Using constrained mixture design method for optimizing the properties of organoclay filled ethylene-octene copolymer nanocomposites

Abstract

The mixture design method was used to model the physical and mechanical properties of ethylene-octene copolymer (EOC) nanocomposite containing organically modified montmorillonite (OMT) which were cross-linked dynamically by various amounts of dicumyl peroxide (DCP). A mixture design technique with three components was employed to assess the correlations between the selected properties of the nanocomposites and the component values. For this purpose, EOC, OMT and DCP content were selected as the components. The influences of these components were studied on the tensile strength, modulus at 100% strain, strain at break, x-ray peak intensity and the initial slope of the logarithm of storage modulus versus the logarithm of angular frequency of the nanocomposites prepared. The regression equations of the models as well as contour plots were generated for the properties studied. Good agreements were found between the experimental results and those predicted by the models. The contour plots of each property were overlaid within the applied constraints to discover the combination of factor ranges that provided the nanocomposite with optimal performance.

Full text

Mater. Res. Express 7(2020)015321 https://doi.org/10.1088/2053-1591/ab61aa
PAPER
Using constrained mixture design method for optimizing the
properties of organoclay filled ethylene-octene copolymer
nanocomposites
Masoud Tayefi, Mohammad Razavi-Nouri
1
and Alireza Sabet
Iran Polymer and Petrochemical Institute, PO Box: 14975-112, Tehran, Iran
1
Author to whom any correspondence should be addressed.
E-mail: Tayefimasoud@gmail.com,m.razavi@ippi.ac.ir and a.sabet@ippi.ac.ir
Keywords: ethylene-octene copolymer (EOC), organoclay, nanocomposite, optimization, mixture design method
Abstract
The mixture design method was used to model the physical and mechanical properties of ethylene-
octene copolymer (EOC)nanocomposite containing organically modified montmorillonite (OMT)
which were cross-linked dynamically by various amounts of dicumyl peroxide (DCP). A mixture
design technique with three components was employed to assess the correlations between the selected
properties of the nanocomposites and the component values. For this purpose, EOC, OMT and DCP
content were selected as the components. The influences of these components were studied on the
tensile strength, modulus at 100% strain, strain at break, x-ray peak intensity and the initial slope of
the logarithm of storage modulus versus the logarithm of angular frequency of the nanocomposites
prepared. The regression equations of the models as well as contour plots were generated for the
properties studied. Good agreements were found between the experimental results and those
predicted by the models. The contour plots of each property were overlaid within the applied
constraints to discover the combination of factor ranges that provided the nanocomposite with
optimal performance.
1. Introduction
In conventional one-factor-at-a-time approach, only one variable is altered while the others are kept constant.
Therefore, using this method is time consuming and expensive. Moreover, the technique is unable to identify
interactions among the factors [1,2]. Design of Experiment (DoE)methods, which are now extensively applied
to analyze the influence of multi factors on the responses, allows us to study a number of factors simultaneously
[3,4]. Experimental designs are classified in three general types, namely; screening method, response surface and
mixture design. Screening designs such as factorial design (FD)and fractional factorial design (FFD)are mainly
employed to identify the most significant factors affected on a response [5,6]. It is worth to mention that the
traditional experimental design methods, such as FD, assume that all variables can be controlled and operated
independently of one another. In a composite material experiment, this could be attained if a researcher is only
concerned in varying one or two components engaged [7]. The response surface methods (RSM)such as Box-
Behnken design, central composite and mixture designs will initially aid users to study the effects of the factors
on each property of a material by carrying out an affordable number of experiments [4,8–13]. Moreover, the
interactions between the factors can be evaluated by the method [14–16]. In addition, the designed experiments
could help an experimenter to find the nonlinear relationships between the factors and properties studied
[17,18]. Besides process optimization, RSM methods are capable of producing an approximate continuous
surface and contour plots for evaluating the interactions [18–20]. It should also be mentioned that while central
composite and Box–Behnken designs are largely carried out when the factors are process variables, mixture
design is employed for formulation factors [15]. In mixture design, the sum of all ingredients of a mixture is
assumed to be 100% or 1 except when any constant mixture factors are present. The shape of the experimental
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26 October 2019
REVISED
27 November 2019
ACCEPTED FOR PUBLICATION
12 December 2019
PUBLISHED
6 January 2020
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zone will be a simplex if all mixture factors change from 0 to 100% [21]. In many cases, the preparation of the
mixtures in the whole range of proportion values of their components, may not be of any interest or even
impossible to be studied. Thus, it could be suitable to establish low and high constraints for a number of
components [22]. Where the factors have different limitations, the restricted experimental zone turns into an
irregular polyhedron within the simplex [21]. The mixture design method has extensively been employed to
study and model the influence of compositions on different properties of a variety of mixtures such as polymers
and other materials [15–17,22–25].
