Get Complete Project Material File(s) Now! »

## From the physics of polymer to reinforced elastomers

**Introduction**

Elastomers are very interesting polymeric materials due to their particular ability to deform reversibly. They are naturally soft extensible materials but suffer from intrinsically coupled mechanical properties. For example, conventional polymer networks suffer from a tradeoff between stiffness and toughness.

These materials are used in a wide range of applications, from shoe sols to joints or tires, mainly under modified reinforced forms. There is still a growing demand for stiffer and tougher materials for some specific applications like artificial actuators and soft electronics. This is why it is of interest to develop new designs of elastomers to meet the criteria of such applications.

However, the understanding of soft material fracture is not complete yet. We mainly lack on molecular scale models that bridge the gap between mechanics and chemistry. Better knowledge on the fracture mechanics of simple conventional soft polymer networks would be a great help in the development of better targeted reinforcement strategies to toughen elastomers.

**Introduction to polymer network physics**

In this part of the chapter, several models used throughout the years to describe the behavior of polymeric materials (from the molecular point of view) will be presented. They are all covered in a reference textbook from Colby and Rubinstein that has been used in this part of the chapter [1].

**The isolated polymer chain**

A polymer chain results from the assembly of monomers and can be modeled as an assembly of N bonds of length . In a situation where all the bonds are aligned, the maximum length of the chain, also known as contour length, corresponds to: = × Eq. 1

In practice, the bonds are not aligned but they each have an angle θ with their neighbors. This angle is called valence angle. The maximal length of chain becomes: = × × cos( ) Eq. 2

In reality, a more representative length of this chain is the average end to end distance. It represents the average chain length at thermodynamic equilibrium. In order to maximize the conformational entropy, the chain is considered a random walk of monomers which means it is composed of randomly linked and oriented segments. An additional parameter called structure factor C∞ refines a bit more the model and accounts for the correlation between adjacent monomers that are not randomly oriented in reality.

The real length of a polymer chain is below its maximal length, so the polymer chain is coiled. The chain has access to different conformations but not all are equally accessible. Assuming that the chain is attached at one end at the origin of a cartesian coordinate system, the other end has a probability to be in the vicinity of a position (x,y,z). This probability calculated by Kuhn, and Guth and Mark, corresponds to a Gaussian function [2]. So the chain conformations follow a Gaussian statistic.

In theory, the polymer chain can be extended by applying a force on one of its end up to its maximal extensibility. This means the chain would unfold and change conformations to deform which requires less energy than bond breakage. That explains why compared to crystalline material where deformation implies bond breakage, polymeric materials are easier to deform.

Figure 3 Schematic of a single molecule force spectroscopy experiment. Upon retraction of the cantilever, the polymer chain is stretched while the force and elongation are recorded.

An experimental study done by Oesterhelt et al. [3] on the mechanical characterization of a single polymer chain of poly(ethylene glycol) using an atomic force microscopy (AFM) tip has witnessed two regimes during the stretching of the chain.

At the beginning, the force remains low as the polymer chain unfolds. The response is not representative of the monomer properties but is characteristic of the statistic behavior of the chain. When extending a chain, the loss of accessible conformations decreases the entropy of the system which encourages the chain to return to its original conformation. The force displacement plot is linear.

At a certain deformation, depending on the number of monomers composing the chain or the maximal length of the chain, the force increases non linearly due to non-linear entropic elasticity. When the chain has reached its extended state, further elongation requires the distortion of the bonds such as the decrease of the valence angle. This regime is enthalpy controlled and depends mainly on the monomer properties.

It is possible to derive the physics of a conventional polymer network above its glass transition temperature (Tg) from the isolated chain behavior.

Conventional polymer networks are defined as polymer chains crosslinked via permanent covalent bond into a network, in which entanglements, physical crosslinks and reversible crosslinks of the polymer chains are negligible. The way the chains would interact with each other cannot be neglected in the final mechanical behavior of a real elastomer.

### Rubber elasticity

Kuhn and Flory [2] developed the affine model to describe a soft network where there is no physical interaction between chains and friction is negligible.

