a reactive extrusion process for the free radical grafting of silanes onto polypropylene: effects of...

A Reactive Extrusion Process for the Free RadicalGrafting of Silanes Onto Polypropylene: Effects ofProcessing Conditions and Properties of WaterCross-Linked Silane-Grafted Polypropylene

Zheng-Hui Li,1 Guo-Hua Hu,1 Jean-Pierre Corriou,1 Sandrine Hoppe,1 Christian Fonteix,1

Richard Laine,1 Jean Habimana,2 Damien Deheunynck31 CNRS-Universte de Lorraine, Laboratoire Reactions et Genie des Procedes, UPR3349,ENSIC, 1 rue Grandville, BP 20451, Nancy F-54000, France

2 Multibase S.A., Z.I. Chartreuse-Guiers, Saint Laurent du Pont F-38380, France

3 Dow Corning Europe S.A., Rue Jules Bordet, Seneffe B-7180, Belgium

Silane grafting and water cross-linking of polypropyl-ene (PP) are a recent method to modify its properties,such as melt strength, heat, and chemical resistance.This work aims at grafting silanes onto PP by reactiveextrusion. The occurrence of the grafting of silane ontoPP was confirmed by Fourier transform infrared (FTIR)and a method based on FTIR was developed to quan-tify the amount of polymerized silane and that of silanegrafted onto PP. The molar mass of the silane-graftedPP and its melt viscosity were also measured. A multi-objective optimization strategy was used to study theeffects of processing conditions on the quality of thesilane-modified PP. It was concluded that to maximizethe amount of silane grafted on PP and minimize theamount of polymerized silane and the decrease in PPchain scission, screw speed and barrel temperatureshould be low and feed rate high. POLYM. ENG. SCI.,53:1571–1581, 2013. ª 2013 Society of Plastics Engineers

INTRODUCTION

Cross-linking is an important method for improving

properties of polyolefins, such as melt strength, heat, and

chemical resistance. A variety of ways can be used to

cross-link them, which includes radiation cross-linking,

peroxide cross-linking, and silane-water cross-linking

[1–3]. Among them, peroxide cross-linking is the most

widely used method, especially for cross-linking polyeth-

ylene (PE) [4]. Unfortunately, peroxide cross-linking

induces serious scission of polypropylene (PP) chains,

leading to a decrease in molar mass. Radiation cross-link-

ing is limited by material thickness and requires an inert

atmosphere, resulting in a high and complicated manufac-

turing technology.

Silane grafting and then water cross-linking of polyole-

fins are a recently developed alternative approach [5–23]

that has gained both academic and industrial interest

because of its various advantages, such as easy processing,

low capital investment, and favorable properties of proc-

essed materials. Compared with peroxide cross-linked

polymers, silane cross-linked ones are found to exhibit

better thermal stability and higher energy storage capacity

[1]. The usual procedure for silane water cross-linking pro-

cess involves the preparation of a silane-grafted polymer

through a free-radical reaction of peroxide and silane. After

being shaped into products, the silane-grafted polymer is

cross-linked by exposure to a humid environment.

Most studies in this field are concerned with PE and

ethylene copolymers [5–15]. Those concerning the silane-

grafting and cross-linking of PP are relatively scarce [16–

22], most likely because of the nature of PP chain struc-

ture. As a matter of fact, if not well controlled, some of

the properties of PP could be highly deteriorated by the

silane-grafting step. In addition, those studies are mainly

focused on the water cross-linking step of silane-grafted

PP [18–22]. Few studies deal with the silane-grafting pro-

cess of PP, despite the fact that it may have a significant

effect on the properties of water cross-linked

silane-grafted PP. The silane grafted on PP is the func-

tional group to cross-link. Its amount directly affects the

cross-linking rate and its degree.

Correspondence to: Guo-Hua Hu; e-mail: [email protected]

Contract grant sponsor: DOW Corning Corporation and China Scholar-

ship Council (CSC to Z.H.L.).

DOI 10.1002/pen.23392

Published online in Wiley Online Library (wileyonlinelibrary.com).

VVC 2013 Society of Plastics Engineers

POLYMER ENGINEERING AND SCIENCE—-2013

In this work, the grafting of silane onto PP by reactive

extrusion is investigated. Fourier transform infrared

(FTIR) is used to confirm the grafting of silane onto PP

and quantify the amounts of grafted and polymerized

silane. The molar mass of the resulting silane-grafted PP

as well as its melt viscosity is also measured. The effects

of the reactive extrusion conditions on the quality of the

silane-grafted PP are discussed. The ultimate goal is to

maximize the amount of the silane grafted on PP while

minimizing the amount of the polymerized silane and PP

chain scission. For this purpose, a multiobjective optimi-

zation strategy is used.

