Antioxidant diffusion in polyethylene hot-water pipes.

INTRODUCTION

This paper is a continuation of a series of papers (1-8) dealing with the phenomenology and the underlying mechanisms of the failure of polyolefin hot-water pipes. Yielding, craze fracture, and chemical degradation are in competition in internally pressurized pipes (9). Yielding and craze fracture are favored by extensive mechanical stresses whereas chemical degradation caused by thermal oxidation occurs at low internal pressures. The creep rupture curve is divided into three stages where the high stress level fractures, Stage I fractures, are ductile when large defects are absent and it has been established that chemical effects in the polymer are insignificant under these conditions (9). Stage II fractures, occurring at intermediate stress levels, are brittle and are dominated by slow crack growth fracture with crazing (9). Stage Ill fractures appear at the lowest stresses and they are brittle, the material in the failure initiation region is chemically degraded and the brittleness of this material is demonstrated by the weak stress dependence of the stage III lifetime (9).

Previous papers (2, 3) reported changes in antioxidant concentration and polymer structure as functions of temperature, hoop stress, exposure time and location in the pipe wall in pressure-tested pipes of a medium-density polyethylene. Two distinctly different antioxidant loss Regimes were identified. The initial antioxidant concentration in these pipes was greater than the solubility at the temperature of the pressure test and this caused rapid loss in effective stabilizer concentration by blooming and internal precipitation of antioxidant (3, 8). This mechanism, denoted Regime A, led to a rapid loss of a substantial portion of the initial content of effective antioxidant (3, 8). Modeling of data by oxidation induction time measurements and scanning electron and optical microscopy indicated that antioxidant in the precipitated particles is not effective and is not assessed in the oxidation induction time measurements (8). Regime B involves loss of antioxidant through migration to the surrounding media and by chemical consumption (3). This paper also presented a model, hereafter referred to as the Regime B model. Further information about this model is given in the section Regime B model. A general feature valid for MDPE and polybutylene pipes was that the antioxidant loss due to chemical reaction was insignificant compared to the loss by migration to the surrounding media (3, 4). The pronounced skewness of antioxidant concentration profiles in the MDPE pipes exposed to internal water and external air was due to a very significant variation in the diffusion coefficient for antioxidants with radial position (3). It was proposed in this paper that the radial dependence of the diffusivity was due to the water gradient through the pipe wall. Plasticization or more probably, competition between water and antioxidant molecules about adsorption sites on the filler particles were the proposed mechanisms (3). At a certain stage, the stabilizing system reached depletion at some part of the pipe wall and a rapid, autocatalytic oxidation of the polymer occurred. This last stage of the lifetime was termed Regime C and it constituted only a minor part of the total lifetime of the pipe (5).

This paper presents oxidation induction time data for a series of polyethylene pipes with known antioxidant systems before and after hydrostatic pressure testing in internal water/external air at elevated temperatures. These data are used for modeling purposes using the Regime B model. Information is presented concerning the transition between Regimes A and B, the diffusivities of used antioxidants and their radial dependence and the parameters that characterize the boundary conditions.

EXPERIMENTAL

The antioxidants used in the so-called model (M) materials were 4,4[prime]-thiobis(3-methyl-6-tertbutylphenol) (Santonox R, Monsanto; melting point: 161 [degrees] C) and 2,2[prime]-thiobis(4-methyl-6-tertbutylphenol) (Irganox 1081, Ciba; melting point: 81-86 [degrees] C). These antioxidants are hereafter referred to as AO-1 and AO-2; their structures are shown in Fig. 1.

The medium density polyethylene used to make the model materials has previously been studied (5, 6), and the compositions of the studied compounds are shown in Table 1. The compounding was carried out in a Clextral BC 21 twin-screw extruder at a rate of 9.8 kg/h. The temperature profile over the screws ranged from 180 to 190 [degrees] C. The string obtained from the extruder die was rapidly cooled in water and cut to granules. Pipes with a wall-thickness of 2.1 [+ or -] 0.2 mm and an outer diameter of 32.3 [+ or -] 0.1 mm were extruded from the granules in a Battenfeld extruder at a rate of 50 kg/h. The temperature profile over the screw was uniform at 190 [degrees] C.

