ELONGATIONAL VISCOSITY AND POLYMER FOAMING PROCESS. Αλέξανδρος Δ. Γκότσης Πολυτεχνείο Κρήτης Χανιά

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1 10 ΠΑΝΕΛΛΗΝΙΟ ΕΠΙΣΤΗΜΟΝΙΚΟ ΣΥΝΕΔΡΙΟ ΧΗΜΙΚΗΣ ΜΗΧΑΝΙΚΗΣ, ΠΑΤΡΑ, 4-6 ΙΟΥΝΙΟΥ, ELONGATIONAL VISCOSITY AND POLYMER FOAMING PROCESS Αλέξανδρος Δ. Γκότσης Πολυτεχνείο Κρήτης Χανιά ABSTRACT The manufacturing of polymeric foam is an example of a process that employs mainly extensional flow. In the present work foaming is used to illustrate the capabilities of theories that predict the effects of long chain branching on the rheology of the material. The numerical simulation of the process uses an approximate implementation of the theory. The study aims to generate knowledge and understanding of the physics of polymer foaming and assist the manufacturers to optimise their process and produce low density and high cell integrity closed cell foam for commercial thermal insulation applications. INTRODUCTION Polymeric structural foams are used in a wide variety of applications due to their low weight, thermal insulation and sound- and shock-absorbing properties. Polymeric foams can be produced from thermoplastic, thermohardening or vulcanisable polymers. In the first case the foaming takes place in the molten state of the material and solidification is obtained by cooling. In thermosetting systems the solidification by polymerisation/vulcanisation reaction takes place simultaneously with the generation of the bubbles of the foam. The incorporation of the bubbles in polymers (to build the foam is achieved by using a foaming agent. Two types of foaming agents are used in thermoplastic or thermosetting/elastomeric polymers. The first involves a gas, or a volatile solvent, which is forced under high pressure to dissolve into the liquid material. Upon lowering the pressure the gas/vapour comes out of solution, it is released within the material in the form of bubbles and results in the foam. The second method involves the chemical decomposition of an explosive substance which has been dispersed in the liquid. This happens when the temperature of the mixture becomes higher than the decomposition temperature of the foaming agent. The products of the decomposition are the gasses that form the bubbles in the foam. There is also the possibility that the gas is generated by the polymerisation reaction, e.g. the water vapour during the condensation polymerisation reaction of polyurethane. The growth of the bubble involves radial flow in the surrounding viscoelastic fluid. This is an extensional flow. The viscoelasticity of the fluid determines the relation between stresses and deformation rates in the flow, and, thus, the growth rate of the bubble. The viscoelastic properties are determined, in turn, mainly by the molecular structure of the polymeric fluid, i.e. by the degree and type of long chain branching [1]. If a cross-linking reaction also takes place, then the structure will evolve with time. This evolution will result in changing viscoelasticity and bubble growth rate. The quality of the produced foam is a function of the foam density, the size of the bubbles and the open or closed cell structure. Many studies have shown that the quality depends on the evolving rheological properties of the material during foaming, which, in turn depend on its (evolving molecular structure. An example is shown in Fig. 1, as SEM photos of foams produced from polypropylene samples that had increasing degrees of long chain branching (Bn : average number of branches per chain on their backbone chains. Many polymeric melts show strain hardening of their viscosity during start-up of elongational flow: The resistance to flow increases as strain is accumulated during this type of flow. Strain hardening of the elongational melt viscosity is necessary in the production of high quality polymeric foam and it is related to the molecular structure of the polymer. It is greatly enhanced by the presence of long chain branches grafted on the main chain [3]. The relation of rheology with structure has been the subject of extensive studies and reliable theories have been developed to describe the viscosity in simple elongational flows as a function of time, strain and molecular structure [4]. Density: kg/m3 Figure 1: Scanning electron micrographs of foam that was made using ipp with Bn = 0, 0.2, 0.4, 0.5, 0.6 [2].

2 σ = t µ(t t h(λ 1 (t, t dt. σ µ h(λ λ 1 h h h(λ h(λ = λ n. µ(t n n = 0 n > 1 n U i = e i x i i = e 1 + e 2 + e 3 = 0 = e e 2 0 = ε m ε e (m + 1 ε m 0.5 m 1 m = 0.5 m = 0 m = 0.5 m = 1

3 n n u /n p /n b = 1.0/1.6/1.8 { } m n n u ( n n 1 n B n n u 2 { a 3 } B n. B n a 2.9 B n > 1 a 1.5 a µ(t 1 (t λ { t 1 (t = 0 λ 2m 0 λ = {ε} = 0 0 λ 2(m+1 0 } dt ε(t. k g i µ(t = { t/τ i } τ i=1 i g i τ i i = N 1 (t η + E (t = N 1(t/ ε ε η + E = k i=1 g i ( (2 nτi {[(2 n ετ i 1]t/τ i } 1/ ε (2 n ετ i 1 (2m + n + 2τ i {[( 2m + n + 2 ετ i 1]t/τ i } + 1/ ε (2m + n + 2 ετ i + 1 m = 1 η + b = k i=1 g i ( (2 nτi {[(2 n ετ i 1]t/τ i } 1/ ε (2 n ετ i 1 (4 + nτ i {[n ετ i 1]t/τ i } + 1/ ε. (4 + n ετ i + 1 t = 0 T 0 R(t r ϕ θ

4 R d (t r u(r, t = Ṙ(tR2 (t u(r, t r 2 ε rr (r, t = = 2Ṙ(tR2 (t r r 3 ε = 2 r 3 t 0 ṘR 2 dt. r R R 0 r R(t ρr 2 0 [ d 2 R dt 2 R ( ] 2 dr σ rr (r, t σ θθ (r, t = P (t, r + 2 dr dt R r r P r = 2σ rr σ θθ r P = P (t, r = R P (t, R = P 0 R 3 (t, P 0 µ dr(t dt A R(t 2 = 0, A = P 0 /(4µ R = 1 t = 0 R(t = At R d (t Rd η b ṘR 2 P (t = 12 R r 4 dr.

5 η 0 = µ B n R m = R d (t = t m r η b = η b (t, ε(t, r = R(t dr dt = A(t R(t 2 = P 0 4η b (tr 2.

6 τ i g i n V 0 ϕ = nv grain /V 0 = n(4π/3r0/v 3 0 t V (t = V 0 + nv bubble = V 0 + ϕv 0 V grain V bubble = V 0 ( 1 + ϕ V bubble V grain, ρ(t = ρ 0 ( 1 + ϕr 3 (t 1 R = R m 0.74V m R m = ϕ = 1.418ϕ 1/3!!

7 ρ 0 V m a 0 σ E = P bubble a(t [ a(t = a P t m T (t ] 1/3 [ T m t m η b (ε, t dt R eq(t = R m 1 + P t m T (t ] 1/3 T m t m η b (ε, t dt P m T m t m