EXPERIMENTAL

Materials

Two PP homopolymers were used. One was supplied by

Total Petrochemicals with a tradename PPH7060. It was in

the form of solid pellets and had a melt flow rate (MFR) of

12 g/10 min (2.16 kg/2308C). The other was supplied by

Accurel systems with a tradename Membrana Accur-

el1XP100. It was in the form of porous pellets and had an

MFR of 2.1 g/10 min (2.16 kg/2308C). It was used for

absorbing liquid ingredients (silane, peroxide, comonomer)

before compounding. The silane used was 3-acryloyloxy-

propyltrimethoxysilane (ATM) of Dow Corning1Z-6530.

Antioxidants were tris-(2,4-di-tert-butylphenyl)phosphate

with a tradename Irgafos1168 and tetrakis [methylene3-(3,

5-di-tert-butyl-4-hydroxyphenyl-propionate)] methane phe-

nolic with a tradename Irganox11010. Both were supplied

by Ciba and used as received in their white powder form.

Peroxide 2,5-dimethyl-2,5-di-(tertbutylperoxy) hexane per-

oxide was either Arkema Luperox1101 or Akzo-Nobel

Trigonox1101. They were used as received in the liquid

form. Ethylsorbate (a coagent for grafting), xylene, and ac-

etone were purchased from Sigma–Aldrich Corporation

and used as received without further purification.

Free Radical Grafting of Silane Onto PP byReactive Extrusion

The silane-grafting process was carried out in a coro-

tating twin-screw extruder of type Clextral BC21. Two

screw profiles were used (Fig. 1). The PP was fed at the

hopper. The liquid mixture of required amounts of silane,

coagent, peroxide, and antioxidant additive were added to

the extruder at Zone 4. The main difference between

those two screw profiles was that the second kneading

zone of the first screw profile (Fig. 1a) was closer to the

injection port of the small molecules than the second one

(Fig. 1b). For all runs, the composition of the grafting

system was the same, as shown in Table 1.

Extrudates obtained at the extruder exit were in the form

of pellets of 2 mm in diameter and 4 mm in length. They

are denoted as original silane-grafted PP samples. The lat-ter were dried in a vacuum oven at 1208C for 20 h to

remove nonreacted silane and coagent residues. They are

designated as dried silane-grafted PP samples. The latter

were dissolved in boiling xylene and precipitated in ace-

tone at room temperature to further remove the nonreacted

silane and coagent residues and polymerized silane. The

latter are denoted as extracted silane-grafted PP samples.

FTIR Characterization of Silane-Modified PP

A FTIR spectrophotometer of type Thermo-Nicolet Ava-

tar 370 FTIR spectrometer was used. Samples were pressed

to films of about 70 lm thick using a hot press at 2108C.

Molar Mass of Silane-Modified PP

A high-temperature gel permeation chromatography

(GPC) of Viscotek 350A HT-GPC System with triple

detectors was used to measure the molar mass of silane-

modified PP. Samples were dissolved in 1,2,4-trichloro-

benzene at a concentration of 0.1 wt% and were measured

at 1358C with a flow rate of 1 ml/min. The GPC was

equipped with two TSK-GEL and one GMH-H(S)HT col-

umns. The retention times were calibrated at 1358Cagainst monodisperse polystyrene standards.

Rheological Properties

A rheometer of type Rheometric Scientific RDAIII

strain controlled rheometer with parallel plate geometry

(diameter of 15mm and gap of 1.0 mm) was used to char-

acterize the rheological properties of silane-modified PP

in the melt. Small amplitude oscillatory shear was

performed in the frequency range from 0.01 to 100 rad/s

FIG. 1. Screw profiles used for the grafting of silane onto polypropylene.

TABLE 1. Composition of the grafting system.

Type Nature Reference Physical wt% Parts

Polymer 1 PP PPH760 Pellets 85.59 100.00

Polymer 2 Porous PP Accurel XP100 Pellets 8.56 10.00

Antioxidant Aox Iragox 1010 Powder 0.43 0.50

Antioxidant Aox Irgafos 168 Powder 0.43 0.50

Silane ATM Dow Corning Z-6530 Liquid 3.00 3.50

Coagent Sorbate Ethylsorbate Liquid 1.82 2.13

Additive Peroxide Luperox 101 Liquid 0.17 0.20

1572 POLYMER ENGINEERING AND SCIENCE—-2013 DOI 10.1002/pen

at 1808C. A strain of 1% was used. It ensured that all

samples were in the linear viscoelastic regime.

