MP10: Process Modelling MPhil Materials Modelling Dr James Elliott 0.1 MP10 overview 6 lectures on process modelling of metals and polymers First three lectures by JAE Introduction to polymer rheology and processing Industrial aspects of polymer processing Modelling of liquid crystalline polymer processing Final three lectures by HKDB Fusion welding Steel microstructures Weld microstructure model Building on courses MP3 (Monte Carlo and Molecular Dynamics) and MP5 (Meso and Multiscale Modelling, but also involving aspects of many others (MP1, MP4, MP6) 1
0.2 Reminder about online teaching resources You can find copies of the overheads and handouts for this lecture at: www.cus.cam.ac.uk/~jae1001/teaching http://www.msm.cam.ac.uk/teaching/mphil/index.html There are copies of the handouts for this course (MP10) and most others in the MPhil Online communities sage.caret.cam.ac.uk Course MP10 Lecture 1 Introduction to polymer rheology and processing How to make polymers behave when things get hot and sticky Dr James Elliott 2
1.1 Introduction In this lecture, we will review some essential concepts of rheology and fluid dynamics, before moving on to consider the basic elements of polymer processing. The following lecture will study in detail some industrial polymer processing methods, and look at process modelling using the finite element package C-MOLD. Finally, we will discuss how to model the processing of liquid crystalline polymers using a Monte Carlo lattice director model. We will see how this model can be used to address specific problems associated with orientation effects in moulded polymers. 1.2 Basic concepts of rheology Newtonian fluids Usually pure atomic or diatomic fluids, e.g. water, with relatively simple intermolecular interactions. Gives a simple shear strain response to an applied shear stress. Shear strain rate proportional to shear stress for Newtonian fluid. e τ12 = 2η t 12 v = η x 1 2 v + x Constant of proportionality η is the shear viscosity, measured in Pa s, or sometimes Poise (1 Poise = 0.1 Pa s). For water, η = 10 3 Pa s, but most liquids rather lower. Not to be confused with bulk viscosity (irrelevant for subsonic flow) or kinematic viscosity, η/ρ, measured in Stokes. [1 Stoke =1 cm 2 s 1 ] 2 1 3
1.3.1 Basic concepts of rheology Reynolds number A dimensionless measure of the ratio of inertial to viscous forces acting in the system R e = ρvd / η Low R e («1) means that viscous forces dominate. Flow is generally non-turbulent (or laminar), e.g. ball bearing falling through treacle. High R e (»1) means that inertial forces dominate. Flow is generally unstable to turbulence, e.g. turning on a tap. ρv = 10 3 kg s 1 m 2, d = 10 2 m, η = 10 3 kg m 1 s 1, so R e = 10 4 and we expect viscous forces to be negligible. Remember that the Reynolds number is an attribute of the flow not the fluid! 1.3.2 Basic concepts of rheology Non-Newtonian fluids Non-linear relationship between shear stress and shear strain rate, i.e. the shear viscosity is dependent on shear stress. We call the viscosity measured at a particular shear stress the apparent viscosity. Most commonly occurring fluids are non-newtonian! Examples: cornflower/water mixture, polymers, paints, shampoos, quicksand, etc. Most demonstrate shear thinning (thixotropy) or more rarely shear thickening. Non-linear shear stress response occurs for a number of different reasons: Chain entanglement and orientation in polymers. Depletion forces in colloidal suspensions. Protein unfolding and gelation in food products. 4
1.4.1 Viscoelasticity The non-newtonian aspects of polymer flow are described well by theory of viscoelasticity. Viscoelastic fluids show a response which is a mixture of both viscous and elastic behaviour, and is dependent on the timescale over which perturbing force is applied. The shear modulus is replaced by a complex shear modulus G*, made up from the storage modulus G and the loss modulus G. G * = G' + ig' ' G measures the elastic response. G measures the viscous response. 1.4.2 Viscoelasticity The elastic component of the complex modulus leads to memory effects, as the fluid tries to recover its conformation prior to deformation. A measure of the extent to which elastic restoring forces are significant in a flow is given by the Deborah number, N deb, which is defined as the longest relaxation time of a material involved in a process η/e divided by the timescale of that process t proc. N deb = η E t proc The mountains flowed from before the Lord. Judges, ch. 5:5. 5
