Polymer Processing HANSER. Modeling and Simulation. Tim Osswald Juan P. Hernández-Ortiz. Hanser Publishers, Munich Hanser Publications, Cincinnati

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1 Tim Osswald Juan P. Hernández-Ortiz Polymer Processing Modeling and Simulation Sample Chapter 2: Processing Properties ISBNs HANSER Hanser Publishers, Munich Hanser Publications, Cincinnati

2 CHAPTER 2 PROCESSING PROPERTIES Didyou ever consider viscoelasticity? Arthur Lodge 2.1 THERMAL PROPERTIES The heat flow through a material can be defined by Fourier s law of heat conduction. Fourier s law can be expressed as q x = k x T x (2.1) where q x is theenergy transport perunitarea in the x direction, k x thethermal conductivity and T/ x thetemperaturegradient. At theonset of heating, thepolymerresponds solely as aheat sink, and the amount of energy per unit volume, Q,stored in the material before reaching steady state conditions can be approximated by Q = ρc p T (2.2) where ρ is thedensity of thematerial, C p the specific heat, and T thechange in temperature. Thematerial properties found in eqns.(2.1) and(2.2) areoften written as onesingleproperty,

3 38 PROCESSING PROPERTIES Table2.1: ThermalProperties for Selected Polymeric Materials Polymer Specific Specific Thermal Coeff. Thermal Max gravity heat conduc. therm. diffusivity temp. expan. kj/kg/k W/m/K µ m/m/k (m 2 /s)10 7 o C ABS CA EP PA PA66-30% glass PC PE-HD PE-LD PET PF PMMA POM copom a PP PPO b PS PTFE upvc c ppvc d SAN UPE Steel a Polyacetal copolymer; b Polyphenylene oxide copolymer; c Unplasticized PVC; d Plasticized PVC namely the thermaldiffusivity, α, which for an isotropicmaterial is defined by α = k ρc p (2.3) Typical values of thermal properties for selected polymers are shown in Table 6.1 [7, 17]. For comparison, the properties for stainless steel are also shown at the end of the list. It should bepointed out that the material properties ofpolymers are not constant and may vary with temperature, pressure or phase changes. This section will discuss each of these properties individually and present examples of some of the most widely used polymers and measurement techniques. For amore in-depthstudy of thermal propertiesofpolymers the reader is encouraged to consult theliterature [24,46, 66] Thermal Conductivity When analyzing thermal processes, thethermal conductivity, k, isthe mostcommonlyused propertythat helpsquantifythe transport of heat through amaterial. Bydefinition,energy is transported proportionally to the speed of sound. Accordingly, thermal conductivity

4 THERMAL PROPERTIES PET Thermal conductivity, W/m/k PIB NR PMMA PVC PBMA Temperature (K) Figure 2.1: Thermal conductivity of various materials. follows the relation k C p ρul (2.4) where u is the speed of sound and l the molecular separation. Amorphous polymers show an increase inthermal conductivity with increasing temperature, up to the glass transition temperature, T g.above T g,the thermal conductivity decreases with increasingtemperature. Figure 2.1 [24] presents the thermal conductivity, below the glass transition temperature, for various amorphous thermoplasticsasafunctionoftemperature. Due to the increase in density upon solidification of semi-crystalline thermoplastics, the thermal conductivity is higher in the solid state than in the melt. In the melt state, however, the thermal conductivity of semi-crystalline polymers reduces to that of amorphous polymers as can be seen in Fig. 2.2 [40]. Furthermore, it is not surprising that the thermal conductivity of melts increases with hydrostatic pressure. This effectis clearly shown in Fig. 2.3[19]. As long as thermosetsare unfilled, their thermal conductivity is very similar to amorphous thermoplastics. Anisotropy in thermoplastic polymers also plays asignificant role inthe thermal conductivity. Highly drawn semi-crystallinepolymer samplescan have a muchhigher thermalconductivityas a resultofthe orientationofthe polymerchains in thedirectionofthe draw. For amorphous polymers, the increase in thermal conductivity in the direction of the draw is usually not higher than two. Figure 2.4[24]presents the thermal conductivity in the directionsparalleland perpendicularto thedrawforhighdensity polyethylene, polypropylene, and polymethyl methacrylate. Asimple relation exists between the anisotropic and the isotropic thermal conductivity [39]. This relation is written as = 3 k k k (2.5) where thesubscripts and represent thedirections paralleland perpendicular to thedraw, respectively.

5 Thermal conductivity k, W/m/K 40 PROCESSING PROPERTIES HDPE 0.3 LDPE PA 6 PC 0.2 PS PP Temperature, T ( C) Figure 2.2: Thermal conductivity of various thermoplastics. Variation in thermal conductivity(k/k 1bar ) T = 230 C PP HDPE LDP E PS PC Pressure, P(bar) Figure 2.3: Influence of pressure onthermal conductivity of various thermoplastics.