Ethylene-1-octene copolymers (EOCs)are made from the copolymerization of ethylene and 1-octene [26].
The new class of the copolymer is now synthesized by metallocene single site catalysts [27]. EOCs show
outstanding physical and mechanical properties such as resistance to solvents, high elongation at break,
environmental resistance and dielectric properties. Therefore, they can be used in the form of blends and
composites in a variety of applications [28].
The preparation of polymer-based nanocomposites has comprehensively been investigated during the past
two decades due to their unique physical and mechanical properties. Among the nanofiller used, polymer/
organoclay (OMT)nanocomposites have received widespread attentions because of their attractive behaviors
and potential applications in many areas [29,30].
Cross-linking (curing)is also widely used to improve the thermal and mechanical properties of some
polymers, especially elastomers. Peroxides are among the cross-linking agents and have numerous advantages
compared to other curatives or curing agents [31].
The main objective of this work is to analyze the significance of EOC, dicumyl peroxide (DCP)and OMT
content on different physical and mechanical properties of the EOC-based nanocomposites using a constrained
mixture design approach. Five responses of tensile strength, modulus at 100% strain, strain at break, XRD peak
intensity and the initial slope of the logarithm of storage modulus (G′)versus the logarithm of angular frequency
(ω)at the terminal zone were selected for this evaluation. The results were employed to determine the mixture
design models for the properties investigated. The contour plots were then applied to understand the effect of
each component on individual responses and finally the optimum formulation was proposed.
2. Experimental
2.1. Materials
EOC (LC370)with 38 wt% octene content, density of 0.87 g cm
−3
and MFI of 3.0 g/10 min (190 °C/2.16 kg)
was supplied from LG Chem Ltd (South Korea). Cloisite 15 A (C15A)with the interlayer spacing of 31.5 Å,
density of 1.66 g cm
−3
and cation exchange capacity of 125 meq/100 g was purchased from Southern Clay
Products (USA). Dihydrogenated tallow dimethyl ammonium was used as a modifier for C15A. DCP with the
purity of 99% was supplied from Concord Chemical Ind. Co. Ltd (Taiwan)and used as a cross-linking agent.
2.2. Sample preparation
In order to remove any traces of moisture, C15A was dried in a vacuum oven at 80˚C for 24 h before being mixed
with the other materials. The nanocomposites were prepared using a two-step melt mixing method. All of the
mixtures were prepared using a Brabender internal mixer (Germany)equipped with a pair of roller blades.
EOC/C15A masterbatch containing 10 wt% OMT was first prepared at 60 rpm and 100 °C for 10 min. The
mixture was then diluted with the addition of appropriate amounts of EOC to prepare the nanocomposites
containing 1 to 5 wt% C15A. The rotor speed, temperature and mixing time were set to be 60 rpm, 150 °C and
5 min, respectively. Thenanocomposites were alsodynamically cured using0.5 and 1 wt%DCP. Reachingthe
mixing torque to its maximum value was selected as criteria to finish the cross-linking stage and remove the
nanocomposites from the mixer. Square plaques with the thickness of 1 mm were then prepared by compression
molding of the materials at 150°C and 15 MPa for 5 min using a Toyosiki Mini Test Hydraulic Press (Japan).
2.3. Low angle X-ray diffraction
The X-ray analysis of each sample was carried out using a X’pert PRO MRD X-ray diffractometer (PANalytical,
The Netherlands)operated with CuKαradiation (λ=1.542 Å)at 40 kVand 40 mA. The2θangles in the range
0.7°–10°with an exposure time of 2 s and scanning rate of 0.02 °/s was selected to scan the samples.
