A New Approach to Model Thermal Expansion of Semi Crystalline Polymers E. RAMAKERS-VAN DORP1,2, T. Haenel1,2, J. STEINHAUS1, D. REITH3, O. BRUCH3,4, B. HAUSNEROVÁ2 and B. MOEGINGER1
Mod-108 Objectives
The Coefficient of Thermal Expansion (CTE) affects dimensional changes and warpage of extrusion blow molded parts. The aim of the study is to develop an new materials model of CTE for semi-crystalline HDPE to improve the FEM simulations of extrusion blow molded parts. Results and Discussion
The final properties of extrusion blow molded parts depend on the processing parameters such as melt temperature, mold temperature, extrusion speed and wall thickness. For bottles and containers shrinkage and warpage are important phenomena which affect geometry and performance. Thus they have to be taken into account in part design and process development. The amount of shrinkage and warpage depend on the CTE as FEM simulations show, Fig.1. The classical method is a linear description of the CTE which is only valid in a certain temperature range and for homogenous materials. In reality the CTE is a function which strongly depends on local crystallinity and molecule orientation. A description of the CTE based on the anisotropic morphology of semi-crystalline polymers does not exist yet.
The thermal expansion of the part is lower in ||-direction than in -direction and increases in a non-linear manner with temperature while crystallinity decreases, Fig. 2. Assuming mechanical coupling of
0,020 Model
60
250
50 dL/um_MD∥ dL/um_MD⊥ Xcr [%]
200
40
150
30
100
20
50
10
0
0 20
40
60
80
100
120
VP10 ∥
rel. Expansion
300
0,015
0,010
0,005
Crystallinity [%]
Expansion [µm]
Introduction
140
Temperature [°C]
Fig. 2: Measured thermal expansion || and to MD and the change in crystallinity
amorphous and crystalline phase, thermal expansion of a semi-crystalline polymer has to depend on crystallinity, orientation as well as stiffness and CTE of both phases.
0,000 20
40
60
80
100
120
140
Temperature [°C]
Fig. 3: Comparison of measured (red) and calculated (blue) thermal expansion || to MD
The calculated thermal expansion || and to MD is in good agreement with the measured ones until melting starts at 110°C, Fig. 4. This deviation can be explained by the fact that the stiffening effect of the crystalline phase decreases below crystallinities of 30%. A minimal crystallinity seems to be required which can be estimated by the chain lengths of olefins melting around 100°C. 0,020
α(T) = α(T,xcr,Ecr||,Ecr,Eam,αcr||,αcr,αam)
Model
The calculation of this anisotropic two phase model yields for the CTE || MD: 𝛂∥ 𝐓 =
Fig. 1: FEM-Simulation of shrinkage and warpage 𝟏 + 𝛂𝐜⊥,𝐲 ∗ 𝚫𝐓 𝟏− 𝟏 + 𝛂𝐚,𝐲 ∗ 𝚫𝐓 𝟏 on a blow molded fuel tank 𝛂∥ 𝐓 = ∗ 𝚫𝐓
𝟏 ∗ 𝚫𝐓
𝟏−
𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓 (𝟏 + 𝛂𝐜⊥,𝐲 ∗ 𝚫𝐓) 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 𝟏+ ∗ ∗ (𝟏 + 𝛂𝐚,𝐲 ∗ 𝚫𝐓) 𝐄𝐚𝐦 𝐓 ∗ (𝟏 − 𝐱𝐜𝐫 ) 𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓 𝟏−
𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓 (𝟏 + 𝛂𝐜⊥,𝐲 ∗ 𝚫𝐓) 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 𝟏+ ∗ ∗ (𝟏 + 𝛂𝐚,𝐲 ∗ 𝚫𝐓) 𝐄𝐚𝐦 𝐓 ∗ (𝟏 − 𝐱𝐜𝐫 ) 𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓
∗ 𝟏+
𝟏 + 𝛂𝐜⊥,𝐲 ∗ 𝚫𝐓 𝟏 + 𝛂𝐚,𝐲 ∗ 𝚫𝐓
𝟏 + 𝛂𝐜⊥,𝐱 ∗ 𝚫𝐓 𝟏 + 𝛂𝐚,𝐱 ∗ 𝚫𝐓
𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓 𝟏 + 𝛂𝐜⊥,𝐱 ∗ 𝚫𝐓 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 𝟏+ ∗ ∗ 𝐄𝐚𝐦 𝐓 ∗ 𝟏 − 𝐱𝐜𝐫 𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓 𝟏 + 𝛂𝐚,𝐱 ∗ 𝚫𝐓
rel. Expansion
VP10 ⊥ 0,015
𝟏 + 𝛂𝐜⊥,𝐱 ∗ 𝚫𝐓 𝟏 −0,010 𝟏 + 𝛂𝐚,𝐱 ∗ 𝚫𝐓
∗ 𝟏+ 𝟏+
𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓 𝟏 + 𝛂𝐜⊥,𝐱 ∗ 𝚫𝐓 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 ∗ ∗ 𝐄𝐚𝐦 𝐓 ∗ 𝟏 − 𝐱𝐜𝐫 𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓 𝟏 + 𝛂𝐚,𝐱 ∗ 𝚫𝐓
∗ 𝐱𝐜𝐫 ∗ (𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓)
0,005
∗ 𝐱𝐜𝐫 ∗ (𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓)
0,000
The aim of the study is to develop a new model to describe thermal expansion of parts based on CTE of amorphous and 𝟏 + 𝛂 ∗ 𝚫𝐓 𝐄 𝐓 ∗𝐱 ∗ ∗ 𝚫𝐓 ∗ (𝛂 −𝛂 𝐓 ∗ (𝟏as − 𝐱 ) crystallinity. 𝟏 as 𝐄well 𝟏 + 𝛂 ∗ 𝚫𝐓 crystalline phase + ∗ 𝐜𝐫,⊥
𝐚𝐦
𝚫𝐓
𝟏+
𝐜∥,𝐳
𝐜𝐫
𝐜𝐫
𝐚,𝐳
𝐜⊥,𝐲
𝟏 + ∗ 𝚫𝐓
𝐚,𝐲 )
𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓 (𝟏 + 𝛂𝐜⊥,𝐲 ∗ 𝚫𝐓) 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 ∗ ∗ (𝟏 + 𝛂𝐚,𝐲 ∗ 𝚫𝐓) 𝐄𝐚𝐦 𝐓 ∗ (𝟏 − 𝐱𝐜𝐫 ) 𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓
Materials and Methods
∗ 𝟏+
60 80 100 𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓 𝟏 +20 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓 40 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 ∗ ∗ 𝚫𝐓 ∗ (𝛂𝐜⊥,𝐲 − 𝛂𝐚,𝐲 ) ∗ ∗ 𝚫𝐓 ∗ 𝛂𝐜⊥,𝐱 − 𝛂𝐚,𝐱 𝐄𝐚𝐦 𝐓 ∗ (𝟏 − 𝐱𝐜𝐫 ) 𝐄𝐚𝐦 𝐓 ∗ 𝟏 − 𝐱𝐜𝐫 𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓 𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓 Temperature [°C] ∗ 𝟏+ ∗ 𝟏 − 𝐱𝐜𝐫 ∗ (𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓) 𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓 (𝟏 + 𝛂𝐜⊥,𝐲 ∗ 𝚫𝐓) 𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓 𝟏 + 𝛂𝐜⊥,𝐱 ∗ 𝚫𝐓 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 𝟏+ ∗ ∗ 𝟏+ ∗ ∗ (𝟏 + 𝛂𝐚,𝐲 ∗ 𝚫𝐓) 𝐄𝐚𝐦 𝐓 ∗ (𝟏 − 𝐱𝐜𝐫 ) 𝐄𝐚𝐦 𝐓 ∗ 𝟏 − 𝐱𝐜𝐫 𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓 𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓 𝟏 + 𝛂𝐚,𝐱 ∗ 𝚫𝐓
𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 ∗ ∗ 𝚫𝐓 ∗ 𝛂𝐜⊥,𝐱 − 𝛂𝐚,𝐱 𝐄𝐚𝐦 𝐓 ∗ 𝟏 − 𝐱𝐜𝐫 𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓 𝟏+
𝟏 + 𝛂𝐜∥,𝐳 ∗ 𝚫𝐓 𝟏 + 𝛂𝐜⊥,𝐱 ∗ 𝚫𝐓 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 ∗ ∗ 𝐄𝐚𝐦 𝐓 ∗ 𝟏 − 𝐱𝐜𝐫 𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓 𝟏 + 𝛂𝐚,𝐱 ∗ 𝚫𝐓
∗ 𝟏 − 𝐱𝐜𝐫 ∗ (𝟏 + 𝛂𝐚,𝐳 ∗ 𝚫𝐓)
Material: HDPE Lupolen 4261 AG from LyondellBasell. Samples were taken out parallel () and perpendicular () to the extrusion direction (MD). The thermal expansion was measured with a Netzsch DMA 242 C in TMA mode with heat rate of 5 K/min between 25°C to 150°C and an initial load: 2 N. The crystallinity was measured with a Perkin Elmer DSC 8000 with heat rate of 20 K/min from -40°C to 180°C in N2-gas.