1.5.1 Viscoelasticity in polymers Polymer melts are viscoelastic because of entanglements between chains, which constrain their movement and therefore their response to mechanical deformation. The entanglements can be thought of as long bars (for example O 1 and O 2 ) which pin the chain in 2 dimensions, and from it can only escape by diffusing along its length. [Taken from de Gennes, Cornell University Press (1979).] 1.5.2 Viscoelasticity in polymers This diffusive wriggling motion, called reptation, governs the time taken by the polymer chains to respond to any applied stress, leading to a non-linear response. This can be seen most clearly by measuring the creep compliance of the polymer J(t). J ( t) = e( t) / δτ T t is known as the terminal time (J 0 e ) 1 is known as the plateau modulus [Taken from de Gennes, Cornell University Press (1979).] 6
1.5.3 Viscoelasticity in polymers This non-linear behaviour can be explained to an extent by a simple scaling argument based on the time taken for a chain to escape a fictitious sock-like tube representing the confining effects of the entanglements. This escape time is identified with the terminal time. [Taken from de Gennes, Cornell University Press (1979).] 1.5.4 Viscoelasticity in polymers If the each segment of the chain has mobility µ 1 then the mobility of the chain moving longitudinally in the tube is given by: µ tube = µ 1 / N Therefore the tube diffusion coefficient can be calculated from an Einstein relationship (see MP3, lecture 9) as: D = µ T N D / N tube 1 / = 1 And so the time taken for the chain to diffuse over a length comparable to the length of the tube is: T 2 2 t L Dtube = NL / / D N 1 3 7
1.5.5 Viscoelasticity in polymers The prediction that the terminal time (and therefore the shear viscosity) scales with N 3 is in reasonable agreement with experiment (N 3.3 ). Also, assuming a value for the segmental mobility of order 10 11 seconds leads to a terminal time of order 10 seconds for a chain with 10 4 segments. This is entirely plausible for real polymers, and implies that any process which acts over similar timescale to this will need to take into account orientation effects induced by the flow field. 1.6 Molecular weight dependence of viscosity Wide range of chemically and structurally dissimilar polymers display the same molecular weight dependence of viscosity Abrupt change in the scaling exponent at a certain critical molecular weight 8
1.7 Diffusion of polymers in a melt In the entangled melt, molecular weight dependence of diffusion coefficient is very strong 1.8.1 Flow through a circular pipe Also known as Hagen-Poiseuille flow, this is a very important process to study as it relates to a wide variety of systems (including physiological ones), and is a method for experimentally determining viscosity. 2R L dv v = = 2πr dr v p 4Lη 2 2 ( R r ) [Newtonian fluid] dv 2 3 ( R r r ) dr 4 π πr = p V = p 2Lη 8Lη 9
1.8.2 Flow through a circular pipe Note that the volume flow rate is proportional to the tube radius to the forth power, directly proportional to the pressure drop and inversely proportional to the fluid viscosity. For non-newtonian fluids, there is no simple relationship between the flow rate and the radius. However, for shearthinning fluids, there is a tendency towards plug-flow. For example, think of toothpaste being squeezed down a tube. 1.9 Polymers as power-law fluids Although there is no general analytical expression for the flow curve of a polymer, it is possible to approximate the flow over a small range of log(τ) to a straight line. Newton s relationship is thus modified to: e τ = C t n where C is called the consistency and n = 0.2-0.5 for polymers at high shear rate (note that n = 1 for a Newtonian fluid). Such fluids are known as power-law fluids. 10
1.10 Polymer processing Let us now look at more realistic processes for turning raw polymers into useful objects. The industrial processes will be covered in more detail in the next lecture, but for the moment we will be concerned with a schematic model of the process. Thermoplastics (PE, PET, etc.) Heat to melt Change shape Cool to solidify Thermosets (rubbers) Mix and heat Change shape Chemical reaction Cool 1.11 General model of polymer processing Overview of the various stages: Mixing Addition of additives to alter physical, mechanical or aesthetic properties of the finished product. Melting Liquefy polymer and destroy memory of previous chain configurations. Melt flow Flow and shaping of the liquid by mechanical processes Cross-linking Formation of chemical cross-links to induce dimensional stability Cooling and solidification Transfer of heat away from polymer, crystallisation or glass formation. 11
Lecture 1 summary In this lecture, we started by reviewing some basic concepts of rheology, including Newtonian and non- Newtonian fluids and the Reynolds number. We then discussed viscoelasticity in polymers, and sketched out a basis for the molecular mechanism which gives rise to this behaviour. In particular, the prediction of reptation theory is that the terminal time is proportional to the cube of the molecular weight. Finally, we reviewed the basic elements of a typical polymer processing scheme. In the next lecture, we will study some real industrial processes in more detail. 12