6 THERMAL PROPERTIES PE-HD k /k iso k /k iso PMMA PMMA PP PP PE-HD Draw ratio Figure 2.4: Thermal conductivity as afunction ofdraw ratio in the directions perpendicular and parallel tothe stretchfor various oriented thermo-plastics. 0.7 Thermal conductivity, k (W/m/K) PE-LD + Quartz powder 60% wt PE-LD + GF (40% wt) ll -orientation PE-LD + GF (40% wt) -orientation PE-LD Temperature, T ( C) Figure 2.5: Influence of filleronthe thermal conductivity of PE-LD. The higher thermal conductivity of inorganicfillers increases the thermal conductivity of filledpolymers. Nevertheless, asharp decreaseinthermal conductivity around the melting temperature of crystalline polymers can still be seen with filled materials. The effect of filler on thermal conductivity for PE-LD is shown in Fig. 2.5 [22]. This figure shows the effect of fiber orientationaswellasthe effect of quartzpowder on thethermal conductivity of lowdensity polyethylene. Figure 2.6demonstrates theinfluence of gascontent on expanded or foamed polymers, and theinfluence of mineral content on filled polymers. There are various models available to compute the thermal conductivity of foamed or filled plastics[39,47, 51]. Aruleofmixtures,suggestedbyKnappe [39], commonlyused

7 Thermal conductivity, kw/m/k 42 PROCESSING PROPERTIES Polymer in metal Closed cells Opencells Metal in polymer Foams Polymer/meta Volume fraction Volume fraction of gas of metal Figure 2.6: Thermal conductivity of plastics filledwith glass or metal. to compute thermal conductivity of composite materials is written as k c = 2 k m + k f 2 φ f ( k m k f ) 2 k m + k f + φ f ( k m k f ) k m (2.6) where, φ f is the volume fraction offiller, and k m, k f and k c are the thermal conductivity of thematrix, filler and composite,respectively. Figure 2.7compares eqn. (2.6) with experimental data[2] for an epoxy filled with copper particlesofvariousdiameters. Thefigurealso comparesthedatatotheclassicmodelgiven by Maxwell [47] whichiswritten as k f 1 k c = 1+3φ k m f k f k m (2.7) +2 k m In addition, amodel derivedbymeredith andtobias [51] applies toacubicarray of spheres inside amatrix. Consequently, itcannot be used for volumetric concentration above 52% since the spheres will touch at that point. However, their model predicts the thermal conductivity very well up to 40% by volume of particle concentration. When mixing several materials the followingvariation ofknappe s model applies 1 n i =1 2 φ k m k i i 2 k k c = m + k i 1+ (2.8) n i =1 2 φ k m k i i 2 k m + k i where k i is thethermal conductivity of thefiller and φ i its volume fraction. Thisrelationis useful for glassfiberreinforced composites(frc) with glassconcentrations up to 50% by volume. Thisisalso valid for FRC with unidirectional reinforcement. However,one must differentiate between the direction longitudinal to the fibers and that transverse to them.

8 THERMAL PROPERTIES Thermal conductivity, k(w/m/k) Experimental data (Temperature 300K) 100 µm 46 µm 25 µm 11 µm Knappe Maxwell Concentration, φ (%) Figure 2.7: Thermal conductivity versus volume concentration ofmetallic particlesofanepoxy resin. Solid lines represent predictions using Maxwell and Knappe models. For high fiber content, one can approximatethe thermal conductivity of the composite by the thermal conductivity of the fiber. The thermal conductivity can be measured using the standard tests ASTM C177 and DIN A new method currently being balloted (ASTM D20.30) is preferred by most peopletoday Specific Heat The specific heat, C,represents the energy required to change the temperature of a unit mass ofmaterial by one degree. It can be measured at either constant pressure, C p,or constant volume, C v. Since the specific heat at constant pressure includes the effect of volumetric change, it is larger than the specific heat at constant volume. However, the volume changes of a polymer with changing temperatures have a negligible effect on the specific heat. Hence, one can usually assume that the specific heat at constant volume or constant pressure are the same. It is usually true that specific heat only changes modestly in therange of practical processing and design temperatures of polymers. However,semicrystalline thermoplastics display a discontinuity in the specific heat at the melting point of the crystallites. This jump or discontinuity in specific heat includes the heat that is required to melt the crystallites which is usually called the heat of fusion. Hence, specific heat is dependenton the degreeof crystallinity. Valuesof heat of fusionfor typical semi-crystalline polymers are shownintable 2.2. The chemical reaction that takes place during solidification ofthermosets also leads to considerable thermal effects. In a hardened state, their thermal data are similar to the ones of amorphousthermoplastics. Figure2.8showsthespecific heatgraphsforthethreepolymer categories.

