Rubber Elast 2

Chapter 3. Rubber elasticity 3.1. Introduction Natural rubber is obtained by coagulation of a latex from a tree called Hevea Brasiliensis. It consists predominantly of cis-1,4-polyisoprene. Fossilised natural rubber discovered in Germany in 1924 stems back about 50 million years. Columbus learned during his second voyage to America about a game played by the natives of Haiti in which balls of an elastic ‘tree-resin’ were used. The word ‘rubber’ is derived from the ability of this material to remove (rub off) marks from paper, which was noted by Joseph Priestley in 1770. Rubber materials are not restricted to natural rubber. They include a great variety of synthetic polymers of similar properties. An elastomer is a polymer that exhibits rubber elastic properties, i.e. a material that can be stretched to several times its original length without breaking and which, upon release of the stress, immediately returns to its original length. Rubbers are almost elastic materials, i.e. their deformation is instantaneous and they show almost no creep. The unique character of rubber was discovered in 1305 by John Gough who described his experiments and findings as follows: ‘Hold one end of the slip of rubber …. between the thumb and forefinger of each hand; bring the middle of the piece into slight contact with the lips; …. extend the slip suddenly; and you will immediately perceive a sensation of warmth in that part of the mouth that touches it … For this resin evidently grows warmer the further it is extended; and the edges of the lips possess a higher degree of sensibility, which enables them to discover these changes with greater facility than other parts of the body. The increase in temperature, which is perceived upon extending a piece of Caoutchouc, may be destroyed in an instant, by permitting the slip to contract again; which it will do quickly by virtue of its own spring, as oft as the stretching force ceases to act as soon as it has been fully exerted.’ Gough made the following comment about a second experiment: ‘If one end of a slip of Caoutchouc be fastened to a rod of metal or wood, and a weight be fixed at the other extremity ….; the thong will be found to become shorter with heat and longer with cold.’ To convince yourself, please make the experiment. You will only need a strip of rubber, a weight and a hair-dryer. Gough presented no good explanation to the unexpected findings, i.e. that the stiffness increases with increasing temperature and that heat is evolved during stretching. It took almost 50 years before the thermodynamics of the rubber elasticity was formulated.

59

Figure 3.1. Entropy-driven elasticity of rubber materials Rubbers exhibit predominantly entropy-driven elasticity. This was concluded already by William Thomson (Lord Kelvin) (1857) and James Prescott Joule (1859) through measurements of force and specimen length at different temperatures. They discovered the thermo-elastic effects: (a) a stretched rubber sample subjected to a constant uniaxial load contracts reversibly on heating; (b) A rubber sample gives out heat reversibly when stretched. These observations were consistent with the view that the entropy of the rubber decreased on stretching. The molecular picture of the entropic force originates from theoretical work during the 1930’s. It was suggested that the covalently bonded polymer chains were oriented during extension. Further theoretical development occurred during the 1940's and the stress-strain behaviour was traced back to the conformational entropy. The view that the long chain molecules are stretched to statistically less favourable states still prevails (Fig. 3.1). You can make a very simple demonstration with a piece of rope, which will act as a model of the polymer chain. Take the ends of the rope with your two hands. If you keep your hands close, the rope can take many different shapes. If you separate the ends of the rope, fewer shapes are possible. Hence, the number of shapes the rope can take decreases with the displacement of the ends. The force acting on the polymer molecule is equal to the slope of the free energy vs. displacement curve (Fig. 3.1). The instantaneous deformation occurring in rubbers is due to the high segmental mobility and thus to the rapid changes in chain conformation of the molecules. The energy barriers between different conformational states must therefore be small compared to the thermal energy (RT). The reversible character of the deformation is a consequence of the fact that rubbers are lightly crosslinked (Fig. 3.2). The crosslinks prevent the chains from slipping past each other. The chains between adjacent crosslinks contain typically 60

several hundred main chain atoms. The crosslinks are covalent bonds in conventional rubbers.

Figure 3.2. Crosslinked rubber. The crosslinks are indicated by filled circles From a historical perspective, the accomplishment by Charles Nelson Goodyear in 1839 of a method to vulcanize natural rubber with sulphur was a crucial breakthrough. Sulphur links attached to the cis-1,4-polyisoprene molecules formed the network structure which is a prerequisite for obtaining elastic properties (Fig. 3.3). Goodyear got the idea from Nathaniel Hayward, who found that natural rubber became less sticky by mixing it with sulphur. Goodyear found accidentally that natural rubber and sulphur reacted at elevated temperatures: a coated piece of fabric with rubber, sulphur and a few other ingredients was hang close to a stove and the part in direct contact with the stove had greatly changed its properties. Goodyear understood the importance of his finding and continued the experimenting. The discovery of vulcanization led to a large increase in production of natural rubber from 750 tons at 1850 to 6000 tons in 1860. Scotland’s John Dunlop made use of Goodyear’s discovery in1888 by placing an air-filled rubber tube on his son’s bicycle. It worked and was the start of a new era.

Figure 3.3. Sulphur bridges vulcanized cis-1,4 -polyisoprene

in

Figure 3.4. Structure thermoplastic elastomers

of

61

The efforts in developing methods to make synthetic rubber started even before Staudinger’s formulation of the macromolecular concept. Production of synthetic rubber dates back to before World War I in Germany. The synthetic rubber research program was then restarted in Germany in 1926. The development in USA initiated some years later. Many synthetic rubbers were made in the succeeding decades. Later development of rubber technology has involved peroxide crosslinking and thermoplastic elastomers. The latter consists of block-copolymers with hard segments (physical crosslinks) and flexible segments (Fig. 3.4). The crosslinkdomains can be either in glassy amorphous or semicrystalline states. These materials can be processed by conventional thermoplastic processing methods at temperatures above the glass transition temperature or above the crystal melting point of the hard segment domains. 3.2. Thermo-elastic behaviour and thermodynamics: energetic and entropic elastic forces Fig. 3.5 shows the classical data for sulphur-vulcanized natural rubber of Anthony, Caston and Guth (1942). At small strains, typically less than ! = L L0 < 1.1 (L and L0 are the lengths of the stressed and unstressed specimen, respectively), the stress at constant strain decreases with increasing temperature, whereas at λ values greater than 1.1, the stress increases with increasing temperature. This change from a negative to a positive temperature coefficient is referred to as thermoelastic inversion. Joule observed this effect much earlier (1859). The reason for the negative coefficient at small strains is the positive thermal expansion and that the curves are obtained at constant length. An increase in temperature causes thermal expansion (increase in L0 and also a corresponding length extension in the perpendicular directions) and consequently a decrease in the true λ at constant L. The effect would not appear if L0 was measured at each temperature and if the curves were taken at constant λ (relating to L0 at the actual temperature). The positive temperature coefficient is typical of entropy-driven elasticity as will be explained in this section. The reversible temperature increase that occurs when a rubber band is deformed can be sensed with your lips, for instance. It is simply due to the fact that the internal energy remains relatively unchanged on deformation, i.e. dQ=-dW (when dE=0). If work is performed on the system, then heat is produced leading to an increase in temperature. The temperature increase under adiabatic conditions can be substantial. Natural rubber stretched to λ=5 reaches a temperature, which is 2-5 K higher than that prior to deformation. When the external force is removed and the specimen returns to its original, unstrained state, an equivalent temperature decrease occurs.

