3.5.1 Elasticity
The rheology of thermoplastic blends provides a concept for the relationship between microscopic structure and their processing capability. Figure 5 and Fig. 6 show the storage modulus (G') and the loss modulus (G'') versus angular frequency (ω) for the specimens, respectively.
Regarding the increase of the dynamic modulus with increasing frequency, it can be said, the loss modulus reflects the strength of the internal friction, which becomes more drastic as the shear rate increases. On the other hand, two factors affect the storage modulus: the number of oriented chain segments will increase obviously as the shear rate increases and, simultaneously, the oriented chain segments will have less time for disorientation 31. Therefore, at high frequencies, entangled chains have a short time to relax and increase the amount of storage modulus 32. It was also observed that the storage modulus (G') and the loss modulus (G'') increase with an increasing weight percentage of HDPE and TS in the blends.
The values G ' and G'' in the specimens shows a crossover point. The frequency (ω) at which the point of intersection occurs is called the crossover frequency ωc, 1/ωc is the relaxation time 33,34. The location of the crossover in oscillatory data )where G' = G") is often regarded as the boundary between solidlike and liquidlike behavior 35. At frequencies higher than the crossover point, the material is unable to relax sufficiently quick within the timescale of the applied oscillations and exhibits significant elastic behaviour (G' dominates over G"). Conversely, at lower frequencies, the relaxation behaviour is faster than the applied oscillations and the material exhibits viscous characteristics 36. According to Table 4, the crossover frequency for HDPE is lower than that of LDPE which shows that the elastic region of HDPE starts at lower frequencies. So, the increment in HDPE content in LDPE/HDPE/TS blends has increased λ. The addition of starch except for sample LD85/HD15/TS30 increased the crossover frequency (observing Fig. 7 and Fig. 8 for a better comparison). In the following, we also use the generalized Maxwell model to compare with the times obtained from crossover point.
Table 4
Gc and λ resulting from the intersection point of the LDPE/HDPE/TS film blends.
sample

ω c [rad/s]

λ [s]

GC [Pa]

LD

7.063

0.141

1.80\(\times\)104

HD

6.308

0.158

5.39\(\times\)104

LD95/HD5/TS15

9.28

0.107

2.13\(\times\)104

LD85/HD10/TS15

7.05

0.141

1.59\(\times\)104

LD85/HD15/TS15

6.94

0.144

1.87\(\times\)104

LD85/HD15/TS20

7.55

0.132

2.08\(\times\)104

LD85/HD15/TS25

8.91

0.112

2.73\(\times\)104

LD85/HD15/TS30

5.45

0.183

1.98\(\times\)104

Maxwell's generalized model is defined as follows, Eq. (5):
$$G\left(t\right)={\sum }_{i=1}^{n}\text{G}\text{i} \left[\text{exp}\left(\frac{t}{{\lambda }\text{i}}\right)\right]$$
5
Since all linear viscoelastic behaviour is governed by the Boltzmann superposition principle, which is based on the single material function, G (t), it is possible to relate the response to any sufficiently small or slow deformation to the linear relaxation modulus. In the case of small amplitude oscillatory shear, for example, it can be shown that G'(ω) and G"(ω) are the Fourier sine and cosine transforms of the relaxation modulus as Eq. (6) and Eq. (7), respectively:
$${G}^{{\prime }}\left(\omega \right)=\omega {\int }_{0}^{{\infty }}G\left(s\right)\text{sin}\left(\omega s\right)ds$$
6
$${G}^{{\prime }{\prime }}\left(\omega \right)=\omega {\int }_{0}^{\infty }G\left(s\right)\text{cos}\left(\omega s\right)ds$$
7
If a generalized Maxwell model is used to represent the relaxation modulus, the resulting functions are:
$${G}^{{\prime }}\left(\omega \right)=\sum _{i=1}^{N}\frac{{G}_{i}(\omega {\lambda }_{i}{)}^{2}}{\left[1+(\omega {\lambda }_{i}{)}^{2}\right]}$$
8
$${G}^{{\prime }{\prime }}\left(\omega \right)=\sum _{i=1}^{N}\frac{{G}_{i}\left(\omega {\lambda }_{i}\right)}{\left[1+(\omega {\lambda }_{i}{)}^{2}\right]}$$
9
Experimental values of storage or loss modulus can also be used to determine a set of Maxwell parameters. In this case suggests that the parameters be chosen such that:
\(\sum _{k=1}^{m}\left[{(G}^{{\prime }}\left({\omega }_{k}\right){G}_{k}^{{\prime }}{)}^{2}+({G}^{{\prime }{\prime }}\left({\omega }_{k}\right){G}_{k}^{{\prime }{\prime }}{)}^{2}\right]\) = minimum (10)
where \({G}^{{\prime }}\left({\omega }_{k}\right)\) and \({G}^{{\prime }{\prime }}\left({\omega }_{k}\right)\) are calculated by means of Equations (8) and (9). It was started by selecting the \({\lambda }_{i}\) equal to integer powers of ten from 10− 4 s to 103 s. Then the \({G}_{i}s\) were estimated by use of Eq. (10). \({G}_{k}^{{\prime }}\) and \({G}_{k}^{{\prime }{\prime }}\) are experimental data. Additional information is available in the book of John M. Dealy et al 37. By writing the code using MATLAB, we can achieve the desired results.
Tables 5 and 6, show the results including statistical parameters and the relaxation module as well as the relaxation times of the generalized Maxwell model for PEs and LD/HD/TS film blends which were obtained by fitting the experimental data of storage modulus and loss modulus versus angular frequency. Figures 9 and 10 show λ and G resulting from the Maxwell model with increasing the percentage of HDPE and TS for the LDPE/HDPE/TS film blends, respectively). The mean relaxation time (\(\stackrel{}{\lambda }\) ) was extracted from the relaxation spectra calculated using \({G}^{{\prime }}\) and \({G}^{{\prime }{\prime }}\) plot.
Table 5
λ and G resulting from the Maxwell model with increasing the percentage of HDPE for the LDPE/ HDPE/ TS film blends. (Comparison with experimental data of storage and loss modulus).
sample

