**4. Healing model parameters**

A parametric analysis performed by Kringos et al. (2012) investigated the effect of the various model parameters on the resulting phase configuration. In the simulations a mesh of 50 m x 50 m is simulated, similar to the size of the bitumen scans under the AFM. The time and space increments are normalized with respect to each other.

The initial t0 configuration is represented by a homogenous mixture with a random perturbation, also known as a spinodal composition. Depending on the chosen parameters, different phase separation patterns will form. In the following, the influence of these parameters is shown by varying their values.

For the Cahn Hilliard parameter of 2 0.0005 and an average start concentration of 0.5 with random fluctuations of zero mean and no fluctuation greater than 0.01, results in the configurations at different time steps shown in Fig.6.

Fig. 6. Spinodal composition resulting in polymer-like phase separation (Kringos et al., 2012)

Keeping the parameters constant, but now changing the initial configuration to a maximum fluctuation of 0.05 results in a different end configuration as is shown in Fig. 7, which resembles better the phase configuration of bitumen as is seen under the AFM scans.

The Cahn Hilliard parameters are often expressed as which is related to the 2 via 2 2 . The effect of this parameter is shown in Fig. 8.

(2009b). The force versus displacement diagram is shown for a simulation of a fatigue test using this model. It can be seen from the graphs that in the case on the 'healing beam' the material has lost its memory of the previous loading cycle, whereas in the case of the 'no-

A parametric analysis performed by Kringos et al. (2012) investigated the effect of the various model parameters on the resulting phase configuration. In the simulations a mesh of 50 m x 50 m is simulated, similar to the size of the bitumen scans under the AFM. The

The initial t0 configuration is represented by a homogenous mixture with a random perturbation, also known as a spinodal composition. Depending on the chosen parameters, different phase separation patterns will form. In the following, the influence of these

with random fluctuations of zero mean and no fluctuation greater than 0.01, results in the

Fig. 6. Spinodal composition resulting in polymer-like phase separation (Kringos et al., 2012)

Keeping the parameters constant, but now changing the initial configuration to a maximum fluctuation of 0.05 results in a different end configuration as is shown in Fig. 7, which

t0 t1 t2

t3 t4 t5

which is related to the

2 via

resembles better the phase configuration of bitumen as is seen under the AFM scans.

The Cahn Hilliard parameters are often expressed as

. The effect of this parameter is shown in Fig. 8.

2

 

2

0.0005 and an average start concentration of 0.5

healing beam' the material is considerably weaker after the first loading cycle.

time and space increments are normalized with respect to each other.

**4. Healing model parameters** 

parameters is shown by varying their values.

configurations at different time steps shown in Fig.6.

For the Cahn Hilliard parameter of 2

Fig. 7. Spinodal composition resulting in bee structure formation (Kringos et al., 2012)

Fig. 8. End configurations with varying Cahn Hilliard parameter (Kringos et al., 2012)

Defining the matrix, the distinct phases (the bees) and the interfaces (IF) as the three fractions in the end configuration, the normalized fractions are plotted as a function of the varying Cahn Hilliard parameter, Fig. 9. As can be seen from the graph, an increased Cahn Hilliard parameter causes an increased interface (IF) fraction. Since this parameter controls the gradient energy distribution this result seems certainly logical. Interestingly thought, it can also be seen that both the matrix and the bees seem reduced. Which means that with an increased gradient distribution coefficient fewer bees are formed with thicker interfaces.

Changing the configurational free energy function whereby normalizing the energy barrier as shown in Fig. 10 (a) has the effect shown in Fig. 10 (b).

Atomic Force Microscopy to Characterize the Healing Potential of Asphaltic Materials 219

*bee-properties,* 

, *Eb b* 

*matrix properties, Em m* ,?

> ! !

> > ! !

! !

Equi.stiffness ratio @ t=0.08s

Fig. 11. Properties of bees and surrounding matrix (Kringos et al., 2012)

Fig. 12. Simulation of constant displacement test (Kringos et al., 2012)

C Phase separated 0 0.5*E* <sup>0</sup> 0.1

From Table 1 the chosen matrix properties and the calculated equivalent stiffness at time 0.08 s is shown. In Fig. 13 the equivalent stiffness over the entire direct tension test is

> *m*

<sup>0</sup> 1.00

<sup>0</sup> 1.06

<sup>0</sup> 0.82

<sup>0</sup> 1.02

<sup>0</sup> 1.00

0.56

*!"# \$*

Case Configuration *Em*

A Homogeneous *E*<sup>0</sup>

B Phase separated *E*<sup>0</sup>

D Phase separated 0 0.5*E*

E Phase separated 0 0.9*E*

F Phase separated 0 0.85*E*

Table 1. Determined parameters

plotted.