Introducing values taken from literature the CTE-model in MD reproduces well the measured thermal expansion || to MD over the total temperature range, Fig. 3.
References
Affiliations
[1] [2] [3] [4] [5]
C. L. Choy et al., J. Polymer Science, Vol. 22 (1984), 835-846 H.-G. Elias, Polymere, 1996 C. L. Choy et al., J. Polymer Science, Vol 22 (1984), 979-991 J. L. Kardos et al., Polym Eng. Sci., Vol 19 (1979), 1000-1009 N. J. Capiati et al., J. Polymer Science, Vol 15 (1977), 1427-1434
140
Fig. 4: Comparison of measured (red ) and calculated (blue) thermal expansion to MD
CTE MD: 𝟏 𝛂⊥ 𝐓 = ∗ 𝚫𝐓
120
𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 𝟏 + 𝛂𝐜∥ ∗ 𝚫𝐓 𝟏 + 𝛂𝐚 ∗ 𝚫𝐓 + ∗ ∗ (𝟏 + 𝛂𝐜⊥ ∗ 𝚫𝐓) 𝐄𝐚𝐦 𝐓 ∗ 𝟏 − 𝐱𝐜𝐫 𝟏 + 𝛂𝐚 ∗ 𝚫𝐓 𝐄𝐜𝐫,⊥ 𝐓 ∗ 𝐱𝐜𝐫 𝟏 + 𝛂𝐜∥ ∗ 𝚫𝐓 𝟏+ ∗ 𝐄𝐚𝐦 𝐓 ∗ 𝟏 − 𝐱𝐜𝐫 𝟏 + 𝛂𝐚 ∗ 𝚫𝐓
−𝟏
αcrǁ,x,y,z= CTE crystalline phase || to MD (-10*10-6K-1) [1,5] αcr⊥,x,y,z= CTE crystalline phase to MD (130*10-6K-1) [1] αa,x,y,z= CTE amorphous phase (300*10-6K-1) [2,3] Ecr⊥(T)= E-Modulus crystalline phase to MD (3400MPa) [2] Eam(T)= E-Modulus amorphous phase (10MPa) [3,5] xcr= Crystallinity
Conclusions This study shows that the new approach to model of CTE || and to MD yields a good correlation to the experimental results. This provides the opportunity to predict the shrinkage and warpage of the final products by FEM if the model is more refined. Acknowledgements This study was supported by the German Ministry of Education and Research. Grant No.: 03FH051PX4. The author also thanks the Graduates Institute, Bonn-Rhine-Sieg University of Applied Sciences for supporting this work by granting a scholarship.
Contact
e-mail:
[email protected]
1. Department of Natural Science, Bonn-Rhein-Sieg University of Applied Sciences, Germany 2. Faculty of Technology, Centre of Polymer Systems, Tomas Bata University in Zlín, Czech Republic 3. Department of Electrical/Mechanical Engineering and Technical Journalism, Bonn-Rhein-Sieg University of Applied Sciences, Germany 4. Dr. Reinhold-Hagen Stiftung, Bonn, Germany