LD

HD

LD95/HD5/TS15

LD90/HD10/TS15

LD85/HD15/TS15

Rsquare

0.9946

0.9941

0.9970

0.9970

0.9972

λ 1 [s]

0.0013

0.0011

0.0008

0.0009

0.0011

λ 2 [s]

0.0217

0.0151

0.0127

0.0147

0.01601

λ 3 [s]

0.2102

0.2256

0.2349

0.2547

0.2554

G1 [Pa]

174611

489580

239847

202999

207636

G2 [Pa]

56217

198995

67990

56591

61497

G3 [Pa]

21113

72496

28699

23025

27014

\(\stackrel{}{{\lambda }}\) [s] (\(\frac{\sum {{\lambda }}_{i}^{2}{G}_{i}}{\sum {{\lambda }}_{i}{G}_{i}})\)

0.1631

0.1877

0.2043

0.2186

0.2189

Table 6
λ and G resulting from the Maxwell model with increasing the percentage of TS for the LDPE/ HDPE/TS film blends. (Comparison with experimental data of storage and loss modulus).
Sample

LD85/HD15/TS15

LD85/HD15/TS20

LD85/HD15/TS25

LD85/HD15/TS30

Rsquare

0.9972

0.9965

0.9971

0.9938

λ 1 [s]

0.0011

0.0008

0.0009

0.0011

λ 2 [s]

0.0160

0.0119

0.0119

0.0121

λ 3 [s]

0.2554

0.2390

0.1700

0.3335

G1 [Pa]

207636

221206

262720

232499

G2 [Pa]

61497

77067

87492

73910

G3 [Pa]

27014

29791

36691

30174

\(\stackrel{}{{\lambda }}\) [s]

