x-ray modulus and strain distribution in single fibers of polyethylene

X-Ray Modulus and Strain Distribution in Single Fibers of Polyethylene

KESHAV PRASAD* and DAVID T. GRUBB, Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14853

Synopsis

Synchrotron x-ray radiation was used to study structural changes during deformation of a single high-modulus, high-strength fiber of polyethylene (Spectra 1O00, Allied Corp.) 20-40 pm in diameter. The high brightness of the synchrotron beam allowed x-ray diffraction patterns to be obtained from 0.5 pg fiber samples in ca. 1 min. The (002) chain axis reflection shifted and broadened when the fiber was loaded in tension. When a model of crystals and disordered material in series was used for the fiber, the measured lattice strain a t stresses up to 1 GPa gave a crystal modulus of 250 GPa. At stresses above 1 GPa the (002) reflection no longer shifted with increasing load. A noncrystalline part of the material deforms and takes up the load this effect has been observed by Raman spectroscopy. The (002) peak also broadens under stress. When fiber bundles were used as samples, broadening could be due to uneven loading of fibers, but with a single fiber sample, one can be sure that the loading a t the fiber level is uniform. Broadening of the (002) reflection then indicates inhomogeneities within the fiber. Deconvolution with the line profile of the fiber a t zero stress should give the distribution of crystal strains. The strain distributions are symmetric and during loading the full width a t half maximum (fwhm) is approximately equal to the mean strain. With an assumed constant crystal modulus this describes the stress distribution as extending from zero to twice the mean stress.

INTRODUCTION

Polymer fibers have been produced which have very high tensile moduli, ca. 300 GPa, and a very high strengths, ca. 4 GPa; they are generally called high- performance fibers. In these fibers strongly bonded molecular chains are well aligned along the fiber axis direction. Weaker secondary molecular bonds in the lateral direction create highly anisotropic materials. The fibers are brittle, generally weak in axial compression, and weaker still transversely, fibrillating and breaking up along the fiber axis very easily.

In one method used in the production of high-performance fibers, polymers such as polyethylene, which have flexible chains, are hot drawn to an extension ratio of 10-100 times.'.' Using the process of gel drawing or ~ p i n n i n g ~ - ~ it has become possible to prepare ultrahigh-molecular weight polyethylene ( PE ) fibers and tapes with Young's modulus of 100-200 GPa and tensile strength of 2-5 GPa.6,7 These values for modulus and strength are very high considering that

1. The theoretical Young's modulus of a PE crystal along the chain direction E, is about 300 GPa,' the transverse tensile moduli, El are only 2-3 GPa, and the theoretical strength of a molecule is about 35 GPa.'

2. Polyethylene is not perfectly crystalline, but generally contains crystals and disordered amorphous regions of lower density and much lower stiff-

* Present address: IBM General Technology Division, Hopewell Jn., NY 12533.

Journal of Polymer Science: Part B: Polymer Physics, Vol. 28, 2199-2212 (1990) 0 1990 John Wiley & Sons, Inc. CCC 0887-6266/90/01202199-014$04.00

2200 PRASAD AND GRUBB

ness and strength. The crystallinity X , of the high-modulus fibers as normally measured is in the range 85-9576.

The anisotropy in the properties of PE chains8V1' implies that any imperfection in the orientation of the crystals reduces the tensile modulus of the fiber. Fiber properties depend on the orientation of the crystals and the crystallinity of the fiber, but it is found that fibers can have the same highly perfect crystal orientation and the same value of X,, but very different moduli and strengths.'lJ2 Other factors must therefore be involved in controlling the mechanical properties of the fiber. If the semicrystalline fiber is considered as a composite containing crystalline and disordered phases, its properties will depend on the arrangement of these two very different phases as well as on their amount and orientation. The arrangement of the phases controls how the stress is distributed within the material, affecting the stiffness. The failure of the fiber will be related to the presence of local stress concentrations. Measurement of local stress distribution within a high-modulus fiber can thus help relate mechanical properties to the structure and predict fiber strength.