The pipes based on the model materials were hydrostatically pressure-tested with internal stagnant de-ionized water at 95 [degrees] C and external moderately circulating air at 95 [degrees] C at sufficiently low stresses to cause Stage III failure. The average failure times in hydrostatic pressure testing for the different model pipes were M-1: 22,323 h, M-2: [greater than]24,200 h, M-3: 12,235 h and M-4: 8853 h (average values based on lifetime (Stage III) data obtained at hoop stress levels between 0.84 and 3.04 MPa). For one of the pipe specimens of each pipe type, the pressure testing was interrupted for a period of 15 min on several occasions to enable material to be sampled after different exposure times.

Pipes based on two commercial pipe grades were also studied. Pipe C-1 was based on a carbon-black-filled MDPE stabilized with 2,2[prime]-thiodiethylbis-[3-(3,5-di-tertbutyl-4-hydroxyphenyl)-propionate; (Irganox 1035, Ciba; melting point: 63-67 [degrees] C here denoted AO-3) and pipe C-2 was based on a peroxide-crosslinked polyethylene stabilized with octadecyl-3-(3,5-di-tertbutyl-4-hydroxyphenyl)-propionate (Irganox 1076, Ciba; melting point: 50-54 [degrees] C, here denoted AO-4). The structures of these antioxidants are shown in Fig. 1. The pipes based on the commercial pipes grades were pressure tested in a manner similar to that of the pipes based on the model materials: Pipes of C-1 were pressure-tested at 80 and 95 [degrees] C and pipes of C-2 were pressure-tested at 95 and 110 [degrees] C. The average failure times in hydrostatic pressure testing for C-1 was 8138 h at 95 [degrees] C (Stage III lifetime; hoop stresses: 1.47-2.51 MPa) and 27,869 h at 80 [degrees] C (Stage III lifetime; hoop stresses: 1.49-3.07 MPa, and for C-2 19,882 h at 110 [degrees] C (Stage III lifetime; hoop stresses: 2.54-3.29 MPa). The estimated lifetime of C-2 at 95 [degrees] C assuming an activation energy for Stage III failure of 100 kJ/mol (3) is 72,500 h. The pressure testing of several pipe specimens was terminated prior to failure to enable samples to be taken for thermal analysis.

Blends of medium-density polyethylene and antioxidants AO-1 to AO-4 at antioxidant concentrations of 0.05, 0.1, 0.15, 0.20, 0.25 and 0.30 wt% were prepared in a Brabender mixer at 190 [degrees] C. Films with an approximate thickness of 100 [[micro]meter] were prepared by compression molding at 190 [degrees] C for the different blends. These films were used to study the relation between the oxidation induction time (OIT) and the antioxidant concentration.

The OIT measurements were carried out in a Mettler TA-3000 system equipped with a DSC 20 Standard Cell and a TC10A TA Processor. The method is basically according to a ASTM standard method (10) with a minor modification according to Refs. 2 and 11. Each sample consisted of three to four sections, 0.1 mm thick and 5 mm in diameter with a total mass of 5 [+ or -] 0.5 mg which were enclosed in a standard aluminum pan with three holes in the cover. The analyses were performed by heating the samples to the test temperature at a rate of 100 [degrees] C/min in a nitrogen atmosphere at a flow rate of 50 ml/min. The samples were then allowed to rest for 2 min before the atmosphere was switched to pure oxygen. The samples were maintained at the constant temperature and the exothermal heat associated with oxidation was recorded. The OIT was obtained as the intersection between the isothermal base line and the tangent to the curve at the point which deviates 1 mW from the isothermal base line. The testing temperature was adjusted from 180 [degrees] C to 230 [degrees] C in order to keep the induction times between 2 and 30 min. All OIT data were finally shifted to a common temperature, 190 [degrees] C, using the Arrhenius equation. The activation energies for antioxidant consumption for the different studied materials were as follows: M-1: 156 kJ/mol: M-2: 165 kJ/mol; M-3:142 kJ/mol; M-4:144 kJ/mol; C-1: 202 kJ/mol; C-2:206 kJ/mol. These values were obtained by measuring the OITs at different temperatures between 180 and 230 [degrees] C of samples taken from the centers of unexposed pipes. The OIT measurements were carried out on the molded films of known antioxidant concentration and on samples die punched from the pipe wall and subsequently microtomed to 0.1 mm sections to give samples from different radial positions in the pipe wall.

