Rheology

Rheology Basics

  • Rheology

    • Rheology is the science of flow and deformation of matter and describes the interrelation between force, deformation and time. The term comes from Greek rheos meaning to flow. Rheology is applicable to all materials, from gases to solids.

    Fluid rheology is used to describe the consistency of different products, normally by the two components viscosity and elasticity. By viscosity is usually meant resistance to flow or thickness and by elasticity usually stickiness or structure.
     

  • Viscosity

    • Viscosity is a very important rheological property that measures the material resistance to deformation. It characterizes the flow behavior of the material.
  • Dynamics Viscosity
    • Given two parallel plates separated by a distance d and with a fluid between them, keep one stationary while
    moving the other at a slow speed u0. A common situation is that the force F required to keep the second plate moving
    is proportional to its area A and to u0/d. A fluid in which these quantities are proportional, called a Newtonian fluid,
    therefore exhibits shear stress  F/A proportional to u0/d, giving
    Here,  is a constant for the given fluid called the dynamic viscosity.
    It can also be written as
    [shear stress]= [strain rate]
    where l is the length scale and u is the velocity scale.  For a Newtonian fluid,
  • Kinematics  Viscocity
    • Dynamic viscosity is related to kinematic viscosity  by
    where is the fluid density.
  • Viscosity Model[C-MOLD Documentation]

    • Viscosity of polymer melts varies with shear rate, pressure, and temperature. Therefore viscosity models should account for the variation with shear rate, pressure, and temperature . Such viscosity models include Cross-Exp and Cross-WLF models as so called in C-Mold.

    (1) CROSS-EXP POLYMER VISCOSITY MODEL

    Description:

    This data set specifies the five constants of the Cross-exp polymer viscosity model, which accounts for the
    effect of temperature, shear rate, and pressure on the viscosity.

    Most polymers exhibit two regimes of flow behavior: Newtonian and shear thinning. Newtonian behavior
    occurs at low shear rates, when the shear-stress-to-shear-rate relationship is linear. At higher shear rates,
    the viscosity decreases as the shear rate increases; this behavior is called shear thinning.

    In the Cross-exp model, the transition between the Newtonian and shear thinning regimes is characterized
    by the parameter, . represents the shear stress at which the onset of shear thinning behavior
    occurs.
    The value of (1 - n), where n is a power-law coefficient in this model, represents the slope of the shear
    thinning curve. The remaining constants are used to model the zero-shear rate viscosity, .
    The parameter, Tb , characterizes the temperature sensitivity of . This tends to depend on temperature,
    especially near Tg . However, in the filling stage, the bulk temperature is usually far higher than Tg , and
    because of this, the Cross-exp model is adequate for C-MOLD Filling.

    Equation:
    t > ttrans, 
    t< ttrans,
    is the zero-shear-rate viscosity, is the shear rate, t is the temperature, p is the pressure.
    n, , B, Tb , and are model constants.
    Limitation of this model:
    The value of Tb, which represents the temperature sensitivity of , is a strong function of temperature. However, this model corresponds to a constant value of Tb. It is inappropriate for modeling the behavior over a large temperature range when the polymer melt undergoes substantial cooling throughout the cavity.
    (2) CROSS-WLF POLYMER VISCOSITY MODEL

    Description:

    This model accounts for the effect of temperature, shear rate, and pressure on the viscosity, over a wide temperature range.

    This model still represents the shear-thinning behavior in the same manner as the Cross-Exp model; however,
    the zero-shear-rate viscosity is represented by a more extensive model that is based on the WLF functional
    form.

    The Cross-WLF model is more appropriate for C-MOLD Post-Filling, because the temperature and pressure
    sensitivities of the zero-shear-rate viscosity are better represented.

    Equation:


     
  • t >= Tg,
    • ,
    where A2 is a function of p
     
  • t < Tg,
    • Tg is the glass transition temperature of the material.
    n, , D1, D2, D3, A1 are model constants.

    Flow and Heat Model Used in Starch-based Polymer Research

  • Introduction

    • Tracking the filling front during polymer molding operations can be categorized as a free surface problem. In many cases, a set of differential equations must satisfy conditions on a domain boundary, and one or more of the boundary conditions must be determined as part of the solution.  This can be achieved by solving the so called "Volume of Fluid" (VOF) equation. [J. Luoma, 1999]
     
  • Mathematical model

    • Based on Cross-WLF model, James Luoma introduced a method which unifies many of the existing VOF based schemes. In his thesis,by the introduction of a pseudo compressibility, he transformed the basic VOF equation into an equation that matches the well known enthalpy formulation for solid/liquid phase change. This leads to a fully explicit scheme of the governing filling equations that does not require the solution of a system of equations. If the pseudo compressibility is small enough, the explicit solution is close to the solution obtained with a direct solution of the incompressibility VOF equation. Further , in some cases, the explicit scheme is computationally efficient. Applications of this method are demonstrated on the problems of resin transfer molding, non-isothermal injection molding and include the calculation of weld lines.
     
  • Simulation result

    • Simulation of injection molding(for non-Newtonian polymers) has been done, using the explicit enthalpy method and C-Mold, the comparision are as following:

    (1) Finite Element Mesh for Explicit Enthalpy Algorithm and C-Mold













    (2) Filling Front predicted by  Explicit Enthalpy Algorithm and C-Mold


    (3) short shot experimental results, compared with the prediction by different methods

  • Weld-line prediction

    • When molding a complex geometry, it sometimes is impossible to avoid  polymer-filling fronts from joining in the mold cavity. When this occurs the resulting interface is call a weld line. There are a variety of reasons for predicting the location of weld lines in the molded part. They include cosmetic appearance, structural integrity, and durability of the produced part. In a problem with multiple input ports, the proposed explicit scheme can be readily extended to predict the weld line along which the polymer from two  ports joins. This is achieved on "color-coding" the polymers from a given port and then tracking the movement of that polymer by-in addition to the main solution-solving an advection equation of the form

          (6)

    for each color of polymer, where Gcolor and Fcolor are connected through a relationship in the form of Eq. (3). In this way the evolution of the weld line can be found by simply plotting the color fields of the polymer in the mold. Note that the simplicity of this solution rests in the fact that with the proposed filling algorithm an explicit time integration of Eq. (6) can be used. A sample weld-line simulation is given in Fig. 4.

      Figure 4. Weld-Line Simulation [Quick-Time Click to Run]
    (If can't run Quick-Time ,please click HERE to view the pictures)

    The mold in the simulation is filled from two gates, one in the corner the other half way up the left-hand wall. The pressure on the wall is ¼ the value of the corner pressure. Each gate is assumed to be the width of the corresponding element. A  Newtonian flow is assumed and a finite-difference solution with a grid of 61x61 is used.
     

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