Stresses applied to any crystalline solid lead to elastic strain of the crystals, i.e., to changes in lattice spacing, though if the material is weak the strain at failure or plastic yield may be small. In most polymer samples this is true, but the high-performance fibers can have a tensile elastic strain to failure of 1-2%. In PE, the diffraction angle 28B of the (002) reflection in wide-angle x-ray diffraction (WAXD) measures a spacing in the stiff direction, the direction of the molecular chain as dooz = 2 sin B B / X . The diffraction angle changes by 68 when the fiber is loaded in tension, measuring the elastic strain of the crystals as t, = -cot 8868. This peak shift has been used in earlier experiments described as x-ray crystal modulus determination; however, to determine the lattice modulus of the crystals some assumption has to made about the way the load is carried in the fiber.

Assumptions are required because the experiments measure the strain t, and the stress applied to the fiber, uf. The stress u, acting on the crystals is not known. In early articles, 13,14 a homogeneous stress distribution (series model) was assumed. In this case ur = u,, and the crystal modulus E, = uj /cc . The plot of crystal strain versus applied stress is usually linear at low stresses, and the slope gives E, N 200 GPa for PE. There are now several more sophisticated mechanical models for high-modulus PE fibers which can be used to relate u, to uf.15-20 One of these has been applied to melt-drawn high-modulus PE fibers, l6

giving E, = 250 GPa. The discrepancy between this and the accepted value of 300 GPa implies that the model or its parameters should be adjusted to make u, larger. The apparent lattice modulus thus gives information about the way load is carried in the fiber and the fiber structure.

X-ray experiments to determine crystal modulus have in the past required large samples, bundles of hundreds of fibers, and long data collection times. X- ray data obtained from such bundles is not very satisfactory, as the orientation of fibers in a bundle and the mechanical loads on them cannot be made completely uniform. Figure 1 compares the WAX diffraction pattern from a bundle of Spectra 1000 fibers with that from a single fiber of Spectra 1000. The mean misorientation derived from the equatorial reflections, after correcting for the instrumental and sample-size effects using standard methods is 2.2' for the

X-RAY MODULUS AND STRAIN DISTRIBUTION 2201

Fig. 1. WAX diffraction pattern from a bundle of Spectra 1000 fibers ( a ) and a single fiber of Spectra 1000 ( b ) .

bundle and 1.6" for the single filament. The difference is due to nonuniformity in the bundle construction. With use of a bundle, variations in structure and orientation within highly perfect fibers will not be detectable. The load, the local crystal size, and the degree of orientation are better defined with a single- fiber sample.


The use of a microbeam camera on a normal x-ray source is difficult and time consuming; but worse than that, the PE fibers are not perfectly elastic at high loads at room temperature. They tend to creep or stress relax so that long exposures cannot be made at constant stress and strain. The high intensity and low divergence of the monochromatic x-ray beam at the Cornell High Energy Synchrotron Source (CHESS) 21 has allowed high-quality x-ray diffraction patterns to be obtained from single fibers with exposure times of ca. 1 min. The very small sample size of about 5 X mm3 or 0.5 pg in the x- ray beam allows the study of local variations in fiber structure.

A WAXD reflection is made unsharp by instrumental effects, and by the crystal orientation distribution, crystal strain distribution, and the limited size and perfection of the crystals. The (002) linewidth increases when a bundle of PE fibers is loaded, but an unknown part of this may be due to uneven loading of fibers within the bundle. This problem has discouraged detailed analysis of the linewidth changes on fiber loading. We now observe large increases in the (002) linewidth on loading a single fiber, where the problem of uneven fiber loading is avoided. Instrumental effects will be unaffected by stress, so the increase in linewidth on loading relates to the fiber structure.

An analogous effect is observed in infrared ( IR) and Raman spectroscopy of Kevlar, PE, and other polymers where spectral lines shift and broaden under load.22-24 In a previous report, we used Raman spectroscopy to study the mi- cromechanics of high-strength polyethylene fibers. The band position and shape of the narrow 1063 cm Raman band corresponding to the C - C asymmetrical stretching mode was studied as function of stress. This band is normally at- tributed to the crystalline phase as the spectrum of the melt shows only a broad scattering region centered at 1080 cm-'. The 1063 cm-' band may also come from all-trans chain sections without lateral order as the line position is sensitive to chain conformation, but not to intermolecular interactions.