Table 1. Compositions of the Model Materials.

Material
Code          AO-1 (wt%)      AO-2 (wt%)      Carbon Black (wt%)

M-1              0.3              0                   0
M-2              0.3              0                   2.5
M-3              0                0.3                 0
M-4              0                0.3                 2.5

THE REGIME B MODEL

A model that describes the time evolution of the antioxidant concentration profile in an exposed pipe during Regime B was developed and presented in Ref. 3. The details of this model are found there. A summary is repeated here to make easy reading of this paper. The model assumes the validity of Fick's second law. In cylindrical coordinates, assuming axial symmetry, the rate of change of antioxidant concentration at any radial point in the pipe wall is at constant temperature conditions given by:

[Delta]C(r, t)/[Delta]t = (1/r) [multiplied by] ([Delta]/[Delta]r(rD(r) [multiplied by] [Delta]C(r, t)/[Delta]r)) - H(C, r) (1)

where C is the concentration of antioxidant, r is the radial distance, t is the time, D is the diffusivity and H is a term quantifying the chemical consumption of the antioxidant. It is assumed that D and H are not explicit functions of time and that D is not a function of concentration. Both D and H are functions of temperature.

It is necessary to determine the form of D(r). The application of the Regime B model to experimental OIT data showed that it was appropriate to express D as a linear function of r (3):

D(r) = [D.sub.0](1 + [Lambda](b - r/d)) = [D.sub.0](1 + [Lambda]b/d) - ([D.sub.0][Lambda]/d) [multiplied by] r

= [D.sub.1] - [D.sub.2](r) (2)

where [D.sub.0] is the diffusivity at the outer pipe wall, [Lambda], is a constant, b is the outer radius and d is the thickness of the pipe wall. It was found that a strong radial variation of D only appeared when the inner (water) and outer media (air) were different (3). It was argued by Smith et al. (3) that the water gradient through the pipe wall was the reason for the observed strong radial dependence, i.e. the high [Lambda], value. The water concentration ([C.sub.H2O]) through the pipe wall is according to Crank (12) given by:

[C.sub.H2O] = [C.sub.H2O,1] ln(b/r) + [C.sub.H2O,2] ln(a/r)/ln(b/a) (3)

where [C.sub.H2O,1] is the water concentration at the inner wall, [C.sub.H2O,2] is the water concentration at the outer wall, a is the radial position of the inner wall and b is the radial position of the outer wall. In the particular case with dry air as external medium, [C.sub.H2O,2] [approximately equal to] 0, which simplifies Eq 3 to:

[C.sub.H2O] = [C.sub.3] ln (b/r) (4)

where [C.sub.3] is a constant. Insertion of the geometries of the pipes used in this study in Eq 4 yield an approximately linear relationship between [C.sub.H2O] and r. The maximum relative deviation between the straight line and the actual logarithmic function is always less than 5%. Trankner et al, (13) reported an approximately 5% radial variation in crystallinity from a maximum value near the inner wall and the lowest crystallinity at the outer wall that will moderately affect [C.sub.H2O] = f(r). However, the approximately linear character of [C.sub.H2O] = f(r) still holds.

Plasticization by the dissolved water can be handled according to the Cohen-Turnbull-Fujita model (14, 15):

D [infinity] [e.sup.-B/f(r)] (5)

where f is the relative free volume and B is a constant. It is reasonable to assume that f is proportional to the water content, hence f should be proportional to r and D should be related to r according to an exponential law; D should thus decrease with increasing r with internal water and external air. However, several D-r functions were tested by Smith et al. (3) including constant, linear, parabolic, exponential and logarithmic functions on OIT data for a MDPE pipe and the goodness of the fit was best for the linear D-r function. Association between water and antioxidant molecules (16), competition between water and antioxidant molecules about adsorption sites on filler particles (e.g., carbon black) (3, 17-19) are possible causes for a r (implicit water concentration)-dependent antioxidant diffusivity. These ideas are basically in accordance with the dual mobility model (20) with the rapid transport of dissolved molecules and slower motion of adsorbed species. The data presented in our paper indicate that water-induced plasticization plays a less important role for the radial dependence of D.

