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DEFECT DIFFUSION MODELS FOR INTERNAL
FRICTION PROCESSES IN POLYETHYLENE
D. Reneker, J. Mazur
To cite this version:
JOURNAL DE PHYSIQUE
Colloque CIO, supplément au n012, Tome 46, décembre 1985 page CIO-499
DEFECT DIFFUSION MODELS FOR INTERNAL FRICTION PROCESSES IN POLYETHYLENE
D.H. RENEKER AND J-MAZUR
National Bureau of Standards, Gaithersburg, Maryland, 20899, U.S.A.
ABSTRACT
Defect loops that encircle one chain in a polyethyene crystal are characterized in terms that are used to describe dislocation loops in metals. In a polyethylene crystal the defect loop which requires the least energy for ,creation belongs to the relatively unknown class of defects called dispirations. Dispirations can be thought of as defects in the helical symmetry of the polyethylene molecule in the crystal. The processes by which the diffusion of dispiration loops can contribute to relaxation processes in polyethylene are described.
INTRODUCTION
Interna1 friction plays an important role in the properties of polymers. It has been studied experimentally in a variety of waysl. Mechanical and dielectric experiments reveal a spectrum of relaxation peaks. This paper describes an atomic scale modeling approach to the molecular motions which produce the internal friction phenomena.
The relaxation processes that can result from the motion of a
crystallographic defect in polyethylenez, called a dispirationa, are described here. The first part of the paper describes the
dispiration. Dislocations and disclinations were also modeled in polyethylene. The dispiration has the lowest energy of the three. It has the property that it both turns the part of the molecule along which it passes by 180° and translates that part by one half a repeat unit.
The second part of the paper treats the dispiration as a point defect that can diffuse along the polymer chain by Brownian motion and shows how data from dielectric relaxation experiments and from nuclear magnetic resonance can be interpreted by ascribing a
diffusion coefficient to the dispiration. The value of the diffusion coefficient is then used to analyze a mode1 for an internal friction process.
JOURNAL DE PHYSIQUE
THE DISPIRATION
Many aspects of a dispiration in polyethylene are described in 485,and in papers referenced there. Aere the relationships between dispirations, disclinations, and dislocations will be described. Al1 three of these classes of crystallographic defects are characterized by a defect line. The defect line either closes upon itself or terminates at a surface.
For an edge dislocation, the line lies at the edge of the extra plane of atoms that is associated with the dislocation. Motion of the dislocation line is associated with slip in the crystal. A vector, called a Burger's vector, is associated with the dislocation line. This vector is revealed in a "walk" from atom to atom in the crystal along a path that would be a closed circuit if the lattice were perfect. If this "walk", or circuit, encircles the dislocation line the last step does not return to the starting point. A vector from the end of the circuit to the starting point is the ilurger's vector. It gives the magnitude and the direction of the displacement
associated with the dislocation.
A similar line is associated with a disclination. Motion of a disclination line is associated with rotation of one part of a crystal with respect to the other. Disclinations are not common in crystals of low molecular weight solids, but are found in liquid crystals, and in biological membranes. Calculations of the energy required to produce a disclination in polyethylene4 suggest that they occur in polyethylene. A circuit, analogous to the Burgeres circuit, and called a Nabarro circuit3 reveals the amount of rotation
associated with the disclination line. A direction for the "walker" to face is defined with reference to the immediate environment of the "walker". A Nabarro circuit around a disclination line results in a cumulative change in the direction the "walker" is facing.
A dispiration can be described as the combination of a partial dislocation with a partial disclination4. Motion of the dispiration produces both slip and rotation of one part of a crystal with respect to the other. Not surprisingly, a circuit around a dispiration line
leaves the "walker" some distance from his starting point and facing in a different direction.
Figure 1 shows how a dislocation loop can be created in a crystal block by mechanical deformation.. A circular depression with a depth of one lattice repeat is created by applying an appropriate set of forces to the end of the crystal. This creates an edge dislocation loop in the immediate vicinity of the depression. The loop, which can move through the.crysta1, is shown in Figure 1 near the center of the crystal block. A Burger's circuit that passes through the loop does not close by a vector that is equal to the depth of the
depression at the end of the crystal because there is an "extra" plane of atoms enclosed by the loop.
