AN INVESTIGATION OF THE MELTING TEMPERATURE EFFECT ON THE RATE OF SOLIDIFICATION IN POLYMER USING A MODIFIED PHASE FIELD MODEL

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1 International Journal of Technology (7) 7: 3-38 ISS IJTech 7 A IVESTIGATIO OF THE MELTIG TEMPERATURE EFFECT O THE RATE OF SOLIDIFICATIO I POLYMER USIG A MODIFIED PHASE FIELD MODEL Mochaad Chalid, Arbi Irsyad Fikri, Hanindito Haidar Satrio, Muhaad Joshua Y.B., Jaka Fajar Fatriansyah,* Departent of Metallurgy and Materials Engineering, Faculty of Engineering, Universitas Indonesia, Kapus UI Depok, Depok 644, Indonesia (Received: June 7 / Revised: July 7 / Accepted: oveber 7) ABSTRACT A phase field odel has been successfully constructed to siulate the behavior of the seicrystalline polyer solidification phenoenon. It is a odel that has been widely and successfully utilized to siulate solidification phenoena in etals. However, the non-conserved phase field equation can be extended to include unique polyer paraeters that do not exist in etals; for exaple, polyer elt viscosity and the diffusion coefficient. In order to extend this odel, we incorporate free energy density and non-local free energy density based on the Harrowell-Oxtoby and Ginzburg-Landau theores for polyer. By using the expansion principle for the higher order of binary phase field paraeter, a full odified phase field equation can be obtained. The solidification phenoenon in polyer is very iportant to optiize the final properties of the products. Here, we use our odified equation to investigate the effect of elting teperature on the rate of solidification. It was found that the rate of solidification is correlated with elting teperature in a non-straightforward anner. Keywords: Phase field; Polyer; Polyer product; Solidification. ITRODUCTIO The deand for geoetrically coplex plastic products has been rising over the past years, and this trend ay continue for a further years. This preference for the use of plastics is partially due to their light weight (Corinaldesi et al., 5) and the relative ease with which they can be anufactured (Atzeni et al., ) in coparison to other aterials. In order to produce geoetrically coplex plastic products, it is very iportant to control the solidification and orphology of the elted plastic in both the injection/extrusion achine and old. To control the solidification and orphology of elted plastic, the basic echanis by which elted plastic becoes solid should be well understood. The basic echanis of plastic solidification involves a connection between coplicated conservation and transport laws as well as the icro phase transforation phenoena that govern the plastic/polyer elt flow and solidification/crystallization. However, it is very difficult to describe these connections analytically. Advances in coputational odeling can address this difficulty and enhance the prediction of solidification and orphology during the product anufacturing process. Solidification in polyers differs fro that in etals due to the higher elt viscosity and lower theral properties (such as elting teperature). The high elt viscosity, low elting teperature, and partial crystallization characteristics of polyers all have a ajor effect on * Corresponding author s eail: fajar@etal.ui.ac.id, Tel: , Fax: Peralink/DOI:

2 3 An Investigation of the Melting Teperature Effect on the Rate of Solidification in Polyer using a Modified Phase Field Model solidification, and orphological evolution in the anufacturing process of polyer products. Furtherore, the coplicated kinetics involved in the solidification of polyers due to their olecular chain folding adds further difficulty to the task of physically describing the, when copared to etals. Various techniques have been used to odel solidification in polyers. Lovinger et al. described single crystal growth fro Poly (trifluoroethylene) in the elted phase using a diffusion echanis explanation (Lovinger & Cais, 984). Micheletti and Burger () exained the non-isotheral solidification of polyers using a stochastic birth-and-growth-based algorith. Raabe and Godara (5) exained the kinetics and topology of polyer solidification using a three-diensional cellular autoaton odel. Xu et al. (5) discussed the solidification of isotactic polystyrene single crystals using the phase field ethod. In this paper, intrigued by the siplicity of the phase field odel, we odify a phase field odel that has been established in etal counterparts and calculate the solidification rate in polystyrene. Phase field theory has been used in diverse probles involving the icrostructural evolution of aterials. Phase field theory has been successfully eployed to calculate the otion of interfaces and phase boundaries without explicitly tracking those interfaces (Warren et al., 3); for exaple, it has been successfully used to odel solidification in pure aterials/copounds (Collins & Levine, 985) and alloys (Wheeler et al., 99). Warren et al. developed a twodiensional phase field odel of grain boundary statics and dynaics for polycrystalline aterials (Warren et al., 3). In this paper, we extend the phase field odel and siulate solidification in sei-crystalline polyer under the effect of elting teperature using phase field theory.. EXPERIMETAL.. Landau Theory To describe the Ginzburg-Landau (GL) theory, it is convenient to begin with the Ising odel of agnetis (Provatas & Elder, 5). According to this odel, the energy of a second icroscopic syste can be described in ters of a collection of agnetic spins. The syste has a doain of atos, each of which carries a agnetic spin with the value s i = ± that shows whether an ato s agnetic oent is pointing up or down. The energy of the syste can be written as where J and B, respectively, are the coupling constant or pure energy that adheres to each spin and its interactions, and the external agnetic field (Ruderan & Kittel, 954; Kittel, 5). The first ter of the equation sus up all of the energies that are caused by the interaction of each spin ( i ) with all other adjacent spins ( j ), while the second ter adds the energy due to the interaction of each spin with an externally iposed agnetic field (B). This agnetic field in soe cases can also be considered as the cheical potential of the syste. and v Esi Jsi s j Bsi i j i () J K, (.b) B h kt B, (.c). (.d)

