czornyj25 and r is the density. The three material properties r, cp and k are functions of T and w, whereas for the degree of crystallinity itself w ˆ w…T ; sgn T_ †, because of the melting/crystallization asymmetry. The DSC technique allows measurement of c0p for heating and cooling, and this was used (via integration of the second of equations (14) through the melting exotherm and crystallization endotherm) to determine the temperature-dependence of w for heating and cooling. Results used in the present work were those for a cooling rate of 5 K min¡1 : they are shown in Figure 3. The initial crystallinity of the as-received powder was 76%, with a peak melting temperature of 141¯ C: after heating to 180¯ C and cooling in the DSC at 5 K min¡1 the peak crystallization temperature was 118¯ C and the nal crystallinity was 43%26. Remote from the phase transition, measured values of cp agreed closely with those of Wunderlich and Gaur27 for semicrystalline UHMWPE and molten polyethylene, and the reported values were used for cp in the solid and molten states respectively. The density was calculated as a function of temperature, for heating and cooling, using additivity of Trans IChemE, Vol 80, Part A, July 2002
Figure 3. Temperature dependence of the effective speci c heat c0p for the UHMWPE powder during heating and cooling. Symbols refer to data reported by Wunderlich and Gaur (Ref. 27).
¡1 speci c volumes r¡1 c and ra of crystalline and amorphous polyethylene: ¡1 r¡1 ˆ wr¡1 c ‡ …1 ¡ w†ra
…15†
where temperature-dependent speci c volumes and r¡1 a were obtained from the published data of Chiang and Flory28 and Gubler and Kovacs29 respectively. The resulting variations of density are shown in Figure 4 for heating and cooling. Thermal conductivity k has been reported for polyethylene by several authors30–33. In the solid (semicrystalline) state it increases with decreasing temperature and increasing crystallinity (and hence density). There is also some evidence for separability of the dependences on temperature and density33: k ˆ f …r†g…T †
r¡1 c
…16†
In the melt (amorphous) state k appears constant to within the usual experimental scatter, at least to 200¯ C30,32,33.
Figure 4. Temperature dependence of calculated density for the UHMWPE powder during heating and cooling: full line and dash-dotted line respectively.
WU et al.
428
In the present work, the initial powder can be deduced from its degree of crystallinity to have a density of 961 kgm¡3 at 20¯ C. In its highly compressed state during heating to the melting range, it was assigned the thermal conductivity function k1 …T † given by Hoechst32, for a linear polyethylene of density 950 kgm¡3 —the nearest available data covering the temperature range required. After crystallization, the measured density was close to 940 kgm¡3 at 20¯ C26. Exploiting equation (16), therefore, the thermal conductivity was obtained from k2 ˆ k1 …k…940; 20†/ k…961; 20††, where the ratio of conductivities at 20¯ C was 0.82, as obtained from the data of Sheldon and Lane31. In the melt state, the thermal conductivity was taken to be constant with value ka . Quoted values for this from several sources30,32,33 lie in the range 0:20¡0:26 Wm ¡1 K¡1 . In the present work, it was treated as the only adjustable parameter, and its value found by optimizing the t between the calculated evolution of temperature at the centre of the moulding and its measured counterpart. The best t was obtained with ka ˆ 0:16 Wm¡1 K ¡1 . At temperatures in the ranges of melting or crystallization, k was obtained by linear interpolation with respect to degree of crystallinity, between ka and k1 or k2 respectively. The resulting dependence of k on temperature is shown in Figure 5, for heating and cooling. Finite Element Implementation The continuum nite element (FE) solver employed in the present work was ABAQUS/Standard 5.7 running on a single processor of the Oxford supercomputer OSCAR. The model outlined above was implemented in ABAQUS by use of a user-de ned material FORTRAN subroutine UMATHT (User Material Thermal model). For each time step, the UMATHT was called by ABAQUS at each iteration, for each integration point, to up-date the speci c internal energy U, heat- ux vector f, effective speci c heat c0p , thermal conductivity k, effective time x and maximum ^ , for a given trial up-date of the reptated molecular mass M temperature T. The latter four variables were carried within
the main program as solution-dependent state variables. They were stored at the end of each time increment and retrieved during the initialization of the UMATHT routine at the next time increment. The main program employed iteration to achieve energy balance, to within the required convergence condition. The Appendix details the steps executed within the UMATHT. To illustrate the application of the model, simulations were carried out of the compression moulding process, for the conditions of the experimental study reported previously from this laboratory7,26. The mould simulated was of hardened tool steel and of cylindrical form, with a close tting steel plunger by which pressure was applied to the polymer. The nished cylindrical mouldings had dimensions: 30 mm diameter and 15 mm length. In use, the mould was placed between the temperature-controlled platens of a manually-operated hydraulic press. The mould was instrumented for measurement of pressure and