A microstructural flow-induced crystallization model for film blowing: validation with experimental data

Abstract

The two-phase microstructural/constitutive model for film blowing of Doufas and McHugh (D-M) (J Rheol 45:1085–1104, 2001a) is validated against online film data of a linear low-density polyethylene (LLDPE) at a variety of processing conditions. The D-M model includes the effects of thermal and flow-induced (enhanced) crystallization (FIC) coupled with the rheological response of both the melt and semicrystalline phases under fabrication conditions. The model predictions of bubble radius, velocity, and crystallinity profiles are in quantitative agreement with available experimental data over a wide range of blow-up ratios (BUR), take-up ratios (TUR), and bubble cooling rates using the same set of material/model parameters. The model naturally predicts the location of the frost line as a consequence of system stiffening due to crystallization overcoming the pitfalls of traditional modeling approaches that impose it as an artificial boundary condition. For a wide range of processing conditions, it is found that key film mechanical properties including elongation to break, yield stress, tensile modulus, and tear strength correlate well with predicted locked-in extensional stresses and molecular orientation at the frost line enabling development of quantitative structure-process-properties relationships that are useful in product and process development. The D-M model for film blowing is physics-based including elements of molecular rheology (polymer kinetic theory), suspension, and nucleation theories as well as irreversible thermodynamics principles, yet being tractable for continuum-based numerical simulations with practical industrial applicability. The FIC enhancement factor of the model is shown to be proportional to $\exp \left (\lambda _{\text {eff},\textnormal {w}}^{2} -1\right )$, where λ _eff,w is a molecular chain stretch ratio of the whole chain and proportional to exp (λ ² − 1), where λ is the stretch ratio of the remaining (uncrystallized) amorphous chain, consistent with fundamental kinetic Monte Carlo simulations of flow-induced nucleation of Graham and Olmsted (Phys Rev Lett 103:115702-1–115702-4, 2009).

Appendix: Dimensionless variables and numbers

Dimensionless variables

The subscript “o” used in the definition of dimensionless variables and numbers below refers to the value of the respective variable at the die exit.

$$ \text{ Axial distance:}{\kern4pt}z^{\ast} =z/L $$

(27)

While the axial distance z is traditionally nondimensionalized with respect to the bubble radius at the die R _o leading to a computational domain of z∗ > 1, the nondimensionalization of z with respect to L(distance from die exit to nip rolls) first proposed by Doufas and McHugh (2001a) is computationally more efficient, since it restricts the domain z* ∈ [0, 1].

$$ \mathit{Bubble\;radius} :R^{\ast} =R/R_o $$

(28)

$$ \mathit{Bubble\;thickness:}{\kern4pt} H^{\ast} =H/H_o $$

(29)

$$ \mathit{Bubble\;temperature:}~T^{\ast} =T/T_o $$

(30)

$$ \mathit{Bubble\;axial\;velocity:}{\kern4pt} \textnormal{v}^{\ast} =\textnormal{v}/\textnormal{v}_o $$

(31)

The bubble axial velocity at the die exit v _o is calculated from the mass flow rate W, the values of H _o , R _o and density ρ via the continuity equation

$$ \mathit{Amorphous\;conformation\;tensor:}{\kern4pt} \mathbf{c}^{\ast} =\mathbf{c} K_{o} /k_{B} T $$

(32)

where K _o is Hookean spring constant of the initially amorphous chains corresponding to N _o statistical links and k _B is the Boltzmann constant. We should note though that the tensor c* represents the remaining amorphous chains [N _o (1 − x) statistical segments] upon inception of crystallization through the correction factor 1 − x in the evolution equations (4–6), the amorphous relaxation time expression (15) and the calculation of the nonlinear force E factor (7).

$$ \mathit{Extra\;stress\;tensor:}{\kern4pt} \boldsymbol{\uptau}^{\ast} =\boldsymbol{\uptau} /G $$

(33)

where G is a characteristic modulus of elasticity more representative of the long relaxation time tail (Doufas 2006). While G is taken constant in our calculations, the stiffening of the semicrystalline system is simulated through dependence of the relaxation times on crystallinity and temperature (14, 15) and increase of chain rigidity through the nonlinear force factor E (7).

Dimensionless numbers

Dimensionless number D ₁ used in the momentum equations

$$ D_{1} =\frac{\Delta P R_{\mathrm{o}} }{G H_{\mathrm{o}}} $$

(34)

Dimensionless numbers D ₂ –D ₅ used in the energy equation

$$ D_{2} =\frac{2 \pi R_{\mathrm{o}} L U}{W C_{p}} $$

(35)

where C _p is the heat capacity

$$ D_{3} =\frac{2 \pi R_{\mathrm{o}} L \sigma_{\mathrm{B}} \varepsilon T_{\mathrm{o}}^{3} }{W C_{p}} $$

(36)

where σ _B is the Stefan-Boltzmann constant, and ε is the emissivity.

$$ D_{4} =\frac{G}{\rho C_{p} T_{\mathrm{o}}} $$

(37)

$$ D_{5} =\frac{\Delta H_{f} \varphi_{\infty}}{C_{p} T_{\mathrm{o}}} $$

(38)

where ΔH _f is the heat of fusion and ∅ _∞ is the ultimate (maximum achievable) crystallinity with the absolute crystallinity ∅ given by ∅ = x∅ _∞.

Dimensionless number D _c used in the evolution equations of c* tensor

$$ D_{\mathrm{c}} =\frac{2 \pi \rho R_{o} H_{o} L}{W \lambda_{\mathrm{a}} \left({x, T} \right)} $$

(39)

where λ _a (x, T) is calculated by Eq. 15.

Dimensionless numbers DS ₁ and DS ₂ used in the evolution equations of S tensor

$$ DS_{1} =\frac{2 \pi \rho R_{\mathrm{o}} H_{\mathrm{o}} L \sigma }{W \lambda_{\text{sc}} \left({x,T} \right)} $$

(40)

where λ _sc (x, T) is calculated by Eq. 14.

$$ DS_{2} =\frac{\lambda_{\text{sc}} \left({x,T} \right) W}{2 \pi \rho R_{\mathrm{o}} H_{\mathrm{o}} L} $$

(41)

Dimensionless number D _x used in the evolution equations of relative crystallinity x

$$ D_{x} =\frac{2 \pi \rho R_{\mathrm{o}} H_{\mathrm{o}} L m K_{\text{av}}}{W} $$

(42)

where K _av is given by Eq. 16.