Thermodynamic Aspects of Brittleness in Glassy Polymers

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1 1 Thermodynamic Aspects of Brittleness in Glassy Polymers Downloaded via on June 19, 2018 at 21:22:40 (UTC). See for options on how to legitimately share published articles. S. MATSUOKA Bell Laboratories, Murray Hill, N. J A shortening in relaxation time in the critically strained region makes some materials tough. The shift of relaxation time is attributed to strain-induced dilatation and can reach as much as five decades. Thermal history, on the other hand, dictates the initial state from which this dilatation starts and may be expressed in terms of excess entropy and enthalpy. The excess enthalpy at T g is measurable by differential scanning calorimetry. Brittle to ductile transition behavior is determined by the strain-induced reduction in relaxation time, the initial amount of excess entropy, and the maximum elastic strain that the material can undergo without fracturing or crazing. T^ailure properties are dictated by events in the limited local regions and are more difficult to handle and to interpret molecularly than average properties such as modulus, viscosity, and heat capacity. We propose shortening of relaxation times in material in the critical regions as a mechanism responsible for the toughness of solid plastics. The stress tends to dilate most solids, and the relaxation time can be shortened by several orders of magnitude by this type of dilatation. This process is quantitatively describable in terms of excess entropy. The initial amount of the excess entropy before stress is a function of thermal and mechanical history, e.g. annealing and drawing. How much the relaxation time can be shortened is determined by the maximum strain attainable in the material before fracture or craze formation occurs. The shortening of the relaxation time is a local plasticization process, and the mechanical energy is more readily dissipated. 3

2 4 TOUGHNESS A N D BRITTLENESS OF PLASTICS Strain Energy as the Strength Criterion In plastics the strength of a material is often unrelated to the average properties such as the modulus. ABS resins, which are essentially rubberreinforced SAN resins, are stronger in most applications than the unmodified SAN resins, even though both the modulus and the elastic stress limit for SAN are higher than that for ABS. Berry (1) noted that much of the mechanical energy required to propagate a crack through the bulk of a material is dissipated during plastics deformation near the tip of the crack; this energy can be several orders of magnitude greater than the actual surface energy for separating the two surfaces. Thus plastic deformation in microscopic scale is perhaps the most important factor in making a material tough. Tensile elongation indicates the ability of a material to deform before breaking; it is a more important design factor in choosing a proper material than many of the average properties referred to above. An impact modifier is a rubber phase dispersed in particulate form throughout the matrix of a polymer solid. Unlike plasticizers, the rubber particles retain their intrinsic properties as a separate phase. The glass transition temperature of the parent matrix is not lowered by the addition of an impact modifier. The rubber particles do two things to the parent matrix phase (2,3,4): they act as stress concentrators (i.e., a large strain will start in the matrix near the interface) and they enhance the multiaxiality in stress. As multiaxial tensile strength near the interface further enhances dilatation, which shortens the mechanical relaxation time, the otherwise brittle polymer solid of the matrix will undergo plastic deformation in the vicinities of the rubber particles. Shift of Relaxation Time under Strain The concept of stress-induced dilatation affecting the relaxation time or rate has been suggested by others (5, 6, 7, 8). The density of most solids decreases under uniaxial stress because the lateral contraction of the solid body does not quite compensate for the longitudinal extension in the direction of the stress, and the body expands. The Poisson ratio, the ratio of such contraction to the extension, is about 0.35 for many polymeric solids; it would be 0.5 if no change in density occurred, as in an ideal rubber. The volume increase, AV, accompanying the tensile strain of c, can be described by the following equation: AV/V = e(l-2m), (1) where V is the specific volue, and /x is the Poisson ratio (8). The volume increases linearly with the strain up to the elastic limit according to this

