Theoretical correlation of linear and non-linear rheological symptoms of long-chain branching in polyethylenes irradiated by electron beam at relatively low doses

Abstract

Dynamic and transient shear and elongation flow experiments along with gel permeation chromatography (GPC) and differential scanning calorimetry (DSC) analysis are performed on linear low-density polyethylenes (LLDPEs) irradiated at doses below 25 kGy. GPC data indicate no changes in the molar mass distribution, and there are almost no changes in melt and crystallization temperatures, likewise. Contrary, dynamic shear rheological behavior including thermorheological complexity, type of reduced van Gurp-Palmen curves, and zero shear-rate viscosities all disclose growing levels of long-chain branching with irradiation dose. An inverse tube model is developed for binary blend of linear and star chains and used to extract the fraction of the branched components. Modeling results reveal progressive increase in the length and fraction of star chains, as evidenced by appearance of an anomalous double overshoot in the transient shear viscosities. Detection of strain hardening in extensional stress growth coefficient data, well-quantified by molecular stress function model, is also in agreement with the predictions of tube model.

Appendices

Tube-based model for linear rheology

A tube-based model was developed to predict the linear rheological behavior of irradiated samples. It was assumed that irradiation at low doses as used here can lead to formation of symmetric star chains and fraction of more complicated structures is negligible.

Symmetric stars can mainly relax by contour length fluctuations according to the following probability:

$$ {p}_{\mathrm{fluc}}\left(x,t\right)= \exp \left(\frac{-t}{\tau_{\mathrm{fluc}}(x)}\right) $$

(a.1)

The specific lifetime for fluctuations is determined by free Rouse motions for the segments near extremities, while deeper segments need to overcome the potential penalty to retract:

$$ {\tau}_{\mathrm{early}}(x)=\frac{9{\pi}^3}{16}{\tau}_e{\times}^4{Z}^4 $$

(a.2)

$$ {\tau}_{\mathrm{late}}(x)={\tau}_0 \exp \left(\frac{U(x)}{kT}\right) $$

(a.3)

$$ U(x)=\frac{3kT}{2N{b}^2}{\left({L}_{eq}x\right)}^2+\mathrm{cons} $$

(a.4)

Since the relaxation times of deep segments are exponentially separated in time space, they can use the already relaxed part of the chain as solvent to dilate the confining tube according to dynamic dilution. Transition between the early and late fluctuations modes happens at the segment where the potential barrier is equal to the system’s thermal energy.

Besides contour length fluctuations, the linear chains can also reptate at the specific reptation lifetime:

$$ {p}_{\mathrm{rept}}\left(x,t\right)={\sum}_{i,\mathrm{odd}}\frac{4}{i\pi } \sin \left(\frac{i\pi x}{2}\right) \exp \left(\frac{-{i}^2t}{\tau_{\mathrm{rept}}}\right) $$

(a.5)

$$ {\tau}_{\mathrm{rept}}=3{\tau}_e{Z}^3{\phi}_{\mathrm{active}}^{\alpha }(t) $$

(a.6)

where φ _active(t) is the fraction of already relaxed segments that reptation can use to dilate the confining tube based on the Graessley’s criterion and α = 1 is the dilution exponent (Van Ruymbeke et al. 2005a).

The fraction of oriented chains can be calculated at each time, from the fraction of segments that are relaxed neither by reptation nor by fluctuations:

$$ \phi (t)={\sum}_i{\varphi}_i{\int}_0^1\left({p}_{\mathrm{rept}}\left(x,t\right){p}_{\mathrm{fluc}}\left(x,t\right)\right)dx $$

(a.7)

where ϕ _i is the weight fraction of chain i. As the other chains present in the system can also relax, the confining tube also relaxes by constraint release process. The probability of tube relaxation is equal to the probability of chain relaxation, while it cannot be faster than what lateral Rouse movements may allow:

$$ {\phi}_{CR}\left({t}_i\right)\ge {\phi}_{CR}\left({t}_{i-1}\right){\left(\frac{t_{i-1}}{t_i}\right)}^{0.5} $$

(a.8)

The transient modulus can be calculated by inclusion of high frequency fast modes and longitudinal slow modes of Rouse motion:

$$ G(t)/{G}_N^0=\phi (t){\phi}_{CR}^{\alpha }(t)+{F}_{\mathrm{Rouse},\mathrm{long}}(t)+{F}_{\mathrm{Rouse},\mathrm{fast}}(t) $$