62

Figure 3.5.. Stress at constant length as a function of temperature for natural rubber. The extension ratio (L/L0) referring to a universal value for L0, is shown adjacent to each line. Drawn after data of Anthony, Caston and Guth (1942).

It is a matter of importance to separate the elastic force into entropic and energetic contributions. A stress acting on the rubber network will stretch out and orient the chains between the crosslink joints. This will decrease the entropy of the chains and hence give rise to an entropic force. The change in chain conformation is expected to change the intramolecular internal energy. The packing of the chains may also change affecting the intermolecular-related internal

energy.

Both the

intra-

and

intermolecular potentials contribute to the energetic force. The following thermodynamic treatments yield expressions differentiating between the entropic and energetic contributions to the elastic force. According to the first and second laws of thermodynamics, the internal energy change (dE) in a uniaxially stressed system exchanging heat (dQ) and deformation and pressure volume work (dW) reversibly is given by: dE = T dS - p dV + f dL

(3.1)

where dS is the differential change in entropy, p dV is the pressure volume work and f dL is the work done by deformation. It is appropriate to point out that the force is obviously a vector (denoted f) but in this treatment is treated as a scalar (denoted f; being the absolute value of the vector). The Gibbs free energy (G) is defined as: G = H-TS = E + pV - TS

(3.2) 63

where H is the enthalpy. Differentiating Eq. (3.2) gives: dG = dE + p dV + V dp - T dS - S dT

(3.3)

Insertion of Eq. (3.1) in Eq. (3.3) gives: dG = f dL + Vdp - SdT

(3.4)

The partial derivatives of G with respect to L and T are: " !G % $$ '' = f # !L & p,T

(3.5)

" !G % $$ '' = ( S # !T & L, p

(3.6)

G is a state function, which means that the order of derivation is unimportant: " " " " % % % % $ ! $ !G ' ' = $ ! $ !G ' ' $ !T $# !L '& ' $ !L $# !T '& ' # # p,T & p, L L, p & p,T

(3.7)

By combining Eqs. (3.5-3.7), the following expression is obtained: " !f % " !S % $$ '' = ( $$ '' # !T & L, p # !L & p,T

(3.8)

The partial derivative of G with respect to L at constant p and constant T (from Eq. (3.2) is : " !G % " % " % $$ '' = $$ !H '' ( T $$ !S '' # !L & p,T # !L & p,T # !L & p,T

(3.9)

64

Eq. (3.8) in Eq. (3.9): " !H % " !f % '' + T $$ '' f = $$ # !L & p,T # !T & p,L

(3.10)

The derivative of H with respect to L at constant p and constant T (from Eq. (3.2): " !H % " % " % $$ '' = $$ !E '' + p$$ !V '' # !L & p,T # !L & p,T # !L & p,T

(3.11)

Experiments show that the volume is approximately constant during deformation,

( !V !L )

p,T

" 0 . Hence,

" !H % " % $$ '' = $$ !E '' # !L & p,T # !L & p,T

(3.12)

and " !E % " !f % f = $$ '' + T $$ '' # !L & p,T # !T & p,L

(3.13)

The first term, (!E !L ) p,T , is associated with the change in internal energy accompanying deformation at constant pressure and temperature. The second term originates from changes in entropy by deformation; note that (!f !T )L, p = "(!S !L) p,T . The entropy- and internal energy-components of the elastic force are not only associated with the orientation of the chains. An additional and important contribution originates from the change in volume: " !E % " % " % " % $$ '' = $$ !E '' + $$ !E '' $$ !V '' # !L & p,T # !L & T,V # !V & T ,L # !L & p,T

(3.14)

Typical of rubbers is that (!V !L ) p,T is small. The change in internal energy associated with volume change (!E !V )T , L is however substantial. Eq. (3.13) can be 65

applied to stress-strain data taken at constant pressure and the separation into entropyand internal energy-components of the elastic force is readily made. The drawback is that internal energy-related force component has two parts, one is due to the energy change associated with the change of conformation and the other is due to change in volume on deformation.

Figure 3.6. Entropic and energetic contributions of the tensile elastic force at constant as a function of extension ratio (λ) for natural rubber. Drawn after data from Wood and Roth (1944). The method used to resolve these components is shown in the insert figure.

Physically more interesting is to consider deformation at constant volume in order to view only the direct effects of orientation on entropy and internal intramolecular energy. An analogous expression to Eq. (3.13) can be derived for constant volume conditions: " !E % " !f % f = $$ '' + T $$ '' # !L & V ,T # !T & V ,L

(3.15)

This equation is difficult to apply in an experiment with an uniaxially stretched rubber specimen, because the hydrostatic pressure has to be adjusted to keep the volume constant to counteract changes in volume caused by the stress-strain work. However, such an experiment was carried out by Allen et al. (1963).