Water Cross-Linking of Silane-Grafted PP andTensile Strength

Dried silane-grafted PP samples were melted in a

15-cm3 twin-screw microcompounder (DSM Xplore, The

Netherlands) at 1908C and then injected into a mold to

obtain dumbbell-shaped bars for tensile testing (the cross-

section in the middle was 10 mm 3 4 mm and the length

of the testing part was 80 mm). The latter are denoted as

uncross-linked specimens. They were cross-linked in boil-

ing 1% acetic acid aqueous solution for 24 h. The result-

ing samples are designated as cross-linked specimens. Atensile testing machine of type Zwick/Roell Z020 was

used to measure the tensile strength at 238C, with a cross-

head speed of 50 mm/min.

RESULTS AND DISCUSSION

Evidence of the Grafting and Polymerization ofSilane Onto PP

Free radical grafting of vinyl monomers onto polymers

including reaction mechanisms have received intense

investigations [24–27]. Basically, the main reactions

involved in a free radical-grafting process can be summar-

ized as follows (see Fig. 2). Pathway A: decomposition of

the peroxide into primary free radicals RO�. The latter

may react either with an unsaturated silane molecule to

form a saturated free radical RM� which may propagate

with more silane molecules leading to polymerized silane

(pathway B), or with a hydrogen atom of the polymer

(PP), and preferentially a tertiary one, to form the

corresponding macroradical PP� (pathway C). The latter

may follow either pathway D: fragmentation to two

shorter segments by the so-called b-scission or pathway

E: grafting with silane. In short, the desired grafting of

silane onto PP is in competition with its polymerization

and PP chain scission, two main side reactions.

Figure 3 compare the IR spectra of the original, dried

and extracted silane-grafted PP samples with those of the

virgin PP. Table 2 indicates the main peak assignments. The

spectra of the extracted silane-grafted PP display new peaks

at 1191, 1086, and 770 cm21, respectively, indicating that

silane was indeed grafted onto PP. The peak at 1060 cm21

characteristic of Si��O��Si linkage [22] is not significant,

implying that no significant cross-linking reaction had hap-

pened. This is further corroborated by the fact that when the

original silane-grafted PP samples were dissolved in boiling

xylene, the insoluble fraction was very small.

Figure 4 shows typical GPC traces of the virgin PP,

dried and extracted silane-grafted PP samples using a 908right angle light scattering detector. Those obtained from

a low angle light scattering are similar. There is an addi-

tional and small peak for the dried sample, corresponding

to molecules of large molar masses. It disappears after the

extraction, indicating that these large molecules can be

dissolved in xylene or acetone. As such, they are not the

virgin PP or silane-grafted PP but polymerized silane.

Shear rheology can be sensitive to the topological

structure of macromolecules. Figure 5 shows the complex

viscosity curves of the virgin PP, representative dried and

extracted silane-grafted PP samples in the molten state.

Compared with the virgin PP, the dried silane-grafted PP

sample exhibits a higher complex viscosity at low fre-

quency, indicating that it contains molecules of large

molar masses. This is in agreement with the GPC trace of

FIG. 2. Possible reaction scheme of the radical grafting of silane on PP.

FIG. 3. FTIR spectra of the virgin PP and silane-grafted PP samples (Q

¼ 10 kg/h, T ¼ 2008C, N ¼ 100rpm, and screw Profile 1).

TABLE 2. Assignment of FTIR spectra.

Wavenumber (cm21) Group Remark

770 Si��O��CH3 Si��O��CH3 (CH3 rocking)

1086 Si��O��C O��C stretching vibration of

reacted or unreacted silane

1191 Si��O��C O��CH3 rocking vibration

1732 ��C¼¼O C¼¼O stretching vibration

2722 ��C��H ��C��H stretching vibration

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2013 1573

the dried silane-grafted sample described above (Fig. 3).

The molecules of large molar masses are attributed to

polymerized silane. Thus, dried silane-grafted samples are

composed of polymerized silane, silane-grafted PP (PP-g-ATM), and PP without silane grafted onto it if there is

any. After the extraction, polymerized silane is removed.

The extracted silane-grafted PP sample exhibits a much

lower complex viscosity except in the low frequency

region. The much lower complex viscosity is indicative of

PP chain scission during the free radical-grafting process.

The fact that the complex viscosity is very high in the

low frequency is likely related to a limited amount of

water-induced cross-linking and/or PP branching which

might have occurred during the extraction process.

