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Antioxidant efficiency loss by precipitation and diffusion to surrounding media in polyethylene hot-water pipes.



INTRODUCTION

This paper is a continuation of a series of papers (1-7) dealing with the deterioration and failure of polyolefin hot-water pipes. In this application there is a competition between yielding, craze fracture and chemical degradation (8). The former two are favored by extensive mechanical stresses whereas the chemical degradation is caused by thermal oxidation In microfabrication, thermal oxidation is a way to produce a thin layer of oxide (usually silicon dioxide) on the surface of a wafer (semiconductor). The technique forces an oxidizing agent to diffuse into the wafer at high temperature and react with it. . The creep rupture curve [ILLUSTRATION FOR FIGURE 1 OMITTED] may be divided into Stages I to III where Stage I fractures are ductile ductile /duc·tile/ (duk´til) susceptible of being drawn out without breaking.

duc·tile
adj.
Easily molded or shaped.



ductile

susceptible of being drawn out without breaking.
 when large defects are absent and it has been established that chemical effects in the polymer are insignificant (8). Stage II fractures are brittle and are dominated by slow crack growth fracture with crazing (8). Stage III fractures are brittle and the material in the failure initiation region is heavily chemically degraded. It appears that the failure time is almost independent of the applied stress in the stage III region, which demonstrates the extreme brittleness of the degraded polymer (8).

Previous papers (2, 3) reported changes in antioxidant antioxidant, substance that prevents or slows the breakdown of another substance by oxygen. Synthetic and natural antioxidants are used to slow the deterioration of gasoline and rubber, and such antioxidants as vitamin C (ascorbic acid), butylated hydroxytoluene  concentration and polymer structure as functions of temperature, hoop stress Hoop stress is mechanical stress defined for rotationally-symmetric objects being the result of forces acting circumferentially (perpendicular both to the axis and to the radius of the object). , exposure time and location in the pipe wall in pressure-tested pipes of a medium-density polyethylene. When the fractional antioxidant loss was plotted as a function of the square root of time in pressure testing, it was possible to identify two antioxidant loss regimes, designated Regimes A and B. One-third to one-half of the initial antioxidant content of the unexposed pipes was lost during a relatively short period of time corresponding to Regime A (3). It was suggested that the initial antioxidant concentration was greater than the solubility solubility

Degree to which a substance dissolves in a solvent to make a solution (usually expressed as grams of solute per litre of solvent). Solubility of one fluid (liquid or gas) in another may be complete (totally miscible; e.g.
 limit at the testing temperature, leading to blooming (exudation exudation /ex·u·da·tion/ (eks?u-da´shun)
1. the escape of fluid, cells, and cellular debris from blood vessels and their deposition in or on the tissues, usually as the result of inflammation.

2. an exudate.
 to the pipe wall surfaces) and possibly internal phase separation in the pipe wall (3). Regime B starts at the end of Regime A and ends when the antioxidant concentration drops below an effective level and autocatalytic au·to·ca·tal·y·sis  
n. pl. au·to·ca·tal·y·ses
Catalysis of a chemical reaction by one of the products of the reaction.



au
 oxidation of the polymer begins to occur. A model that describes the time evolution of the antioxidant concentration profiles in exposed pipes during Regime B was developed and presented in Ref. 3. The model, hereafter referred to as the Regime B-model, contains five adjustable parameters corresponding to the diffusion coefficient at the outer pipe wall surface, the variation in the diffusion coefficient through the pipe wall, coefficients describing the outer and inner wall boundary conditions, either in contact with water or air, and the chemical consumption of the antioxidant. A third regime, corresponding to the time period from the beginning of the autocatalytic oxidative degradation to the failure of the pipe, was termed Regime C.

Blooming of antioxidants Antioxidants
Substances that reduce the damage of the highly reactive free radicals that are the byproducts of the cells.