66

The difficult problem of eliminating the effect of volume changes on the internal energy was tackled much earlier by Elliot and Lippmann and Gee. They showed that it was possible to derive the change in internal energy at constant volume from stressstrain measurements obtained at constant pressure according to the following expression:

" !E % " !f % $ ' ( f ) T$ ' # !L & V ,T # !T & p,*

(3.16)

where ! = L L0 , L0 being the length of the specimen at zero stress and temperature T. The application of Eq. (3.16) to the early experimental data for natural rubber obtained by Anthony et al. and Wood and Roth (1944) showed that internal energy contribution to the elastic force at constant volume is small at λ < 2.7 (Fig. 3.6). Treloar (1975) and Mark (1984) have collected f e f data (fe is the force component relating to the change in internal energy at constant volume and f is the total force) for natural rubber: f e f =0.18±0.03 (λ ≤2). Mark concludes from gathering data from literature that f e f is not influenced by dilution, i. e. swelling of the network polymer in a low molar mass solvent (Table 3.1). Thus, f e f intramolecular

energetics,

i.e.

the

energy

is controlled by the

differences

between

different

conformational states. Table 3.1 shows results from experiments on a number of polymers. Both negative and positive values of f e f are found. Table 3.1. Energetic stress ratio of a few polymers (Mark, 1984) Polymer

Diluent

v2a

Polyethylene

none

1.00

-0.42

n-C30H62

0.50

-0.64

n-C32H64 none

0.30 1.00

-0.50 0.17

" "

n-C16H34

0.34-0.98

0.18

" "

decalin

0.20

0.14

none

1.00

0.25

isoprene)

none

1.00

-0.10

"

decalin

0.18

-0.20

" " Natural rubber

( fe f ) b

Poly(dimethylsiloxane) Trans (1,4 - poly-

a

Volume fraction of polymer in network

b

Obtained at constant volume

67

Polyethylene shows a negative f e f value (-0.42). During stretching of crosslinked (molten) polyethylene obviously a large entropy force builds up and the internal energy (at constant volume) decreases because many gauche conformers are transferred into trans states. The energetic force must then be negative. Other polymers such as natural rubber and poly(dimethyl siloxane) exhibit positive f e f values, i.e. the extended conformation is of higher energy than the unstrained structure. The low-energy conformation of poly(dimethyl siloxane) is all-trans, but this gives the chain a non-extended (‘circular’) form due to the difference in bond angles for O-Si-O and Si-O-Si. The ratio f e f at constant volume is thus related to the intramolecular energy of the polymer chains and it can be shown [details about the derivation are presented by Treloar (1975)] that:

! d ln r 2 ! fe $ # ## && = T# # " f % V = constant dT "

(

0

) $& && %

(3.17)

The temperature coefficient of the dimension of the unperturbed polymer molecules, d ln r2 0 dT obtained from stress-strain data for a range of crosslinked

((

) )

polymers is in accordance with estimates from viscometry of the chain dimensions in theta solvents at different temperatures. Polyethylene shows negative f e f and

(d( ln r ) dT ) . The trans-content in polyethylene becomes progressively lower with 2

0

increasing temperature and hence the size of the random coil decreases with increasing temperature. 3.3. The statistical mechanical theory of rubber elasticity The early molecular-based statistical mechanical theory was developed by Wall and Flory and Rehner, with the simple assumption that chain segments of the network deform independently and on a microscopic scale in the same way as the whole sample (affine deformation). The crosslinks are assumed to be fixed in space at positions exactly defined by the specimen deformation ratio. This model is referred to as the ‘affine network model’. James and Guth allowed in their ‘phantom network model’ a certain fluctuation of the crosslinks about their average affine deformation positions. These two theories are in a sense ‘limiting cases’ with the affine network model giving an upper bound modulus and the phantom network model theory the 68

lower bound. It is important to emphasize that both models assumes that the chains of the network behave like phantom chains, i.e. the dimensions of these chains are unperturbed by excluded-volume effects. This assumption has later been confirmed by small-angle neutron scattering (SANS) of labelled (deuterated) amorphous samples. Figure 3.7 shows schematically the difference between the affine network model and the phantom network model. The affine network model assumes that the junction points (i.e. the crosslinks) have a specified fixed position defined by the specimen deformation ratio (λ). The chains between the junctions points are however free to take any of the great many possible conformations. The junction points of the phantom network are allowed to fluctuate about their mean values (shown in Fig. 3.8 by the unfilled circles) and the chains between the crosslinks to take any of the great many possible conformations. We will derive an equation relating stress and strain in a rubber on the basis of the affine network model. This model is based on the following assumptions: (1) The chains between crosslinks can be represented by Gaussian statistics of phantom chains. (2) The free energy of the network is the sum of the free energies of the individual chains. (3) The positions of the crosslinks are changed precisely according to the macroscopic deformation, i.e. deformation is affine. (4) The unstressed network is isotropic. (5) The volume remains constant during deformation.

Figure 3.7. Schematic representation of the deformation of a network according to the affine network model and the phantom network model. The unfilled circles indicate the position of the crosslinks assuming affine deformation (phantom network).

69

Figure 3.8. Affine deformation of a single chain from unstressed state r0 =(xo,y0,z0) to stressed state r =(λ1x0,λ2y0,λ3z0). The words ‘chains of network’ or ‘Gaussian chains of network’ are frequently used in this section. The two are synonymous and mean the piece of the network between two adjacent crosslinks. The derivation of the stress-strain equation goes through some elementary steps. The Gaussian distribution function for the end-to-end distance expresses the probability of finding the end of a chain at a certain position (x,y,z) with respect to the other chain end found at (0,0,0). This equation expresses, to phrase it differently, the number of conformations a chain can take provided that the chain ends are in (0,0,0) and (x,y,z). It is then possible to calculate the Helmholtz free energy (the free energy associated with constant volume) for a single chain and by adding the contributions from all individual chains of the network, also for the network. The stress-strain equation is finally obtained by taking the derivative of the Helmholtz free energy with respect to length. Let us start by showing the Gaussian function describing the distribution of the chain end position: 32

( " 3 % P(r ) = $ e ' 2 # 2! r 0 &

where r2

0

3 x2 +y2 +z

(

2 r2

) (3.18)

0

is the average end-to-end distance of the phantom chains. The Helmholtz

free energy (A) is related to the conformational partition function Z(r) for a given chain with the end-to-end vector (r) according to the following equation:

(

A(r ) = ! kT ln Z(r ) = ! kT ln P( r)Z

)

(3.19)

70

where Z is the partition for the unconstrained chain (taking all possible r values). The following expression is obtained by inserting Eq. (3.18) in Eq. (3.19): 2 2 2 ) 3 # 3 & 3kT ( x + y + z ) ,. A(r ) = ! kT + ln Z + ln% ! ( +* 2 $ 2" r 2 0 ' 2 r2 0 .-

(3.20)

The end-to-end vector r0 = ( x 0 , y 0 , z0 ) characterizes the unstressed state of a single chain in the network (Fig. 3.8). It may be noted that energetic effects are considered by the inclusion of r 2 0 . The end-to-end vector r = ( x, y, z) corresponding to the stressed state of the sane single chain is related to r0 through the deformation ratios (!1 , !2, !3 ) according to:

x = !1 x 0 ; y = ! 2 y0 ; z = ! 3 z0

(3.21)

The free energies of the chain before (A0) and after (A) the stress has been applied are: " 3 (x 2 + y 2 + z 2 % ( 0 0 0 )' A0 = C + kT $ $ ' 2 r2 # & 0

(3.22)

!