Tensile Properties of Water-Induced Cross-Linking ofSilane-Grafted PP

Figure 6 shows the stress–strain curves of the virgin PP,

representative uncross-linked specimen (from dried silane-

grafted sample; feed rate of the PP Q ¼ 5.5 kg/h, barrel

temperature of the extruder T ¼ 2008C, and screw speed

of the extruder N ¼ 300 rpm), and cross-linked one. The

uncross-linked specimen shows a higher tensile stress at

yield than the virgin PP. This is because it contains poly-

merized silane whose molar mass is very high. The water

cross-linking further increased the tensile strength at yield

of the dried silane-grafted PP, as expected.

Quantification of the Amounts of Polymerizedand Grafted Silane

FTIR was used to quantify the amount of polymerized sil-

ane and that of grafted one based on the Beer–Lambert law:

A ¼ ecl

where A is the absorbance, e the extinction coefficient, cthe concentration of the entity, and l is the path length. The

peak at 1191 cm21 was used to measure the amount of sil-

ane grafted onto PP [7, 28] in the extracted silane-grafted

PP samples and the one at 2722 cm21 characteristic of PP

was used as an internal reference. Therefore, the ratio

A1191/A2722 represents the ratio between the mass of the

grafted silane and that of PP.

If it is assumed that there is no loss in silane during

the reactive extrusion process and during the preparation

of films by compression in a hot press at 2108C of the

original samples, the values of the ratio A1191/A2722 of all

the original samples should be the same. They should cor-

respond to the maximum silane content in the PP, namely,

3.5 g silane/100 g PP (see Table 1). Table 3 shows the

properties of the samples obtained from screw Profile 1 in

terms of the percentages of polymerized, grafted, and

reacted (sum of the polymerized and grafted silane) silane

as well as the complex viscosity at a very low frequency.

It is interesting to see that with one or two exceptions,

the values of A1191/A2722 for the original silane-grafted PP

FIG. 5. Complex shear viscosity of the virgin PP (triangles), dried sil-

ane-grafted PP sample (squares), and extracted silane-grafted PP sample

(circles) (Q ¼ 10 kg/h, T ¼ 2008C, N ¼ 100rpm, and screw Profile 1).

FIG. 6. Stress–strain curves of the virgin PP, uncross-linked specimen

(Q ¼ 5.5 kg/h, T ¼ 2008C, N ¼ 300rpm, and screw Profile 1), and

cross-linked one.

FIG. 4. Typical GPC traces of the virgin PP, and dried and extracted

silane-grafted PP samples (Q ¼ 10 kg/h, T ¼ 2008C, N ¼ 100 rpm, and

screw Profile 1) from a 908 right angle (RALS) detector.


samples are indeed almost the same. The maximum of

A1191/A2722 of the original silane-grafted PP sample

should be equal to 1.346, the value of A1191/A2722 of the

initial grafting system. The ratio between the value of

A1191/A2722 of a dried or extracted silane-grafted PP

sample and 1.346 is a measure of the percentage of poly-

merized and/or grafted silane.

Effects of Process Conditions on the Grafting

The ultimate goal of this study is to optimize the qual-

ity of the silane-grafted PP by maximizing the percentage

of grafted silane and minimizing that of polymerized sil-

ane as well as the PP chain scission. Because the compo-

sition of the grafting system is fixed, this work mainly

considers the effects of the following four process param-

eters on the quality of the silane-grafted PP: screw profile,

feed rate Q, barrel temperature T, and screw speed N.Table 4 compares the two screw profiles in terms of

the percentages of grafted and polymerized silane. The

main difference between those two screw profiles was that

the second kneading zone of screw Profile 1 was closer to

the injection port of the small molecules than screw Pro-

file 2. It is seen that screw Profile 1 systematically outper-

forms screw Profile 2 in terms of the percentages of

grafted and polymerized silane except for Sample 3.

These results show that the quality of local mixing

between the PP melt and the liquid reagents is a key to

the free radical-grafting process. To ensure good local

mixing, it is important that the location at which the

liquid reagents are injected be as close to the downstream

mixing block as possible. This is in agreement with the

literature [29, 30].

Concerning the effects of Q, T, and N, they may be

highly coupled. For example, a low screw speed provides

a long resident time. The latter favors the grafting of

silane but aggravates PP chain scission. A high screw

speed increases mixing intensity but shortens the resi-

dence time. The former aggrevates PP chain scission,

whereas the latter reduces it.

To simplify the problem, experiments were carried out

with a given specific throughput which is the ratio

between the feed rate and screw speed, Q/N. The latter

characterizes, to some extent, the overall degree of fill

TABLE 4. Comparison between Profiles 1 and 2.