Insertion of Eq 1 in Eq 2 yields the expression:

[Delta]C/[Delta]t = ([[Delta].sup.2]C/[Delta][r.sup.2]) [multiplied by] ([D.sub.1] - [D.sub.2](r)) - (1/R) [multiplied by] ([Delta]C/[Delta]r) [multiplied by] ([D.sub.1] - 2[D.sub.2](r)) - H (6)

where it is understood that C = C(r, t). The evaporative boundary condition at the outer wall is prescribed according to Crank (12) and Calvert and Billingham (21):

D [multiplied by] ([Delta]C/[Delta]r) -[Beta]C (7)

where [Beta] is a parameter that depends on the solubility of the antioxidant, the vapor pressure of antioxidant and the thickness of the diffusive boundary layer. Assuming that the bulk antioxidant concentration in the water phase is in equilibrium with the polymer surface concentration and that the antioxidant is consumed in the water phase according to first order kinetics (22, 23), the flux boundary condition at the inner wall is given by

DA [multiplied by] ([Delta]C/[Delta]r) = K[V.sub.0] [multiplied by] ([Delta]C/[Delta]t) + [V.sub.0][k.sub.r]KC (8)

where K is the ratio of antioxidant solubility in water to that in the polymer, [V.sub.0] is the volume of water, A is the surface area of the polymer/water interface, and [k.sub.r] is the rate constant for antioxidant consumption. Since [k.sub.r]C [much greater than] ([Delta]C/[Delta]t), Eq 8 can be simplified to:

DA [multiplied by] ([Delta]C/[Delta]r) = [V.sub.0][k.sub.r]KC (9)

The differential equations were solved by first transforming the quantities into dimensionless analogues and then using the method of lines (24) to transform the parabolic partial differential equation into a system of first order ordinary differential equations in time. Application of the boundary conditions yields a system of n + 1 ordinary differential equations where n is the number of equal radial divisions of the pipe wall in the grid. The system of differential equations contains five physical chemical parameters ([D.sub.1], [D.sub.2], [Alpha], [Beta] and H) that are adjustable in the model. Parameter values at each temperature were determined by obtaining the best fit of the model to isothermal antioxidant concentration profiles using standard non-linear least-squares procedures and a solving routine for ordinary differential equations (3).

RESULTS AND DISCUSSION

It may be assumed that the consumption of phenolic antioxidants at high temperatures obeys zero-order kinetics:

dc/dt = -[k.sub.0] exp(-[Delta]E/RT) [square root of [p.sub.[0.sub.2]]] (10)

where c is the concentration of efficient antioxidant, t is the time, [k.sub.0] is a constant, [Delta]E is the activation energy, R is the gas constant, T is the temperature and [p.sub.[0.sub.2]] the partial pressure of oxygen. The induction time (OIT) is obtained by solving this differential equation (Eq 10) under the assumption that [p.sub.[0.sub.2]] is constant during the measurement:

OIT = [c.sub.0] [multiplied by] [k[prime].sub.0] [multiplied by] [e.sup.[Delta]E/RT] (11)

where [c.sub.0] is the initial concentration of antioxidant and [k[prime].sub.0] is a constant. Howard (25) showed in the 1970s the validity of Eq 11 for 4,4[prime]-thiobis(3-methyl-6-tertbutylphenol) (AO-1) in polyethylene. Figure 2 shows that the OIT data of the specially prepared films with known antioxidant contents indeed conformed to Eq 11. The OIT data for compounds containing AO-1, AO-2 and AO-4 were proportional to the antioxidant concentration, whereas the compound containing AO-3 showed a moderate deviation from linearity. Ultraviolet spectroscopy showed, however, that this was due to discrepancies between the nominal and real antioxidant concentrations in the films containing AO-3. In the remainder of this paper, all the OIT data have been shifted to 190 [degrees] C, and they are thus proportional to the concentration of effective antioxidant.