9 44 PROCESSING PROPERTIES Table2.2: Heat of Fusion of Various Thermoplastic Polymers [66] Polymer λ (kj/kg) T m ( o C) Polyamide Polyamide Polyethylene Polypropylene Polyvinylchloride Specific heat, C p (KJ/kg) Polystyrene Polyvinyl chloride Polycarbonate a) Amorphous thermoplastics UHMWPE HDPE LDPE b) Semi-crystalline thermoplastics 2.4 Before curing 1.6 After curing 0.8 c) Thermosets (phenolic type 31) Temperature, T ( C) Figure 2.8: Specific heat curves for selected polymers of the three general polymer categories.

10 THERMAL PROPERTIES 45 3 Specific heat, Cp (KJ/kg/K) 2 1 PC PC + GF 0 % 10% 20 % 30 % ( C ) Temperature, T Figure 2.9: Generated specific heat curves for afilledand unfilledpolycarbonate. Courtesy of Bayer AG,Germany. For filled polymer systems with inorganic and powdery fillers, arule ofmixtures 1 can be written as C p ( T )=(1 ψ f ) C p m ( T )+ψ f C p f ( T ) (2.9) where ψ f represents the weight fractionofthe filler and C p m and C p f the specific heat of the polymermatrixand the filler, respectively. As an exampleofusing eqn. (2.9), Fig. 2.9 shows aspecific heat curveofanunfilledpolycarbonateand its corresponding computed specific heat curves for 10%,20%,and 30% glassfiber content. Inmostcases,temperature dependence of C p on inorganicfillers is minimal and need not be takenintoconsideration. The specific heat of copolymers can be calculated using the molefractionofthe polymer components. C p copolymer = σ 1 C p 1 + σ 2 C p 2 (2.10) where σ 1 and σ 2 are the mole fractions of the comonomer components and C p 1 and C p 1 thecorresponding specific heats Density The density or its reciprocal,the specific volume,is a commonly used property forpolymeric materials. The specific volume is often plotted as a function of pressure and temperature in what is known as a pvt diagram. Atypical pvt diagram for an unfilled and filled amorphous polymer isshown, using polycarbonateasanexample, in Figs and 2.11 The two slopes in the curves represent the specific volume of the melt and of the glassy amorphous polycarbonate, separated by the glasstransition temperature. Figure 2.12 presents the pvt diagramfor polyamide66 as an example of a typical semicrystallinepolymer. Figure2.13showsthe pvt diagramforpolyamide66filled with 30% 1 Valid up to 65% filler content by volume.

11 Specific volume, cm /g 3 46 PROCESSING PROPERTIES Pressure, P (bar) Specific volume, V( cm 3 /g) PC Temperature, T ( C) Figure 2.10: pvt diagram for a polycarbonate. Courtesy of Bayer AG, Germany Pressure, P (bar) PC + 20% GF Temperature, T ( C) Figure 2.11: Germany. pvt diagram forapolycarbonate filledwith 20% glass fiber. Courtesy of Bayer AG,

12 THERMAL PROPERTIES Pressure, P ( bar ) Specific volume, v ( cm 3 /g) 0.95 PA Temperature, T ( C) Figure 2.12: pvt diagram for apolyamide 66. Courtesy of Bayer AG,Germany. glass fiber. The curves clearly show the melting temperature (i.e., T m 250 o Cfor the unfilledpa66cooled at 1bar,which marks thebeginning of crystallizationasthe material cools). Itshouldalso comeasno surprisethattheglass transitiontemperaturesarethesame for the filled and unfilled materials. When carrying out die flow calculations, the temperature dependence of the specific volume must often be dealt with analytically. At constant pressures, the density of pure polymers can be approximated by ρ ( T )=ρ α t ( T T 0 ) (2.11) where ρ 0 is thedensity at reference temperature,t 0,and α t is thelinear coefficientofthermal expansion. Foramorphouspolymers,eqn. (2.11) is valid onlyforthe linearsegments(i.e., below orabove T g ), and for semi-crystalline polymers it is only valid for temperatures above T m.the density of polymers filled with inorganicmaterialscan be computed at any temperature usingthe following ruleofmixtures ρ c ( T )= ρ m ( T ) ρ f ψρ m ( T )+(1 ψ ) ρ f (2.12) where ρ c, ρ m and ρ f are the densities of the composite, polymer and filler, respectively, and ψ is theweightfractionoffiller.

Plastics Testing and Characterization Industrial Applications

Alberto Naranjo C., Maria del Pilar Noriega E., Tim A. Osswald, Alejandro Rojan, Juan Diego Sierra M. Plastics Testing and Characterization Industrial Applications ISBN-10: 3-446-41315-4 ISBN-13: 978-3-446-41315-3