Sample 1 2 3 4

Feed rate (kg/h) 5.5 5.5 5.5 5.5

Temperature (8C) 200 220 240 220

Screw speed (rpm) 300 100 300 500

Profile 1

Percentage of grafted silane (%) 37.76 19.85 30.37 25.72

Percentage of polymerized silane (%) 14.88 41.80 24.69 31.17

Percentage of reacted silane (%) 52.64 61.64 55.07 56.89

Profile 2




Difference between Profiles 1 and 2




TABLE 3. A1191/A2722 ratios and properties of the original silane-grafted PP samples in terms of the percentages of grafted and polymerized silane as

well as complex viscosity at a frequency of 1.0 s1 of dried silane-grafted PP samples obtained at different operating conditions and using screw

Profile 1.

Sample

Operating conditionsProduct quality

Q (kg/h) T (8C) N (rpm) A1191/A2720

Percentage of silane (%)

g1.0-extracted (3103 Pa�s)Grafted Polymerized Reacted

PP — — — — — — — 9.930

1 10 200 100 1.228 44.74 2.33 47.07 3.052

2 10 200 300 1.102 16.91 22.57 39.48 1.791

3 10 200 500 1.131 11.85 22.26 34.12 1.530

4 10 220 100 1.143 21.47 22.26 43.73 2.169

5 10 220 300 0.780 14.89 15.38 30.27 1.596

6 10 240 100 1.034 20.86 23.18 44.03 2.536

7 10 240 500 1.037 16.71 27.93 44.64 1.656

8 5.5 200 300 1.208 37.76 14.88 52.64 2.408

9 5.5 220 100 1.332 19.85 41.80 61.64 1.850

10 5.5 240 300 1.281 30.37 24.69 55.07 3.010

11 5.5 220 500 1.161 25.72 31.17 56.89 0.943

12 1 200 100 1.228 13.98 49.62 63.60 3.163

13 1 200 300 1.141 21.97 24.79 46.76 0.689

14 1 200 500 1.227 29.76 37.96 67.72 0.549

15 1 220 100 1.181 25.21 32.52 57.73 0.994

16 1 220 300 1.280 40.19 19.98 60.16 0.944

17 1 240 100 1.169 25.82 26.40 52.22 0.469

18 1 240 500 1.273 18.83 33.50 52.33 0.676


and intensity of mixing in a twin screw extruder [31–33].

Table 5 shows the percentages of grafted, polymerized

and reacted silane as well as the complex viscosity of the

dried PP-g-ATM products for a ratio Q/N of 1.83 3 1022

kg/screw turn and screw Profile 2. In this screw profile,

the quality of silane-grafted PP is more sensitive to proc-

essing parameters. It is seen that within experimental

errors, a high feed rate together with a high screw speed

results in a low percentage of grafted silane and that of

reacted silane. By contrast, the percentage of polymerized

silane follows more or less an opposite trend.

When Q/N is fixed, the overall degree of fill and the

intensity of mixing are fixed, whatever Q and N. An

increase in Q with a concomitant increase in N does not

change the degree of fill or the intensity of mixing but

shortens the residence time [34]. As a result, the time

available for the grafting and polymerization of silane is

shortened and the percentages of grafted and polymerized

silane are reduced. This indicates that apart from the local

mixing between PP and monomers at the injection

location, residence time is another important process

parameter.

Experimental Design and Multiobjective Optimization

Because the effects of the operating conditions (Q, T,and N) on the quality of the silane-grafted PP in terms of

the percentages of grafted and polymerized silane as well

as the complex viscosity are highly coupled, it would be

very difficult, if not impossible to experimentally search

for the optimum process conditions which allow obtaining

the highest percentage of grafted silane with the lowest

percentage of polymerized silane and smallest degree of

PP chain scission. In this work, an attempt was made to

develop computer aided simulation models and experi-

TABLE 5. Effect of Q and N on the percentages of grafted,

polymerized, and reacted silane for a given ratio Q/N of 1.83�102 kg/

screw turn by screw Profile 2.

Sample

Q

(kg/h)

T

(8C)N

(rpm)

Percentage of silane (%) g1.0-extracted(31023

Pa�s)Grafted Polymerized Reacted

1 5.5 220 300 26.83 34.76 61.59 0.735

2 8 220 440 19.10 24.21 43.31 0.848

3 10 220 545 16.40 27.68 44.07 0.741

TABLE 6. Fischer–Snedecor test.

Validation/

identification F ¼ r21/r22 (n1; n2)

1/F0.025

(n2, n1)F0.025

(n1, n2) Validation

Percentage of

grafted silane

1.390 (5; 5) 0.198 5.050 Yes

Percentage of

polymerized silane

1.826 (5; 9) 0.210 3.482 Yes

g1.0-extracted 1.255 (5; 5) 0.198 5.050 Yes

TABLE 7. Standard deviation and confidence interval of the polynomial model coefficients for the polynomial models with a risk of 5%.