Figure 3 shows the development of the OIT profiles with time at 80 [degrees] C for the C-1 pipe. The initial OIT profile was not parabolic, as was found earlier for another MDPE pipe (3). The spread of the data points about the fitted curves were small. The OIT values at the inner and outer boundaries were relatively large indicating that the loss of antioxidant in these cases is partly controlled by the boundary coefficients [Alpha] and [Beta]. The OIT profiles appeared skewed on extended exposure [ILLUSTRATION FOR FIGURE 3 OMITTED], which indicates that the diffusivity is highest near the inner wall. The OIT profile data obtained at 95 [degrees] C were basically in accordance with the data presented in Fig. 3. The data presented in Fig. 2 and the OIT data obtained at 95 [degrees] C were used to obtain the average OIT for the whole pipe cross-section as a function of the square-root of the exposure time [ILLUSTRATION FOR FIGURE 4 OMITTED]. The distinct changes in slope coefficient after 2000 h (95 [degrees] C) and 4500 h (80 [degrees] C) suggest that the mechanism for decrease in antioxidant content changed during the course of the pressure testing from Regime A to Regime B, which is principally in accordance with earlier data of Gedde et al. (3, 8). The earlier data and the data reported in this paper are compared in Fig. 5. The MDPE grade studied by Viebke et al. (8) is identical with material M-1. The MDPE grade studied by Smith et al. (3) was stabilized with a phenolic antioxidant and it contained carbon black. It should be noted that the OIT data presented in Fig. 5 were normalized with respect to the initial OIT values. A break in the curves was found in all cases, which further substantiates the generality of the antioxidant loss mechanism transition, i.e. the Regime A-B transition. There is however a substantial variation in the time associated with the transition point ranging from approximately 100 h for the MDPE pipe reported by Smith et al. (3) to 2000 h for C-1. The transition occurred for M-1 after 1200 h which corresponds to 5.4% of the Stage III lifetime. This is of the same order of magnitude as for the majority of the pipe grades: M-2: [less than]1.4%; M-3: 7.1%; M-4: 7.3%; C-1: 24.6%; C-2: 0%. Grade C-1 thus shows an unusually prolonged Regime A. This is an important feature. Lifetime predictions based on the Regime B model would be possible for the majority of these pipes using data from 10-30% of the Stage III lifetime.

In previous work (3) it was shown that the chemical consumption of antioxidant under these conditions and in similar materials is insignificant compared with the loss of antioxidant to the surrounding media. No improvement in the fit of the model to the experimental data was obtained, in any of the cases presented in this paper, by allowing the reaction term H to vary and to be different from zero; H was thus constrained to zero in all other cases.

Application of the Regime B model to the OIT profiles of C-1 obtained at 95 [degrees] C showed that successful fitting was obtained only when the initial conditions were defined at 1997 h or at longer exposure times. This is in accordance with the data presented in Fig. 4, showing a breaking point at approximately 2000 h. It may thus be proposed that the breaking point constitutes the transition from Regime A to Regime B. Figure 6 shows the variation in diffusivity across the pipe wall obtained from the fitting of the Regime B model to the OIT data using different initial conditions within the Regime B period, i.e. for exposure times [greater than or equal to]1997 h. The diffusivity at the inner wall is approximately independent of the initial conditions, whereas the diffusivity at the outer wall shows a significant dependence on the initial conditions used in the modeling. This can be explained by the fact that Regime A mechanisms caused a more rapid loss of antioxidant than the Regime B mechanism. The values obtained for the outer wall diffusivity thus suggest that part of the loss of antioxidant efficiency at the outer wall was due to the Regime A mechanism, even after 1997 h of exposure, i.e. after exposure times longer than the time associated with the breaking point in Fig. 4. A plausible explanation of the radial dependence of the time of the termination of Regime A mechanism is that the water gradient in the pipe wall caused a radial variation in the solubility and diffusivity of the antioxidant. The more extended Regime A period at the outer wall was due to the fact that a lower water content leads to a lower solubility of antioxidant (8).