In the all-trans PE chains there is a linear relationship between the local strain in the molecular chain direction and the vibrational frequency of the 1063 cm -1 Raman band. Figure 8 of Ref. 22 shows that the peak shift is linear a t low applied stresses, but at high stresses, the peak position does not change with stress and the band becomes asymmetric. A t the highest applied stress, 85% of the area in the band remains in a symmetric peak, but the other 15% is in a long tail extending to low wavenumbers. This corresponds to a highly strained material. The stress at which the peak shift becomes nonlinear was called the break point. It was found that above this break point, the fibers undergo significant plastic flow. On unloading the fiber to zero stress, the wave number of the peak was higher than the wave number at the start of the loading process, indicating that material was in compression.

These results were explained by considering the polyethylene fibers to have two load-bearing phases. A t low stresses one phase carries most of the load. At high stresses, the load on this phase no longer increases and all the extra load is now carried by the second phase which is seen in the asymmetric tail of the 1063 cm-' Raman band. The second phase is not detected by WAXD, and is therefore noncrystalline or disordered. The chains in this phase must be all- trans, both because they carry a significant load and because they are detected by the 1063 cm -' Raman band. The chains must also be well aligned parallel


to the fiber axis or they would not be mechanically effective. The order within this region is thus similar to that of a nematic liquid crystal.

Noncrystalline, all-trans chains in fibers which carry load are not a new idea; they are generally called taut tie-molecules. Molecular chains in this phase must have approximately the same modulus as the crystals and some of them are strained to 5%. They thus carry a load equivalent to about 15 GPa. The second phase is only 15% of the sample and carries up to half the load when the applied stress is 2 GPa. On unloading, the crystals go into compression, implying that the tensile stresses in the taut tie-molecules are slow to relax and have residual tension.

EXPERIMENTAL

The sample used in this study was Spectra 1000, a gel-spun high-strength polyethylene fiber, obtained from Allied Corporation. It has a molecular weight of 2.5 million, strength of 3.1 GPa, and a modulus of 180 GPa measured at a strain rate of 0.02 s-'. The average diameter of the fiber is about 27 pm. The cross-sectional area of each fiber was determined using the vibroscope technique, ASTM D1577 and this area, determined for individual fibers, was used in cal- culating the stress on the fiber.

Wide-angle x-ray scattering studies were performed at the Cornell High Energy Synchrotron Source on the A-1 beam line, which uses a wiggler magnet and a double-focusing monochromator to produce a highly parallel beam of x- rays, wavelength 1.5 A, with 5 X 10l1 photons/s in a 0.3 X 1 mm2 spot at the detector plane. In these experiments (Fig. a ) , a 500 pm pinhole collimator was used.

The ( 002) meridional reflection was recorded using a Braun position-sensitive detector placed at an angle of 28B = 75" to the direction of the incident beam.

Storage Ring CHESS

Computer < Position isitive

vcLector

I I

I Motor Controller

Fig. 2. Experimental setup for WAXD at the Cornell High Energy Synchrotron Source.


The sample to detector distance was 304 mm, corresponding to a detector resolution of 0.2 pad . PE fiber cannot be mounted by normal tabbing methods, so special capstan inserts were used.25 The capstan inserts were mounted in a small strain frame with the fiber direction at an angle OB - 90 = -52.8' to the direction of the incident beam, to put the (002) planes in the diffracting position. Strain was applied in staircase fashion; that is, the fiber was rapidly extended to a particular strain value and allowed to stress relax until the stress became stabilized. At high stresses, relaxation was considerable and the step of the strain was increased to overstress the fiber. Stress was considered stable when it remained constant within k 60 MPa over a data collection time of 120 s. The experimental details were explained in a previous report on Raman spectroscopy of PE fibers."

The data were transferred to a CONVEX computer. A polynomial baseline was drawn, and the subtracted data set was fitted to a Gauss-Lorentz sum function, using a nonlinear least-squares fitting routine. This gave the peak position and full width at half-maximum ( fwhm) of the reflection. To get the mean position (center of mass) Z (position X height) / Z (height) was calculated. To ascertain if there was any asymmetry in the reflection, only the right-hand side of the data was fitted to a Gauss-Lorentz sum function and the function was superimposed on the left-hand side of the data.