This model suggests that each individual chain deforms proportionally to the macroscopic deformation of the material. Unless there is bond breakage, in this model, all deformation is fully reversible due to the entropic nature of the elasticity.

Defining λi the relative deformation in the direction i, if the materials deforms in the direction x, y and z, the affine model defines that the chains deform by the same ratios in the respective directions. Then an expression of the change in free energy ΔF can be given as: ΔF = ( 2 + 2 + 2 − 3) Eq. 4

Where n is the number of elastic strands in the network, k is the Boltzmann constant and T the temperature.

Assuming that the material is incompressible, we have = 1. In the case of uniaxial extension in the x direction, at a stretch λ, we have = and = = √1 .

The value from the nominal stress σ can be derived from Eq. 4, relatively to the uniaxial stretch λ:

σ = =( − 1 ) = ( − 1 ) Eq. 5

Where is the force, 0 is the initial cross-section of the material in the transverse direction to the traction direction, is the number density of elastic strands in the network and Gx is the shear modulus of the material.

The shear modulus is linked to the Young’s modulus by: = × (1 + ) where ν is the Poisson’s ratio of the material. In the case of an incompressible material, = 0.5 which gives =3 .

Where is the density of the material, R is the perfect gas constant, Mx is the molar mass between crosslinks, M0 is the molar mass of one monomer and Nx is the number of monomers between two crosslinks.

From Eq. 6 we can see that the stiffness of an elastomer increases with increasing temperature which is the sign of the entropic nature of rubber elasticity. We can also see the dependence with, Nx, the chain length between two crosslinks. The number of chains per unit volume (equivalent to the number of crosslinks per unit volume, with a prefactor) does affect the modulus. The longer the chain between crosslinks is, the softer the final elastomeric network will be. For temperatures well above Tg, when friction is negligible, the nature of the chemical bonds does not affect the modulus.

The affine network model supposes that σ ⁄( − 12) is a constant, equal to Gx, that only depends on the crosslinking density of the network. But this is without counting on the question of entanglements.

**The importance of entanglements**

Entanglements are defined as physical interactions between polymer chains that are long enough to constrain one another through knots and loops. These limit the chains motion and add transient rigidity to the material. In fact, the entanglements can slide upon deformation in opposition to chemical crosslinks that are permanent. The effect of entanglements on the modulus is strain dependent and should decrease as the strain increases, which results in the softening of the material.

The simplest way to take entanglements into account is to consider these as transient crosslinks. To do so, we can introduce a density of entanglements νe and add it to the contribution of the shear modulus = + = ( + ) . However, this does not reflect the strain dependence.

Mooney and Rivlin [4,5] gave an empirical approach to the affine model by considering that the reduced stress = / ( − 12) evolves linearly with 1/λ: = =2 + 2 2 = Eq. 7

Where C1 and C2 are two empirical parameters experimentally accessible.

This simple model describes quite well the behavior of crosslinked and entangled materials and gives a fairly simple formula to take into account the softening with increasing strains, typically observed as in Figure 4 [6].

Figure 4 : (a) Typical nominal stress versus uniaxial stretch plot for a PSA and (b) Mooney representation of the same data as a function of 1/λ and comparison with an elastic gel and an elastic rubber (reproduced from [6])

Note that, this expression of the model is limited to uniaxial extension, and the parameters C1 and C2 aren’t representative of any physical characteristic of the material.

Rubinstein and Panuykov [7] proposed more recently a molecular model which takes into account the entanglements. The entanglements are modeled as proposed by Edwards’ slip tube model [8]: a potential is applied along the chains and is unfavorable to the movement of monomer perpendicular to the chain. The chain behaves as if constrained in a tube as represented in Figure 5.

Rubinstein and Panuykov precise that the tube fluctuates randomly and that the amplitude of the fluctuation evolves with regards to the strain tensor. In uniaxial extension, their model gives a semi-analytical stress-strain equation: = = 1 ( + ) Eq. 8 ( − 1 ) 3 0.74 + 0.61 −1 − 0.35

Where Ex and Ee are respectively the contribution of the crosslinks and the entanglements to the Young’s modulus E such as = + .