0.2189

0.2080

0.1425

0.3003

by comparing the relaxation modulus and the relaxation time resulting from the intersection point, also the relaxation modulus and the relaxation times resulting from the model for the LDPE and HDPE, it can be seen that parameter values for HDPE values are significantly higher than LDPE. The change in storage modulus is directly related to the orientation and disorientation of the chain segments, which are affected by frequency, temperature, molecular weight and molecular weight distribution 31,38. The occurrence of branching is often accompanied by a broadening of the molecular weight distribution (LDPE has a broad molecular weight distribution than HDPE). Physically, the introduction of long branches has two opposing effects. First, the radius of gyration Rg is decreased compared to that of a linear chain of the same molecular weight. The decreased Rg results in fewer entanglements and lower viscosity. The second effect of branching occurs when the branch length is sufficiently long to be entangled, i.e., when the molecular weight of the branch becomes comparable to the critical molecular weight for entanglement of a linear chain (Me). The overall entanglement network then has a much longer lifetime than that of a linear polymer network. The viscoelastic relaxation spectrum is extended to much longer relaxation times. This behaviour is consistent with the reptation model. A linear chain relaxes by diffusing out of its entanglement "tube." A branchedchain is attached at its branch points to at least two other chains, each in their own tubes. It cannot, therefore, diffuse out independently, and relaxation can occur only by processes requiring much longer times 37,39
According to the MFI, the molecular weight of HDPE is higher than that of LDPE. When the molecular weight is above Mc (the critical molecular weight for entanglements) the relaxation time of fractions with different molecular weights are separated widely. The fractions with a low molecular weight can relax quickly, while the fractions with a high molecular weight contribute to the dynamic moduli because of their slower relaxation. Polymer chains with a higher molecular weight form a more stable entanglement network which makes it difficult for the orientation of the chain segments along the direction of the shear flow. Thus, regarding the hindrance effect of the entanglements of the fractions with high molecular weight, the number of the oriented chain segments will not increase as much as the fractions with low molecular weight do with the shear rate increases. Therefore, the dynamic moduli of the polymers with a higher molecular weight will have a weaker dependence on the frequency than the polymers with lower Mw. So, the relaxation times and moduli for HDPE are higher when compared to LDPE 31,40.
Increasing the weigh percent of HDPE in the blends, regarding that HDPE, does not have a long side branch and has a higher molecular weight than LDPE, causes pack closer of matrix chains and creates a better entanglement network, so it increases the relaxation time resulting from the crossover point and λis resulting from the model. According to Table 4, the relaxation time for the sample LD90/HD10/TS15 is very close to that of the LDPE and for the LD95/HD5/TS15 specimen is less than that of the LDPE. In the LD95/HD5/TS15 specimen with the lower content of HDPE the greater the effect of starch particles results in a weaker entanglement network, so less relaxation time was observed.
Except for LD85/HD15/TS30, increasing starch loading, has reduced the relaxation time obtained from the crossover point and the generalized Maxwell model. This reduction can be attributed to the dispersion of starch particles within LDPE/HDPE chains and reduction of the close packing in the polymeric chains. But at higher loading of the filler, where the particleparticle interactions are stronger than the particlematrix ones, the agglomeration of particles may occur and results in the immobilization of the polymeric chains which retardates the stress relaxation rate. Therefore, the melt behavior tends to be more solidlike 2,7. in the LD85/ HD15/TS30 specimen (see Fig. 8 and Fig. 10), relaxation time was increased. The storage modulus of thermoplastic starch is higher than the loss modulus in the whole frequency range of 0.01–100 rad/s, so in this blend the behaviour of TS is dominant and the crossover point occurs at low frequencies 41.
It should be noted that calculated mean relaxation times are in accordance with frequency crossover data. With respect to the small statistical errors in fitting the generalized Maxwell model to rheological data, with comparison of the relaxation time and modulus values of LDPE/HDPE/TS blends more knowledge about the structure of the blends can be achieved.
3.5.2 Viscosity
Figure 11 shows the relationship between complex viscosity ɳ* and frequency (ω) for the samples. It is clear that the complex viscosity of all samples decreases with increasing angular frquency, which indicates the shearthinning behaviour of the prepared melts 42. According to the Fig. 11a and b it is observed that with increasing the content of HDPE and TS in the composite films, the complex viscosity of their melts increases which is more obvious at lower frequencies. The increasing complex viscosity with an increment TS loading is due to the attractive interaction between matrix and themoplastic starch. To obtain zero shear viscosity and other flow properties of the prepared samples with considering CoxMerz rule (the term CoxMerz rule generally refers to the (near) equality between the shear rate \(\dot{\gamma }\) dependence of the nonlinear steadystate shear viscosity \(ɳ\) and the angular frequency dependence of the linear complex viscosity η*) 43 complex viscosity data are fitted by the Cross (see Eq. (11)) and Power law models (see Eq. (13)) which are given in the following.
$$ɳ=\frac{{ɳ}_{\infty }+\left({ɳ}_{0}{ɳ}_{\infty }\right)}{1+(\lambda \dot{\gamma }{)}^{c}}$$
11
where ɳ is the viscosity at the shear rate \(\dot{\gamma }\), the parameters \({\text{ɳ}}_{0}\)and \({\text{ɳ}}_{{\infty }}\) are the limit values of the viscosity at \(\dot{\gamma }=0\)and \(\dot{\gamma }={\infty }\)and λ is a characteristic time constant which is related to the relaxation time of the chains. For polymer melts it is\({\text{ɳ}}_{0}\gg {\text{ɳ}}_{{\infty }}\) so we have\(\frac{{\text{ɳ}}_{{\infty }}}{{\text{ɳ}}_{0}}\ll 1\) and the Eq. (11) can be converted to the Eq. (12):
$$\text{ɳ}=\frac{{\text{ɳ}}_{0}}{1+({\lambda }\dot{\gamma }{)}^{c}}$$
12
c = 1n which n is the Power law behavior index 44,45.
Table 7
Cross model parameters for the LDPE/HDPE/TS film blends.
sample