3 Fatriansyah et al. 33 According to the Landau theory, the partition function of the syste can be forulated fro the Ising odel of agnetis. The partition function of the syste is given by. (3) The average agnetization ( ) of the syste can be considered as the order paraeter Ψ, which is defined as (Ruderan &Kittel, 954) s s. (4) Using Equation 3, we can copute the free energy function as follows, (5) v F = k B T(J i s i s j ) k B T (B i s i ). (6) The free energy, F, is called the Landau free energy. We can obtain the free energy of each spin by dividing Equation 5 by the nuber of spins, (Kittel, 5): j. (7) Fro here, we can use the partition function to calculate the free energy per spin by introducing a new ter:. (8) where F is the free energy function derived fro the partition function. Substituting this into Equation 6 yields: (, ) ft (, T) F T fi fe(, T). (9) If we consider the jup to be sall, then f e (, T) can be expanded using a Taylor series. However, f e (, T) is an even function, thus we only need to consider the even ters. Adding this back into our equation, we obtain f t Z J e si v sis j B si i j i. () '' '''' 4 F(, T) fe fe ft (, T) fi fe, T.... () 4 In our paper, we are interested only in the perturbation or noise that occurs in the syste. Thus, we can oit the f i + f e (, T ) ter. One thing that needs to be considered is that first-order transitions typically occur along phases of different topological syetry. For instance, there will be soe broken syetry. Correspondingly, a first-order transition terinates at a critical point, i.e., the point at which two co-existing phases erge into one. We can break this syetry by adding to the equation a cubic-order ter with a negative sign. B i i F k T ln Z i v i j i F kbt J sis j kbt B si f kbt F '' '''' 4 fe fe fe, T fe, T... 4

4 34 An Investigation of the Melting Teperature Effect on the Rate of Solidification in Polyer using a Modified Phase Field Model F, T f f f ft, T fi fe, T 3 4. ().. Ginzburg-Landau Free Functional Energy According to the GL theory, the interaction energy between eleents can be assued to be spatially dependent, and it varies between any two eleents. Let ϵ be the separation between two eleents (i and j), then the ean internal energy U can be defined as (Ginzburg & Landau, 95): U ij ` xi x j i j ò. (3) i ji '' ''' 3 '''' 4 e e e Equation 3 can be elaborated using an algebraic identity, / i j i j i j i s. (4) Assuing that ε ij is negligible for any j > v and substituting Equation 3 back into equation U, we obtain: U ò ij i j j i ij 4 ò. (5) i ji i ji We can add another assuption to siplify the equation, which is the interaction energy per particle ε ij approaching ε ij. ε v i is the isotropic ean energy over. 4 4 a i R i T i L i B i j (6) ji a a a a 4 i j a xi (7) ji F, T Wo f x, T x (8) V.3. Phase Field Model A phase field odel describes phase transforation using a crystal order paraeter. The crystal order paraeter distinguishes the two phases of liquid and solid in ters of. is defined as and for liquid and solid respectively. The solidification process can be described as a continuous increase in the ψ value fro zero to one. The solidification process of polyers can be odeled based on GL theory based on the fact that GL theory incorporates the change in free energy on the boundary to explain the transforation of phase. This theory explains the solidification of polyer under the effect of decreasing teperature and distance fro the nuclei. In this theory, one needs to define the anisotropy coefficient to induce solidification. In etal solidification, the anisotropy coefficient (ε) will be greater than zero. However, in polyer solidification, the anisotropy coefficient (ε) is zero. When the value is greater than zero, the icrostructure of the aterial becoes dendritic, while a zero value leads to the icrostructure becoing spherulitic. In this section, we derive phase field theory fro the definition of free energy. A phase field odel begins with the definition of free energy. A classical phase field odel of the solidification of pure aterial/copound has free energy that can be written as follows:

5 Fatriansyah et al. 35, F f T dv local local,, f T f T dv grad f, T dv. local The free energy in the syste basically consists of local free energy flocal, T free energy f, T. This non-local free energy f, T grad grad (9) and non-local can be represented using a gradient ter. can be interpreted by phase, but here let us generally interpret as the crystal order paraeter. The evolution of the crystal order paraeter can be described as a standard GL approach: r, t F t r, t where is the obility coefficient. This coefficient is defined as inversely proportional to the elt viscosity. To accoodate the dynaics of the interfacial surface, obility is introduced as a function of the order paraeter. In the case of the dynaic of the crystal (solid)-liquid interface, the functional for of () would be ore reasonable (Harrowel & Oxtoby, 987). They derived the one-diensional odel of the crystal-liquid interface where the tie and space evolution of phase can be written as follows: d df d K dt d dx where K is the kinetics. If coupling of the otions between different tiescales is weak, then the interface between the two paraeters and can be written as: d v d dx dx d () d f f,, (.a) d, d v d f dx dx d. (.b) The siplest for of f and related f can be written as 3 4, f a a (3.a) f a a (3.b) Equation 5.b is the functional for of free energy that is used in this paper. For polyer, Equation 5.b can be transfored to:, (4) f, local T W where W is the height of the energy barrier of nucleation, is the unstable energy barrier, and is the stable solidification potential. In our calculation, we used three polyers/plastics: polypropylene (PP), polyethylene (PE), and polystyrene (PS) with the cheical structure shown

6 36 An Investigation of the Melting Teperature Effect on the Rate of Solidification in Polyer using a Modified Phase Field Model in Figure and the physical properties shown in Table. According to (Xu et al., 5), one can substitute Equations and into Equation 9 to obtain: ( rt, ) Γ( W ψ ψ). (5) t Assuing the process occurred in one diension and was observed fro a oving frae reference with a unifor velocity of υ = ψ/ t, Equation 5 can be re-written as: d d f dx Γdx. (6) According to (Harrowel & Oxtoby, 987), by setting the boundary condition of ψ ζ as x and ψ as x + ), we obtained ψz. (7) W exp( z Figure The cheical structure of: (a) polypropylene; (b) polystyrene; and (c) polyethylene Table The therophysical paraeters of polyers (elting teperature and crystallization teperature) Polyer T ( C) T ( C) T c ( C) PS PE PP METHODOLOGY For each polyer, we input a set of different ν values into Equation 7 and used the change of solidification surface at tie t to plot Figure, which could be plotted using a standard hoe PC. In this calculation, it is assued that the obility value of all polyers is 5 because at this stage we are interested only in the theral gap between the elting teperature and the crystallization teperature. All of the therophysical paraeters can be found in a polyer database. The therodynaic paraeters such as kinetic coefficient, density, free energy, etc are taken fro Xu et al. (5). 4. RESULTS AD DISCUSSIO Let us discuss the results obtained fro Equation 7. Figure shows the change of solidification surface in respect of tie. The dots are obtained through calculations and the lines are fitted by assuing that the growth of crystal is linear at a sall value of t. Our calculation shows that PP has the highest solidification rate (fro the gradient), followed by PS and PE. This can be