Mentioned in: Aging, Nutritional Supplements

antioxidants,
n.
 on the surface of polyolefins was first reported by Bair (9), and has since then been the subject of several papers (e.g., 10-12). Several authors also proposed the possibility of precipitation, either by crystallization Crystallization

The formation of a solid from a solution, melt, vapor, or a different solid phase. Crystallization from solution is an important industrial operation because of the large number of materials marketed as crystalline particles.
 or by agglomeration ag·glom·er·a·tion  
n.
1. The act or process of gathering into a mass.

2. A confused or jumbled mass:
 of antioxidants in the bulk of polyolefins (9, 11-14). The antioxidant is effective as radical scavenger only when it is properly dissolved in the polymer matrix (13). Precipitation should thus have an adverse effect on the efficiency of the antioxidant in preventing thermal oxidation. There have been no reports of direct observations of internal precipitation of antioxidants, although Bair (9) reports fusion experiments Experiments directed toward developing fusion power are invariably done with dedicated machines which can be classified according to the principles they use to confine the plasma fuel and keep it hot.

The major division is between magnetic confinement and inertial confinement.
 indicating two separate phases in a low-density polyethylene low-density polyethylene
n. Abbr. LDPE
A form of polyethylene having many side branches off the main carbon backbone and a less closely packed structure than that of high-density polyethylene.
 containing 0.10 wt% antioxidant. Calvert and Billingham (15) were unsuccessful in detecting precipitation of additives in polypropylene, using X-ray analysis, thermal analysis Thermal analysis is a branch of materials science where the properties of materials are studied as they change with temperature. Techniques include:
  • Differential scanning calorimetry
  • Dynamic mechanical analysis
  • Thermomechanical analysis
, and ultraviolet microscopy, even though they claim indirect evidence that it may occur in some cases.

The purpose of this work is to characterize the Regime A antioxidant loss and to rationalize observations into a mathematical model
Note: The term model has a different meaning in model theory, a branch of mathematical logic. An artifact which is used to illustrate a mathematical idea is also called a mathematical model and this usage is the reverse of the sense explained below.
. A material consisting of medium density polyethylene Medium Density Polyethylene, or MDPE is a type of Polyethylene defined by a density range of 0.926 - 0.940 g/cc. It is less dense than HDPE, which is more common. MDPE can be produced by chromium/silica catalysts, Ziegler-Natta catalysts or metallocene catalysts.  (MDPE MDPE Medium Density Polyethylene
MDPE Mobile Dual-Phase Extraction (technique for environmental cleaning)
MDPE Mobile Dual-Phase Extraction (USA)
MDPE Maximum Permissible Dose Equivalent
) and 4,4[prime]-thiobis(3-methyl-6-tertbutylphenol) (Santonox R) was compounded and extruded into pipes, which were subjected to hydrostatic pressure hydrostatic pressure  

The pressure exerted by a fluid at equilibrium at a given point within the fluid, due to the force of gravity. Hydrostatic pressure increases in proportion to depth measured from the surface because of the increasing weight of fluid
 testing at elevated temperatures and subsequently analyzed by differential scanning calorimetry Differential scanning calorimetry or DSC is a thermoanalytical technique in which the difference in the amount of heat required to increase the temperature of a sample and reference are measured as a function of temperature.  for antioxidant concentration. Direct observations of internal precipitates are reported. Based on these observations a model describing the evolution of profiles for effective antioxidant concentration due to internal precipitation has been developed.

EXPERIMENTAL

The antioxidant selected for this study was 4,4[prime]-thiobis(3-methyl-6-tertbutylphenol), which has a melting point melting point, temperature at which a substance changes its state from solid to liquid. Under standard atmospheric pressure different pure crystalline solids will each melt at a different specific temperature; thus melting point is a characteristic of a substance and  of 161 [degrees] C and a molar mass Molar mass, symbol M,[1] is the mass of one mole of a substance (chemical element or chemical compound).[2] It is a physical property which is characteristic of each pure substance.  of 358 g/mol. The structure of the antioxidant is shown in Fig. 2.