# 3 ( "2 x 2 + "2 y 2 + "2 z 2 & ( 1 0 2 0 3 0 )( A = C + kT % % ( 2 r2 $ ' 0

(3.23)

!

) 3 # 3 & ,. + where C = !kT ln Z + ln % ( , which is independent of +* 2 $ 2" r 2 0 ' .-

(!1 , !2 , !3 ) .

The

difference in free energy between the two states is: % $2 #1 x 2 + $2 #1 y 2 + $2 #1 z 2 ( ( 1 ) 0 ( 2 ) 0 ( 3 ) 0* "A = A # A0 = 3kT ' ' * 2 r2 & ) 0

!

(3.24)

The change in free energy of the network (∆AN) is the sum of the contributions of all chains of the network. It is here assumed that deformation is affine, i.e. all chains are equally deformed according to the deformation ratios (!1 , ! 2 , !3 ) : 71

N N N % 2 ( 2 2 2 2 x 0 + (# 2 $ 1)" y0 + (# 3 $ 1)" z20 * N ' ( #1 $ 1)" 1 1 1 !AN = " !A = 3kT ' * 2 2r 0 1 '' ** & )

(3.25)

It is assumed that the undeformed network is isotropic, i.e.: N

N

!x

2 0

1

N

= ! y 02 = ! z20 1

(3.26)

1

The sum of the squares of the x, y and z components must be equal to: N

N

N

N

! x 02 + ! y 02 + ! z02 = ! r02 1

1

1

(3.27)

1

Insertion of Eq. (3.27) in Eq. (3.26) yields: 2 1 N 2 N r !1 x = !1 y = !1 z = 3 !1 r0 = 3 N

N

2 0

N

2 0

2 0

0

(3.28)

where N is the number of chains of the network. Insertion of Eq. (3.28) in Eq. (3.25) gives:

!A =

1 NkT( "21 + " 22 + "23 # 3) 2

(3.29)

Eq. (3.29) is general and is not restricted to any particular state of strain. Let us derive a stress-strain equation for a rubber specimen subjected to constant uniaxial stress. The extension ratio along the stress vector is denoted ! 1 = ! . It may also be assumed that the transverse deformation ratios are equal: ! 2 = ! 3 = ! t . The transverse deformation ratio can be calculated assuming constant volume (∆V= V0–V= 0) on deformation:

!V = "1 x 0 # " 2 y 0 # "3 z0 $ x 0 # y 0 # z0 = 0 72

! 1! 2 ! 3 " x 0 y 0 z0 # x 0 y0 z0 = 0 (3.30)

! 1! 2 ! 3 = 1

considering that ! 1 = ! and ! 2 = ! 3 = ! t :

! " !2t = 1 # ! t = 1

(3.31)

!

(

Hence, the following deformation ratios are obtained: ! , 1

!, 1

)

! . The force

(f) is obtained by inserting the deformation ratios in Eq. (3.29) and then take the derivative of the free energy with respect to the length (L):

# ! ("A)& # ! ( "A) & # !) & f =% ( =% ( $ !L ' T ,V $ !) ' T ,V $ !L ' T ,V

(3.32)

! $& NkT f = !" &% 2

(3.33)

$ 2 2 '' $ ' $ ' && " + # 3)) )) * ! && L )) = NkT && " # 12 )) % " ( ( !L % L0 ( L0 % " (

# A & nRT # & %% " ) 12 (( f = ! %% 0 (( = $"' L0 $ " '

(3.34)

where σ is the real stress, A0 is the original specimen cross-sectional area, L0 is the original length of the specimen parallel to the stress, n is the number of moles of Gaussian chains and R is the gas constant. After simplification, the following stressstrain equation is obtained:

!=

nRT $ 2 1 & 1 " # ' = Ne0 RT $% "2 # &' % V0 " "

(3.35)

where V0 is the volume of the specimen, Ne0 = n V0 is the number of moles of Gaussian chains in the network per unit volume of rubber. The stress at a given extension ratio is thus proportional to Ne0 . Another way of expressing the crosslink density is:

73

n ! nMc $ ! 1 $ ! m0 $ ! 1 $ ( =# & '# & = # & '# &= V0 " V0 % " Mc % " V0 % " Mc % Mc

(3.36)

where M c is the number average molar mass of the Gaussian chains of the network and ρ is the density. The equation relating the true stress and the extension ratio becomes after insertion of Eq. (3.39) in Eq. (3.38):

!=

" RT %' 2 1 (* '# $ * Mc & #)

(3.37)

The ‘modulus’, !RT M c , is proportional to the temperature (in Kelvin). This is typical of entropy-elastic materials. The other important aspect of eq. (3.37) is that the modulus is inversely proportional to Mc . Rubbers with a high crosslinking density, i.e. low M c , are stiff. Fig. 3.9 shows the thermoelastic behaviour of the ideal entropyelastic rubber material. All lines meet in the origin at 0 K.

Figure 3.9. Stress as a function of

Figure

temperature

representation of a tetrahedral

at

constant

λ according the affine network model.

3.10.

Schematic

network.