The wide-angle x-ray peak under load h ( 8 ) is broadened by several factors: camera geometry, crystal size, and crystal strain distribution. If the first two factors are taken to be unaffected by stress, and g ( 8 ) is the peak shape at zero stress when the fiber is fully relaxed with no internal strains, we have

Here * indicates the convolution operation, and f ( 8) is the crystal strain distribution (in terms of angle). In order to deconvolute g ( 8 ) from h( 8) we take the Fourier transform of both sides of eq. ( 1) , divide and take the inverse transform.

Routines using fast Fourier transform algorithms make this deconvolution procedure straightforward. In practice, resolution is limited by the finite number of data points. This truncates the useful range of the transform and is equivalent to a boxcar apodization, giving rise to large sidelobes in the deconvoluted result. Various other apodization windows were used to reduce the sidelobes, but these also reduce the resolution of the peak. The choice of the apodization window made little difference as they had similar effects on the width of the deconvoluted peak.

RESULTS

Figure 3 shows the x-ray detector output for the (002) reflection of PE as a function of fiber stress, and curves fitted to the data. In this figure, the (002)


73.0 73.5 74.0 74.5 75.0 75.5 76.0 208, degrees

Fig. 3. WAXS data for the (002) meridional reflection of a single fiber of Spectra 1000.

peak shifts to lower angles as dooz increases with stress until about 1 GPa. In this region it also becomes broader. At high stresses above 1 GPa, neither the line shape nor the line position changes significantly. The solid line is a symmetric function fitted only to the data on the right-hand side of the peak of each curve. It is merely superimposed on the data on the left-hand side of each peak. As can be observed, the line shape is symmetric at both low and high stresses. As another measure of the symmetry of the x-ray reflection line shape, we measured the peak position and the first moment (mean position) of the (002) reflection. The peak position corresponds to the most probable value of dmz in the fiber and the first moment to the average value of dm2. The peak and mean positions are very close, with an average separation of only 0.015' and no systematic trend to this separation. The small differences between the peak and the mean are due to noise in the background, which makes measurement of the first moment less certain.

X-ray data for dhkl are usually plotted as a function of applied stress. For better comparison with mechanical testing, stress is plotted as a function of crystal strain t, in Figure 4. This is the form of a regular stress-strain curve, and it shows that the data for t, derived from the (002) reflection are clearly odd. A t stresses greater than 1.2 GPa the crystal modulus apparently goes to infinity, which is not reasonable. On unloading the fiber the peak position does not follow the original loading curve; there is considerable hysteresis. The unloading curve is above the loading curve in Figure 4 so that stresses are higher at a given strain and it appears that the crystal modulus is higher. However, it is more reasonable to assume that the crystal modulus is essentially constant, so that the crystal strain t, is a measure of the (T,, the stress on the crystals. For a given applied fiber stress cf this is always lower on unloading than it was on loading. This means that the load-bearing material not detected by WAXD,


20e, degrees 74.66 74.53 74.40 74.27 74.14

2.0 a Loading a

f a I

i i 2*o

Crystal Strain, %

Fig. 4. Applied fiber stress as a function of capital strain.

which is not activated until E , = 0.5% on loading, remains partially active and load-bearing throughout unloading.

The hysteresis is also clear when the width of the (002) reflection is plotted as a function of stress (Fig. 5 ) . It may be noticed that the data point corresponding to zero applied stress is not represented in this plot. There was some slack in the fiber at zero stress and this led to misalignment and a broader peak. An estimate of the line shape at zero stress was obtained by extrapolating the position, width, and intensity of the (002) line at higher stresses within the linear range to zero stress. Using these parameters in the same shape function that fitted all the real data, the line at zero stress was simulated. The shape of this simulated line at zero stress was deconvoluted from the shape of ( 002 ) line at higher stresses. The deconvoluted data shown are therefore only estimates of the true values. Theoretically the deconvolution of a curve with itself should give a delta function. Due to the finite number of data points one gets a deconvoluted line with large side lobes. To reduce the side lobes, a triangular apodization window was used giving rise to a line of finite width. Thus the width of 0.05' for the deconvoluted data at zero stress is entirely a numerical artifact which does not depend on the exact form chosen for the zero-stress data. To get the true strain distribution, the internal strains, if any in the unstressed fiber have to be taken into account; however, owing to the limited resolution of the deconvolution process, the width due to any residual stresses is much less than the uncertainty of the deconvoluted result.