7 Fatriansyah et al. 37 explained as follows. The solidification rate is dependent on the ratio between the difference of elting teperaturecrystallization teperature and equilibriu elting teperature ( T T T ). This paraeter is related to the heat dissipation that connects to the growth of crystal. PP has a higher value for this paraeter copared to the other two polyers. This shows that in PP, heat dissipates faster and thus yields a high solidification rate. It can thus be seen that PP has a high relative difference between its elting teperature and crystallization teperature. Thus, to achieve equilibriu, PP needs to both dissipate heat faster and crystallize faster. Figure The change of solidification surface, x, in respect of tie, t The relation between the solidification rate and elting teperature is shown in Figure 3. Generally, a higher elting teperature yields a higher solidification rate, except for PS. However, the ratio between the difference of elting teperature - crystallization teperature and equilibriu elting teperature ( T T T ) is the one that relates directly to the solidification rate as this variable reflects the heat dissipation rate for each polyer. For this ratio, PP has a uch higher value than the other two polyers and thus yields a high solidification rate despite its low elting teperature. In this calculation, we did not consider elt viscosity/obility coefficient, which we suspect ay change the results. However, it is difficult to predict the obility coefficient as it ay only be accessible via an MR experient. By assuing that the value of obility is siilar for every polyer, it is reasonable to use our approach. Figure 3 The difference in velocity of solidification, v, in respect of elting teperature, T

8 38 An Investigation of the Melting Teperature Effect on the Rate of Solidification in Polyer using a Modified Phase Field Model 5. COCLUSIO We calculated that PP has the highest solidification rate (fro the gradient), followed by PS and PE. This is due to the relatively large difference between the elting teperature and crystallization teperature for PP. Thus, to achieve equilibriu, PP needs to dissipate heat faster and crystallize faster. Generally, a higher elting teperature yields a higher solidification rate, except for PS. This shows that the elting teperature is not directly related to the solidification rate. There is a direct relation, however, for the ratio between the difference of elting teperaturecrystallization teperature and equilibriu elting teperature T T T. 6. ACKOWLEDGEMET We are grateful to and thank DRPM UI for the PITTA 7 research grant that ade it possible to conduct this research. 7. REFERECES Atzeni, E., Iuliano, L., Minetola, P., Sali, A.,. Redesign and Cost Estiation of Rapid Manufactured Plastic Parts. Rapid Prototyping Journal, Volue 6(5), pp Collins, J.B., Levine, H., 985. Diffuse Interface Model of Diffusion-liited Crystal Growth. Physical Review B, Volue 3(9), pp Corinaldesi, V., Donnini, J., ardinocchi, A., 5. Lightweight Plasters Containing Plastic Waste for Sustainable and Energy-efficient Building. Construction and Building Materials, Volue 94, pp Ginzburg, V.L., Landau, L.D., 95. The Theory of Superconductivity. Zh. Eksp. Teor. Fiz, Volue, pp Harrowell, P.R., Oxtoby, D.W., 987. On the Interaction between Order and a Moving Interface: Dynaical Disordering and Anisotropic Growth Rates. Journal of Cheical Physics, Volue 86(5), pp Kittel, C., 5. Introduction to Solid State Physics. Wiley Lovinger, A.J., Cais, R.E., 984. Structure and Morphology of Poly(trifluoroethylene). Macroolecules, Volue 7(), pp Micheletti, A., Burger, M.,. Stochastic and Deterinistic Siulation of on-isotheral Crystallization of Polyers. Journal of Matheatical Cheistry, Volue 3(), pp Provatas,., Elder, K., 5. Ising Model of Magnetis. In: Phase-Field Methods in Material Science and Engineering, Wiley-VCH, ew York, pp. 4 Raabe, D., Godara, A., 5. Mesoscale Siulation of the Kinetics and Topology of Spherulite Growth during Crystallization of Isotactic Polypropylene (ipp). Modelling and Siulation in Materials Engineering, Volue 3, pp Ruderan, M.A., Kittel, C., 954. Indirect Exchange Coupling of uclear Magnetic Moents by Conduction Electrons. Physical Review, Volue 96(), pp. 99 Warren, J.A., Kobayashi, R., Lobkovsky, A.E., Carter, W.C., 3. Extending Phase Field Models of Solidification to Polycrystalline Materials. Acta Materialia, Volue 5(), pp Wheeler, A.A., Boettinger, W.J., McFadden, G.B., 99. Phase-field Model for Isotheral Phase Transition in Binary Alloys. Physical Review A, Volue 45(), pp Xu, H., Matkar, R., Kyu, T., 5. Phase-field Modelling on Morphological Landscape of Isotactic Polystyrene Single Crystal. Physical Review E, Volue 7(), 84