What is the relationship between the number of chains and the number of crosslinks? A particular simple and common case is the tetrahedral network with four chains meeting in each junction point (Fig. 3.10). The number of chains in the network with ! = N " N junctions is Ne0 = N ! N (vertical) + N ! N ( horisontal) . Hence, for an infinite tetrahedral network the following relationship holds:

74

! N e0 = 1 2

(3.38)

It can be shown that the following general equation relates the number of crosslinks (ν) to the number of chains and the crosslink functionality (ψ= the number of chains emanating from a junction point):

!=

2N e0 "

(3.39)

which after insertion in Eq. (3.35) gives:

!=

"#RT &( 2 1 )+ ($ % + 2 ' $*

(3.40)

The other early statistical mechanical theory, the phantom network model of James and Guth permits fluctuations of the junction points about average positions prescribed by the macroscopic deformation. The derivation of the free energy-strain equation is here complicated by the fact that the chains of the network are coupled and the probability function for the network is the product of the probability functions of the individual chains. The lengthy derivation is not shown here but some important results of the derivation are highlighted. The average positions of the junctions are deformed affinely and the average fluctuation of any given junction is independent of strain and can be described by a Gaussian probability function. The average force on a chain is the same as they were fixed at their most probable position. The forces exerted by the network are the same whether any given junction is treated as free, or as fixed at its most probable position. In all, these findings justify much of the assumptions made in affine network model and it is no surprise that the free energy equations and the derived stress-strain equations are similar for the two models. The free energy-extension ratio expression derived by James and Guth is:

$ 2 ' NkT 2 !A = && 1 " )) (*1 + *22 + *23 " 3) % #( 2

(3.41)

which, for an infinite tetrahedral network (ψ=4) becomes:

75

$ 2 ' NkT 2 1 "A = &1# ) *1 + *22 + *23 # 3) = NkT ( *12 + *22 + *23 # 3) ( % 4( 2 4

!

(3.42)

which is precisely half of the value predicted by the affine network model (cf. Eq. (3.29). The following true stress-extension ratio equation is obtained for the phantom network in the case of a uniaxial stress:

$ 2 ' #*RT ! = && 1" )) % #( 2

$ 2 1' && + " )) % +(

(3.43)

For the tetrahedral network with the crosslink functionality (ψ) of 4, the phantom network model predicts a modulus equal to ! RT , which is 1/2 of the modulus predicted by the affine network model ( 2!RT ; insert ψ=4 in Eq. 3.40). It is common in the literature to show the stress-strain equation in another form: nominal stress (σN) defined as force divided by the original cross-sectional area. Provided that the volume is constant during deformation, the nominal stress and the true stress (σ, force divided by actual cross-sectional area) are related as follows:

"N =

!

" #

(3.44)

which after insertion in eq. (3.35) gives: % 1( N e0 RT' #2 $ * % 1( & #) "N = = N e0 RT' # $ 2 * & # # )

(3.45)

(3.37, 3.40 and 3.43). The nominal stress is approximately proportional to λ at high λ !

because then ! >> 1 ! 2 , It may be also noted that the force is always proportional to the nominal stress and hence also the force is also proportional to (! " 1 !2 ) according to the statistical mechanical theory. 3.4. Comparison of predictions made by theory and experimental data Fig. 3.11 shows how theory compares with experimental nominal stress-extension ratio data for natural rubber and silicone rubber. The modulus G = KNe0 RT - for tetrahedral network is K=1 (affine network model) and 1/2 (phantom network model) – is basically used an adjustable parameter fitting the theoretical equations to the

76

experimental data. The theoretical equation captures the trend in the experimental data in compression (λ<1) and at low extension ratios (λ<1.2). At higher extension ratios the experimental data fall below the theoretical curve. This behaviour is quite general to rubber materials and it indicates a flaw in the fundamental statistical mechanical theory.

Figure 3.11. Nominal stress-extension ratio data for natural rubber and poly(dimethyl siloxane) collected by Higgs and Gaylord (1990). The data are scaled to fit the relation [! ] = (" # "#1 ) near λ=1.

Figure 3.12. Mooney diagram (reduced modulus as a function of the reciprocal extension ratio) for natural rubber swollen in benzene. The volume fraction of polymer (v2) in the swollen network is shown adjacent to each line. The insert figure shows C2 (constant in the Mooney equation) as a function of volume fraction of rubber 77

for natural rubber, butadiene-styrene rubber and butadiene-acrylonitrile rubber. Drawn after data from Gumbrell et al. (1953).

Figure 3.13. Schematic Mooney diagram showing the upper bound (affine network) and lower bound (phantom network) and the gradual change between the two according to the constrained-junction model. The trend in experimental data is displayed with the thick line. Drawn after Mark (1993). The Mooney equation, which was derived from the assumption that Hooke’s law is obeyed in simple shear, is in accordance with the experimental data in this particular extension ratio range (1<λ<2): % 1 (% C ( " N = 2' # $ 2 *'C1 + 2 * & # )& #)

!

(3.46)

where C1 and C2 are empirical constants. It may be noted that the Mooney equation is identical with the equations derived from statistical mechanics when C2=0. In a Mooney diagram, " N 2( # $1/ #2 ) plotted versus 1/λ, C2 is obtained as the slope

[

]

coefficient and C1+C2 is the intercept at 1/λ=1. The reduced stress or modulus E* = " N v1/2 3 ( # $1/ #2 ) is often used. The factor v21/ 3 considers that the number of

[

]

load-bearing ! chains is reduced in a swollen system. The modulus changes from 2(C1+C2) at λ=1 to 2C1 at λ=∞. The ‘built-in’ gradual decrease in modulus of the

!

Mooney equation makes it very useful in capturing the experimental data (Fig. 3.12). Gumbrell et al. showed for a range of rubbers that C1 depended on the crosslink density basically according to the statistical mechanical theory, whereas C2 remained approximately constant. The stress-strain behaviour of rubbers swollen in organic solvents shows some interesting general features (Fig. 3.13): C1 is practically independent of the degree of swelling, whereas C2 decreases with increasing degree of swelling approaching C2 = 0 at a volume fraction of rubber in the swollen system of 78