Solutions of the antioxidant with ethanol were prepared and subsequently precipitated on microscope slides during evaporation evaporation, change of a liquid into vapor at any temperature below its boiling point. For example, water, when placed in a shallow open container exposed to air, gradually disappears, evaporating at a rate that depends on the amount of surface exposed, the humidity  of the solvent. The precipitated antioxidant was studied in a Polyvar optical microscope optical microscope

See under microscope.
 for the assessment of the lateral shape of the antioxidant crystals. The miscibility miscibility (miˈ·s·biˑ·l  of the antioxidant with polyethylene was studied in a Leitz Ortholux II Pol-BK microscope with crossed polarizers, equipped with a Mettler GP 82 HT Hot Stage and a Mettler FP 90 Central Processor.

The MDPE selected for manufacture of the model pipe material has previously been thoroughly studied (5, 6). A compound containing 0.3 phr ([approximately]0.3 wt%) antioxidant was produced on a Clextral BC 21 twin screw extruder at a rate of 9.8 kg/h. The temperature profile over the screws ranged from 190 to 180 [degrees] C. The string obtained from the extruder die was rapidly cooled in a water bath and subsequently cut to granules Granules
Small packets of reactive chemicals stored within cells.

Mentioned in: Allergic Rhinitis, Allergies
.

Pipe with a wall-thickness of 2.1 [+ or -] 0.2 mm and an outer diameter of 32.3 [+ or -] 0.1 mm was extruded from the granules in a Battenfeldt extruder at a rate of 50 kg/h. The temperature profile over the screw was 190 [degrees] C flat. The pipe specimens produced were subjected to long-term hydrostatic pressure testing with stagnant de-ionized water as internal medium and moderately circulating air as external medium at 95 and 105 [degrees] C. The average Stage III failure time at 105 [degrees] C pressure testing based on eight pipe specimens was 12,050 h. At 95 [degrees] C, six specimens are still running in testing after 21,200 h. For one pipe specimen at each temperature, the pressure testing was interrupted for [approximately]15 min on several occasions to enable sampling of material for further analysis.

Pipe wall samples were studied by scanning electron microscopy electron microscopy

Technique that allows examination of samples too small to be seen with a light microscope. Electron beams have much smaller wavelengths than visible light and hence higher resolving power.
 (SEM) in a JEOL JEOL Japan Electron Optics Laboratory  JSM JSM Journal of Sexual Medicine
JSM Just Shoot Me (sitcom)
JSM Journal of Sport Management
JSM Journal of Software Maintenance
JSM Jabber Session Manager
JSM John Sidney McCain
JSM JEOL Scanning Microscope
 6300 equipped with a Tracor Voyager unit for the energy dispersive dispersive /dis·per·sive/ (-per´siv)
1. tending to become dispersed.

2. promoting dispersion.
 spectroscopy (EDS (Electronic Data Systems, Plano, TX, www.eds.com) Founded in 1962 by H. Ross Perot (independent candidate for the President of the U.S. in 1992), EDS is the largest outsourcing and data processing services organization in the country. ) analyses and in a JEOL JSM 8820 equipped with a LINK AN-1000 EDS-unit. The samples for the SEM/EDS-studies were prepared by cutting through the pipe wall in the radial plane with a scalpel blade to generate surfaces of the cross section of the pipe wall. The surfaces were coated with conductive carbon. Other techniques evaluated for generation of the cross-section surfaces were breaking after cooling to liquid nitrogen Noun 1. liquid nitrogen - nitrogen in a liquid state
atomic number 7, N, nitrogen - a common nonmetallic element that is normally a colorless odorless tasteless inert diatomic gas; constitutes 78 percent of the atmosphere by volume; a constituent of all living
 temperature and grinding. The breaking in liquid nitrogen yielded rugged fracture surfaces which made observation of particles difficult. The grinding introduced SiC particles in the surface which could be mistaken for particles present in the pipe wall prior to sample preparation. Thus scalpel cutting was the preferred method used for the generation of the cross-section surfaces.