The Regime B model could be fitted to the experimental OIT profiles shown in Fig. 3 using initial conditions at exposure times of 1000 h or longer. The breaking point for average OIT vs. square root time appeared at approximately 4500 h [ILLUSTRATION FOR FIGURE 6 OMITTED], which is longer than the minimum time for applicability of the Regime B model. Figure 7 shows the variation in diffusivity across the pipe wall as Indicated by the fitting of the Regime B model to the OIT data using different initial conditions within the Regime B period, i.e., for exposure times [greater than or equal to] 1000 h. The diffusivity at the inner wall was in this case practically independent of the initial conditions. The diffusivity at the outer pipe wall showed an overall decreasing trend with increasing time for the initial condition, and decreased with the time of exposure of the selected initial profile. It is important to note that the diffusivities for the outer pipe wall, when using 4516 h and 6260 h as the initial profiles, are reversed. This may indicate that a pure Regime B mechanism prevails from 4516 h and onwards, which is in agreement with the time for the breaking point of the plot shown in Fig. 4. The diffusivity at the inner wall is more than one order of magnitude greater than the diffusivity at the outer wall [ILLUSTRATION FOR FIGURES 6 and 7 OMITTED], which is consonant with earlier reported results (3) from another MDPE grade also containing an phenolic antioxidant and carbon black, which was tentatively explained by Smith et al. (3) to be due to competition between antioxidant and water molecules about adsorption sites on carbon black particles.

The pipe based on C-2 exhibited a behavior different from that found for the pipes based on C-1. The OIT profiles displayed in Fig. 8 conform to a more parabolic shape even after short exposure times. The scatter in the OIT data was more pronounced for C-2 than for C-1 but it did not mask the trends in the data. Even after short exposure times the boundary values were close to zero, which indicates that the loss of antioxidant was controlled by diffusion. On extended exposure, the OIT profiles became skewed towards the inner wall, indicating that the diffusivity increased towards the outer pipe wall (air side). The lack of a clear breaking point in the data shown in Fig. 9 indicates that there is no pronounced transition between Regimes A and B. The Regime B model was also capable of describing all the OIT profiles, using the virgin pipe as the starting condition. The diffusivity profiles [D = f(r)] were also very similar for all initial conditions except when using 0 h as initial profile. The latter "improper" initial condition for Regime B led to a higher diffusivity at the inner wall due to a moderate influence initially from the Regime A mechanism. This supposition is further substantiated by the data presented in Fig. 10. The 5029 h OIT profile showed extensive scatter and could not be used as an initial condition. It is believed that initial concentration of antioxidant in the C-2 pipes is relatively low and that the Regime A period becomes very short in this case. It has been suggested that the peroxide crosslinking process should chemically bind the antioxidant molecules to the polymer chains and that the long Stage III lifetime of crosslinked polyethylene pipes is due to resulting inhibited antioxidant migration. The antioxidant (AO-4) used in grade C-2 has only one hydroxyl functionality and it is difficult to see how the stabilizing effect would remain after a bonding to the polymer chain. Also, the antioxidant diffusivities obtained for this material are of the same order of magnitude as those for the MDPE material. It is suggested that the long Stage III life of C-2 is attributed to the greater wall thickness of this grade and to the fact that crosslinked polyethylene exhibits a very high fracture toughness.

Initial conditions defined at 1003 h or longer (C-2; 110 [degrees] C) yielded "stable" values for the inner and outer wall diffusivities: [D.sub.m] = 4.1 x [10.sup.-10] [cm.sup.2]/s; [D.sub.out] = 3.3 x [10.sup.-9] [cm.sup.2]/s. This is indeed an important result, particularly in view of the data obtained for the carbon-black-containing pipe C-1 and these data in conjunction indicate that plasticization by the dissolved water cannot be the major cause for the radial dependence D. A of possible explanation for the lower antioxidant diffusivity at the water-rich side was provided by Poidevin (16), who suggested that antioxidant and water molecules formed loosely bonded clusters. These clusters should be less mobile in the PE matrix than isolated antioxidant molecules.

