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4 Stress–strain relation and SIC

4 Stress–strain relation and SIC

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The effect of SIC on the physical properties of NR



145



Murakami et al., 2002, Trabelsi et al., 2003a, Tosaka et al., 2004, 2006,

2012). During uniaxial deformation, stress increases with strain and stress

shows an upturn at large deformation following the inverse Langevin equation

of the classic theory of rubber elasticity; SIC occurs at around 200% strain,

and stress increases until, at the final stage, breakage occurs. The hysteresis

in stress–strain relations and birefringence of vulcanized NR at 25°C under

uniaxial deformation are shown in Fig. 5.10 (Treloar, 1947). Birefringence,

expressed as n1-n2, measures total molecular orientation of amorphous and

crystal molecules. During extension, stress (tension (kg/cm2)) increases with

strain (extension ratio a). During retraction, stress decreases much lower

than the stress during extension, but the molecular orientation decreases

much higher than that during extension. The hystereses of these are opposite

behaviors.

WAXD intensity of the (120) plane of the crystal cell of vulcanized NR

and IR – and the stress–strain relation – are measured simultaneously by a

conventional X-ray instrument (Toki et al., 2000). The behavior of WAXD

intensity on the (120) plane shows the same hysteresis as described above for

birefringence. Crystal diffraction intensity increases with strain levels above

approximately 200% during extension. The stress during retraction decreases

much less than during extension; therefore, the hysteresis of birefringence

and WAXD intensity are caused mainly by SIC.

Selected WAXD patterns at strains during extension and retraction with

the stress–strain relation of vulcanized NR by synchrotron X-ray are shown

in Fig. 5.11 (Toki et al., 2002, 2003, 2004a). Oriented amorphous and crystal

fractions increase with strain during extension; both decrease during retraction,

where the fractions during retraction are larger than during extension. When



10



20



A



5



15



0



10

B



(n1-n2) ¥ 103



Tension (kg/cm2)



15



5

0



1



2



3

Extension ratio



4



5



5.10 Hysteresis phenomena as shown by tension (A) and

birefringence (B) in vulcanized NR at 25°C (reproduced from Treloar,

1947).



146



Chemistry, Manufacture and Applications of Natural Rubber



5

NR 25°C



Stress (MPa)



4

3

2

1

0





0



1



2



3

4

Strain



5



6



7



5.11 The stress–strain relation and selected WAXD patterns collected

during extension and retraction of sulfur vulcanized NR at 25°C.

Each image was taken at the average strain indicated by the arrows

(reproduced from Toki et al., 2003, with permission of Elsevier).

20

Anisotropic fraction (%)



Crystalline fraction

15

10

5

0



Oriented amorphous





0



1



2



3

Strain



4



5



6



7



5.12 The variations of the crystal fraction and oriented amorphous

fraction of NR-S at 25°C during stretching (reproduced from Toki et

al., 2004b, with permission of the Rubber Division of the ACS).



the crystal fraction reaches around 20%, the oriented amorphous fraction is

less than 10%, and more than 70% of molecules are un-oriented amorphous,

as shown in Fig. 5.12 (Toki et al., 2003). In Fig. 5.11, the stress hysteresis



The effect of SIC on the physical properties of NR



147



closes at around strain 2.0; in Fig. 5.12, the crystal fraction also comes close

to zero at around strain 2.0. These imply that the hysteresis of stress and

crystal fraction share the same origin. The phenomena are also reported in

Toki et al. (2000, 2002), Trabelsi et al. (2003a) and Tosaka et al. (2004).

Figure 5.13 (Trabelsi et al., 2003a) marks the following: the onset of

2



s [MPa]



Sample I

Nc = 335



30



C



Sample I

Nc = 335



E



20



B



A



c [%]



D



1

Ds



O



1



4



3

4

Draw ratio l



5



0



6



20

s [MPa]



Sample II

Nc = 238



3



A



2



2



s [MPa]

4



2



3



4

5

Draw ratio l



c [%]



6



Sample II

Nc = 238



15



C



D



B



10



Relation 14



5

D



E



0

1



C



Ds



O



1



A

B



D



2



Dc



10



E



0



C



3

4

Draw ratio l



Sample III

Nc = 145



E



A



Dc

B



0

5



6



2



3



4

5

Draw ratio l



c [%]



C



6



Sample III

Nc = 145



10



C



D

Ds



2



5



O



E

E



0

1



2



D



3

4

Draw ratio l

(a)



5



6



0



A



Dc

B



2



3



4

Draw ratio l

(b)



5



6



5.13 Stress (a) and crystallinity (b) curves of NR (1.2 g sulfur)

during cyclic deformations at room temperature (V = 0.035 mm/s).