temperature at the mould wall, and of temperature at the centre of the moulding. The process was simulated using a mesh of 4-node linear, diffusive heat transfer, axisymmetric elements (ABAQUS type DCAX4) for the cylinder of compressed polymer. In view of the relatively high thermal conductivity of steel, and short distance between mould temperature measuring point and the mould wall …2 mm†, the measured mould temperature represented well the outer surface temperature of the polymer during compression moulding, provided the surfaces were in intimate contact. The steel mould was, therefore, treated as providing a uniform boundary temperature, equal to that measured in the mould wall. Slight nonaxisymmetry of the real mould, arising from the pressure transducer port, was neglected. Thus, the three dimensional problem was reduced to a two-dimensional one in radial and axial coordinates r, z. Exploiting symmetry of the problem, an 81-element mesh was used …9 £ 9†, covering one quadrant of a plane axial cross-section of the real moulding. Re nement of the mesh to 162 elements was examined, but found to offer no further convergence improvement in the results of interest, but more computing time was required. Boundary conditions applied to the quadrant were of two types. The axis and transverse mid-plane of the cylinder were treated as insulated, in accordance with the symmetry of the problem. The two outer surfaces received an imposed temperature–time sequence, equal to that measured in the mould wall. The thermal model was validated by simulating the benchmark problem of a long cylinder with constant properties and no internal heat source, subject to a step in boundary temperature. The predicted evolution of temperature versus time and position was indistinguishable from that published by Schneider34. RESULTS AND COMPARISON WITH EXPERIMENTS
Figure 5. Temperature dependent thermal conductivity input data for the thermal model. Circles and triangles refer to the heating and cooling process, respectively. Solid line refers to the published data from Ref. 32 for linear polyethylene with density of 950 kg cm¡3 .
A comparison of simulation results with in-situ temperature measurements during compression moulding is shown in Figure 6, for a maximum mould temperature of 176 § 1¯ C. The calculated temperature shows reasonable agreement with experimental data: for the solid state before melting, in the melt state and during crystallization. Within the melting (plateau) region however, the measured temperature Trans IChemE, Vol 80, Part A, July 2002
MODELLING THE DECAY OF FUSION DEFECTS
Figure 6. Comparison of measured UHMWPE temperature data (symbols) with the thermal model simulations (lines) during compression moulding. Filled and opened symbols show measured temperature in the mould wall near the polymer outer surface, and in the polymer at the centre of the moulding, respectively. The dashed line shows calculated temperature at the centre of the moulding during a simulation of the process, with the same boundary temperature sequence (full line).
was higher than that calculated, with a maximum temperature difference of 6:3 K. This is probably because of the effect of pressure on melting temperature, since the measurements of w used in calculating c0p for the model were obtained by DSC at atmospheric pressure. However, in the compression moulding process, melting takes place at higher pressure e.g. 20 MPa7. The rate of change of melting temperature with respect to pressure P is given by the Clausius-Clapeyron equation: dTf Tf¤ …1†Dn¤ ˆ dP Dh¤f
…17†
where Tf¤ …1† is the melting temperature of equilibrium polyethylene with in nite molecular weight, which is 414:6 K25. Dn¤ is the increase in speci c volume during melting for equilibrium polyethylene, and is equal to ¡1 ¡4 3 r¡1 m kg¡1 , employing the data of a ¡ rc ˆ 2:36 £ 10 28 Chiang and Flory and Gubler and Kovacs29 mentioned previously. For a change of pressure from atmospheric to 20 MPa, equation (17) indicates a rise in the melting temperature Tf of 6:7 K. This is satisfactorily close to the maximum temperature difference 6:3 K between the simulations and the measured data. Furthermore, it is interesting to note that the measured temperature indicates a broader melting region than the simulations (see Figure 6). This may be attributed to the pressure build-up during melting. As shown previously7 the pressure rose during melting, and then remained constant until cooling. Clearly, the simulation of temperature could be improved by coupling the thermal model to an FE analysis of pressure evolution during moulding. However, this lay beyond the scope of the present work. For the solid state after crystallization, the numerical simulations indicated a faster cooling than measured experimentally, see Figure 6. The most likely explanation is an insulating air gap developing between the polymer surface and the mould wall during polymer contraction after crystallization. To investigate the importance of the inherent timedependence of crystallization, a set of simulations was carried out using different data sets for the function c0p , Trans IChemE, Vol 80, Part A, July 2002
429
Figure 7. Result from a simulation of compression moulding of UHMWPE with a dwell of 10 minutes at 175¯ C: 3-D plot showing the development of ^ with time and radial position, at the transverse mid-plane of the M cylindrical mould.
