Thermal stresses and birefringence in quenched cylinders

18 February 2011
Avraam I. Isayev and Antonio J. Carillo
During quenching from melt to solid state, the contribution of thermal stresses to birefringence is minimal for polystyrenes and significant for polycarbonates.

Thermal stresses and birefringence in polymeric materials are caused by the combined effects of nonequilibrium density changes and visco-elastic polymer behavior during cooling.1,2 They play an important role in determining the final product properties.3,4 Increasing demand for polymeric products for special applications, such as for optical5 and medical6 devices (where low levels of birefringence and residual stresses are desirable), has triggered significant interest in residual birefringence and stresses.

Thermal residual stresses caused by rapid temperature changes were first studied in inorganic glasses,7–9 but volume relaxation during cooling was not considered.10 Several techniques can be applied to measure residual stresses in polymeric materials.2 The layer-removal method is most commonly used,11 but it encounters some difficulties as regards application to polymers,12 in particular in measuring the curvature of the removed layer. An alternative method uses birefringence. However, correlating residual birefringence and stress is not straightforward, since birefringence depends on the thermal history and not just on the final state of stress. Nevertheless, the relative ease with which birefringence can be measured in transparent polymers validates our efforts to attempt a quantitative determination.

We fabricated rods with lengths and diameters of 60 and 10mm, respectively, from polystyrene (PS: 615 APR 26-W/Dow) and polycarbonate (PC: Lexan 123/GE). To obtain tubular samples, we axially drilled the rods to a diameter of 5mm. To eliminate the residual birefringence introduced during sample preparation, we annealed the samples above the glass-transition temperature (Tg), followed by slow cooling. Subsequently, we heated the samples to above Tg and quenched them in water. We measured the birefringence distribution, Δn and nθθnrr, where (z, r, θ) represent the axial, radial, and circumferential directions, respectively, using a polarized optical microscope and a compensator.

Figure 1 shows the simulated radial, circumferential, and axial transient thermal stresses for a PS tube. For these calculations, we adopted the visco-elastic material properties listed in Table 1, as well as Young's modulus and the strain-optical-coefficient master curves.13,14 At 0.35s, the outer layers were in tension, while the inner layers were compressed. When the temperature near the surface dropped below Tg, the polymer solidified and its contraction rate decreased. The high temperature near the inner wall allowed the stresses to relax faster than near the outer wall, where the temperature is low. As cooling progressed, each layer compressed. Layers near the outer wall already solidified, with compressive and tensile axial stresses near the outer and inner walls, respectively. During solidification, the stresses increased throughout the tube's wall until the temperature became homogeneous. Subsequently, volume relaxation caused the stresses to partially relax: see Figure 1(a), (b), and (c) for a time of 2×105s. They did not relax completely, since long-range molecular motion is hindered at low temperatures.

Simulated (a) radial (σrr), (b) circumferential (σθθ), and (c) axial (σzz) transient thermal stresses for a polystyrene (PS) tube quenched in water from 120 to 25°C, as a function of normalized radius. R: Maximum radius.

Figure 2 shows the simulated transient birefringence, Δn, for a PS tube and rod quenched from 120 to 25°C. At the onset of quenching, Δn is negative near the outer wall and positive in an extensive region toward the inner wall, since the thermal stresses are tensile at the surface and compressive near the inner wall. As cooling progresses, the thermal stresses become compressive at the surface and tensile near the inner wall. Therefore, the birefringence sign changes: Δn becomes positive near the outer wall and negative near the inner wall.

Simulated, transient thermally induced birefringence for (a) a PS tube and (b) rod, both quenched in water from 120 to 25°C.

Figures 3 and 4 show the measured and simulated residual birefringence Δn and nrrnθθ for quenched tubes and rods. For PS samples, both quantities were positive near the outer wall. They decreased and passed through zero, to negative values, in the core region. The location where the birefringence changed from negative to positive values shifted toward the outer wall with increasing initial temperature. For PC samples, Δn and nrrnθθ were negative near the outer wall. They increased, passing through zero to positive values. The location where the birefringence changed from negative to positive values also shifted toward the outer wall.

Simulated (lines) and measured (symbols) thermally induced residual birefringence Δn and nrrnθθ for PS tubes quenched to 25° from (a) 120 and (b) 135°C, and PS rods quenched to 25° from (c) 120 and (d) 135°C.

Physical properties of PS used in our simulations. βl, βg: Thermal expansion coefficients in rubbery, glassy states. α: Thermal diffusivity. WLF: Williams-Landel-Ferry. C1, C2: Constants. Tr, T2: Reference, variable temperature. τr: Relaxation time.

α (m2/s)6.13×10−87.31×10−8
Poisson ratio0.330.41
WLF equation:
C2 (K)57.040.2
Tr (K)370.0420.5
T2 (K)313.0380.0
τr(s)0.04 at 97°C0.3s at 147°C

Simulated (lines) and measured (symbols) thermally induced residual birefringence for PC tubes quenched to 25°C from (a) 170 and (b) 180°C, and PC rods quenched to 25°C from (c) 170 and (d) 180°C.

In summary, we developed a theoretical and visco-elastic scheme15 to calculate the thermally induced residual stresses and birefringence in quenched tubes and rods. Our birefringence measurements and simulations showed that the magnitudes of the various components increase with initial temperature. We also found that the contribution of the thermally induced stresses to the birefringence of PS and PC samples was, respectively, minimal and significant. We will next extend our visco-elastic approach for calculation of thermally induced birefringence and stresses to 2D and 3D heat-transfer configurations, similar to those generally found in polymer processing. We will also aim at obtaining the overall frozen-in birefringences and stresses developed in manufactured polymer products. This will be a very important step for forecasting their mechanical and optical performance.


Avraam I. Isayev
Department of Polymer Engineering, The University of Akron

Avraam Isayev obtained an MSc in chemical engineering from the Azerbaijan Institute of Oil and Chemistry, a second MSc in applied mathematics from the Institute of Electronic Machine Building in Moscow (Russia), and a PhD in polymer engineering and science from the Moscow Institute of Petrochemical Synthesis of the Russian Academy of Sciences. He is currently a Distinguished Professor.

Antonio J. Carillo
Institute of Polymer Engineering, The University of Akron

Antonio Carillo obtained his BSc and MSc degrees in mechanical engineering from the Institute of Technology in Mérida and the Research Center for Advanced Materials in Chihuahua (Mexico), respectively, and his PhD in polymer engineering from the University of Akron. Presently, he is research and development scientist at Americhem Inc. in Akron.


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DOI:  10.2417/spepro.003420