OIT profiles of pipes based on the model materials (M-1 to M-4) were taken after different times of pressure testing at 95 [degrees] C. Figures 11-14 present OIT profiles before and after pressure testing obtained from pipes made of these compounds. The initial OIT profiles used for the modeling deviate markedly from the parabolic shape. It was argued in a previous paper (8) that the higher concentration of water near the inner wall increases the solubility of the antioxidant. The OIT profile data displayed in Figs. 11-14 were integrated to yield the average OIT data, which in Fig. 15 is plotted as a function of the square root of the exposure time. All four materials exhibited breaks in the curves after approximately 1000 h of exposure. The Regime B model was applied to the OIT profiles for exposure times longer than times corresponding to the breaking points. It was found that the Regime B model could be fitted to the experimental data but the diffusivities obtained depended strongly on the initial condition selected as was the case for C-1 and C-2. It thus seems that the loss of antioxidant efficiency between 800 h and 3000 h exposure was due to the combined action of Regimes A and B loss mechanisms. When OIT profiles from pipes tested for 3000 h or longer were used as initial conditions, the "stable" diffusivity values displayed in Table 2 were obtained.

M-1 preserves much of the shape of the original OIT profile even after extensive pressure testing [ILLUSTRATION FOR FIGURE 11 OMITTED]. The boundary concentration near the inner wall remained relatively high. It is also obvious from the data presented in Fig. 11 that diffusion occurred more readily near the outer wall. The Regime B modeling showed that the diffusivity at the outer wall was almost 40 times greater than that at the inner wall (Table 2). The almost linear water gradient through the pipe wall (maximum value at the inner wall) is believed to be of major importance; the higher the water concentration the lower is the diffusivity. It is suggested that the interaction between the polar sites of the antioxidant molecules and water molecules retards diffusion because of the formation of clusters involving antioxidant and water molecules, as was suggested by Poidevin (16). M-2 [ILLUSTRATION FOR FIGURE 12 OMITTED] showed a considerable scatter in the OIT data and no meaningful modeling can be performed on the basis of these data.

The OIT profiles of M-3 and M-4 were closer to the parabolic shape, particularly after prolonged exposure [ILLUSTRATION FOR FIGURES 13 and 14 OMITTED]. The scatter in the OIT data is limited and proper conclusions can be drawn based on the results of the modeling. It was necessary to use profiles from pipes M-3 and M-4 exposed for more than 3000 h (95 [degrees] C) as the initial conditions to obtain "stable" diffusivities. The initial (3067 h) OIT-profile of M-3 was skewed towards the inner wall and this skewness remained after long-term exposure [ILLUSTRATION FOR FIGURE 13 OMITTED]. The radial dependence of the diffusivity for M-3 was negligible (Table 2). This is in sharp contrast to the behavior of M-1, which is particularly interesting because of the structural resemblance of the two antioxidants (AO-1 and AO-2). The inclusion of carbon black (M-4) led to a distinct change in the shape of the OIT profiles [ILLUSTRATION FOR FIGURE 14 OMITTED]. The skewness towards the outer wall indicates that the diffusivity is position-dependent with a maximum value near the inner wall, a fact that was also revealed by the modeling (Table 2). It is also interesting to note that C-1 showed a similar behavior, i.e., with a minimum in diffusivity at the inner wall. A similar but stronger radial trend in the [TABULAR DATA FOR TABLE 2 OMITTED] diffusivity data was reported by Smith et al. (3) for the carbon-black containing medium-density polyethylene pipes stabilized with hindered phenols. These phenomena were explained by Smith et al. (3), who suggested that water absorption on the carbon black particles may interfere with the absorption of antioxidant molecules. Hawkins et al. (17) reported as early as 1960 that antioxidants may be adsorbed onto filler particles, and Roe et al. (18) made similar observations. It is interesting to compare the fracture times for the pipes based on M-3 (without carbon black) and on M-4 (with carbon black): The lifetimes associated with Stage III failure were 12,200 h (M-3) and 8900 h (M-4); the shorter lifetime being attributed to the higher diffusivity of antioxidant near the inner wall in the carbon-black-containing material.

Figure 16 shows the most energetically favorable conformations of AO-1 and AO-2. It is obvious that AO-1, for steric reasons, has a greater potential for intermolecular interactions via its hydroxyl groups than AO-2. This greater potential for AO-1 for interaction is also demonstrated in the considerably higher melting point of AO-1 (161 [degrees] C compared with 81-86 [degrees] C for AO-2). This may explain the weaker radial dependence of the diffusivity exhibited by M-3 than by M-1 (Table 2).