Because the width of the reflection increases by a factor of three to four times on loading the fiber, the effect of deconvolution on the line shape of the


(002) reflection is small. Even though the data in Figure 3 appear good, the small effect of deconvolution cannot be accurately calculated. This is due the presence of large side lobes in the deconvoluted data and when apodization windows are used to reduce the side lobes, the resolution is reduced. The only effect of deconvolution is a slight decrease in the linewidth but the form of the curve in Figure 5 does not change and the exact shape of the stress distribution cannot be determined. The basic qualitative information can be seen from the raw data; the linewidth increases rapidly on loading from an estimated 0.075" to ca. 0.4" at 1 GPa and then remains constant. On unloading the width drops rapidly at first, then levels off. On complete unloading it remains much larger than its initial value, indicating that there are residual stresses in the fiber.

Comparison of the values of linewidth in Figure 5 with the peak position in Figure 4 shows that, on loading, the full-width at half-maximum is close to the peak shift. The peak shapes are roughly triangular (Fig. 3) so the strain in the crystals takes all values from zero to twice the mean strain. Again, the deconvolution process does not affect this basic result. Figure 6 shows the line shape of the deconvoluted curves. During loading the low-strain tail of the distribution is around zero stress; it is not expected that crystals should go into compression during tensile loading. On unloading to a low stress (curve 6 of Fig. 6 ) , the width remains high as the peak position returns close to zero strain; and it is clear that a significant fraction of the crystals do go into compression.

DISCUSSION

The x-ray diffraction experiments were carried out in order to compare the results with those from Raman spectroscopy under load. The 1063 cm-' band

0.4

2 0.2 is Q

m Deconvoluted Data I Loading Unloading

0.0 ' I I 0.0 0.5 1 .o 1.5 2.0 2.5

Stress, GPa

Fig. 5. Change in width of the (002) meridional reflection with stress.


1

I - - Unloading I 7 - 0.00 GPa

- Loadina I

2 - 0.56 GPa 3 - 1.14 GPa 4 - 2.05 GPa 5 - 1.15 GPa 6 - 0.00 GPa

-1 .o -0.5 0.0 0.5 1 .o 1.5 Strain, %

Fig. 6. True line profile of the (002) meridional reflection as a function of stress.

in the Raman spectrum of PE corresponds to the C-C asymmetric stretching mode. In Spectra 1000 fibers, the peak position of this band shifts linearly and broadens until it is ca. 1 GPa. At stresses above this ‘break point,’ hardly any change occurs in the peak position or in the full width at half-maximum. These features are identical to the WAXD results, which show more directly that the strain in the crystals does not increase at fiber stresses above 1.2 GPa. However, the mean position of the Raman band, which is very close to the peak position at low stresses, continues to shift with stress, linearly all the way to fracture. There is a significant amount of asymmetry in the Raman band at high stresses due to a tail at longer wavelengths, which corresponds to greater strains. Al- though the full width at half-maximum remains constant, the integral breadth of the peak increases. These results are quite different from the WAXD results. The Raman spectral line positions and x-ray reflection angle were initially expected to show the same behavior, following E ~ , the elastic strain in the PE crystals. This is why so much attention has been paid to demonstrating an apparently obvious conclusion-that the x-ray (002 ) reflection remains symmetric in shape, with no tail.