0.2. Hence, highly swollen rubbers obey the stress-strain behaviour according to the fundamental statistical mechanical theories. The gradual decrease in modulus with increasing extension ratio at moderate extensions captured by the Mooney equation suggests that the response of the network mediates between affine deformation at low extension ratios to phantom-network-type deformation at high extension ratios (Fig. 3.13). The two models, affine network and phantom network may be considered as limiting cases. It seems thus that the fluctuations of the junctions are suppressed at low extensions and that further extension makes the junctions more mobile to fluctuate. Flory and coworkers proposed the constrained junction model, which introduced a constraint parameter (! ), which at high extension takes the value 0 (phantom network) and at low extensions (λ≈1) takes the value ∞ (affine network). This development of the fundamental statistical mechanical theories bringing one additional adjustable parameter is thus capable of describing the stress-strain data in the intermediate extension ratio range (1.2<λ<2). An even more critical test of the theory would be to see if the modulus predicted by the theory is according to the experimental data. The simplicity (and beauty) of the statistical mechanical theory is that the modulus is related to only a single material parameter ( Ne0 ) . The problem is that this quantity is seldom known by independent measurements. In fact, the crosslink density is often determined by applying the statistical mechanical theory to experimental stress-strain data. An obvious approach to the problem is use ‘selective’ chemistry to produce model networks of known crosslink density (exact networks). The issue is still not resolved. Some model networks show moduli according to the basic statistical mechanical theories; the stress-strain characteristics of real networks is between the limits set by the affine network (upper limit in modulus) and phantom network (lower limit in modulus) with the closer resemblance to the affine network at low extension ratios. Other researchers have reported experimental moduli higher than those predicted by the statistical mechanical theory. A tentative explanation to discrepancy between theory and experimental data is the presence of trapped chain entanglements not accounted for in the basic statistical mechanical theory. Let us return to Fig. 3.11. At very high extension ratios (λ>3) the experimental nominal stress – extension ratio data show an upturn which is clearly inconsistent with the fundamental statistical mechanical theory which predict a linear relationship between these quantities at high extension ratios. Two different explanations to the upturn have been proposed:

79

b)

The Gaussian probability function that describes the chain statistics is a reasonable accurate approximation at low extension ratios but certainly not when the chains of the network become highly extended. The refinement of the rubber elastic theory considering the non-Gaussian statistics is discussed in section 3.6.

(b)

The orientation of the chains molecules of the network reduces their entropy (Smelt) and the equilibrium melting point ( Tm0 = !H !S , where ∆H is the heat of fusion and ∆S is the change in entropy on melting) should increase because !S = Smelt " Scrystal is reduced by orientation. Hence, the effective degree of supercooling, !T = Tm0 " T increases with increasing extension ratio and the tendency for regular polymers like natural rubber to crystallise increases with the extension ratio. Goppel showed that natural rubber crystallised at λ≈4 when stretched at room temperature. Treloar recorded the rate of crystallisation by measurement of the density of stretched natural rubber at 0°C and found that crystallisation occurred at all extension ratios, but the rate of crystallisation greatly increased in the highly stretched rubbers. It is reasonable to assume that a semicrystalline rubber is stiffer than its fully amorphous analogue and in fact, Flory suggested that the upturn in the force-extension ratio curve was due to crystallisation. The minor change in the force-extension ratio curvature on heating natural rubber to 100°C, where basically no crystallinity remains, reported by Wang and Guth and the X-ray diffraction data of Smith et al. suggest, however, that the upturn in the force-extension ratio curve is dominated by the non-Gaussian effect.

3.5. Deviations from classical statistical theories for finite-sized and entangled networks In the classical theories, it is assumed that the network is infinite, i.e. that no loose chain ends exist. Loose chain ends transfer stress less efficiently than the other parts of the network and it may be assumed that they have no contribution to the elastic force. In the classical treatments it assumed that the chains are phantom chains and that they can take any conformation possible to a single phantom chain in vacuum. This is not a true picture because chain entanglements may be present. Two different kinds of junctions may thus exist: chemical crosslinks and trapped chain entanglements. Let us start by considering the effect of chain ends and to see how it affects the stress-strain function. Each linear molecule prior to crosslinking has two chain ends and these form cilia in the network structure even after vulcanisation (Fig. 3.14). The 80

number of chain ends (cilia) per unit volume in the network is Nce = 2 !N A M , where NA is the Avogadro number and M is the number average molar mass of the polymer prior to crosslinking. The number of load-bearing chains (Ne) per unit volume is: Ne = Ne0 ! Nce =

"NA 2 "N A ! Mc M

(3.47)

where Ne0 is the number per unit volume of load-bearing chains in an infinite network. The ratio Ne Ne0 becomes:

Ne !N A Mc " 2 !NA M 2Mc =1 " 0 = Ne !NA Mc M

(3.48)

Figure 3.14. Chain ends before and after vulcanization. The stress-strain equation for a uniaxially stressed rubber can be modified to take into account that only the fraction Ne Ne0 of the Gaussian chains carries the load:

"N % 1 " 2Mc % " 2 1 % * RT " 2Mc % " 2 1 % ! = Ne0 $ e0 ' RT "# (2 ) %& = Ne0 RT $ 1 ) ' ( ) &= $ 1) ' ( ) & # # Ne & ( M &# ( Mc # M &# ( (3.49) The second factor, i.e. (1 ! 2Mc M ) , becomes important when M is of the same order of magnitude as Mc . Other types of network defects also exist, physical crosslinks and intramolecular crosslinks (loops), as displayed in Fig. 3.15. The physical crosslinks may be permanent with a locked-in conformation (Fig. 3.15; case a) or temporary by entanglement (Fig. 3.15, case b). The presence of the latter type leads to visco-elastic 81

behaviour, i.e. to creep and stress relaxation. Intramolecular crosslinks decrease the interconnectivity of the network and reduce the number of load-carrying chains (Fig. 3.15, case c).

Figure 3.15. Imperfections in real networks: (a) trapped chain entanglement; (b) temporary chain entanglement; (c) intramolecular crosslink.