The maximum draw ratios lmax are indicated by arrows. The

characteristic draw ratio lE is independent of lmax. Nc is a number of

monomers between adjacent cross-links (reprinted from Trabelsi et

al., 2003a, with permission of the American Chemical Society).



148



Chemistry, Manufacture and Applications of Natural Rubber



crystallization and the start of the stress plateau point (A); upturn of the

stress starting point (B); the maximum strain and stress point (C); the start

of flat stress during retraction (D); and the onset of stress decrease from the

flat stress during retraction (E). Nc is the number of monomer units between

network points and the top figure is for lowest network density of sulfur

vulcanized NR. During (A–B), SIC increases and stress does not increase.

During (B–C), stress hardening is caused by the super network of SIC.

During (C–D), the stress decreases with decreasing strain, and the crosslink

density decreases by the proportion of SIC that melts. During (D–E), the

stress curve is flat, since the melting of SIC does not occur. At (E), SIC

suddenly disappears. Trabelsi et al. suggest that the discrepancy with Toki’s

data (Toki et al., 2000, 2002) was caused by different network densities,

since the bottom figures of highest network density in Fig. 5.13 show no

plateau of stress and similar behavior of Figs 5.10 and 5.11. An onset strain

of SIC is around 200% strain (draw ratio is 3.0) and the melting of crystal

is correlated with the strain where the hysteresis of stress–strain closed.

Since strain increases by stretching, stress should increase, but eventually

stress ceases to increase continuously and the stress level plateaus during

stretching.

The stress–strain relation of vulcanized IR shows a shoulder at around

400% strain (at an elongation ratio of 5.0) and the retraction curve almost

coincides with the extension one below strain 2.0 in Fig. 5.14 (Miyamoto

et al., 2003). The pattern of maximum strain in stres–strain relations shows the

onset strain of SIC, the decrease of stress (to minimum levels), followed by

stress increases at higher strain. In the case of vulcanized IR, the onset strain

of SIC is around 400% (elongation ratio is 5.0) by WAXD measurements

(Toki et al., 2003, 2004a). Miyamoto discussed the possibility that SIC is

caused by contractive activity in the extended matrix, rather than by the

decrease of entropy. The phenomenon of a shoulder, plateau, or minimal

stress at large strain is observed in several papers (Trabelsi et al., 2003a,

Rault et al., 2006; Toki et al., 2000, 2005).

The phenomenon of stress decrease has not often been reported, since it

depends on the balance of SIC and increase of strain with temperature, strain

rate and network densities (Trabelsi et al., 2003a; Miyamoto et al., 2003). A

schematic model of stress decrease during extension between network points

is proposed in Fig. 5.15 (Toki et al., 2004b). In this figure, the molecule

between the adjacent network points possesses a random coil state before

deformation (A). In the deformed state, the molecule is stretched, forming

an oriented amorphous state (B). If a fraction of the oriented amorphous

chain crystallizes, the overall length between the adjacent network points is

able to increase because the length of extended crystal chains is longer than

oriented amorphous chains (C). As a consequence, the amorphous segment

has more freedom to choose the conformation, leading to decreased stress.



The effect of SIC on the physical properties of NR



149



6



Nominal stress (MPa)



5

4

Crystallization

3

Melting

2

1

0

1



2



3



4

5

6

Elongation ratio



7



8



5.14 Maximum strain dependence of stress–strain relations at 29°C

at a rate of deformation of 0.07 s–1 (reprinted from Miyamoto et al.,

2003, with permission of the American Chemical Society).



C



B



A



5.15 Schematic model of stress decrease during extension between

network points: (A) in the random coil state before deformation,

(B) in the deformed state (oriented amorphous), (C) in the partially

crystallized state (reproduced from Toki et al., 2004b, with permission

of the Rubber Division of the ACS).



