10. References

11.1.1 Viscosity-Enhancement-Factor Model

Injection molding simulation needs to consider both molten flow and solidification of the ma-terial. To achieve accurate results in a flow analysis, it is important to predict the solidification layer thickness accurately. There are two ways in which the solidification layer thickness can be determined:

1. Using the concept of a no-flow temperature, which indicates the temperature at which the liquid-solid phase transition takes place.

2. Using a solidification criterion based on viscosity increase: the region where the viscosity increases by one order of magnitude will determine the solidification layer.

For amorphous polymers, the no-flow temperature is related to the glass transition tempera-ture T_g. The T_g is not very dependent on cooling rate and flow. This fact allows the use of a single temperature as a criterion to determine the melt and solid phases. Therefore, the con-cept of the no-flow temperature provides a useful simplification in the simulation of injection molding for amorphous materials (Section 5.3.5).

Solidification behavior of semi-crystalline polymers is more difficult to model. The polymers solidify due to crystallization above the glass transition temperature. The crystallization rate is influenced by both thermal history and flow. It is well known that a semi-crystalline poly-mer under higher cooling rates solidifies at lower temperatures. It is obvious that, for semi-crystalline polymers, there is no single value of a “no-flow temperature” that can be identified.

To simulate the solidification behavior, it would be better to use the viscosity increase criterion.

Thus, detailed knowledge of material rheology during crystallization is important.

To consider the effect of crystallization kinetics on rheology, one needs to build a relationship between the viscosity and the relative crystallinity of the material. The simplest method to construct such a relationship is to introduce an enhancement factor f_η(α) into existing con-stitutive equations. With this idea, Pantani et al. [283] adopted the following modified Cross model:

η = η0

1 + (η0γ/τ˙ ^∗)¹⁻ⁿf_η(α), (11.1)

whereη0is the zero-shear-rate viscosity of the amorphous phase,τ^∗is a constant related to the shear stress at the transition between Newtonian and power-law behavior, and n is the power-law index, a measure of the degree of the shear-thinning behavior. The enhancement factor, f_η(α) is assumed to have the following empirical form (Titomanlio et al. [365]):

f_η= 1 + β exp

Tanner [357] considered the rheology of semi-crystalline polymers in the linear viscoelastic strain regime, based on suspension theory. He proposed using two separate models for low and high concentrations, respectively, and used an interpolation between solutions of the two models to ensure a continuous transition at intermediate volume fractions (crystallinities). If the complex shear moduli at low and high concentrations are G₀^∗and G₁^∗, respectively, the overall complex shear modulus G^∗is

G^∗(ω,α) = fG(α)G^∗₀(ω) + hG(α)G^∗₁(ω), (11.3)

whereω is the frequency. Tanner [357] determines fG(α) and hG(α) directly by fitting them to the oscillatory shear data of Boutahar et al. [44] for a polypropylene melt containing different volume fractions of spherulites.

For flows in the non-linear viscoelastic regime, Tanner and his co-workers [357, 359] proposed using a Phan-Thien-Tanner (PTT) model [286, 293] in the following modified form:

λa∆τ

where∆τ/∆t represents the upper convected derivative of τ, the extra stress tensor, defined as

△τi j

△t =∂τi j

∂t + uk

∂τi j

∂xk − Li kτk j− Lj kτki, (11.5)

ε is a parameter, trτ is the trace of τ, η0is the zero-shear viscosity of the amorphous phase, and D is the rate of deformation tensor. The function f_ηis expressed as follows, based on the suspension theory of Metzner [248]:

f_η= 1

(1 − α/A)², (11.6)

where A is a parameter representing the geometrical effect. For smooth spheres, A ≈ 0.68. For rough compact crystals, A ≈ 0.44. By varying A, Equation 11.6 can also be applied to non-spherical shapes.

Pantani et al. [281] provide a review of more models with several different enhancement fac-tors. Generally, these models predict a sharp upturn of viscosity with increasing relative crys-tallinity. Some models, such as Equation 11.6, show that viscosity turns up and goes to infinity at reasonably low crystallinity, while others, such as Equation 11.2, show that the viscosity turns up first and then levels off, reaching a finite value.