The oxidation induction time (OIT OIT Organización Internacional del Trabajo (Spanish: International Labor Organization)
OIT Organisation Internationale du Travail (International Labour Organization)
OIT Office of Information Technology
) measurements were carried out in a Mettler TA-3000 system equipped with a DSC (1) (Digital Signal Controller) A microcontroller and DSP combined on the same chip. It adds the interrupt-driven capabilities normally associated with a microcontroller to a DSP, which typically functions as a continuous process. See microcontroller and DSP.  20 Standard Cell and a TC10A TA Processor. 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 first 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. After reaching the measurement temperature, the samples were allowed to rest for 2.00 rain before the atmosphere was switched to oxygen at a gas flow rate of 50 ml/min. The samples were then maintained at the constant temperature and the exothermal exothermal /exo·ther·mal/ (ek?so-ther´mal) exothermic.

exothermal, exothermic

marked or accompanied by the evolution of heat; liberating heat or energy.
 heat associated with oxidation was recorded. The OIT was obtained as the intersection between the isothermal i·so·ther·mal
adj.
Of, relating to, or indicating equal or constant temperatures.



isothermal, isothermic

having the same temperature.
 base line and the tangent tangent, in mathematics.

1 In geometry, the tangent to a circle or sphere is a straight line that intersects the circle or sphere in one and only one point.
 to the curve at the point which deviates 1 mW from the isothermal base line. The testing temperature was adjusted from 180 to 230 [degrees] C in an attempt to keep the induction times between 20 and 60 min. All OIT data were then shifted to a common temperature, 190 [degrees] C, using the Arrhenius equation The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of a chemical reaction rate, more correctly, of a rate coefficient, as this coefficient includes all magnitudes that affect reaction rate except for concentration. . The activation energy activation energy, in chemistry, minimum energy needed to cause a chemical reaction. A chemical reaction between two substances occurs only when an atom, ion, or molecule of one collides with an atom, ion, or molecule of the other.  for the antioxidant consumption was determined to be 155.5 kJ/mol by measuring OIT of samples taken from the center of the unexposed pipe at different temperatures between 180 and 230 [degrees] C. OIT profiles were measured on samples die punched from the pipe wall and subsequently microtomed to 0.1 mm sections to give samples from different radial positions.

RESULTS AND DISCUSSION

Figure 3a shows that the antioxidant crystals precipitated from an ethanol solution have a hexagonal hex·ag·o·nal  
adj.
1. Having six sides.

2. Containing a hexagon or shaped like one.

3. Mineralogy
 shape. Figure 3b shows the antioxidant crystals in SEM as obtained directly from the manufacturer. Some of these crystals basically have the same shape as the precipitated crystals shown in Fig. 3a. Figure 3c presents an EDS spectrum from one of the crystals shown in Fig. 3b and it is obvious that the sulfur atom present in the antioxidant is readily detectable by this method. Several particles were found in the pipe wall which were shown to contain sulfur by EDS-analysis, i.e. they were identified as antioxidant. Figures 4a and 4b show the shape of particles found in the bulk of the pipe wall of a pipe pressure tested for 883 h at 95 [degrees] C and Fig. 4c shows a typical EDS spectrum for this kind of particles, indeed proving them to be rich in antioxidant content. The particle in Fig. 4b exhibits a shape with striking resemblance to the crystals precipitated from ethanol solution shown in Fig. 3a. The EDS spectrum in Fig. 4c indicates little contamination from other substances. However, a significant fraction of the antioxidant containing particles exhibited a variety of contaminating con·tam·i·nate  
tr.v. con·tam·i·nated, con·tam·i·nat·ing, con·tam·i·nates
1. To make impure or unclean by contact or mixture.

2. To expose to or permeate with radioactivity.

adj.
 elements when analyzed by EDS. It is hence possible that contaminants such as catalyst residues or metal particles from processing equipment act as nucleating agents for antioxidant agglomeration and crystallization.