We have already suggested an interpretation of this discrepancy between WAXD and Raman results:” Raman spectroscopy is sensitive to all-trans chains whether or not they are in a crystal, whereas the WAXD is sensitive on!y to the crystals. Thus the highly strained material detected in the Raman spectrum but not in the WAXD of fibers loaded in tension is not crystalline, although it is in the all-trans conformation and is aligned parallel to the fiber axis. Such material is commonly called ‘taut tie-molecules.’ The location and amount of


taut tie-molecules depends on the draw ratio of the fiber and its molecular weight. In the fibers studied here, there is a large amount of this material, and it can carry half the load on the fiber. The degree of order apparently present in this material makes it likely that models based on two phases which are perfect crystals and fully disordered material will be incorrect. Indeed models for high-modulus fibers do allow for partial order in the noncrystalline phase, either explicitly as a fraction of taut t i e - m o l e ~ u l e s ~ ~ ~ ~ , ~ ~ or by allowing the modulus of that phase to be higher than that of normal amorphous PE.

If simple two-phase models of this kind are discarded, and partially ordered material considered, we can consider what level of order has to be present for the x-ray experiment to detect the material. One measure of order is the di- mensions of a perfect crystal with the same diffraction properties. An isolated taut tie-molecule is one extreme possibility; the other is a crystal, similar to those in the fiber. These are not very large, a t least 100 nm long in the fiber direction but only about 20 nm wide.

How will the detected intensity depend on the diameter W of a small cylin- drical crystal? If the diameter is very small, the intensity is spread out and most will not fall on the detector, which is only 10 mm wide. A t some large diameter, the intensity will not rise further because the diffraction conditions become more restrictive, and crystals misaligned even by a small degree from the fiber axis will not contribute. We are not concerned primarily with misorientation from side to side, which causes arcing of the diffraction spot. The detector collects diffracted intensity from crystals misaligned in this plane by angles up to k 1" from the fiber axis direction. This range of 2" is about twice the mean misorientation of the crystals in the fiber; therefore, almost all the arc falls on the detector. It is the misorientation in the plane of the x-ray beam, moving the crystals off the exact Bragg conditions, that will affect intensity. Since the sample is a fiber, the misorientation in this plane also has a full width at half-maximum of 1.2". The range of orientations that give rise to diffraction is about A / ( W sin 0,). This is 2" when W = 7.3 nm.

Thus for the collection geometry used, crystals 10 nm or so in diameter appear to be large, and are detected with good efficiency. The intensity from smaller crystals will begin to fall, and by W = 3 nm the (002) intensity collected from the same volume of material would be only 10-15%. Since there is no sign of an asymmetric tail in the x-ray data, any crystals in that part of the fiber could not be much larger than 3 or 4 nm across, containing only about 50 molecules.

The apparently infinite crystal modulus E, at stresses above 1.2 GPa (Fig. 4 ) confirms the view that it is better to assume a constant value for E, and interpret the observed t, as a measure of the actual (T,. The results can be summarized as follows: the crystals see a local stress that can vary from zero to twice the mean stress on the crystals. This rises with the applied stress until plastic flow sets in at about 1 GPa. Then although the macroscopic fiber strain can be 4-5% and the stress on the fiber continues to rise to 2 GPa the mean crystal strain remains at 0.596, implying a constant mean stress on the crystals of about 1.2 GPa.

How does this new information affect mechanical or structural models of high-modulus PE fibers? Although a model based on two load-bearing phases was used to explain the Raman results,22 it cannot tell us what structural model


or class of model to use, for there are still many unknowns. For example, there is no absolute intensity calibration in WAXD that would allow the determination of X , or the fraction of taut tie-molecules. Standard methods for deter- mining X,, such as calorimetry or density, rely on a two-phase model with a completely disordered noncrystalline phase, which is known to be false. When the noncrystalline material is more ordered, more like the crystal in its properties, X , determined by the simple two-phase methods will be an upper limit for the fraction of crystals.

If a series-parallel multiphase approach is chosen to explain the Raman and x-ray results, all the crystals must be in the same mechanical region, and there must be a region mechanically in parallel with the crystals that contains noncrystalline material primarily made up of taut tie-molecules. The constraint of having the standard value of X , for the crystal fraction is removed in a series- parallel model which has some crystals in series with disordered material and some not. But then there would be a bimodal distribution of crystal strain, which is not observed. If only the low strain elastic properties are to be modeled, this is all that can be said. If E, is taken to be 300 GPa, then the crystal strain observed in these fibers, 0.5% at a fiber stress of 1.2 GPa could be modeled by a simple parallel model containing 80% crystals. A series model would give a E, = 250 GPa, and would not meet the requirement for the existence of a second load-bearing system at high stresses.