obtained by DSC by cooling the powder at different rates. There was no discernible difference between the predicted temperatures. Moreover, the cooling rate of the mould during compression moulding was approximately 7 K min¡1 , close to the DSC cooling rate 5 K min¡1 , chosen for providing the input data c0p during crystallization in the model. This con rmed the adequacy of treating the degree of crystallinity w as a unique function of temperature during crystallization, under the conditions of compression moulding modelled here. ^ , for the cylindrical Sample results from calculations of M moulding described above, are shown in Figures 7–9. Figure 7 is a 3-D plot showing the predicted mid-plane maximum reptated molecular weight as a function of radial position and time, for a particular set of process conditions. It reveals how the nite rate of heat transfer causes reptation to proceed more slowly at the interior of the moulding …r ˆ 0†, compared with its surface …r ˆ 15 mm†. Figure 8 shows the predicted spatial variation of nal maximum reptated molecular weight, at the end of a moulding cycle, for moulding prepared with a particular process history of 175¯ C=15 minutes. Simulations for a range of process histories indicated, as expected, that higher temperature,
Figure 8. Result from a simulation of compression moulding of UHMWPE ^ £ 10¡6 at the end of with a dwell of 15 minutes at 175¯ C: contour plot of M a moulding cycle.
430
WU et al. and users of their products that the severity of the fusion defects lies within acceptable bounds? The dilemma is particularly acute for load-bearing prosthesis parts, where the consequences of an in-use fatigue failure are extreme for the patient, and where a given product is produced in a range of sizes to suit different patients. The model presented here provides a means of overcoming this dif culty, as it allows a quanti ed measure of the severity of Type 2 fusion defects to be mapped in advance for a given component, to be subjected to a given process history. CONCLUSIONS
Figure 9. Sample results from a series of simulations of compression moulding, showing the predicted dependence on temperature–time history ^ , for the centre in the melt of the nal maximum reptated molecular mass M of the moulding. Circle and triangle symbols represent different values of melt temperature during the dwell time: 165¯ C and 175¯ C, respectively. The dashed lines indicate the speci ed upper and lower limits on viscosity average molecular mass, for the medical grade UHMWPE used in the current work.
longer dwell time, processes produce mouldings with higher ^ …x; 1†26. The worst case (lowest and more uniform M ^ M …x; 1†) clearly occurs at the centre of the cylindrical mouldings simulated here. Figure 9 summarizes the predicted dependence on dwell time and temperature in the melt state, as obtained from a series of simulations. The effects of temperature and time are clearly apparent. However, the most remarkable feature of this graph is that, ^ …x; 1† come for none of the conditions simulated does M close to the range of viscosity average molecular mass MZ speci ed for the grade of UHMWPE used in the actual mouldings being simulated—indicated by dashed lines in Figure 9. The sample results presented here have demonstrated that, for the process conditions simulated, there is incomplete reptation in UHMWPE. The maximum reptated molecular mass is predicted not to reach the range of the viscosity average molecular mass. This means that the nal solid mouldings retained memory of the original molecular conformations in the powder particles. Since this includes molecules located at particle boundaries, it follows that mouldings also retained a memory of the particle interfaces, in terms of incomplete molecular diffusion across them. Thus, simulations have predicted the existence of Type 2 fusion defects. It is interesting to note, therefore, that these were discovered by the authors experimentally, in the same mouldings whose processing has been simulated here6. As anticipated by the simulations, they were observed even for the longest times and highest temperatures employed during processing, in fully-compacted mouldings that appeared defect-free until tested using the special procedure described before6,7. It is clear that Type 2 fusion defects cannot be avoided in UHMWPE, for reasonable process conditions. However, another prediction of the present work is that the severity ^ , is a strong function of of them, as expressed through M process history, and typically varies signi cantly within a moulding. In this case, how do processors satisfy themselves