A number of samples from different exposure times were investigated by SEM/EDS. All particles containing sulfur according to according to
prep.
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

3.
 the EDS spectrum, regardless of the amount of contaminants, were considered to contain antioxidant. Figure 5 presents the number of antioxidant containing particles found per investigated area as a function of the square root of the exposure time. Figure 5 shows that the number of antioxidant containing particles increases rapidly from zero to a significantly higher level after a short time of exposure. Hence, the antioxidant containing particles are most likely to consist of phase separated and subsequently precipitated antioxidant. Figure 5 also suggest that the number of antioxidant containing particles decrease on prolonged exposure. The decrease may indicate that the particles act as reservoirs when the dissolved antioxidant is depleted de·plete  
tr.v. de·plet·ed, de·plet·ing, de·pletes
To decrease the fullness of; use up or empty out.



[Latin d
. This is in accordance with the suggestions of Billingham and Calvert (11). It should be stressed that the relation between the number of antioxidant containing particles and the amount of precipitated antioxidant is influenced by several factors such as variation in volume between the particles, contamination of the particles, destruction of particles during preparation of cross sections, and so on. The results in Fig. 5 may thus only be viewed as a qualitative measure of the amount of precipitated antioxidant.

The consumption of phenolic phe·no·lic
adj.
Of, relating to, containing, or derived from phenol.

n.
Any of various synthetic thermosetting resins, obtained by the reaction of phenols with simple aldehydes and used as adhesives.
 antioxidants at high temperatures should follow zero-order kinetics zero-order kinetics Therapeutics The in vivo dynamics of drug elimination, which is linear with time, proportional to the concentration of the enzyme responsible for catabolism, and independent of substrate concentration.  (16):

-dc/dt = [k.sub.0] exp exp
abbr.
1. exponent

2. exponential
 ([Delta]E/RT) [square root of [p.sub.[o.sub.2]]] (1)

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. Integration of Eq 1 taking into account that during OIT measurements [p.sub.[0.sub.2]] is approximately constant yield the expression:

[c.sub.0] = [k[prime].sub.0] [exp (-[Delta]E/RT)] [multiplied by] OIT (2)

where [c.sub.o] is the initial concentration of antioxidant and [k[prime].sub.0] is a constant. Equation 2 was confirmed to be valid for 4,4[prime]-thiobis(3-methyl-6-tertbutylphenol) by Howard (17). It is difficult to generate calibration samples of a known antioxidant concentration because there will be a physical and chemical consumption of the antioxidant during the compounding of the calibration samples per se. For the remainder of this paper, we deal directly with OIT results shifted to 190 [degrees] C. These data are thus proportional to concentration of effective antioxidant.

Figure 6 presents OIT as a function of the radial position in the pipe wall for different exposure times in hydrostatic pressure testing at 95 [degrees] C. The corresponding data obtained from OIT measurements on pipe exposed at 105 [degrees] C exhibited a similar trend. The average OIT value corresponding to each exposure time was calculated by integration of the least-squares cubic spline In computer graphics, a smooth curve that runs through a series of given points. The term is often used to refer to any curve, because long before computers, a spline was a flat, pliable strip of wood or metal that was bent into a desired shape for drawing curves on paper. See Bezier and B-spline.  fits represented by the broken lines in Fig. 6. Figure 7a presents the average values for samples taken from pipes pressure tested at 95 and 105 [degrees] C as a function of the square root of the exposure time. The data in Fig. 7a show a trend similar to that earlier reported by Smith et al (3). The change in slope occurring after 1000 h exposure at both 95 and 105 [degrees] C suggests a rapid change in antioxidant depletion mechanism. Over 80% of the initial effective antioxidant is lost within the 1000 h. Figure 7b presents OIT values from different positions in the pipe wall, which indicate that Regime A becomes more pronounced with increasing distance from the inner wall. The solubility of the antioxidant in the polymer is probably influenced by the water gradient in the pipe wall. The solubility parameter of polyethylene is 15.1 to 17.1 [J.sup.1/2] / [cm.sup.3/2] (18) whereas the solubility parameter of the antioxidant is 24.8 [J.sup.1/2] / [cm.sup.3/2] based on group contribution theory and data collected by Van Krevelen (18). A minor inclusion of water in the polyethylene matrix should increase the solubility parameter of the polyethylene/water mixture and thus lead to a greater equilibrium solubility of the antioxidant.