Even at larger deformations than (B), the total stress seems rather smaller

than (B). The mechanism would decrease entropy-stress; thus, the stress

decreases and shows a shoulder or minimum in stress–strain relations. At

constant strain (at a fixed sample length), this mechanism would result in

stress relaxation by SIC. Rault showed almost the same schematic model of



150



Chemistry, Manufacture and Applications of Natural Rubber



the co-existence of amorphous and crystal in Fig. 5.7 (Rault et al., 2006).

In the case of extension, the strain is increased continuously, and the stress

increases again sharply because of the amorphous chain, following inverse

Langevin equation in the classical theory of rubber elasticity (Treloar, 1975)

or the tube model theory of rubber elasticity (Edwards and Vilgis, 1988).

Both theories suggest that the limited extensibility of chains between network

points is responsible for the steep increase in stress.

Real rubber is composed of networks of chains, and network points are

not necessarily distributed homogeneously. Even if vulcanization agents and

accelerators are uniformly distributed in rubber, vulcanization rapidly increases

local viscosity, impeding the free movement of these components. Most of

the vulcanization agents and accelerators are powders, and some of them

melt at vulcanization temperatures; the vulcanization process is a chemical

reaction of solid powder in solid rubber. It is thus impossible to expect the

homogeneous distribution of network points in rubber; this non-homogeneity

seems to induce SIC easily, since higher-oriented molecules may transform

into crystal with higher strain. A schematic model of ideal homogeneous

and real non-homogeneous distributions of networks is proposed in Fig.

5.16 (Toki et al., 2003). Ikeda et al. (2009) reported the non-homogeneity of

network points in vulcanized NR with small angle X-ray scattering (SAXS)

and small angle neutron scattering (SANS). The length of chains between

network points may be variable, with shorter chains being easily oriented

and reached fully in an extended style. These become crystal nuclei, and

then surrounding chains become crystal as shown in the schematic model

in Fig. 5.17 (Tosaka et al., 2004).

An onset of SIC in NR is about half that of the synthetic analogue (cis1,4- polyisoprene (IR)) (Toki et al., 2000, 2002, 2003, 2005; Murakami

et al., 2002; Trabelsi et al., 2003a; Tosaka et al., 2004). The onset strain of

SIC of sulfur-vulcanized NR seems to be independent of network density

(Tosaka, 2009; Tosaka et al., 2004, 2012). Tosaka proposed a theory that

the viscous part of the network may be involved in the onset strain of SIC

that is independent of network density in vulcanized NR. The behavior

of peroxide-vulcanized NR and IR has been compared (Toki et al., 2003;

Ikeda et al., 2008). Ikeda suggests that the onset strain of peroxidevulcanized NR seems to depend on network density. Toki’s data on

peroxide and sulfur-vulcanized NR show almost the same onset strains.

SAXS and SANS show the network structure and non-homogeneity of

sulfur-vulcanized and peroxide-vulcanized NR to differ (Ikeda et al., 2009),

confirmed by NMR (Che et al., 2012). TIC in sulfur-vulcanized NR seems

to occur less readily in comparison to peroxide-vulcanized NR (Gent &

Zhang, 2002).

Filler has not shown any impediment to SIC; inversely, SIC increases

with filler content, as the rubber component of such mixtures deforms more



The effect of SIC on the physical properties of NR



151



Network point

Indeal homogeneous

network

Deformed state



Real inhomogeneous

network

Deformed state



5.16 Schematic models of uniaxial deformed vulcanized

polyisoprenes. Undeformed ideal homogeneous network and

deformed ideal network. Undeformed real inhomogeneous network

and deformed real network (micro-fibrillar structure, crystallites,

network points) (reproduced from Toki et al., 2003, with permission

of Elsevier).

Red

lines



(a)



Crystal

part



(b)



(c)



5.17 Model of nucleation and crystallization in vulcanized NR.

Relatively short chains are drawn as red lines. Filled circles represent

cross-links. (a) Before deformation: cross-links are distributed

uniformly for easy understanding. (b) After deformation. Short

chains are fully stretched. Note that the distribution of cross-links is

no longer uniform to keep many network chains in the random-coillike state. (c) The fully stretched chains act as nucleus of crystallites

(dark parts) (reprinted from Tosaka et al., 2004, with permission of

the American Chemical Society).