The role of the chain entanglements is still under debate. The early work was due to Mullins, who argued that the number of trapped (such which cannot disentangle) entanglements should increase with increasing degree of crosslinking. Mullins proposed that the number of effective crosslinks should be the sum of chemical (νchem) and trapped entanglements (νent):

$ #Mc ' ) ! e = (! chem + ! ent )& 1" % M (

(3.50)

where β is an empirical constant. The chain ends not carrying load are accounted for in the second factor in Eq. (3.50). By inserting Eq. (3.50) in the expression for C1 of the Mooney equation, the following expression was obtained:

$ #Mc ( chem) ' ) C1 = C1Mc (chem) + ! & 1" % M (

(

)

(3.51)

where C1Mc ( chem ) is the C1 value corresponding to the pure chemical crosslinks [described by Mc (chem ) ] and 1/M=0, and α is the term that is related to the entanglements. Eq. (3.51) could be fitted (two adjustable parameters: α and β) to experimental data of crosslink density by chemical methods and stress-strain measurements. The presence of trapped entanglements in networks was further supported by the more recent data of Gottlieb et al. and Dossin and Graessley of the

82

shear modulus as a function of crosslink density display non-zero intercepts, i.e. a certain shear modulus at zero-concentration of chemical crosslinks. 3.6. Large deformations when the Gaussian approximation is not valid The pronounced upturn in the nominal stress-extension ratio (λ) graph appearing at

λ>3 is a general feature characteristic of basically all rubbers (Fig. 3.11). The statistical mechanical theories based on Gaussian chains predict that the nominal stress should be proportional to the extension ratio in this extension ratio range. It is now generally accepted that these early theories neglect the finite extensibility of the network. To phrase it differently: the Gaussian distribution function is a poor descriptor of the chain statistics of highly extended chains. Fig. 3.16 presents a comparison between results obtained by Monte Carlo simulation of a 50-carbon chain and the theoretical (Gaussian) distribution. It should be noted that P(r ) = P (r) ! 4"r 2 , where P(r) is given by eq. (3.21). The Gaussian distribution function predicts too high probabilities for the extended conformations as is illustrated by Fig. 3.16. However, at low extension ratios (below and around the maximum of the P(r) distribution), the theoretical distribution is in accordance with the results obtained by Monte Carlo simulation.

Figure 3.16. Radial distribution function [P(r)] of a 50-carbon-alkane at 400 K obtained by Monte-Carlo simulation using Lattice™ (Nairn, 1998) assuming that the chains are unperturbed by excluded volume represented as a histogram. The Gaussian approximation is shown as the continuous line.

83

The above is not a shortcoming of statistical mechanics but rather of the Gaussian approximation, which is valid only at low chain extension ratios. The problem can be ‘solved’ by considering non-Gaussian chain statistics. Kuhn and Grün treated the problem for a single chain and they showed that the radial distribution function is approximately given by:

"" r % " ( %% '' ' ln P(r) = C ! n$$ $ ' ( + ln $$ # sinh ( & '& # # nl &

(3.52)

where C is a constant and β is a parameter related to the extension of the chain (r/nl): r 1 = coth " # = L (" ) nl "

!

where L is the Langevin function. The parameter β is explicitly given by:

" = L#1 ( r nl)

!

(3.53)

(3.54)

where L -1 is the inverse Langevin function. Eq. (3.52) can be written in a more useful form by making use of series expansion:

( 3 " r % 2 9 " r % 4 99 " r % 6 + ln P(r) = C ! n* $ ' + $ ' + $ ' + .....*) 2 # nl & -, 20 # nl & 350 # nl &

(3.55)

where C is a constant. For small chain extension ratios (i.e. small r/nl values) the higher terms of the series can be neglected and

3r2 ln P(r) = C ! 2nl 2

(3.56)

which is identical with the Gaussian distribution function. The force-extension ratio equation can be obtained by first by transforming P(r) into P(r), calculating the entropy using the Boltzmann entropy equation [S=k ln P(r)] and by differentiating the entropy with respect to r [ f = !T ("S "r ) ]: " kT % " r % f = $ 'L(1$ ' # l & # nl &

(3.57) 84

!

which can be expanded to:

f =

kT l

' !r$ )3# & + )( " nl %

* 9 ! r $ 3 297 ! r $ 5 1539 ! r $ 7 # & + # & + # & + ...., ,+ 5 " nl % 175 " nl % 875 " nl %

(3.58)

The first term in Eq. (3.58) corresponds to the Gaussian approximation, i.e. the linear portion of the f-r curve. The following terms produce a pronounced upturn at higher extensions in the force-extension curve (Fig. 3.17).

Figure 3.17. Force as a function of extension for a single polymer molecule according to the non-Gaussian theory (eq. (3.58). The broken line shows the Gaussian approximation. Drawn after Treloar (1975).

Figure 3.18. Nominal stress (divided by NkT) as a function of extension ratio for the three-chain model with different number of bonds (n) as shown adjacent to each curve The lower line shows the Gaussian approximation. Drawn after Treloar (1975).

85

Figure 3.19. Nominal stress as a function of the extension ratio for natural rubber. The points are experimental data and the continuous line shows the best fit of eq. (3.59) to the experimental data. The lower line shows the Gaussian approximation. Drawn after Treloar (1975). It is much more of a challenge to develop a non-Gaussian theory for polymer networks. It should be remembered that the previous mathematical treatment is concerned with a single molecule. Treloar (1975) stated that the conclusions arrived for networks must be regarded as somewhat uncertain in the quantitative sense. Simplest but still very useful is the three-chain model. The network is here replaced by sets of chains parallel to three orthogonal axes; x, y and z. The entropy of the system is the sum of the entropies of the three chains affinely deformed. The entropy of each chain is obtained by transforming P(r) into entropy and in the simple case of uniaxial deformation inserting the following r values: rx=λr0 and ry=rz=λ−1/2r0 in the entropy equation. The force-extension ratio expression is obtained according to the method used for the simple Gaussian network: + #1% $ ( # 3 #1% 1 (. NkT f = " n -L ' * # $ 2L ' *0 3 & $n )/ , & n)

!