A process-modelling tool has been devised, for predicting particle interface integrity in compression-moulded UHMWPE, processed from as-polymerized powder. This information is central to optimizing performance of UHMWPE bearing surfaces in hip and knee prostheses, since it is known that fatigue failure in-use is associated with fusion defects. Earlier work from the authors’ laboratory identi ed two distinct types of fusion defect: voids (Type 1 defects) and diffusion–de cient interfaces (Type 2 defects). The new quanti er of the extent of self-diffusion introduced ^ —charachere—maximum reptated molecular weight M terizes the degree of severity of Type 2 fusion defects, since it represents the size of the longest molecules that have had time to nd a totally new conformation since particle surfaces were brought together. For a given size and shape of component, it depends on process conditions, speci cally the time-temperature history in the melt state, and hence, on location within the mould. The present model, combining results from reptation theory, extended to the non-isothermal case, with a Finite Element heat transfer ^ solver, allows computation of the spatial variation of M within the nal solid component. The usefulness of the model has been clearly demonstrated by the simulation of compression moulding trials described in the authors’ earlier publication, for one of the two instrumented cylindrical moulds used. Results showed how interface ^ , varied throughout mouldintegrity, quanti ed in terms of M ings, for typical values of temperature and time spent in the ^ melt state during the moulding cycle. In all cases, however, M everywhere remained signi cantly below the range of viscosity average molecular weight for the grade of UHMWPE studied, thus predicting the presence of Type 2 fusion defects. The modelling results explain, therefore, the microscopic evidence for Type 2 fusion defects observed previously in these mouldings6. Given the apparent impossibility of reaching times long enough for full reptation of the entire sample during compression moulding, it must be concluded that UHMWPE components always contain Type 2 fusion defects. In this situation, a model of the form proposed here becomes a useful tool for computer-aided evaluation of the distribution of the degree of severity of the defects, either as a process/product design tool, or as a means of certifying the state of Type 2 defects in existing products35. Although promising, the approach described here needs further development to achieve its potential. In particular, there is a need to include erasure of Type 1 defects, to achieve a full model of the consolidation of UHMWPE during processing. Furthermore, to achieve maximum relevance to in-use performance, the state of Type 1 and Type 2 Trans IChemE, Vol 80, Part A, July 2002
MODELLING THE DECAY OF FUSION DEFECTS defects, as predicted, needs to be correlated with the resulting in-vivo fatigue and wear performance of components. These issues remain as objectives for further work. APPENDIX Steps Executed by the User-De ned Material Subroutine On a call to the subroutine UMATHT at discrete time t ‡ Dt, it was provided with a trial new temperature t‡Dt T ˆ t T ‡ DT and temperature gradient vector t‡Dt HT, together with previous values of the speci c internal energy t U and the solution-dependent state variables. The steps executed were as follows. (1) Update the effective speci c heat to t‡Dt c0p using the new temperature and the sign of DT , and, hence, compute the new speci c internal energy: t‡Dt
U ˆ t U ‡t‡Dt c0p DT
…A1†
The neglect of the distinction between cp and cv is justi ed on the grounds that, at the pressures involved (maximum 20 MPa) the relative difference is only of order 1%. (2) Update the thermal conductivity t‡Dt k using t‡Dt T and the sign of DT , and hence, compute the new heat ux vector: t‡Dt
f ˆ ¡t‡Dt k t‡Dt
t‡Dt t‡Dt
HT
…A2†
t‡Dt 0 t‡Dt cp , k.