3 1. M A T S U O K A Thermodynamic Aspects 5 equation. However beyond this limit the body either yields, crazes, or fractures, so that deformation beyond this point no longer causes an additional dilatation. The maximum amount of such a strain is about 3 X 10" 2 for many polymers. The corresponding fractional volume increase, AV/V, is about 10" 3 for many polymer solids. The WLF equation is an empirical expression for the shift of relaxation time with respect to the temperature change and applies to rubbery materials. It is well known (JO) that the WLF equation can be written as a function of the fractional free volume only: In a T = B(l//-l//g), (2) where a T is the normalized relaxation time; B is a constant and is usually unity; / is the fractional free volume, and the g is the glass transition temperature. The fractional free volume reaches 1/40 at vitrification where the remaining free volume no longer contributes to the relaxation process and presumably remains constant below Tg. If an increase in free volume arising from stress-induced dilatation contributes to the relaxation process in the same manner as dilatation by raising the temperature, we can estimate the shift in relaxation time with Equation 2 by substituting for the fractional free volume, /, / = f e + AV/V = f g + e(l-2/i). 0) (See Curve A in Figure 1.) The maximum possible shift in relaxation time is determined by the limiting elastic strain, beyond which dilatation no longer continues. Effect of Annealing Well-annealed polymeric solids tend to be stiffer and more brittle than unannealed solids. For crystalline polymers annealing increases the degree of crystallinity and crystallite sizes (lamellar thickening). These changes are defined and measured thermodynamically as the decrease in enthalpy and entropy. Annealing of glassy polymers can be thermodynamically treated in a similar way, even though it does not involve crystalline phase transition (11, 12). Annealing a polymer glass will lower the enthalpy. This decrease is approximately equal to the product of the temperature and the corresponding change in entropy since the Gibbs free energy would remain essentially unchanged, as in the case of crystallization of the super-cooled liquid. This excess entropy is associated with the free volume as it becomes available at the glass-to-rubber transition by heating

4 6 TOUGHNESS A N D BRITTLENESS OF PLASTICS or by mechanical dilatation. Thus entropy of mixing of the occupied and the unoccupied sites is AS = Rirtx In vi + n2 In #2), (4) where n is the mole fraction; v is the volumetric fraction, and the subscripts 1 and 2 refer to the occupied and unoccupied sites. When the concentration of the unoccupied sites is much smaller than that of the occupied sites, Equation 4 is simplified to the form AS = -Rf In /, (5) where / is the fractional free volume. We stated before that the unoccupied sites in the glass become available as free volume, fg, at the point of the glass-to-rubber transition. If the glass is subjected to a thermal history which decreases its excess entropy (annealing), the free volume fraction might be less than the average value, f g, for the quenched glass. Differential scanning calorimetry experiments (10, 11) show that a well-annealed glass would have greater endotherm at the glass-to-rubber transition and conversely would require a greater strain to gain a given amount of fractional free volume by dilatation. A number of curves to account for the different thermal history are shown in Figure 1. These additional curves were calculated for materials with different degrees of \ 1 r \ 0.15 CAL/g o -6 _L_ STRAIN Figure 1. The effect of strain-induced dilatation on relaxation time

5 1. M A T S U O K A Thermodynamic Aspects 7 annealing. The initial excess enthalpy is a measurable quantity and decreases with annealing. The points are from the experimental data of Ref. 13. The smaller the excess entropy initially, the greater the strain required before the relaxation time begins to shorten. It follows that annealing impedes a strain-induced plasticization. We have stated that the maximum attainable dilatation is determined by the hmiting magnitude of strain; anything above that would cause cracks or crazes. If a sample is well annealed, this maximum strain is reached before a sufficient shift in relaxation time has occurred (see Figure 1). The material would not be able to dissipate much strain energy and could fail in brittle manner. How much of a shift in relaxation time is required for the tough behavior depends on the rate of straining pertinent to the particular application or testing method in question. Assuming the average relaxation time at / = f g is 1 sec, a shift of two to three decades corresponds to such a requirement. Our data given here for ABS resins hold for other polymers as well since many exhibit similar values for fg, the Poisson ratio, and the limiting stress. For example, the shift of four decades in PVC creep retardation time after annealing was obtained by Turner (13). The maximum excess entropy at T g is similar for many polymers and some inorganic glasses (14). Literature Cited 1. Berry, J. P., J. Appl. Phys. (1963) 34, Wang, T. T., Matsuo, M., Kwei, T. K., J. Appl. Phys. (1971) 42, Wang, T. T., Matsuo, M., Kwei, T. K., J. Polym. Sci. Part A-2 (1972) 10, Matsuoka, S., Daane, J. H., ACS Polymer Preprints 10-2, 1198 (1969). 5. Ferry, Stratton, Kolloid-Z. (1960) 171, Sternstein, J. Macromol. Sci. Phys. (1973) 8, Kovacs, Stratton, Ferry, J. Phys. Chem. (1963) 67, Struik, Rheol. Acta (1966) 5, Daane, J. H., Matsuoka, S., Polym. Eng. Sci. (1968) 8 (4), Ferry, J. D., "Viscoelastic Properties of Polymers," Chap. II, Wiley, New York, Petrie, S. E. B., J. Polym. Sci. Part A-2 (1972) 10, Matsuoka, S., Aloisio,C.J.,Bair, H. E., J. Appl. Phys. (1973) 44, Turner, S., Brit. Plast. (December, 1964), p Matsuoka, S., Bair, H. E., Bull. Amer. Phys. Soc. (1974) 19 (3), 239. RECEIVED October 18, 1974.