2.4. Mechanical properties
A Universal Testing Machine (Santam-SMT20, Iran)was employed to determine the tensile properties of the
nanocomposites. At least five dumbbell-shaped specimens with 35×2×1mm
3
dimensions were punched
out from the plates for each composition and tested with the cross-head speed of 500 mm min
−1
. The results
obtained were then averaged to find the mean and standard deviation values.
2
Mater. Res. Express 7(2020)015321 M Tayefiet al

2.5. Rheological properties
A stress controlled rheometer (Anton Parr MCR501, Austria)was used to study the rheological behavior of the
samples. Disk-type parallel plates with 25 mm diameter and 1 mm gap were employed to determine the dynamic
oscillatory shear responses of the nanocomposites in their linear viscoelastic regions. The values of G′were
obtained in the range 0.01–100 rad s
−1
of ωat the temperature of 110 °C and strain of 0.5% under a nitrogen
atmosphere.
3. Experimental design
According to the mixture design method, the formulations of a series of EOC-based nanocomposites were
considered with three components of EOC, DCP and OMT. In this work, the amount of OMT and DCP were
selected to be in the range 1–5 wt% and 0–1 wt%, respectively. This is because, the physical and mechanical
properties of the nanocomposites were unfavorably decreased or at most remained nearly unchanged beyond
the two values mentioned above.
To model the system, the scale of the mixture design should be modified owing to the existence of some
constraints for the components selected. A lower-bound pseudo-component was considered for our system to
find the fitting model. The lower pseudo-components (X
i
values)were determined using the following equations
[32];
=-
-
XRL
L100
1
i
ii
()
where i=1, 2, 3, ..., n and
å
=
=
LL
i
n
i
1
å
=
=
X1
i
n
i
1
where R
i
is the real value and L
i
is the lower bound of each component. In this study n=3; demonstrated as the
number of mixture variables in this work (X
1
,X
2
, and X
3
). The range of actual components and corresponded
pseudo-components are shown in table 1. Figure 1shows the schematic layout of the design. As it can be seen, a
parallelogram shaped image containing nine design points is constructed due to the applied constraints. For
confirmation of the accuracy of the models obtained by experimental design, one additional point located inside
of the experimental region was also considered. Design-Expert
®
software version 10 was employed to analyze the
effects of the components on the properties evaluated in this study. A regression was performed on the
experimental data wherein the responses were estimated according to practical correlations between the
predicted responses and the components. To fit a model containing the components studied, the least square
technique was utilized. It was carried out by the minimization of the residual error measured via the sum of
square deviations between the real and predicted responses. This includes the approximations of the regression
coefficients, i.e. the coefficients of the models’components and the intercepts. It is worth to mention that, the
statistical significance of the computed coefficients of the equations should then be examined. Therefore, three
examinations were executed to evaluate (1)the significance of the regression model, (2)the significance of each
coefficient of the models and (3)the lack of fit. Moreover, each examination should be carried out to check
whether the model could explain the experimental data. This was achieved, in this work, by calculating the
various coefficients of determination (R
2
)in which their values vary in the range 0 to 1. Additionally, the
capability of the model was also studied by checking of residuals [4]. The term describes the difference between
the predicted and the observed responses. The residuals were checked here using the normal probability plots of
the residuals and the plots of the residuals against the responses predicted by the models. The points on the
Table 1. Real proportions of the components in the mixture and in pseudo-
components.
Range of each
component
(wt%)Pseudo-component value
Mixture components Low High Symbol Low High
EOC content 94 99 X
1
01
DCP content 0 1 X
2
0 0.2
OMT content 1 5 X
3
0 0.8
3
Mater. Res. Express 7(2020)015321 M Tayefiet al

former plots should form a straight line, if the model is satisfactory. Furthermore, the points on the latter plots
should not form any specific structure, i.e., no obvious pattern should be detected.
A three component mixture design method was adopted to find a correlation between the responses which
are the tensile strength, modulus at 100% strain, strain at break, XRD peak intensity and the initial slope of log Gʹ
against log ω. The linear, quadratic and special cubic models were employed to achieve the best accurate model
to fit the experimental results. The models are as follow;
aa a=+ +YX X X 2
11 22 3 3
()
aa a a a a=+ + + + +Y X X X XX XX XX 3
1 1 2 2 3 3 12 1 2 13 1 3 23 2 3
()
aa a a a a a=+ + + + + +Y X X X XX XX X X XX X 4
1 1 2 2 3 3 12 1 2 13 1 3 23 2 3 123 1 2 3
()
where Y represents the responses (dependent variables)and all the α
i
values are the numerical coefficients of the
models.