Coefficients Values or a amin amax aminred amax

red

Percentage of grafted silane: y = a0 + a1Q + a2T + a3N + a22T2 + a33N

2 + a12QT + a13QNa0 0.2755 20.0850 0.6359 0.1151 0.4358

a1 20.0195 20.2023 0.1633 20.2023 0.1633

a2 20.0255 20.2083 0.1573 20.2083 0.1573

a3 20.0353 20.2339 0.1633 20.2280 0.1574

a22 0.0651 20.3657 0.4960 20.1177 0.2479

a33 20.1124 20.5008 0.2760 20.3051 0.0803

a12 20.0249 20.2293 0.1795 20.2293 0.1795

a13 20.0573 20.2617 0.1471 20.2617 0.1471

Percentage of polymerized silane: y = a0 + a1Q + a3N + a12QT + a13QN + a123QTN

a0 0.2250 0.1547 0.2953 0.1558 0.2942

a1 20.0129 20.0874 0.0617 20.0849 0.0591

a3 20.0299 20.1101 0.0502 20.1088 0.0490

a12 20.0174 20.0977 0.0628 20.0963 0.0614

a13 20.0447 20.1250 0.0356 20.1236 0.0342

a123 0.0644 20.0238 0.1526 20.0238 0.1526

g1.0-extracted: y = a0 + a1Q + a3N + a11Q2 + a22T

2 + a33N2 + a12QT + a23TN

a0 1.7620 20.6342 4.1582 0.8758 2.6481

a1 0.4569 20.5534 1.4672 20.5534 1.4672

a3 20.6011 21.7307 0.5285 21.6661 0.4639

a11 20.4920 22.8882 1.9042 21.5023 0.5138

a22 0.9470 21.4492 3.3432 20.0633 1.9573

a33 20.5131 22.9093 1.8831 21.5781 0.5519

a12 0.2721 20.8575 1.4017 20.8575 1.4017

a23 0.4329 20.6967 1.5625 20.6967 1.5625


mental factorial design tools to design, simulate, and

optimize the free radical grafting of silane onto PP by

reactive extrusion. Based on an experimental design, two

polynomial models aiming at quantitatively describing the

relationship between the product properties and the oper-

ating conditions are established. They are:

Model I

y ¼ a0 þ a1Qþ a2T þ a3N þ a11Q2 þ a22T

2 þ a33N2

þ a12QT þ a13QN þ a23TN þ e (1)

Model II

y ¼ a0 þ a1Qþ a2T þ a3N þ a12QT þ a13QN þ a23TNþ a123QTN þ e

(2)

where y is the value of the criterion; Q, T, and N repre-

sent the normalized values of variables. The ais are the

coefficients of the polynomial to be determined. e is the

unknown experimental error.

A decision maker can then use them to obtain the best

process conditions by multiobjective optimization (see

Appendix).

The modeling results are shown in Tables 6 and 7. All

the factors are normalized in the range [21, 1]. Figures

7–9 compares the results predicted by the models with

the experimental ones. Overall, the agreement is satisfac-

tory. Nevertheless, there are obvious even huge disagree-

ments under certain conditions.

In what follows, the polynomial models are used in the

multiobjective optimization methodology to estimate the

objectives (percentages of grafted and polymerized silane

as well as complex viscosity of dried silane-grafted PP).

Keep in mind that the polynomial models are established

in the following ranges: 1.0 � Q �10.0 (kg/h), 200 � T�240 (8C), and 100 � N � 500 (rpm). This will be the

space for the multiobjective optimization. Figure 10

shows the inputs and outputs of the simulation model.

The simulation package is coupled with the optimization

loop for performing the multiobjective optimization.

Figure 11a shows the multiobjective optimization

results with the maximum percentage of grafted silane

and the minimum percentage of polymerized silane as

well as the highest g1.0 of the extracted silane-grafted

samples obtained with screw Profile 1. Figure 11b shows

the corresponding reactive extrusion conditions. It is seen

that to obtain a maximum in the percentage of grafted sil-

ane, a minimum in the percentage of polymerized silane

and a maximum in the complex viscosity, the screw speed

and the barrel temperature should be low and the feed

rate high.

FIG. 7. Comparison between model predictions and experiments in

terms of the percentage of grafted silane.


terms of the percentage of polymerized silane.


terms of the complex viscosity of extracted silane-grafted PP samples at

a frequency of 1.0 s21.

FIG. 10. Inputs–outputs of the simulation model.