The equilibrium solubility of the antioxidant at the OIT analysis temperature is far greater than the 0.3 wt% initially added to the polymer. Thus, the phase-separated antioxidant should, when reaching the analysis temperature, gradually dissolve. The important question is whether full concentration equilibrium is reached within the time required for the analysis. If it is not, then the OIT method would not assess the concentration of effective antioxidant at the pressure testing temperature. Approximate diffusion calculations of the evolution of the antioxidant concentration profile in the vicinity of the internal antioxidant phase indicated that the analysis time and temperature would be sufficient for full equilibrium to be reached. However, hot-stage polarized A one-way direction of a signal or the molecules within a material pointing in one direction.  microscopy showed that an internal antioxidant phase remained stable in a polyethylene matrix for a longer time than the OIT at the temperature concerned. This observation was particularly apparent in the presence of certain impurities, which indeed were found by EDS in many antioxidant particles. It seems that these findings indicate that there is a resistance for the transport of antioxidant through the antioxidant/polyethylene phase boundary.

Figure 8 shows the results of fitting the Regime B model previously developed (3) to the experimental OIT profiles (long exposure times corresponding to Regime B). This made it possible to determine the adjustable parameters of the model (Table 1). Earlier determinations of the diffusion coefficient for 4,4[prime]-thiobis(3-methyl-6-tertbutylphenol) in polyethylene have been performed with the same media at both inner and outer boundaries (19-21) and rendered values of about [10.sup.-7] [cm.sup.2]/s (18) when extrapolating to 95 and 105 [degrees] C. Since previous modeling work, using the Regime B model (3), has shown that alteration of internal and external media of the pipe may change [D.sub.0] with as much as two orders of magnitude at 95 [degrees] C, the values of [D.sub.0] shown in Table 1 are reasonable.

It was impossible to fit the Regime B model to the OIT profiles from shorter (Regime A) and longer (Regime B) exposure times simultaneously. Figure 9 presents an attempt to calculate the OIT profiles at 95 [degrees] C after short exposure times (Regime A), using the parameters (Table 1) obtained from fitting OIT profiles from the longer exposure times (Regime B). These results show that the Regime A antioxidant depletion cannot be explained by diffusion controlled migration of the antioxidant to the surrounding media.

[TABULAR DATA FOR TABLE 1 OMITTED]

THE REGIME A MODEL

Polyethylenes are normally processed at temperatures above 200 [degrees] C, at which temperature the solubility of antioxidant is high. The pipe is rapidly cooled to room temperature after the extrusion, and the antioxidant solubility becomes limited. Under these conditions, there is thus a thermodynamic ther·mo·dy·nam·ic
adj.
1. Characteristic of or resulting from the conversion of heat into other forms of energy.

2. Of or relating to thermodynamics.
 driving force for phase separation. Hydrostatic pressure testing for the evaluation of Stage III performance of polyethylene hot-water pipes is typically conducted at temperatures between 70 and 105 [degrees] C. The solubility of antioxidant increases, but it is still lower than that at the extrusion temperature. The higher mobility of the polyethylene chains at the pressure testing temperatures will allow diffusion of antioxidants towards nucleation sites nucleation sites

the ends of microtubules in the cytoplasmic skeleton; contributes to the growth of protofilaments.
 (small precipitates already formed or impurities). The antioxidant concentration in a small volume in the vicinity of each precipitate drops from the initial value, [C.sub.0], to the equilibrium solubility of antioxidant in the polyethylene matrix at the present temperature, [C.sub.eq]. This is the initial condition of the Regime A model [ILLUSTRATION FOR FIGURE 10 OMITTED]. It is assumed that the diffusivity Dif`fu`siv´i`ty

n. 1. Tendency to become diffused; tendency, as of heat, to become equalized by spreading through a conducting medium.
 is independent of the radial distance from the precipitate and independent of time. The rate of change of antioxidant concentration at any radial point is then given by Fick's second law expressed in spherical coordinates:

[Delta]C/[Delta]t = D([[Delta].sup.2]C/[Delta][r.sup.2] + 2/r [Delta]C/[Delta]r) (3)

where C is the antioxidant concentration, r is the radial distance, t is the time and D is the diffusion coefficient. The method of lines The method of lines (MOL, NMOL, NUMOL) ( Schiesser, 1991; Hamdi, et al., 2007; ) is a technique for solving partial differential equations (PDEs) where all but one dimension is discretized.  (22) was used to transform Eq 3 into a system of first order ordinary differential equations ordinary differential equation

Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function).
 in time. Application of [C.sub.eq] as the boundary condition at the precipitate/matrix interface yields the system of ordinary differential equations which in discretisized form becomes:

[C.sub.1, j+1] = [C.sub.1, j] + D[Delta]t/[Delta]r ([C.sub.2, j] - 2[C.sub.1, j] + [C.sub.eq, j]/[Delta]r + 1/r([C.sub.2, j] - [C.sub.eq, j])) (4)

[C.sub.n, j+ 1] = [C.sub.n, j] + D[Delta]t/[Delta]r ([C.sub.n + 1, j] - 2 [C.sub.n, j] + [C.sub.n-1, j]/[Delta]r + 1/r ([C.sub.n+1, j] - [C.sub.n-1, j])) (5)

i = 2, 3, 4, . . ., n - 1

[C.sub.n, j+1] = [C.sub.n, j] + D[Delta]t/[Delta][r.sup.2] ([C.sub.n-1, j] - 2 [C.sub.n, j] + [C.sub.n-1, j]) (6)

[Delta]r = b - a/n (7)

where n is the number of grid points and [Delta]t is the time increment selected for the solution of the system, [C.sub.i,j] is the antioxidant concentration at the i:th grid point after time j, 2a is the diameter of the precipitate, 2b is the distance between the centers of two adjacent precipitates and D is the diffusivity. Thirty-two grid points were sufficient to obtain an accuracy within the experimental error of the OIT method used. r = b corresponds to an isolated point were [Delta]C/[Delta]r equals zero, hence the form of Eq 6. Integration of the concentration profile over the sphere yields the average antioxidant concentration in the system according to:

[Mathematical Expression A group of characters or symbols representing a quantity or an operation. See arithmetic expression.  Omitted] (8)

Smith et al. (3) showed that large variations in D over the pipe wall are to be expected when water is the internal medium and air is the external medium. They assumed a linear dependence of D with the radius in the pipe wall according to:

D(p) = [D.sub.0] [1 + [Lambda] (q - p / d)] (9)

where [D.sub.o] is the diffusion coefficient at the outer pipe wall, p is the radius, [Lambda] is a constant, q is the outer pipe radius and d is the thickness of the pipe wall. By solving Eqs 4 - 6 using the values for [D.sub.o] and [Lambda] in Table 1 and subsequently integrating, Regime A OIT profiles over the pipe wall were obtained.

The radial dependence of [C.sub.eq] was estimated from OIT data by calculating the intersection of Regimes A and B lines at different radial positions in the pipe wall, as is shown in Fig. 7b. If it is assumed that the average initial concentration of antioxidant in the pipe wall is 0.3 wt%, the average value obtained for [C.sub.eq] will be close to 0.05 wt% at 95 [degrees] C, in accord with data of Roe et al. (20). Solubility data reported by other researchers (19, 21) are at least one order of magnitude A change in quantity or volume as measured by the decimal point. For example, from tens to hundreds is one order of magnitude. Tens to thousands is two orders of magnitude; tens to millions is three orders of magnitude, etc.  greater than the present data, but this discrepancy is probably due to the fact that in previous work measurements have been performed under vastly different boundary conditions.