In metals the diameter of a dislocation loop is generally much larger than the unit cell. Very small dislocation loops are
The anisotropy of the interatomic forces in a polymer crystal favors the occurence and stability of defect loops that encircle a single chain. Figure 1 can be adapted to show a dispiration loop around a single polyethylene chain. This is shown in Figure 2. In this Figure, the edges of the crystal are seen to lie along the axes of the molecules at the corners of the orthorhombic unit ce11 of polyethylene. The dispiration loop encircles only the central chain. The end of the chain at the top of the crystal is depressed by one carbon atom which is half the unit ce11 repeat distance along the chain axis. If this were the only thing that had happened, the chain would not fit into the crystal lattice. It is necessary that the ribbon containing the planar zig-zag is also roteted by 1800. This is evident from the positions of the solid and dotted lines which mark the edges of the ribbon. The relative positions of the solid and dotted lines reversa below the dispiration loop which is shown near the center of the crystal.
To display the walk around a circuit that passes through the dispiration loop it is convenient to make the circuit in one of the planes of molecules that is perpendicular to the a axis of the
orthorhombic polyethylene crystal. Figure 3 shows such a plane after the molecules at the corners of the unit ce11 are removed. The
corners of the rectangles in this Figure show the positions of the molecules that were removed. In Figure 4 the view point is moved so
JOURNAL DE PHYSIQUE
Figure 5 is an enlarged region near the center of Figure 4 seen from nearly the same view point. The la,ttice "walker" starts with his head near a carbon atom in the right chain. The rule that provides a local reference for the direction the "walker" faces is that his right hand should be near the dotted edge of the ribbon and his left near the solid edge of the ribbon. The "walk" proceeds by moving down the chain in the c direction by a number of carbon atoms large enough to reach a position well below the dispiration loop. In this example the "walker" moves 11 carbon atoms or 5.5 unit cells and then steps back to the central chain without turning. He finds that he is still facing in the
FIGURE 5 WAM AROUND
MSPlRATION LOOP
FIGURE 7 BURGER~ URCUIT THROUGH A DISLOCATION LOOP
Figure 6 shows a side view of a dislocation in a chain lying in the (200) plane and Figure 7 shows the Burger's circuit that passes through the dislocation loop. Figure 8 shows a side views of a disclination in a chain lying in the (200) plane and Figure S'shows the Nabarro circuit that passes through-the disclination loop.
NABARRO~ URCUIT THROUGH A DlSCUNATlON LOOP
While the interna1 structure of these defects is interesting, and the ability to relate them to each other in general terms is
satisfying, the most valuable property remains to be discussed. Each defect can now be considered as a single entity with its own
characteristics. A dispiration in a polymer crystal can diffuse by Brownian motion, which is maintained by the thermal energy of the sample. As the defect advances it rotates and translates the parts of the chain along which it passes. This is a kind of scaling mode1 which greatly simplifies description of some properties of
JOURNAL DE PHYSIQUE
minimum energy conformation and motion o f each defect. In the
application of defects t o t h e interna1 friction and relaxation process in polyethylene that follows w e have used the powerful advantage gained from t h e introduction o f the defect and its careful description.
T H E STOCUASTIC DEFECT DIFFUSION MODEL
In the rernainder of t h e paper w e have concentrated on relaxation
processes that result from t h e motion o f dispirations. Three
processes are discussed in detail i n e ; dielectric relaxation in
lightly oxidized polyethylene containing one polar ketone group in
each polyethylene stem7, the disappearance o f certain satellite lines in t h e C-13 NMR spectrum of crystalline polyethylenes, and mechanical relaxation associated with t h e diffusion along t h e stem o f a
dispiration loops.
In each o f these three processes, t h e dispiration must diffuse along a particular route to contribute t o the observable relaxation. I n t h e dielectric relaxation t h e segment o f t h e chain bearing the polar group must be rotated about the chain axis by 1800 at a rate equal t o t h e frequency o f t h e dielectric relaxation peak. T h e rotation is accomplished when a dispiration arrives at the location
o f t h e t h e polar group. The problem then becomes one o f calculating
the average rate at which a dispiration starting at an arbitrary point along t h e stem makes its way, by a random walk, t o t h e position o f t h e polar group. Routes appropriate for this and other relaxation processes are described below.
E s ;R O 8
0
3 2 r5
i3 2 n 0-
0 O NUtlBER Of STEPSFigure 10 s h o w s t h e position along t h e chain o f a defect a s a function o f the number o f steps it has taken in a random walk in one dimension. Bach step carries t h e center of t h e dispiration by t h e distance, in t h e axial direction, from o n e carbon atom t o an adjacent
one. T h e direction o f t h e defect movement along t h e chain is random.