CONCLUSIONS

This work aims at grafting silane onto PP by reactive

extrusion using a free radical mechanism. During the pro-

cess, silane is grafted onto PP, on the one hand; and poly-

merized, on the other hand. The PP is also subjected by

chain scission, as revealed by its complex viscosity at a

very low frequency. Therefore, the quality of the silane-

grafted PP depends very much on the percentages of

grafted and polymerized silane as well as the complex

viscosity at a very low frequency.

For a given composition for the grafting system, it

may be affected by the screw profile of the twin screw

extruder, feed rate, screw speed, and barrel temperature.

The screw profile should ensure good mixing between the

PP melt and the small molecule reagents (silane and free

radical initiator) at the location at which the small mole-

cule reagents are injected. Residence time is also an im-

portant process parameter. For a given screw profile, the

effects of feed rate, screw speed, and barrel temperature

are complex and are highly coupled. Multiobjective opti-

mization is used to search for process conditions leading

to a maximum in the percentage of silane grafted onto PP

and a minimum in the percentage of polymerized silane

and a minimum in PP chain scission. The feed rate should

be high and the screw speed and barrel temperature low.

APPENDIX

The applied procedure of process optimization includes

model definition, analysis, and optimization (Fig. 12).

POLYNOMIAL MODELS

In optimization problems encountered in industry, the

targets (outputs) and the factors (inputs) to be controlled

are determined. The aim is to find quantitative relation-

ships between the outputs and the inputs. When a polyno-

mial used to describe the industrial process, the choice of

the polynomial form is important. The form of the poly-

nomial with three parameters (inputs) is given by Eq. 3

y ¼ a1Aþ a2Bþ a3Cþ a11A2 þ a22B

2 þ a33C2 þ a12AB

þ a13ACþ a23BCþ a123ABCwithAmin < A < Amax;Bmin

< B < Bmax;Cmin < C < Cmax ð3Þ

This type of model comprises first order terms (coeffi-

cients) (type a1) expressing the primary effects, second

FIG. 11. Maximum percentage of grafted silane with the minimum per-

centage of polymerized silane and the maximum complex viscosity (a)

and corresponding reactive extrusion operating conditions obtained by

multiobjective optimization (b).

FIG. 12. Structure of the methodology for multiobjective optimization.


order terms that can be square (type a11) or of interaction(type a12), and a third order interaction term (a123).Though polynomials allow to simulate any set of experi-

mental values with any precision by raising the order of

the polynomials, however, when the order of the polyno-

mials is large, more coefficients should be calculated,

which requires more experiments. Traditionally, the form

of the polynomial is chosen empirically. When the poly-

nomial is determined, the model should be associated

with a domain of application, as nothing allows extrapo-

lating outside of the studied domain. All these factors are

real values (continuous variation between minimum and

maximum values), and must be normalized in the range

[21, 1]. For this, the minimum and the maximum of each

operating conditions take the values 21 and 1, respec-

tively (as Eq. 4)

Ar ¼A� ðAmax þ AminÞ=2ðAmax � AminÞ=2

(4)

Experiment Design

To obtain a model with a correct specification and well

determined parameters, a series of experiments must be

conducted. The D-optimal criterion, which implies the

choice of a mathematical model to represent the responses

versus the experimental factors, is often used when a min-

imal number of experiments are needed [35]. The desired

number of runs is extracted from the candidate design to

give an experimental design which minimizes the stand-

ard error of the estimates of the coefficients of the chosen

model.

When the result Pi of a given experiment xj can be

predicted by Pi ¼ fT(xj)y where y is a vector of coeffi-

cients, the complete set P of experiments can be modeled

by P ¼ Xy, where X is the matrix of the row vectors

fT(xj) and xj the vector of factors which defines the jthexperiment. When the results of the experiments are

obtained, the vector of coefficients can be calculated

using a multilinear regression. Provided the number of

degrees of freedom (NDF) is different from zero, the con-

fidence interval ½hmin; hmax� for the parameter h is based

on Eq. 5

yi � srffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip � F0:05 p; q� pð Þ � XTXð Þ�1

ii

q� yi � yi

þ srffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip � F0:05 p; q� pð Þ � XTXð Þ�1

ii

qð5Þ

It also gives information on the degree of correlation

of the coefficients considered. A reduced confidence inter-

val, ½hredmin; hredmax�, is also calculated considering all the

coefficients at their optimum value.

This interval is used to determine whether the parame-

ter value is significantly different from zero. If 0 belongs

to the confidence interval, the corresponding coefficient is

negligible.