The experimentally determined results are listed in table 2. It is worth to mention that the number of runs
required for the analysis was 10 i.e. 9 for the pseudo-component values used in the analysis and 1 for replication
of the center point. It should also be noted that the figures related to the X-ray diffraction response and
rheological behavior of the materials studied have been reported earlier by the authors [33].
Figure 1. Layout of the mixture design with its constraints for the three components (the values in the bracket define the pseudo-
component of each component at that specific point).
Table 2. The proposed compositions according to the mixture design method and their mechanical properties,XRD peakintensities and
rheological values.
Pseudo-components Dependent variables
Run EOC DCP OMT
Tensile
strength (MPa)
Modulus at 100%
strain (MPa)
Strain at
break (%)
XRD peak
intensity (Counts)
Slope of log G′
versus log ω
1 1 0 0 11.27±1.09 2.03±0.05 1846±171 6464 1.33
2 0.9 0.1 0 6.62±0.01 2.53±0.07 807±79 6421 0.40
3 0.8 0.2 0 8.92±0.39 5.19±0.12 193±32 2823 0.23
4 0.6 0 0.4 14.83±0.77 1.97±0.09 2167±134 15002 1.27
5 0.5 0.1 0.4 8.27±0.30 2.66±0.03 817±36 11369 0.37
6
a
0.5 0.1 0.4 8.00±0.33 2.69±0.02 800±30 11485 0.36
7 0.4 0.2 0.4 9.35±0.47 6.23±0.30 182±35 5221 0.22
8 0.2 0 0.8 15.65±0.57 1.92±0.09 2196±88 22594 1.22
9 0.1 0.1 0.8 8.13±0.29 2.91±0.14 808±61 17549 0.32
10 0 0.2 0.8 9.82±0.35 6.21±0.12 176±7 7014 0.19
a
Replication of the center point.
4
Mater. Res. Express 7(2020)015321 M Tayefiet al

4. Results and discussion
4.1. Analysis of variance for all responses
The analysis of variance (ANOVA)was carried out in order to quantify the effects of selected components and
their interactions on the dependent variables. The significance of each term can be determined based on its
probability value (P-value). Significant term should have the probability value more than 95% (P-value0.05)
and the probability of insignificant term will have the value less than 95% (P-value0.05)[34,35]. Therefore,
the insignificant terms were omitted here from the final analysis and results.
The ANOVA results for the analysis of tensile strength, modulus at 100% strain, strain at break, XRD peak
intensity and the initial slope of log Gʹagainst log ωare tabulated in tables 3to 7, respectively. Table 3reveals that
a special cubic model can be used to fit the tensile strength response. The P-value of the model was calculated to
be 0.0007 (probability of >99%)which is in agreement with the fact that the model was highly significant. On
the contrary, the P-value of lack of fit(0.3958)indicated that the lack of fit was insignificant and the model fitted
the data satisfactorily. The interactions between the components of EOC
*
DCP, EOC
*
OMT, DCP
*
OMT and
EOC
*
DCP
*
OMT were significant with the P-value of 0.0003 (probability of >99%), 0.0150 (probability of
≈99%), 0.0003 (probability of >99%)and 0.0436 (probability of ≈96%), respectively.
Table 4shows that a quadratic model can be suitable to fit the modulus at 100% strain with the P-value of
<0.0001 (probability of >99%). The P-value of lack of fit was found to be 0.1414, revealing its insignificance.
The two-component interactions of EOC
*
DCP and DCP
*
OMT were significant with the P-value of 0.0015 and
0.0023 (probability of >99%), respectively. However, the P-value of EOC
*
OMT was more than 0.05 which
means that the interaction was not significant. Therefore, the interaction was eliminated from the equation.
In order to fit the strain at break, a special cubic model was recommended according to the data presented in
table 5. The analysis of variance for the model showed that the P-value of the model was less than 0.0001
Table 3. Analysis of variance for tensile strength.