Table 2 also presents data for the [Alpha] and [Beta] parameters. The variation in the values obtained for [Beta] for the different materials is small compared to the uncertainty of the data. Relatively large changes in the values of this parameter cause only moderate changes in the goodness of fit. The variation in the values of [Alpha] for the different materials is somewhat larger than it is for [Beta], and this parameter also has a more significant effect on the goodness of fit. Parameter [Alpha] is a compound of the solubility ratio for the antioxidant in polyethylene and in water, and the reaction rate constant for the consumption of the antioxidant in the internal water. Our currently limited knowledge of how these quantities vary individually with the different materials makes any further discussion concerning [Alpha] far too speculative.

It is important to compare the appearance of the antioxidant concentration profiles at the time of failure of the different pipe grades in hydrostatic pressure testing. Naturally, the failure times in pressure testing exhibit a certain statistical variation, hence creating a lifetime interval. Figure 17 shows the antioxidant concentration profiles as extrapolated to the failure time intervals with the Regime B model for M-1, M-3 and M-4. It is very interesting to note that the profiles roughly fall in the same antioxidant concentration range. For geometric reasons the hoop stress in pipes of this dimension is about 16% higher at the inner wall surface than at the outer wall surface. This means that a lower level of degradation is needed for inside initiated fracture to occur than for outside initiated fracture. A survey of the pipe samples that failed in hydrostatic pressure testing shows that all of the specimens based on M-3 and M-4 exhibited fractures initiated at the inner wall, whereas the specimens based on M-1 showed fractures initiated from both the inner and outer walls. The extrapolated OIT profile of M-1 shown in Fig. 17 indicates that the depletion of the antioxidant system should occur first at the outer wall and that the pipes should be most prone to fail by outer-wall-initiated fracture. On the.other hand, there is also a preference for inner wall initiation of the fracture because of the concentration of stress at this location.

Lifetime predications for pipes are typically based on data from hydrostatic pressure testing taken at elevated temperatures. Pressure testing is a "functional" test and extrapolation of lifetime data to lower temperatures is in many cases straightforward according to the Arrhenius law (7). Pressure testing is, however, time-consuming, and it does not permit direct assessment of the condition of a pipe system in service (26). OIT-profiles of pressure-tested pipes involving testing times of only a fraction of the failure time can be used as in-data for lifetime prediction based on the Regime B model (26). The total lifetime, including Regimes A, B and C is then obtained under the assumption that Regime C (polymer degradation period) constitutes [approximately]10% of the total lifetime (5).

CONCLUSIONS

The Regime B model (a model assuming Fickian diffusion of antioxidant to the surrounding and chemical consumption of antioxidant) was successfully applied to oxidation induction time profile data except for a case with a pipe with a substantial spatial scatter in the oxidation induction time data indicative of compositional heterogeneity and in the case of an inappropriate choice of the initial conditions. The use of initial conditions based on insufficiently exposed pipes leads to an overestimate of the antioxidant diffusivities caused by the combined Regimes A (internal precipitation) and B loss mechanisms. The time period associated with Regime A constituted 0% to 25% of the lifetime. Antioxidants with sterically accessible polar groups showed a higher melting point and a greater degree of interaction with dissolved water and carbon black. Obtained radial dependences of the antioxidant diffusivities (D) indicate that the water concentration in the polymer influenced D primarily through cluster formation involving water and antioxidant molecules and by competition between water and antioxidant molecules about adsorption sites on carbon black particles and to a much lesser extent by plasticization. Antioxidant concentration profiles calculated for the failure time interval in pressure testing appeared in the same concentration range, and correlated well with the location of the fracture initiation in the pipe wall.

ACKNOWLEDGMENTS

This study was sponsored by the fund for fundamental research at Studsvik AB, Sweden and by the National Swedish Board for Industrial and Technical Development (NUTEK; grant P5728-2). Borealis Polyeten (Sweden), Monsanto Scandinavia, and Ciba are thanked for the supply of unstabilized MDPE and the antioxidants used. Mr. M. Ifwarson, Studsvik Polymer AB, and Dr. C. DeArmitt, the Institute for Surface Chemistry, Stockholm are acknowledged for valuable discussions. Prof. C.-G. Gustavsson, NTH, Trondheim, Norway, is thanked for making the compounding equipment available and Mr. M. Trang and Mr. J. Rydberg for experimental assistance.

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