152



Chemistry, Manufacture and Applications of Natural Rubber



to compensate for the non-deformation of fillers (Gehman and Field, 1932;

Trabelsi et al., 2003a; Poompradub et al., 2005; Gonzalez et al., 2008a,b;

Toki et al., 2008a). In the case of sulfur-vulcanized NR compounds, the

proper choice of accelerator makes it possible to increase the strain at break,

since sulfur bridges to connect rubber chains are composed of mono-, diand poly-sulfur. The strain at break depends on the fraction of poly-sulfur

bridges and the amount of SIC. The best recipe for sulfur-vulcanized NR

showed a tensile strength of 42.5 MPa. The tensile strength of vulcanized NR

compounds was the highest in the official competition held by the Society

of Rubber Science and Technology, Japan in 2004 (Toki et al., 2008a). The

stress–strain relation of vulcanized NR with filler shows more SIC than pure

vulcanized NR, as may be expected; some fillers also seem to accelerate SIC

(Gonzalez et al., 2008a, 2008b; Weng et al., 2010).

The size of SIC during extension and retraction is calculated using

the Scherrer equation (Trabelsi et al., 2003a, Tosaka et al., 2004). Their

conclusions are very similar; the volume of SIC shows almost no change

during deformation, and the amount of SIC increases with strain. Therefore,

the increase of crystal fraction is an increase in the number of crystals, and

not growth of existing crystals. The crystal form seems to be extended chain

crystal with no folding of chains; no lamellar structure has been observed

(Luch and Yeh, 1973b). New WAXD simulation method confirmed the size

and structure of SIC (Che et al. (2013a)).

The stress–strain relation of vulcanized rubber has been simulated using

non-Gaussian chains in the classical theory of rubber elasticity by the three- or

four-chain network model (Flory et al., 1943; Treloar, 1946; Wang and Guth,

1952) and the eight-chain network model (Arruda and Boyce, 1993; Boyce

and Arruda, 2000). These simulations assumed that chains are composed of

amorphous molecules that act according to the non-Gaussian chain model

while under deformation. Values for modulus and limited extensibility of

chains are introduced in their calculation. Boyce and Arruda (2000) compared

their eight-chain model to experimental data (Treloar, 1944) in three types

of deformation: uniaxial extension, biaxial extension, and pure shear. Their

non-Gaussian chain network model simulated stress–strain relations of

sulfur-vulcanized NR in three deformation modes, from small strain to large

strain to breaking point. The modulus value affects the stress–strain relation

significantly, and the difference between simulation and experimental data

seems to be large at small strain. The value of limited-extensibility determines

strain where stress goes to infinity.

Meissner proposed to improve these simulations by adding a

phenomenological equation using a C2 term in the Mooney–Rivlin equation

(Meissner, 2000). According to the author, the three-chain model and eightchain model are essentially the same, but non-Gaussian chain models and

the van der Waals theory of elasticity and the slip-link model each have



The effect of SIC on the physical properties of NR



153



limits; thus, a C2 term of the Mooney–Rivlin equation is necessary to fit

the theories to the experimental data. The three-chain model and the van der

Waals theory of elasticity are compared with Treloar’s data (Treloar, 1944)

in Fig. 5.18 (Meissner, 2000). A constitutive equation with the non-Gaussian

chain model considering the effect of SIC was recently proposed, but the

calculation did not show any shoulder or minimum of stress (Kroon, 2010),

although the author simulated the experimental data (Toki et al., 2003) that

did not show a shoulder.

Three aspects on the controversial relation between SIC and the upturn

of stress in the stress–strain relation are summarized below:

1. Amorphous molecular chains between network points that cannot be

stretched beyond their limited extensibility. This limited extensibility is

temperature-dependent, since the chains are not in crystal order and not

theoretically at their fully extended length, although the energy modulus

dominates the stress at the strain close to break. Stress increases infinitely

at the limit of extensibility. Mechanical simulation of the classic theory of

rubber elasticity with network models that assume few chains succeeded

in following the experimental data completely. The calculation did not

include the effect of SIC, concluding that the contribution of SIC to

stress is negligible. But the simulations are not perfect, especially at

low strains; the assumed values of modulus and the limited extensibility

also change the result significantly. It is interesting to note that Treloar

started his research on SIC by birefringence and dilatometry extensively

and connected SIC with the stress–strain relations; he then successfully



6



Stress s (MPa)



5

4

3

2

1

0

1



3

5

Extension ratio l



7



5.18 Comparison of the Treloar data on NR networks (points) with

theoretical equations of dashed curve of three chains model and

full line of the van der Waals theory of elasticity (reproduced from

Meissner, 2000, with permission of Elsevier).