Factor SS
a
Df
b
MS
c
F-value P-value
Model 80.09 6 13.35 172.11 0.0007
Linear mixture 38.78 2 19.39 250.03 0.0005
EOC
*
DCP 30.66 1 30.66 395.29 0.0003
EOC
*
OMT 1.98 1 1.98 25.52 0.0150
DCP
*
OMT 30.54 1 30.54 393.78 0.0003
EOC
*
DCP
*
OMT 0.88 1 0.88 11.31 0.0436
Residual 0.23 3 0.078
Lack of fit 0.20 2 0.098 2.69 0.3958
Pure error 0.036 1 0.036
Cor total 80.33 9
a
Sum of Square: Sum of the squared differences between the average values and
the overall mean.
b
Degree of freedom.
c
Mean of Square: Sum of squares divided by the degree of freedom.
F-value: Test for comparing term variance with residual variance.
P-value: Probability of seeing the observed F-value if the null hypothesis is true.
Residual: Consists of terms used to estimate the experimental error.
Lack of fit: Variation of the data around the fitted model.
Pure error: Variation in the response in replicated design points.
Cor total: Totals of all information corrected for the mean.
Table 4. Analysis of variance for modulus at 100% strain.
Factor SS Df MS F-value P-value
Model 0.14 4 0.035 267.25 <0.0001
Linear mixture 0.13 2 0.067 511.59 <0.0001
EOC
*
DCP 5.213E-003 1 5.213E-003 39.66 0.0015
DCP
*
OMT 4.287E-003 1 4.287E-003 32.61 0.0023
Residual 6.573E-004 5 1.315E-004
Lack of fit 6.514E-004 4 1.629E-004 27.71 0.1414
Pure error 5.878E-006 1 5.878E-006
Cor total 0.14 9
5
Mater. Res. Express 7(2020)015321 M Tayefiet al

(probability of >99%)and the P-value of lack of fit was 0.5217. Consequently, the model was convincing from
statistical point of view without any significant lack of fit in the range of the components examined. The
interactions of EOC
*
DCP, EOC
*
OMT, DCP
*
OMT and EOC
*
DCP
*
OMT were effective for the determination
of the values of the strain at break with the P-value of 0.0001 (probability of >99%), 0.0368 (probability of
>96%), 0.0002 (probability of >99%)and 0.0469 (probability of >95%), respectively.
As it can be observed in table 6, a quadratic model with the P-value of <0.0001 (probability of >99%)could
be suggested to fit the XRD peak intensity results. It should be noted that the lack of fit was insignificant with the
P-value of 0.1339. The interactions of EOC
*
DCP and DCP
*
OMT were significant in calculating the XRD peak
intensity values with the P-value of 0.0009 (probability of >99%)and 0.0093 (probability =99%), respectively.
However, the effect of EOC
*
OMT was negligible due to its high P-value which was more than 0.05. Therefore,
this interaction was omitted from the equation.
The analysis of variance showed that a quadratic model could be used to fit the initial slope of log G′against
log ω(see table 7). The P-value of the model was less than 0.0001 (probability of >99%)and that of lack of fit was
0.4048. This was also consistent with the fact that the model was statistically convincing and fitted the results
successfully. The two-component interactions of EOC
*
DCP and DCP
*
OMT were effective in influencing the
initial slope of log G′versus log ωwith the P-value of <0.0001 (probability of >99%)for both. However, the
P-value of EOC
*
OMT was higher than 0.05, indicating that the interaction was not effective in evaluating the
quantity. Therefore, this two-component interaction was eliminated from the results.
The regression coefficients and their standard errors for all the models are tabulated in table 8. The models
are based on the pseudo-component coding.
Table 5. Analysis of variance for strain at break.
Factor SS Df MS F-value P-value
Model 1.69 6 0.28 5530.42 <0.0001
Linear mixture 1.66 2 0.83 16223.25 <0.0001
EOC
*
DCP 0.031 1 0.031 615.30 0.0001
EOC
*
OMT 6.614E-004 1 6.614E-004 12.95 0.0368
DCP
*
OMT 0.027 1 0.027 524.18 0.0002
EOC
*
DCP
*
OMT 5.449E-004 1 5.449E-004 10.67 0.0469
Residual 1.532E-004 3 5.106E-005
Lack of fit 1.115E-004 2 5.574E-005 1.34 0.5217
Pure error 4.170E-005 1 4.170E-005
Cor total 1.69 9
Table 6. Analysis of variance for XRD peak intensity.