U, (3) Return to the FE solver: f, (4) Update the further solution-dependent state variables ^ and return them to the FE solver. From x and M equation (6): µ ³ ´¶ ED 1 1 t‡Dt aT ˆ exp ¡ …A3† R t‡Dt T T ¤ hence: t‡Dt
Dt x ˆ t x ‡ t‡Dt aT
…A4†
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3. Landy, M. M. and Walker, P. S., 1988, J Arthroplasty (Suppl), 3: S73. 4. Blunn, G. W., Joshi, A. B., Lilley, P. A., Engelbrecht, E., Ryd, L., Lidgren, L., Hardinge, K., Nieder, E. and Walker, P. S., 1992, Acta Orthop Scand, 63: 247. 5. Wrona, M., Mayor, M., Collier, J. and Jensen, R., 1994, Clin Orthop, 299: 9. 6. Wu, J. J., Buckley, C. P. and O’Connor, J. J., 2001, Journal of Materials Science Letters, 20: 473. 7. Wu, J. J., Buckley, C. P. and O’Connor, J. J., Biomaterials, in press. 8. De Gennes, P. G., 1983, Physics Today, June: 33. 9. De Gennes, P. G., 1971, J Chem Phys, 55: 572. 10. Wool, R. P., Yuan, B.-L. and McGarel, O. J., 1989, Polym Eng Sci, 29: 1340. 11. Wool, R. P., 1995, Polymer Interfaces: Structure and Strength (Hanser Publishers, Munich, Germany). 12. Haire, K. R. and Windle, A. H., 2001, Computational and Theoretical Polymer Science, 11: 227. 13. Wool, R. P. and O’Connor, K. M., 1981, J Appl Phys, 52: 5953. 14. Pecorini, T., 1997, Polym Eng Sci, 37: 308. 15. Truss, R. W., Han, K. S., Wallace, J. F. and Geil, P. H., 1980, Polym Eng Sci, 20: 747. 16. Gao, P., Cheung, M. K. and Leung, T. Y., 1996, Polymer, 37: 3265. 17. Farrar, D. F. and Brain, A. A., 1997, Biomaterials, 18: 1677. 18. Gao, P. and Mackley, M. R., 1994, Polymer, 35: 5210. 19. Doi, M. and Edwards, S. F., 1986, The Theory of Polymer Dynamics (Clarendon Press, Oxford, UK). 20. Frischknecht, A. L. and Milner, S. T., 2000, Macromolecules, 33: 5273. 21. Klein, J. and Briscoe, B. J., 1979, Proc Roy Soc A, 365: 53. 22. Bartels, C. R., Crist, B. and Graessley, W. W., 1984, Macromolecules, 17: 2702. 23. Lodge, T. P., 1999, Phys Rev Lett, 83: 3218. 24. Tao, H., Lodge, T. P. and von Meerwall, E. D., 2000, Macromolecules, 33: 1747. 25. Wunderlich, B. and Czornyj, G., 1977, Macromolecules, 10: 906. 26. Wu, J. J., 2000, DPhil Thesis (University of Oxford, UK). 27. Wunderlich, B. and Gaur, U., 1981, J Phys Chem Ref Data, 10: 119. 28. Chiang, R. and Flory, P. J., 1961, J Am Chem Soc, 83: 2857. 29. Gubler, M. G. and Kovacs, A. J., 1959, J Polymer Sci, 34: 551. 30. Hansen, D. and Ho, C. C., 1965, J Polymer Sci, Part A, 3: 659. 31. Sheldon, R. P. and Lane, S. K., 1965, Polymer, 6: 205. 32. Anon, 1980, Hostalen (Hoechst Plastics). 33. Kraybill, R. R., 1981, Polym Eng Sci, 21: 124. 34. Schneider, P. J., 1963, Temperature Response Charts (Wiley, New York, USA). 35. O’Connor, J. J., Wu, J. J. and Buckley, C. P., Compression Moulding, UK Patent, led April 6th, 1999 (Application No. 99 07843.8). 36. Flory, P. J., 1989, Statistical Mechanics of Chain Molecules (Hanser Publishers, Munich, Germany).
ACKNOWLEDGEMENTS
and: t‡Dt
^ ˆ g… M
1=…a‡1†
x†
where the constant g is given by: ³ ´ =…a‡ D¤¤ M0 1 1† g ˆ M ¤ 3p 2 C1 jl2 M ¤
…A5†
The authors thank Biomet Merck Ltd for providing the UHMWPE powder. J. J. Wu also thanks the K. C. Wong Foundation and St. Hugh’s College, Oxford for the award of scholarships.
…A6†
ADDRESS
REFERENCES
Correspondence concerning this paper should be addressed to Dr C. P. Buckley, Department of Engineering Science, University of Oxford, Parks Rd, Oxford OX1 3PJ, UK. E-mail: [email protected]
1. Li, S. and Burstein, A. H., 1994, J Bone and Joint Surg, 76-A: 1080. 2. Rose, R. M., Crugnola, A., Ries, M., Cimino, W. R., Paul, I. and Radin, E. L., 1979, Clin Orthop, 145: 277.
The manuscript was received 30 November 2001 and accepted for publication 22 April 2002.
Trans IChemE, Vol 80, Part A, July 2002