After the experiments to identify the model have been

chosen, the centre point of the experimental design is

replicated 3–5 times to characterize the experimental

error, then to obtain the confidence interval of the calcu-

lated coefficients of the model and to observe the repro-

ducibility. Once the coefficients are calculated, it is neces-

sary to verify whether they are meaningful, so that several

experiments should be taken for validation.

Model Reduction

When the form of the model and the experiments for

identification are determined, the coefficients of the model

can be calculated using a multilinear regression [36]. How-

ever, the validation of the model is possible only if NDF is

greater than zero. A reduction of the model is thus neces-

sary, i.e., certain coefficients are to be discarded. The con-

fidence interval determined for each coefficient can be used

to reduce the model. The uncertainty on the estimated

coefficients is all the more important as the interval is cen-

tered on zero. Defining the ratio Y between the minimum

and the maximum of the confidence region

� ¼ Maxðjy redminj; jy

redmaxjÞ

Minðjy redminj; jy

redmaxjÞ

(6)

if h redmin � h red

max < 0 and Y is close to 1, the correspond-

ing coefficient must be discarded. The reduction of the

model was conducted by steps including one coefficient

each time until all the remaining coefficients have a posi-

tive product hredmin � hredmax.

Model Validation

Once the coefficients are calculated, it is necessary to

verify whether they are meaningful or whether the devia-

tions of the data points from a constant value are simply

due to a random variation of the response, because of

measurement errors or drift of uncontrolled factors.

Fisher–Snedecor test and the standard deviation of the

coefficients can be used.

The Fischer–Snedecor test, or F-test, is based on the ra-

tio F of the variances (experiments used for identification,

validation, and replication). To prove that the relationship

is statistically significant, F must be lower than the corre-

sponding value given by Fischer–Snedecor tables F0.025(n1,n2) and higher than 1/F0.025(n2, n1) with a risk of 5% and

degrees of freedom equal to n1 and n2, respectively.Then, to evaluate the prediction capabilities of the

model, by using the value of Student with a risk of 5%

(Eq. 7), a confidence interval is defined to estimate the

uncertainty on the simulated experiments

y� Stu

ffiffiffiffiffis2r

q� y � yþ Stu

ffiffiffiffiffis2r

q(7)

where y and y denote the true and the predicted values,

respectively. Stu is the value from Student distribution

tables with a risk of 5% or 10% and r2r is the experimental

variance.


Multiobjective Optimization

After model validation, the models used for the multi-

objective optimization are defined. The optimization could

be obtained with the aid of a real encoding diploid

genetic algorithm, which is developed in our laboratory

[37, 38], to determine a finite representation of the Pareto

domain. The following steps are performed:

1. By randomly selecting values for each of the several

process inputs within their acceptable ranges, the evolu-

tionary genetic algorithm consists in generating an initial

number of solutions. The process outputs in each case are

then determined using the stacked neural networks [39].

For example, 7000 unique solutions can be calculated to

represent a first approximation to the solution set. Ini-

tially, each point in the solution set is assigned a domina-

tion number equal to 0.

2. A comparison is performed between each pair of

points from the solution set, if one of the two points is

dominated by the other one (one member in the set is said

to be dominated by another if its values for all the optimi-

zation criteria are worse than those of the second mem-

ber), its domination number is incremented by 1. The

entire solution set is thus sorted in ascending order with

respect to the domination number. Then the number of

selected points, N, is given by

N ¼ N0 þ int Fs M � N0ð Þð Þ (8)

where M represents the total number of points in the orig-

inal set and N0 represents the number of dominant

points(those with a domination number equal to 0). FS

referred to as the survival fraction which lies between 0

and 1 and int(X) returns the integer value of X.This subset of N points forms a part of the next gener-

ation estimate of the Pareto domain. Meanwhile, the

dominated points eliminated from the set are replaced by

new points generated through the evolutionary genetic

algorithm as follows. For each eliminated point, a random

pair of points is selected as parents among the vector of

N points that were retained from the previous generation.

A new set of inputs Ikþ1p (generation ¼ k þ 1) is then

determined by a weighted average of the two parent

points, i and j:

Ikþ1p ¼ DpI

kp;i þ ð1� DpÞIkp;j (9)

where the variable Dp is a randomly selected number

lying between 0.2 and 1.2, taking a different value for

each input and each generation.

3. The domination ranking, the solution set reduction

and the replacement process should be repeated to deter-

mine the next generation of the members of the solution

set until all the points contained within the solution set

are dominant points.

By using evolutionary algorithms for multiobjective

optimization, a good approximation of the Pareto’s zone

can be obtained. The decision maker can define his pref-

erences based on his knowledge of the process, for exam-

ple by use of partial aggregation method [40] of rough set

method [39].

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