This model is very useful to extract the contribution of crosslinks and entanglements from non-linear elastic responses of a polymer network at intermediate stains. This model fits the experimental data in uniaxial extension and compression test.

#### End of the Gaussian regime

At large extension, all the previous models become irrelevant as they are all only valid as long as the chains’ extension remained well below the contour length.

The number of possible conformations at high strain becomes so limited that the amount of free energy needed to deform the network further becomes very high. This translates by an increase in rigidity when the chains are stretched close to their limit extensibility, which can be observed experimentally as an increase in the local slope of a stress-strain curve: this phenomenon is called strain-hardening.

The stiffening of a single chain is well described by the Langevin function, and can be introduced into the previous model as in Arruda and Boyce’s model [9] to take into account the finite extensibility of the chains. However, the stiffening of an elastic material cannot be predicted easily from only the single chain as stress concentration effects and interactions between chains become important in highly stretched polymer networks.

Gent’s constitutive model [10] gives an empirical expression that takes into account the finite extensibility of the chains. In this model, the nominal stress σN can be calculated by a fairly simple formula: 1= ( − 2) Eq. 9 1− 1−3

where 1 is the first strain invariant defined by 1 = 2 + 2 + 2 and Jm is the maximum allowable value of 1 − 3.

This model is in good agreement with experiments as long as the effect of entanglements is not dominant.

Even though the mechanics of soft polymers is well understood, the fracture of these materials remains an active field of research. In reality, the imperfect network chain length distribution is difficult to assess and may affect everything. Some shorter chains may break before others and create defects that can later concentrate stress and lead to cracks. The complexity of real polymer networks makes it particularly difficult to understand and model macroscopic failure. Understanding fracture is fundamental to the development of new polymeric material and their use in cutting edge applications.

**Macroscopic failure**

The study of fracture consists usually in determining the energy needed to propagate a crack.

This fracture energy is generally noted Γ or Γc.

**LEFM**

In the framework of Linear Elastic Fracture Mechanics (LEFM), the main assumption is that the bulk material remains elastic everywhere except in a small region around the crack tip. This simplifies the approach to localized dissipative phenomenon occurring to propagate a crack, that is the creation of two new surfaces.

The strain energy release rate G is then defined as the change in total mechanical energy over the change in surface area of a crack.

Griffith [6] proposed that a crack only propagates in a material when G reaches a critical value Gc, where Gc is the minimal energy to create two new surfaces, also known as Dupré energy of adhesion, and the fracture energy is defined as Γ= Gc.

In reality, both the energy to create two new surfaces as well as the energy dissipated during the propagation should be considered in order to propagate a crack.