(a.9)

$$ {F}_{\mathrm{Rouse},\mathrm{long}}(t)={\sum}_i\frac{\varphi_i}{4{Z}_i}{\sum}_{j=1}^{Z_{i-1}} \exp \left(\frac{-{j}^2t}{\tau_{\mathrm{Rouse},i}}\right) $$

(a.10)

$$ {F}_{\mathrm{Rouse},\mathrm{fast}}(t)={\sum}_i\frac{5{\varphi}_i}{4{Z}_i}{\sum}_{j={Z}_i}^N \exp \left(\frac{-2{j}^2t}{\tau_{\mathrm{Rouse},i}}\right) $$

(a.11)

where Z corresponds to the chain’s span, and the specific Rouse relaxation time is:

$$ {\tau}_{\mathrm{Rouse}}={\tau}_e{Z}^2 $$

(a.12)

The molar mass distributions of both linear and star chains were considered to follow log-normal distribution function with similar PDI (Table 2). The mother linear chain had a constant molar mass as shown in Table 3, while the molar mass and fraction of star chains were adjusted using Nelder-Mead simplex method to capture the dynamic moduli measured by frequency sweep. The following error function was defined, and the obtained values are shown in Table 3:

$$ \chi =\frac{1}{n}{\sum}_{i=1}^n\left[\frac{{\left({G}_{exp}^{\prime }(i)-{G}_{the}^{\prime }(i)\right)}^2}{G_{exp}^{\prime }{(i)}^2}+\frac{{\left({G}_{exp}^{"}(i)-{G}_{the}^{"}(i)\right)}^2}{G_{exp}^{"}{(i)}^2}\right] $$

(a.13)

Molecular stress function model

Extensional data were analyzed using molecular stress function (MSF) constitutive equation which is a molecular model based on Doi-Edwards and reptation concept with taking to account the backbone stretching. This model can predict and quantify the strain hardening in extensional flows using two non-linear parameters, β and f _max (Rolón-Garrido 2014). Constant β represents the ratio of entanglements in polymer, Z = Z _b  + N _br  × Z _a , to its backbone, Z _b , and governs the slope of strain hardening. Parameter f _max is related to the maximum stretching of the backbone and determines the steady-state value of the stress growth coefficient, ${\eta}_{E, \max}^{+}$ in extensional flows. Here, we used a version of MSF model which was presented by Abbasi et al. (2012) and verified using different branched polymers in different non-linear deformations (Abbasi et al. 2013):

$$ \boldsymbol{\upsigma} (t)={\varSigma}_{i=1}^N{\int}_{-\infty}^t\frac{g_i}{\tau_i}{e}^{\frac{\left(t-{t}^{\prime}\right)}{\tau_i}}{f}^2\left(t,{t}^{\prime}\right){\mathbf{S}}_{DE}^{IA}\left(t,{t}^{\prime}\right){dt}^{\prime } $$

(b.1)

$$ {\mathbf{S}}_{DE}^{IA}=5\mathbf{S}=5\left[\left(\frac{1}{J-1}\right)\mathbf{B}-\left(\frac{1}{\left(J-1\right){\left({I}_2+13/4\right)}^{0.5}}\right)\mathbf{C}\right] $$

(b.2)

$$ J={I}_1+2{\left({I}_2+13/4\right)}^{0.5} $$

(b.3)

$$ \frac{\partial f}{\partial \varepsilon }=\frac{1}{2}\beta f\left({S}_{11}-{S}_{22}-\frac{f^2-1}{f_{\max}^2-1}\sqrt{S_{11}+0.5{S}_{22}}\right) $$

(b.4)

where bolded elements σ, S, B, and C are stress, measure of strain, Finger, and Cauchy tensors, respectively. I ₁ and I ₂ are the trace (invariants) of B and C, respectively. Relaxation spectra g _i and τ _i are obtained by fitting Maxwell model on linear viscoelastic data G′ and G″. Stretching function, f, shows the stretch of backbone, which causes the strain hardening phenomenon in the polymer chains and is a function of Hencky strain, ε, and non-linear parameters, β and f _max. Equations b.1–b.4 represent an integral time-strain separable constitutive equation, which will be fitted on the uniaxial stress growth coefficient data by adjusting the β and f _max as fitting parameters. These parameters were used as a bridge between the strain hardening criteria and branching content.