The s t e m length in Figure 10 is 100 carbon atoms. A dispiration,
starting at t h e fiftieth carbon atom, diffuses along t h e chain. Only certain aspects o f t h e route a r e important for relaxation processes. Different kinds of boundaries a r e used t o define these aspects and
the diffusion process is unconstrained except at these boundaries. A
reflecting boundary condition is used t o keep t h e dispiration in t h e part o f the stem of interest. An absorbing boundary condition is used t o terminate t h e random walk when t h e defect h a s completed its first
passage along t h e route. (Partially absorbing and reflecting
In Figure 10, the reflecting boundary at the zeroth carbon atom is encountered several times before the absorbing boundary at the hundredth carbon atom is encountered, and the graph ends. For the dielectric relaxation process, since the first passage is completed when the defect arrives at the polar group, in this graph the polar group would be at the hundredth carbon atom. Generally the dielectric route terminates in the interior of the stem. For the NMR
experiment, the route must turn every part of the chain at least once and terminate at one of the boundaries. In both examples the
boundaries must be moved and interchanged in order to include al1 possible routes. This procedure is described in6.
The time required for the defect to complete its route is called the "first passage time". The methods for calculating the first passage times using the Smoluchowski diffusion equation are described in detail in6 as are the methods for averaging over al1 possible defect starting positions. In the dielectric relaxation and the NMR experiment there is no significant driving force applied to the defect, so its motion can be satisfactorily characterized by a diffusion coefficient and the route that is required for a first passage. The average times are different for the two experiments because the routes are different, as explained in detail6. Choice of the correct routes brings the values of the diffusion coefficients determined from the two experiments into satisfactorily close agreement.
Now we describe a process that requires introduction of driving forces on the defect and which produces a mechanical relaxation with a frequency dependence similar to that of the alpha relaxation. This relaxation results from the diffusion of a dispiration in a bent lamellar crystal. Bending the lamellar crystal produces a strain field in which the region near one side of the lamella is compressed and the other is extended. Since a dispiration contains extra
material, its energy is increased in a compressed lattice and reduced in an extended lattice. For modest strains, the energy difference is commensurate with the thermal energy associated with motion of the defect and is sufficiently large to bias the motion. Therefore, in a suddenly bent crystal the randomly distributed defects tend to
diffuse toward the extended side of the lamella where, because of their tendency to expand the lattice, they cause a small increase in the bending for a constant applied bending moment. Estimates of the strength of this relaxation process, that is, the amount by which the modulus is reduced by the diffusion of dispirations, show that the effect is much smaller than the strength of the alpha relaxation observed in polyethylene. This leads to the suggestion that the times observed in the alpha relaxation are associated with the
JOURNAL
DE
PHYSIQUESUMMARY AND CONCLUSIONS
In this paper a correspondence is established between a
dispiration loop that encircles a single polymer chain and the more familiar features of dislocations and disclinations. The lowest energy defect loop in polyethylene is the dispiration which both rotates the chain segment along which it passes by one half revolution and translates it axially by one half a lattice repeat unit. The defect loops can be treated as objects with their own set of properties, which greatly simplifies the treatment of processes that involve motion of a polymer molecule in a crystal.
Interna1 friction processes and other relaxation processes that depend upon the diffusion of dispirations are discussed. Both the dielectric relaxation and the broadening of NMR satellite lines depend upon diffusion of dispirations along the polymer chain. Similar values of the diffusion coefficient are derived from the two sets of data by taking into account the different routes that the defect must diffuse.
The three defect loops are described here as they are manifested in polyethylene, but they can also occur in other cristalline
polymers. In polyethylene the higher energy defects, the disclination and the dislocation, probably play a role in mechanical deformation or in the adjustment of the long period4 of polyethylene fibers.
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3. 4. 5 . Physics, 6. 7. Physics 8."Anelastic and Dielectric Effects in Polymeric Solids", N. G.
B. E. Read and G. Williams, John Wiley and Sons, Ltd. London, D.
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H. Reneker, B.M.
Fanconi, and J. Mazur, J. Applied,
vol. 48, 4032(1977)D. H. Reneker and J. Mazur, Polymer, vol. 23, 401(1982) C. R. Ashcraft and R. H. Boyd, J. Polymer Science, Polymer Edition, vol. 14, 2153(1976)