154



Chemistry, Manufacture and Applications of Natural Rubber



matched non-Gaussian chain models to his own experimental data. After

that, Treloar wrote ‘the effect of SIC is secondary in character, producing

only minor modifications to the stress–strain relation of vulcanized NR’

(Treloar, 1975).

2. SIC initially decreases stress, then beyond a deformation thereshold,

SIC becomes a giant cross-linkage and increases both stress and tensile

strength (Flory, 1947, 1953). The stress upturn is caused by the increase

of network densities by giant SIC networks. This is a majority view;

however, the mechanism of the transformation from SIC that decreases

stress to giant network points which increase the stress is still unclear.

Flory’s comments are notable:

That the net effect of crystallization on the stress–strain relationship

is to increase the stress may seem to violate the principle of Le Châtelier

… The first crystallites formed by elongation probably do contribute

an incremental increase to the elongation (or a decrease in the stress).

They also act as giant cross-linkages each of which binds together many

chains. (Flory, 1953)

3. SIC decreases stress continuously, until the strain at break and the

amorphous fraction decreases with strain. As the sample is stretched

continuously, amorphous parts are extended to the strain of in-extensibility.

Stress increases infinitely, following the theory of rubber elasticity.

This elucidation focuses on the remaining amorphous molecules in an

upturn of stress. SIC is considered as a result of rearrangement of chains

and network points to reduce the local energy level that is induced by

deformation. Since crystal size in SIC does not increase, but the number

of crystals increases with strain, the mechanism creating SIC and the

mechanism of contribution to stress should not change during stretching.

SIC is a rearrangement process of the system under deformation; as a

result, the system is able to extend the limit of extensibility, since the

extended chain in crystal order is longer than the extended amorphous

chain at the given temperature. The higher the SIC, the higher strain at

break; thus, the higher the tensile strength achieved.



5.5



Tear resistance and SIC



Tear resistance has been analyzed extensively, as compounds used in tires

require high tear resistance under large and repeated deformation. There are

two ways of measuring tear or cut resistance:

1. Measuring the stress or energy required to break a piece into two parts

from an initial given cut. The value depends on the length of cut, sample

shape and direction of deformation. The most popular method is cutting

a trouser sample apart by uni-axial stretching.



The effect of SIC on the physical properties of NR



155



2. Measuring the rate of cut growth. The sample is deformed repeatedly

at a certain strain with a certain frequency. The growth of cut with the

number of deformations is measured. The rate is expressed as dc/dn,

where c is the length of cut and n is the number of deformations. It also

depends on the length of cut, strain, frequency and temperature.

NR compounds show a peculiar cut-growth pattern, as the longitudinal

crack grows either ‘up’ or ‘down’ as shown as a schematic model in Fig.

5.19 (Hamed et al., 1996). The sample is a sulfur-vulcanized NR/BR blend

with 50 phr carbon black; its dimensions are 25 mm wide, 120 mm long,

and approximately 2 mm thick, and 2.24 mm was the precut length. Under

uni-axial stretching, the cut grows from the precut to up, down and final

catastrophic failure. It is considered that SIC changes the direction of cut

growth, and requires more energy to reach a failure of NR compounds.

SIC at the tip of the cut is studied using X-ray (Lee and Donovan, 1987;

Trabelsi et al., 2002).

Trabelsi et al. (2002) observed highly deformed crystallized, bulk and

relaxed regions around a tip of cut of sulfur-vulcanized NR as shown in Fig.

5.20. In the relaxed region, rubber chains are not deformed, although the

sample is stretched. In the bulk region, the rubber chains are deformed in the

same way as the whole sample is deformed. In the highly deformed region,

the rubber chains are crystallized even at the whole strain 1.1. Distribution

of the crystalline fraction in the region is shown in Fig. 5.21. The crystalline

fraction near the crack tip depends on the distance from the crack tip. The



Uni-axial stretching

Up-crack

growth

Final

catastrophic

failure line



Precut



Down-crack

growth



5.19 Schematic model of precut specimen of vulcanized NR after

deforming monotonically to catastrophic failure. Up-crack growth,

down-crack growth and final catastrophic failure (adapted from the

photograph in Hamed et al., 1996).



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