Factor SS Df MS F-value P-value
Model 3.489E+008 4 8.723E+007 519.03 <0.0001
Linear mixture 3.050E+008 2 1.525E+008 907.53 <0.0001
EOC
*
DCP 8.241E+006 1 8.241E+006 49.04 0.0009
DCP
*
OMT 2.834E+006 1 2.834E+006 16.86 0.0093
Residual 8.403E+005 5 1.681E+005
Lack of fit 8.335E+005 4 2.084E+005 30.97 0.1339
Pure error 6728.00 1 6728.00
Cor total 3.497E+008 9
Table 7. Analysis of variance for the slope of log G′versus log ω.
Factor SS Df MS F-value P-value
Model 0.97 4 0.24 1319.10 <0.0001
Linear mixture 0.91 2 0.46 2474.08 <0.0001
EOC
*
DCP 0.060 1 0.060 325.44 <0.0001
DCP
*
OMT 0.060 1 0.060 325.05 <0.0001
Residual 9.231E-004 5 1.846E-004
Lack of fit 8.523E-004 4 2.131E-004 3.01 0.4048
Pure error 7.080E-005 1 7.080E-005
Cor total 0.98 9
6
Mater. Res. Express 7(2020)015321 M Tayefiet al

In addition, a brief statistics of the best model for each response is tabulated in table 9. The value of R
2
can
also be used as a criterion in order to evaluate the ability of the models to predict the results. A model can
estimate the result more accurately if the value is much closer to 100%. As it can be observed, the R
2
values
obtained from the analysis are 99.71 for tensile strength, 99.53 for modulus at 100% strain, 99.99 for strain at
break, 99.76 for XRD peak intensity and 99.91 for initial slope of log G′versus log ω. The high values of R
2
for all
the properties investigated are consistent with the fact that the proposed models are highly capable in predicting
the responses in the range studied.
It is worth to mention that R
2
–adj. (adjusted determination coefficient)of all responses was also near to
100%. The values also confirmed that the significance of the models were high. The predicted R
2
(Pred.
R-Squared)values were also in good agreement with the adjusted R
2
(Adj. R-Squared)values.
The appropriate amounts (pseudo-component)of X
1
,X
2
and X
3
were substituted in the models in order to
compare the experimental data, for each response, with those values calculated from the models. For instance,
for sample 4 (0.6, 0, 0.4), the experimental and predicted values were found to be 14.83 and 14.92 MPa for tensile
strength, 1.97 and 1.97 MPa for modulus at 100% strain, 2167 and 2152% for strain at break, 15002 and 14686
counts for XRD peak intensity and finally 1.27 and 1.27 for the initial slope of log G′versus log ω, respectively. As
another example, sample 10 (0, 0.2, 0.8), it was established that the experimental and predicted values were
obtained to be 9.82 and 9.72 MPa for tensile strength, 6.21 and 6.62 MPa for modulus at 100% strain, 176 and
176% for strain at break, 7014 and 7276 counts for XRD peak intensity as well as 0.19 and 0.19 for the initial
slope of log G′versus log ω, respectively. The results showed that there was very good agreement between the
experimental and predicted data in all cases studied.
The normal probability plots of the studentized residuals for the models are shown in figures 2(a)to (e).
These plots can be useful for checking the capability of a model to fit the data set [2,36]. The points located on
the normal probability plots of the residuals should form a straight line if the model is sufficient [37]. All the
figures indicate that the residuals form a straight line in agreement that the errors are normally distributed.
Figures 3(a)to (e)illustrate the plots of the residuals versus the predicted responses. All plots showed that the
residuals did not form any reasonable pattern. These are consistent with the fact that the models suggested for
the responses are quite suitable and there is no reason to be worried about any violation of the independence or
constant variance assumption [3].
Table 8. The estimated regression coefficients and their standard errors for the models obtained.