The fact that the crystal stress u, does not rise above 1.2 GPa is explained by the yield properties of the phases. In the simple parallel model, the crystal phase must yield plastically at constant stress, while the noncrystalline material rapidly strain-stiffens, taking up a greater and greater fraction of the load. If a more complex series-parallel model is used, 15-19,28929 where a small amount of a more compliant material is in series with the crystals, then at stresses greater than the break point, this material can start to flow. As the amount of this material is small, its local strain would have to be large.

These series-parallel models do not allow consideration of local shears between crystals and surrounding compliant disordered material. This shear can be seen as the controlling factor for fiber modulus when the crystals act as reinforcing elements in a disordered m a t r i ~ . ~ ~ ~ ” If the stress exceeds the yield stress for shear of the interface, the crystalline and disordered regions slide past each other. If the flow continues without any hardening of the interface, the fiber will creep at constant applied stress. As the stress applied to the fiber continues to increase, the interface must harden. This should cause the load on the crystals to increase in such a model, which is not observed.

The series-parallel models have no basis for a strain distribution. On the other hand, the strain distribution might arise naturally from fibrous crystals acting as reinforcing elements. Such fibers have zero stress at their ends, and the stress rises as load is transferred by shear of the matrix. Also, if there are soft disordered regions in the fiber, laterally adjacent crystals should carry extra load, and longitudinally adjacent crystals reduced load. This should cause the WAXD peak to broaden.

The models described so far all have at least two phases; another class of models has only one phase, with defects distributed randomly throughout the

Termonia et al.33-36 calculate the properties of such a model by computer simulation. Chain alignment is perfect and the only defects are pairs of

X-RAY MODULUS AND STRAIN DISTRIBUTION 221 1

molecular ends. In this model, the “crystals” are regions with lower than average defect density, and the highly strained “taut tie-molecules’’ are molecules adjacent to molecular ends or clusters of ends. With particular respect to the experimental results in this paper, it is difficult to see how the single-phase model can predict no increase in the strain of the well-ordered regions when the strain applied to the fiber increases by a factor of two. The chains adjacent to defects are overstressed, but they connect directly to well-ordered regions so that extra load should be transferred to them. The predicted strain distribution in the primary chain bonds is much narrower than is observed. A narrow strain distribution is a general feature of models where broken chains are the only defects. The activation energy of a break is large, so for a starting material of high-molecular weight there are relatively few defects before failure. The simulated fibers are also much stiffer and stronger than any real ones, although many features are qualitatively predicted.

It may be that differences between experiment and simulation are due, a t least in part, to computational limitations of the simulation, which is limited to 40,000 nodes and failure times of a few seconds. For the present it seems that the single-phase model should be treated as described33 as a model for perfect PE fibers. Alternatively, since a single-phase model does not explain the appearance of the equatorial streak seen in the SAXS of these fibers, it could be used as a model for microfibrillar crystals. Another hierarchy of simulation would then be required to model the entire fiber.

CONCLUSIONS

Wide angle x-ray diffraction from a single fiber was used to obtain the properties of crystals (ordered regions over 5 nm in diameter) in high modulus polyethylene. The mean chain misorientation of the crystals is only 1.6”. When the fiber is stressed, the motion of the (002) peak shows that the crystals do not deform uniformly. The strain along the chain direction is spread from zero to twice the mean strain. At applied stresses from 1 to 2 GPa the mean strain of the crystals remains at 0.5%. We take it that the crystal modulus is a constant. Then the stress in the crystals is also constant above 1 GPa and the extra load must be carried by material which is not crystalline, or in crystals <5 nm diameter. The models currently used to describe the mechanical properties of high performance fibers have multiple phases or partial order in the amorphous region or a single phase containing distributed defects. None of them can explain all the fiber properties that we have observed.

The authors are grateful to the National Science Foundation-Materials Research Center for financial support of this work. They thank W. W. Adams of AFWAL/MLBP for the loan of the x-ray detector system.

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Received August 7, 1989 Accepted January 26,1990

x-ray modulus and strain distribution in single fibers of polyethylene

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