Fig. 9. Relationship between fractions and Cahn Hilliard parameter (Kringos et al., 2012)

Fig. 10. (a) Normalized .. potential (b) Changing <sup>0</sup> ; =1, 2, 4 and 6 for 1 4 (Kringos et al., 2012)

#### **5. Mechanical properties of the phases**

As was discussed in the previous sections, it is important to know the properties of the bitumen phases with respect to its surrounding matrix, which is also needed for the healing simulation in the next section, Fig. 11. The rheology at various temperatures of the individual phases turned out to be more challenging than anticipated, so to get the parameters needed for the individual phases for the healing simulation, in this section a finite element analyses is performed. In this analysis the overall bitumen properties as determined in the previous section is used.

For each case the equivalent stiffness ratio is calculated from *E E case caseA*

where case A represents an homogeneous mesh without any phase separation. The subscript m refers to the matrix properties and the subscript b refers to the bee structures. Using the phase separated configuration as shown in Fig. 12, the properties of the matrix were varied in the analyses while assuming the bee properties remained constant at 0 2 *E E <sup>b</sup>* and 0 10 *b* .

0 0.00001 0.00002 0.00003 0.00004 0.00005

Fig. 9. Relationship between fractions and Cahn Hilliard parameter (Kringos et al., 2012)

(a) (b)

As was discussed in the previous sections, it is important to know the properties of the bitumen phases with respect to its surrounding matrix, which is also needed for the healing simulation in the next section, Fig. 11. The rheology at various temperatures of the individual phases turned out to be more challenging than anticipated, so to get the parameters needed for the individual phases for the healing simulation, in this section a finite element analyses is performed. In this analysis the overall bitumen properties as

where case A represents an homogeneous mesh without any phase separation. The subscript m refers to the matrix properties and the subscript b refers to the bee structures. Using the phase separated configuration as shown in Fig. 12, the properties of the matrix were varied in the analyses while assuming the bee properties remained constant at 0 2 *E E <sup>b</sup>* and 0 10

=1, 2, 4 and 6 for 1 4 (Kringos

<sup>3</sup> <sup>4</sup>

<sup>1</sup> <sup>2</sup>

*b* .

Cahn Hilliard parameter

matrix IF bees

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

2 4

Fig. 10. (a) Normalized .. potential (b) Changing <sup>0</sup> ;

*c*

**5. Mechanical properties of the phases** 

determined in the previous section is used.

For each case the equivalent stiffness ratio is calculated from *E E case caseA*

et al., 2012)

  fractions

Fig. 11. Properties of bees and surrounding matrix (Kringos et al., 2012)

Fig. 12. Simulation of constant displacement test (Kringos et al., 2012)

From Table 1 the chosen matrix properties and the calculated equivalent stiffness at time 0.08 s is shown. In Fig. 13 the equivalent stiffness over the entire direct tension test is plotted.


Table 1. Determined parameters

Atomic Force Microscopy to Characterize the Healing Potential of Asphaltic Materials 221

(a)

*0.0 0.2 0.4 0.6 0.8 1.0 1.2*

*time [s]* 

*time [s]* 

*time [s]* 

(b)

*0.0 0.2 0.4 0.6 1.2 1.4 1.6 1.8* 

(c) Fig. 15. Fatigue simulation (a) with no rest period (b) with VE unloading during rest period

*0.0 0.2 0.4 0.6 1.2 1.4 1.6 1.8* 

In Fig. 16 and Fig. 17 the evolution of the normal stress and damage development are shown for the simulated time steps, respectively. Damage is hereby defined as equivalent plastic

What is very noticeable from the stress and damage development plots is the inhomogeneous development of the stresses and damage throughout the material. Additional it can be noticed from the damage graphs that the damage originates from the

In Fig. 17 the damage development at a chosen location in the simulated mesh is plotted as a function of the loading cycles and loading time. From these graphs it can be seen that case B has generated less damage than case A after the six loading cycles. By comparing with case C, that also included the phase rearrangement based on the kinetics of the material, it can be

seen that even less damage was generated in comparison with case A and B.

(c) with phase rearrangement during rest period (Kringos et al., 2012)

interface areas between the bee structures and the matrix.

strain, or permanent deformation.

*y* 

*y* 

> *y*

*0.001* 

*0.001* 

*0.001* 

Fig. 13. Calculation of equivalent stiffness for direct tension cases (Kringos et al., 2012)

In the case of the homogeneous configuration, the overall matrix normalized concentration is 0.5 , in the case of the phase separated configuration, the minimum normalized concentration of the bees is -1 and the maximum normalized concentration of the matrix is +1, Fig. 14. Based on this, the following relationship is formed:

Fig. 14. Alpha as a function of normalized concentration (Kringos et al., 2012)

#### **6. Simulation of healing in bitumen**

Using the parameters as described in the previous sections, in this section the developed healing model will be demonstrated. For this, three different cases have been selected, Fig. 15.