Coefficients and their
errors
Tensile
strength
(Modulus at 100%
strain)
−0.5
Log
10
(Strain at
break)
XRD peak
intensity
Log
10
(Initial slope of
log G′versus log ω)
α
1
11.35 0.71 3.27 6460 0.13
Standard error 0.27 9.93×10
–3
6.85×10
–3
355 0.01
α
2
304.45 −4.37 −11.38 −160269 9.02
Standard error 15.27 0.60 0.39 21409 0.71
α
3
14.64 0.72 3.33 27026 0.07
Standard error 0.55 0.01 0.01 462 0.02
α
12
−380.68 4.66 12.19 185300 −15.82
Standard error 19.15 0.74 0.49 26462 0.88
α
13
9.37 0.17
Standard error 1.85 0.05
α
23
−393.07 4.31 11.64 110681 −16.11
Standard error 19.81 0.75 0.51 26953 0.89
α
123
−49.19 −1.23
Standard error 14.62 0.38
Table 9. The statistics of the best model selected for each response.
Response Model R-Squared (%)Adj. R-Squared (%)Pred. R-Squared (%)
Tensile strength Special Cubic 99.71 99.13 93.03
Modulus at 100% strain Quadratic 99.53 99.16 96.76
Strain at break Special Cubic 99.99 99.97 99.82
XRD peak intensity Quadratic 99.76 99.57 98.73
Slope of log Gʹversus log ωQuadratic 99.91 99.83 99.40
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Mater. Res. Express 7(2020)015321 M Tayefiet al

4.2. Contour plots
The influence of combination of the mixture ingredients on the magnitude of responses are depicted in
figures 4(a)to (e)in the form of 2D contour plots. The plots have been divided into several sections in which the
variations of the colors’of different regions reveal the trend of responses. The sections with darker colors are
related to responses with higher magnitudes in comparison with those of brighter colors. Various magnitudes of
responses and coordinating points and thus the proportion of components could be evaluated by passing
Figure 2. Normal probability plots of studentized residuals for (a)tensile strength, (b)modulus at 100% strain, (c)strain at break,
(d)XRD peak intensity and (e)the slope of log G′versus log ω.
8
Mater. Res. Express 7(2020)015321 M Tayefiet al

through different regions in contour plots. Figure 4(a)shows that the high tensile strength values (higher than
14 MPa)can be obtained in the region illustrated by red color at the edge of EOC-OMT. The highest values were
determined for formulations 4 and 8 with the values of 14.83 and 15.65 MPa, respectively. The former was made
of 0.6 EOC, 0 DCP and 0.4 OMT and the latter consisted of 0.2 EOC, 0 DCP and 0.8 OMT (see table 2). However,
the lowest value of tensile strength was found to be 6.62 MPa which was assigned to formulation 2 containing 0.9
EOC, 0.1 DCP and 0 OMT.
Figure 4(b)indicates that the highest values for modulus at 100% strain response (more than 6 MPa)are
found to be 6.23 and 6.21 MPa. The former was obtained for formulation 7 containing 0.4 EOC, 0.2 DCP and 0.4
Figure 3. Plots of the residuals against predicted responses for (a)tensile strength, (b)modulus at 100% strain, (c)strain at break,
(d)XRD peak intensity and (e)the slope of log G′versus log ω.
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Mater. Res. Express 7(2020)015321 M Tayefiet al

OMT and the latter was that of formulation 10 made of 0 EOC, 0.2 DCP and 0.8 OMT. The lowest value
(1.92 MPa)was also obtained for formulation 8.
The contour plot for strain at break (figure 4(c)) shows that the higher values (more than 2000%)were
achieved at the edge of EOC-OMT. The values of 2167 and 2196% were found for formulations 4 and 8,
Figure 4. Contour plots of (a)tensile strength, (b)modulus at 100% strain, (c)strain at break, (d)XRD peak intensity and (e)the slope
of log Gʹversus log ω.
10
Mater. Res. Express 7(2020)015321 M Tayefiet al

respectively, similar to what was already observed for the tensile strength. The lowest value (176%)was obtained
for formulation 10 containing 0 EOC, 0.2 DCP and 0.8 OMT.
As it can be seen from figure 4(d), the highest XRD peak intensity was found for formulation 8 with the value
of 22594 counts. The lowest value (2823 counts)was attained for formulation 3 containing 0.8 EOC, 0.2 DCP
and 0 OMT.