(3.59)

The three-chain model predicts that the onset of curvature in the force-extension ratio diagram occurs at very low extension ratios for short molecules (dense networks) and that this extension ratio progressively increases with increasing molar mass of the chains (Fig. 3.18). Very short chains, e.g. n= 5 or less, the mean chain extension even in the unstrained state already exceeds that for which the Gaussian approximation is valid. For networks of such short chains, a non-Gaussian treatment is essential for the accurate representation stress-strain even at the smallest strains (Fig. 3.18). Fig. 3.19 shows that Eq. (3.59) is capable of fitting experimental data at high extensions (λ>3). 86

3.7. Summary Conventional rubbers are crosslinked amorphous polymers well above their glass transition temperature. The elasticity of rubbers is predominantly entropy-driven which leads to a number of spectacular phenomena. The stiffness increases with increasing temperature. Heat is reversibly generated on deformation. A more detailed analysis shows that the elastic force originates both from changes in conformational entropy and changes in the internal energy. The latter are normally small and at constant volume they relate to changes in conformational energy. Polymers like polyethylene with an extended conformation as their low energy state exhibit a negative energetic force contribution, whereas other rubbers, a notable example is natural rubber, show positive energetic forces. Statistical mechanical models have proven useful in describing the stress-strain behaviour of rubbers. The affine network model assumes that the network consists of phantom Gaussian chains and that the positions of the junctions are prescribed by the macroscopic deformation. The phantom network model assumes that the positions of the junctions fluctuate about their mean positions prescribed by the macroscopic deformation ratio. The change in Helmholtz free energy (∆A) on deformation at constant temperature (T) is due to decrease in the number of possible conformations of the chains of the network: !A = KNkT ("21 + "22 + "23 # 3)

(3.60)

where K is dimensionless number different for the different models, N is the number of chains in the network, k is the Boltzmann constant, T is the temperature (in K) and

λi are the extension ratios along the axes of an orthogonal coordinate system. Eq. (3.60) can be used to obtain the stress-strain equation for different stress states. For the case of a rubber specimen subjected to a constant uniaxial stress, the following true stress (σ)-strain (λ) expression holds: 1 ! = KN e0 RT $% "2 # &' "

(3.61)

where K is dimensionless number different for the different models, Ne0 is the number of moles of Gaussian chains in the network per unit volume of rubber and R is the gas constant. The statistical mechanical theories thus predict that the stiffness of a rubber 87

increases with the crosslink density and the temperature. The theoretical equation captures the trend in experimental stress-strain data of unfilled rubbers in compression (λ<1) and at low extension ratios (λ<1.2). At higher extension ratios the experimental data fall below the theoretical curve. The Mooney equation which was derived from continuum mechanics is however capable of describing the nominal stress-strain data at λ between 1 and 2: % 1 (% C ( " N = 2' # $ 2 *'C1 + 2 * & # )& #)

!

(3.62)

where C1 and C2 are empirical constants. It may be noted that the Mooney equation is identical with the equations derived from statistical mechanics when C2 = 0. Experiments have shown that a range of different rubbers that C1 depends on the crosslink density basically as the modulus does according to the statistical mechanical theory, whereas C2 is approximately constant. Stress-strain data of rubbers swollen in organic solvents analysed by the Mooney equation show interesting results: C1 is practically independent of the degree of swelling, whereas C2 decreases with increasing degree of swelling approaching C2 = 0 at v2 ≈ 0.2. Hence, highly swollen rubbers (v2 ≤0.2) behave according to the statistical mechanical theories. Stress-strain data described by the Mooney equation suggests that the networks at low extension (λ=1.0-1.2) deform near affinely whereas at higher extensions (λ=1.5-2) the fluctuations of the junctions increases (phantom network behaviour). None of Gaussian statistical mechanical theories is adequate to describe the stressstrain behaviour at large strains (λ>3 to 4). The pronounced upturn in the stress-strain curve can be accounted for by the finite extensibility of the network not accounted for by the Gaussian distribution that describes the statistics of the phantom chains. In addition, some rubbers crystallise during extension and a smaller part of the increase in stiffness at high extension ratios is due to the presence of crystallites in the material. Loose chain ends, temporary and permanent chain entanglements, intramolecular crosslinks are complications not directly addressed by the classical statistical mechanics theory. Chain ends and intramolecular crosslinks do not contribute to the elastic force, whereas chain entanglements add more junction points to the covalent network.

88

3.8. Exercises 3.1. The rubber in a blown-up balloon is biaxially stretched. Derive the force-strain relationship under the assumption that the rubber follows the Gaussian statistical theory of rubber elasticity. 3.2. Derive the relationship between the internal pressure (p) and the degree of expansion (α) of the balloon. Assume the validity of the ideal gas law (pV = nRT). 3.3. At what α value has the internal pressure a maximum? 3.4. Suppose the balloon has a small nose. Is it possible to get the nose to expand to the same degree as the rest of the balloon? 3.5. Many rubber materials exhibit time-dependent mechanical properties (Fig. 3.20). Make a list of possible reasons. 3.6. Polyethylene can be crosslinked by decomposition of organic peroxides, hydrolysis of vinyl-silane grafted polyethylene or by high-energy (β or γ) irradiation. Design a suitable experiment to determine the crosslink density and present the relevant equations.

Figure 3.20. Results from measurements of continuous stress relaxation of nitrile rubber (low network density). Drawn after data from Björk (1988).

3.7. Calculate by using the affine network model the modulus at room temperature of natural rubber (ρ ≈ 970 kg m-3) crosslinked with n molar fraction of organic peroxide. Assume that each peroxide molecule results in one crosslink. 3.8. Calculate the temperature increase occurring in natural rubber with M c = 5 000 g mol-1 when it is stretched to λ=5 at room temperature. Use the following data: ρ = 970 kg m-3, cp= 1900 J kg-1 K-1.

89

3.9. References Anthony, P. C., Caston, R. H. and Guth, E. (1942) J. Phys. Chem., 46, 826. Björk, F. (1988) Ph.D. Thesis: Dynamic stress relaxation of rubber materials, Department of Polymer Technology, Royal Institute of Technology, Stockholm, Sweden. Gumbrell, S. M., Mullins, L. and Rivlin, R. S. (1953) Trans. Faraday Soc., 49, 1495. Higgs, P. G. and Gaylord, R. J. (1990) Polymer, 31, 70. Mark, J. E. (1984) The rubber elastic state, in Physical Properties of Polymers, Mark, J. E. (ed.), American Chemical Society, Washington, DC. Mark, J. E. (1993) The rubber elastic state, in Physical Properties of Polymers, 2nd ed., Mark, J. E. (ed.), American Chemical Society, Washington, DC. Nairn, J. A. (1998) Lattice™ 7.0, Random walk simulations of polymer molecules on a tetrahedral lattice, University of Utah, Salt lake City, USA. Treloar, L. R. G. (1975) The Physics of Rubber Elasticity, 3rd ed., Clarendon, Oxford. Wood, L. A. and Roth, F. L. (1944) J. Appl. Phys., 16, 781.

90