**Table of contents :**

**General introduction and motivation of the project **

**Chapter 1: From the physics of polymer to reinforced elastomers**

Introduction

1. Introduction to polymer network physics

1.1. The isolated polymer chain

1.2. Rubber elasticity

1.3. The importance of entanglements

1.4. End of the Gaussian regime

2. Macroscopic failure

2.1. LEFM

2.2. Greensmith approximation

2.3. Lake and Thomas model

3. How to reinforce elastomers?

3.1. General guidelines

3.2. Delay nucleation

3.3. Slow down propagation

3.4. The specifications of elastomers

3.4.1. Viscoelasticity

3.4.2. Fillers

3.4.3. Physical bonds

3.5. Multiple networks

3.5.1. Double network hydrogels

3.5.2. Models for the fracture mechanics of double network hydrogels

3.5.3. Multiple network elastomers

Conclusions

References

**Chapter 2: Nanocomposite elastomers filled with soft interpenetrable particles **

Introduction

1. Results

1.1. Synthesis and characterization of the materials

1.1.1. Particles synthesis

1.1.2. Dispersion and swelling of the particles in ethyl acrylate

1.1.3. Synthesis of a simple matrix as a reference

1.1.4. Synthesis of the nanocomposites

1.2. Mechanical properties of the nanocomposites

1.2.1. Effect of the volume fraction of filler

1.2.2. Effect of the degree of crosslinking of the filler particles

Conclusion

2. Experimental part

2.1. Chemicals

2.2. Synthesis conditions

2.2.1. Soft “Filler Particles” by emulsion polymerization

2.2.2. “Reference matrix” by bulk polymerization

2.2.3. Preparation of the nanocomposite

2.3. Characterization methods

2.3.1. Gravimetric analysis

2.3.2. DLS

2.3.3. DSC

2.3.4. Mechanical tests

References

**Chapter 3: Aqueous route to double network particles and characterization of the films therefrom **

Introduction

1. Results

1.1. Synthesis

1.1.1. Synthesis of the “Single latex” (S) by emulsion polymerization

1.1.2. Swelling of the latex particles in aqueous dispersion by EA

1.1.3. Synthesis of the “Double latex” (D) by seeded emulsion polymerization

1.2. Film formation and characterization

1.2.1. Temperature effect on film formation

1.2.2. Comparison of D latex with the corresponding constituting single networks

1.2.3. Effect of the crosslinking density in the S seeds on S and D films

1.3. Connecting particles

1.3.1. Presentation of the DAAm/ADH system

1.3.2. Effect of DAAm on S film

1.3.3. Connecting the particles through the 2nd network

Conclusion

2. Experimental part

2.1. Synthesis

2.1.1. Chemicals and reagents

2.1.2. Synthesis of the “Single latex” (S) by emulsion polymerization

2.1.3. Synthesis of the “Double latex” (D) by seeded emulsion polymerization

2.1.4. Drying process

2.2. Characterization methods

2.2.1. Gravimetric analysis

2.2.2. DLS

2.2.3. AFM

2.2.4. CryoTEM

2.2.5. Mechanical tests

Appendix

References

**Chapter 4: Imaging and quantification of bond breakage in elastomers using confocal microscopy **

Introduction

1. Sample synthesis

1.1. Single network

1.2. Multiple network

1.3. Mechanofluorescent single network

2. Mechanical test

3. Imaging and quantification of bond breakage

3.1. Confocal set-up

3.2. Image collection

3.3. Imaging analysis

3.3.1. Calibration of fluorescence

3.3.2. Quantification

4. Representativity of the mechanophore’s activation for chain breakage

5. Dependence of DACL activation’s on strain rate and temperature

Conclusions

References

**Chapter 5: Soft network fracture mechanics study using mechano-fluorescence **

Introduction

1. Fracture of notched samples

1.1. Crack propagation

1.1.1. Video analysis

1.1.2. Effect of notch length

1.2. Quantification of chain scission on fracture surfaces

1.3. Discussion

2. Fracture of unnotched samples

2.1. Mechanical reproducibility

2.2. Direct observation

2.3. Effect of the viscoelasticity

Conclusion

Appendix

References

**Chapter 6: Effect of soft networks structures on fracture studied with mechano-fluorescence **

Introduction

1. Experimental methods

1.1. Materials

1.2. Mechanical characterization

2. Results and discussion

2.1. Mechanical properties

2.2. Damage quantification

Conclusion

Appendix

References

**Chapter 7: Quantitative study of molecular bond breakage and load transfer during the necking of multiple network elastomer **

Introduction

1. Materials and methods

1.1. Chemicals and reagents

1.2. Single network synthesis

1.3. Multiple network synthesis

1.4. Mechano-fluorescent network synthesis

1.4.1. DACL incorporation

1.4.2. SP incorporation

2. Confocal imaging of stretched samples

2.1. Home-made tensile test set up

2.2. Confocal imaging

2.3. Systematic image analysis

3. Necking in multiple networks

3.1. Mechanical characteristics of necking

3.2. Damage quantification bulk vs neck regions (in the bulk and the matrix)

3.3. Load transfer between the filler and the bulk (SP in the matrix)

4. What parameters control the necking process?

4.1. Effects of the pre-stretch

4.2. Effects of the filler network crosslinking content

5. Effect of the connectivity between filler and matrix networks

5.1. SNHMA synthesis

5.2. Necking in HMA multiple networks

6. Discussion

Conclusion

Appendix

**References **

**General conclusion**