The 2D plot of the initial slope of log G′versus log ω(figure 4(e)) reveals that higher values (more than 1.2)
are observed at the edge of EOC-OMT. The values of 1.22, 1.27 and 1.33 were found for formulations 8, 4 and 1,
respectively. Therefore, the highest value was found for formulation 1 having 1 EOC, 0 DCP and 0 OMT. The
lowest value of 0.19 was also obtained for formulation 10.
4.3. Optimization
In systems with several responses, it is not realistic to expect having the maximum values for all the responses
investigated. Therefore, we should be satisfied with acquiring the maximum values for the most important or
desired responses. Numerical optimization was accomplished using Design-Expert
®
software to find the
optimum combination of the selected components. The program employs different possibilities for a goal to
make the desirability indices. These are maximize, minimize, target, in range, none (only for responses)and
equal to (for factors only). The optimization process could be carried out by using the software’s numerical and
graphical tools.
At the first stage of optimization, the criteria for the desired nanocomposite should be defined. The
mechanical and rheological properties are the most required responses for our systems. In this work, we would
like to maximize the value of strain at break and reduce the slope of log G′versus log ω. Furthermore, it was
required to have a reasonable tensile strength and modulus at 100% strain at the same time. The values of the
desired responses are tabulated in table 10.
For optimization, the overlay plot was generated by superposition of the contour plots obtained for different
responses [38]. By selecting the desired limits for each response, the area with yellow color represented the
acceptable values as illustrated in figure 5.
The proposed formulation for obtaining the optimum responses along with the predicted and experimental
values is listed in table 11. As it can be observed, the experimental results are in close agreement with the
optimum values determined from the models. However, there are some differences between the predicted and
experimental values, especially for XRD results. For XRD peak intensity, the difference between the values is
originated from a phenomenon that cannot be predicted by the software. It was found that the addition of 0.05
DCP is not sufficient to break down the OMT tactoids due to the small elastic forces created by the cross-linking
agent. The elastic force is produced by cross-linking of some parts of the polymeric chains accommodated
between the OMT layers [39]. This behavior has also been reported by other researchers for other
nanocomposites [40,41].
5. Conclusions
In this paper, the effect of EOC, DCP and OMT content on physical and mechanical properties of dynamically
cured OMT-filled ethylene-octene nanocomposites were studied by constrained mixture design approach. The
tensile strength, modulus at 100% strain, strain at break, XRD peak intensity and the slope of log Gʹversus log ω
at the terminal zone of the materials prepared were selected as the desired responses. From the results obtained,
it could be concluded that the method employed was a very helpful fitting tool for the optimization of the
properties studied. It was found that special cubic was the best model to describe the tensile strength and strain at
break while quadratic model was the best one to express the modulus at 100% strain, XRD peak intensity and the
slope of log G′versus log ω. The results also revealed that two-component interactions of EOC
*
DCP and
Table 10. Constraints of the responses studied for the determination of the optimum composition.
Name Goal Minimum value Maximum value
EOC In therange 0 1
DCP In the range 0 0.2
OMT In the range 0 0.8
Tensile strength (MPa)In the range 7.8 15
Modulus at 100% strain (MPa)In the range 2 6
Strain at break (%)Maximum 1100 2100
XRD peak intensity (Counts)None None None
Slope of log Gʹversus log ωMinimum 0.4 0.75
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DCP
*
OMT were the most important interactions affected all the responses. However, EOC
*
OMT interaction
had no significant effect on three responses of modulus at 100% strain, XRD peak intensity and the slope of
log G′versus log ω. Numerical optimization analysis pointed out that the nanocomposite with optimal
properties should be made of 0.15 EOC, 0.05 DCP and 0.8 OMT in pseudo-component which was equal to 94.75
EOC, 0.25 DCP and 5 OMT in real values. This was in agreement with the results obtained experimentally.
Acknowledgments
The authors would like to thank Iran Polymer and Petrochemical Institute (Grant No. 31761206)for financial
support of this work.
Conflict of interest
There are no conflicts of interest to declare
ORCID iDs
Masoud Tayefihttps://orcid.org/0000-0002-3199-6459
Mohammad Razavi-Nouri https://orcid.org/0000-0001-9699-4510
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