\({ɳ}_{0}\)[Pa. s]

λ [s]

n

Rsquare

AdjRsq

LD

2.217\(\times\)104

1.820

0.3661

0.9971

0.9969

HD

8.242\(\times\)104

1.925

0.2179

0.9827

0.9800

LD95/HD5/TS15

2.241\(\times\)104

2.201

0.4149

0.9999

0.9999

LD90/HD10/TS15

2.517\(\times\)104

4.462

0.4459

0.9995

0.9995

LD85/HD15/TS15

3.387\(\times\)104

10.440

0.5256

0.9985

0.9982

LD85/HD15/TS20

3.237 \(\times\)104

5.683

0.4791

0.9992

0.9991

LD85/HD15/TS25

4.014\(\times\)104

5.525

0.4647

0.9997

0.9997

LD85/HD15/TS30

5.530\(\times\)104

14.350

0.4949

0.9993

0.9992

It is observed from Table 7 that the value of λ for HDPE is slightly higher than LDPE, and increasing the weigh percent of HDPE in the blends increases the λ. In contrast, the increasing weight of TS causes to decrease in the value of λ which is similar to \(\stackrel{}{{\lambda }}\) trend obtained from the generalized Maxwell model.
The value of ɳ0 is higher for HDPE than that of LDPE and increasing the content of HDPE and TS in the blend increases the value of zero shear viscosity. At the zero rate of shear, the chains are in the entanglement state in which the zero viscosity is calculated in this case. Entanglement between polymer chains is the main factor of viscosity dependence on molecular weight. The higher molecular weight of HDPE causes a higher zero shear viscosity and with increasing the weight percentage of HDPE in the blends, an increase in ɳ0 is observed 38,46.
The exponent of c is dependent on molecular weight distribution and approaches an upper limit of unity for a monodisperse linear polymer 44. As can be seen, the value of (n) was increased with increment of HDPE/LDPE ratio from 0.41 to 0.52 and decreased from 0.52 to 0.47 with TS weigh percent in the melts.
The flow curves of the viscosity plotted against the shear rate in logarithmic scales of polymers at various temperatures are approximately equivalent to parallel straight lines. A relationship of straight lines on a logarithmic scale indicates that the viscosity and shear rate can be described by a powerlaw Eq. 47. This equation is the most common model used to express shear thinning behavior, The equation is as follows:
$$\text{ɳ}=\text{m}{\dot{\gamma }}^{n1}$$
13
where \(\text{ɳ}\) is the viscosity, \(\dot{\gamma }\) is the shear rate, m is the consistency index, and n is the powerlaw melt index. This relationship has poor performance in fitting in with data at shear rates close to zero (Newtonian region) 42.
Table 8 shows the Power law model parameters for polyethylene film samples and LDPE/HDPE/TS film blends.
Table 8
Powerlaw model parameters for the LDPE/HDPE/TS film blends.
Sample

n

m [pa.sn]

Rsquare

AdjRsq

LD

0.5046

9191

0.9987

0.9984

HD

0.4342

3.575\(\times\)104

0.9988

0.9985

LD95/HD5/TS15

0.4487

1.123\(\times\)104

0.9992

0.9991

LD90/HD10/TS15

0.4495

9636

0.9995

0.9994

LD85/HD15/TS15

0.4483

1.144\(\times\)104

0.9993

0.9992

LD85/HD15/TS20

0.4432

1.227\(\times\)104

0.9992

0.9991

LD85/HD15/TS25

0.4435

1.482\(\times\)104

0.9992

0.9990

LD85/HD15/TS30

0.4629

1.338\(\times\)104

0.9991

0.9989

According to Table 8, the value of m for HDPE is higher than LDPE, and the addition of HDPE to the mixture has resulted in an increase in m (The value of m for sample LD90/HD10/TS15 is approximately equal to the LDPE). Increasing TS to the mixture has increased the value of m (The presence of glycerol in starch as a plasticizer increases macromolecular mobility and decreased the rheological properties) 30,48.
The value of (n) for the HDPE is lower than that of LDPE. The power law exponent (n) of various polyethylene (PE) grades, is between 0.3\(\text{t}\text{o}\)0.6, and depends on molecular weight and longchain branching 34,49. Linear narrow molecular weight distribution polymers are more viscous (less shear thinning) than their with broad distribution of the same average molecular weight 42. A low melt index means a high molecular weight with a greater sensitivity to shear. The higher molecular weight in HDPE dominates the broad molecular weight distribution of LDPE and has shown more shear thinning behaviour.