The enlargement of the particles during growth was assumed to be insignificant compared to the distance between the particles. Hence, a was set to an average value of 17.44 [[micro]meter] calculated from SEM measurements and b was used as an adjustable parameter to fit Regime A OIT profiles from 95 and 105 [degrees] C. Figure 11 presents the best least-squares fit of the Regime A model to OIT profiles from 95 [degrees] C. Values obtained for b were 274 and 257 [[micro]meter] for OIT data from pressure testing at 95 and 105 [degrees] C respectively. The fit of the model to the experimental data is good in view of the significant spatial variation in initial conditions found earlier in similar pipes (2, 3). Considering that most parameters in the model are uncertain, this should only be considered as an order of magnitude calculation and, hence, the fit obtained is satisfactory. Further development of the Regime A model should include recognition of diffusion of the antioxidant to the surfaces of the pipe. Considering the values obtained for b, this effect may contribute significantly to the overall antioxidant loss. Also, the uncertainty in the determination of [C.sub.eq] is significant, and methods for the more accurate determination of this parameter should be developed. There may also be a need for further studies of the possibly significant reservoir effect of the precipitates, which may call for the need of modification of the Regime B model. Further development of the models along these lines will produce a Regime A model including the adjustable parameters in the Regime B model. Fitting of the Regime A model to short-term OIT profiles may hence give parameter values which can be used for calculation of long-term OIT profiles. This would open new possibilities for development of lifetime prediction methods based on readily accessible short-term data.

CONCLUSIONS

Particles containing high concentrations of a phenolic antioxidant were found in the pipe wall of MDPE pipes that had been hydrostatically hy·dro·stat·ic   also hy·dro·stat·i·cal
adj.
Of or relating to hydrostatics.



hydro·stat
 pressure tested at 95 and 105 [degrees] C. The particles had evolved during the first 1000 h of testing and consisted of antioxidant that had phase separated and precipitated from the polymer matrix because the initial antioxidant concentration was greater than the solubility limit at the pressure testing temperatures. Oxidation induction time measurements of the relative antioxidant content in the pipe wall revealed a loss of more than 80% of the initial antioxidant content during the first 1000 h of pressure testing at both temperatures. A model assuming diffusion of antioxidant towards internally precipitated antioxidant particles was developed. It is suggested that the antioxidant confined in the precipitated particles is not effective and is not assessed in the oxidation induction time measurements. Values for the diffusivity of the antioxidant in different parts of the pipe wall were obtained by fitting long-term antioxidant concentration profiles with a previously developed model. Using these values it was possible to fit short-term antioxidant concentration profiles with the newly developed model.

ACKNOWLEDGMENTS

This study was sponsored by the basic research fund at Studsvik AB, Sweden. Borealis (Sweden) and Monsanto Scandinavia are thanked for the supply of MDPE and antioxidant. Prof. C.-G. Gustavsson, NTH nth  
adj.
1. Relating to an unspecified ordinal number: ten to the nth power.

2. Highest; utmost: delighted to the nth degree.
, Trondheim, Norway is gratefully acknowledged for making the compounding equipment available. We thank M. Trang, J. Rydberg, and S. Arvidsson for experimental assistance.

REFERENCES

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2. K. Karlsson, G. D. Smith, and U. W. Gedde, Polym. Eng. Sci., 32, 649 (1992).

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Title Annotation:International Forum On Polymers - 1996
Author:Viebke, J.; Hedenqvist, M.; Gedde, U.W.
Publication:Polymer Engineering and Science
Date:Dec 1, 1996
Words:4483
Previous Article:Preface - International Forum on Polymers: status report 1996.
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