11.1 Effects of Crystallization on Rheology 157

11.1.2 Two-Phase Model

Doufas et al. [87, 88] proposed a two-phase model in which the semi-crystalline phase is mod-eled as a rigid dumbbell, and the amorphous phase is modmod-eled using a modified Giesekus constitutive equation. The total extra stress tensor of the system is given by an additive rule:

τ = τa+ τsc, (11.7)

where the subscript “a” stands for the amorphous matrix, and “sc” for the partially crystalline, partially amorphous material inside the spherulites, namely, the semi-crystalline phase. The amorphous extra stress tensorτais calculated from a modified Giesekus model, with the re-laxation time depending on the relative crystallinity as

λa(α,T ) = λa(0, T )(1 − α)². (11.8)

Here, the degree of crystallinity is built into the amorphous constitutive equation to account for the loss of chain segments due to crystallization. The semi-crystalline contributed extra stress tensorτscis calculated from a rigid dumbbell model, given by

τsc= µ¡3〈uu〉 − I + 6λscD : 〈uuuu〉¢

(11.9) where µ = n0kBT is the melt shear modulus, with n0 being the number density of the molecules, kB being the Boltzmann’s constant, and T the absolute temperature.λscis the re-laxation time of the semi-crystalline phase, D is the deformation rate tensor, and u is the unit vector of orientation. The angular brackets denote the average with respect to the distribution function of the semi-crystalline phase, and 〈uu〉 and 〈uuuu〉 are the second and fourth orien-tation tensors, respectively. The changes in the orienorien-tation tensor are governed by an equation of the form

λsc

³D〈uu〉

D t − L · 〈uu〉 − 〈uu〉 · L^T+ 2D : 〈uuuu〉´

+ 〈uu〉 =1

3I , (11.10)

which is exactly the expression of the rigid dumbbell model given by Bird et al. [36].

Zheng and Kennedy [418] followed the approach of the two-phase model. In their work, the semi-crystalline phase is also modeled by the rigid dumbbell model, but the amorphous phase is modeled by a FENE-P model for simplicity. The FENE-P model is a non-linear elastic dumb-bell model, in which the dumbdumb-bell is constrained to a maximum allowable length. The model is relatively simple, but it captures most important non-linear rheological properties of poly-mer solutions such as memory effects and shear thinning. Its rheological properties are well known [399]. In Zheng and Kennedy’s approach [418], the effect of crystallinity is not built into the amorphous parameters. Instead, the system is viewed as a suspension of semi-crystalline phase (spherulites or shish-kebabs modeled as rigid rods) in a matrix of amorphous material.

The physical properties of the amorphous phase, such as the viscosityηaand the relaxation timeλa, are independent of the relative crystallinity, while the properties of the whole system is a function of the relative crystallinity.

Let us consider the response of the model when u has a near-equilibrium isotropic distribu-tion. In this case, the leading term of the distribution function that satisfies the Fokker-Planck equation (Equations 4.75 and C.27) in the u space is (4π)⁻¹. Using the integral theorem de-scribed by Brenner [45], one obtains

D : 〈uuuu〉 = 2

15D , (11.11)

and

L · 〈uu〉 + 〈uu〉 · L^T=2

3D . (11.12)

From Equation 11.10, assuming steady state (D〈uu〉/Dt = 0) and using Equations 11.11 and 11.12, we then obtain

3〈uu〉 − I =6

5λscD . (11.13)

Substituting Equations 11.11 and 11.13 into Equation 11.9 gives

τsc= 2µλscD, (11.14)

whereµλscshould be a function ofα. At the limit α → 0, the amorphous phase dominates, andµλsc→ 0. At high values of α, the semi-crystalline phase dominates the response. The dependence ofµλscon the relative crystallinity is approximated as follows [418]:

µλsc=ηa(α/A)^β1

(1 − α/A)^β, (11.15)

where A is the same as defined in Equation 11.6 andβ and β1are empirical constants. We then write Equation 11.14 as

τsc=2ηa(α/A)^β1

(1 − α/A)^βD. (11.16)

In a simple shear flow, Equation 11.16 reduces to τ^(sc)₁₃ = (α/A)^β1

(1 − α/A)^βηaγ,˙ (11.17)

while the amorphous phase contributed shear stress is given by

τ^(a)₁₃ = ηaγ.˙ (11.18)

The total shear stress is then τ13= τ^(a)₁₃+ τ^(sc)₁₃ =

"

1 + (α/A)^β1 (1 − α/A)^β

#

ηaγ.˙ (11.19)

Thus, we can write the shear viscosity function as η =τ13

γ˙ =

"

1 + (α/A)^β1 (1 − α/A)^β

#

ηa, (11.20)

which behaves similarly to the suspension model (Equation 11.6). Clearly, all of the above equations apply only forα < A. When α → A, jamming and cessation of flow take place, and the viscosity increases and tends to infinity.

However, as pointed out by Tanner [357], the stress law of Equation 11.7 oversimplifies the real picture of structure. The additive rule envisages parallel components of amorphous and crys-talline phase at each point. Photographs of crystallizing polypropylene such as those shown by Koscher and Fulchiron [211] do not support this assumption.