In all cases a mesh of 50 m x 50 m was exposed to six displacement controlled tension and compression loading cycles. In case A, this was done continuously without any rest periods between the cycles. In case B and C, a rest period was applied after the first three loading cycles. For case B, during this rest period the material was allowed to visco-elastically unload. In case C, in addition to the visco-elastic unloading, the material was allowed to rearrange its phases, based on the PS model as was shown in the earlier sections. The initial material configuration in all simulation was the phase-separated configuration as was computed in Fig. 7. In the simulation, the mechanical properties for the matrix and the bees as derived earlier in this chapter and the energy based elasto-visco-plastic constitutive model briefly described earlier were utilized.

Stiffness [MPa]

Case A Case B Case C Case D Case E Case F

(a) (b)

In the case of the homogeneous configuration, the overall matrix normalized concentration is 0.5 , in the case of the phase separated configuration, the minimum normalized concentration of the bees is -1 and the maximum normalized concentration of the matrix is

0

Fig. 14. Alpha as a function of normalized concentration (Kringos et al., 2012)


Using the parameters as described in the previous sections, in this section the developed healing model will be demonstrated. For this, three different cases have been selected, Fig. 15. In all cases a mesh of 50 m x 50 m was exposed to six displacement controlled tension and compression loading cycles. In case A, this was done continuously without any rest periods between the cycles. In case B and C, a rest period was applied after the first three loading cycles. For case B, during this rest period the material was allowed to visco-elastically unload. In case C, in addition to the visco-elastic unloading, the material was allowed to rearrange its phases, based on the PS model as was shown in the earlier sections. The initial material configuration in all simulation was the phase-separated configuration as was computed in Fig. 7. In the simulation, the mechanical properties for the matrix and the bees as derived earlier in this chapter and the energy based elasto-visco-plastic constitutive

concentration

0.04 0.05 0.06 0.07 0.08 0.09 0.1 time [s]

Case A Case B Case E Case F

0.5

1

1.5

2

Fig. 13. Calculation of equivalent stiffness for direct tension cases (Kringos et al., 2012)

+1, Fig. 14. Based on this, the following relationship is formed:

0 0.02 0.04 0.06 0.08 0.1 time [s]

Em = E0 ; = (0.84+0.34C) -1

alpha

Stiffness [MPa]

**6. Simulation of healing in bitumen** 

model briefly described earlier were utilized.

Fig. 15. Fatigue simulation (a) with no rest period (b) with VE unloading during rest period (c) with phase rearrangement during rest period (Kringos et al., 2012)

In Fig. 16 and Fig. 17 the evolution of the normal stress and damage development are shown for the simulated time steps, respectively. Damage is hereby defined as equivalent plastic strain, or permanent deformation.

What is very noticeable from the stress and damage development plots is the inhomogeneous development of the stresses and damage throughout the material. Additional it can be noticed from the damage graphs that the damage originates from the interface areas between the bee structures and the matrix.

In Fig. 17 the damage development at a chosen location in the simulated mesh is plotted as a function of the loading cycles and loading time. From these graphs it can be seen that case B has generated less damage than case A after the six loading cycles. By comparing with case C, that also included the phase rearrangement based on the kinetics of the material, it can be seen that even less damage was generated in comparison with case A and B.

Atomic Force Microscopy to Characterize the Healing Potential of Asphaltic Materials 223

Keeping in mind that these plots are made for a given location in the material, the equivalent stiffness of the overall mesh was also determined, Fig. 18. From these graphs it can be seen that case B again showed a lesser reduction of the intial stiffness after the six loading cycles as compared with case A. Case C showed even more restoration of the

0.0000

(a) (b)

From these analyses is can be concluded that the 'healing' propensity of bitumen which is generally quantified by a fatigue test, could be partly due to visco-elastic loading and partly due to the healing mechanism that was presented in this chapter. If this is indeed the case, this would have important implications for the in time development of the healing

Moreover, the two contributions (visco-elasticity and phase rearrangements) are controlled by completely different parameters. Which is important to understand when one wants to

For instance it can be seen from Fig. 18 (b) that by varying the phase mobility, the rate of healing can be changed. This would mean that the applied rest periods in laboratory healing tests or the breaks between traffic loading on asphalt pavements would have important effects on the healing of generated damage. It is very well possible that some bitumen have intrinsic healing capacity, but due to a relatively low mobility of the phases, this is never maximized in practice. This would mean that if bitumen producers could change something in the bitumen blend to increase the mobility of certain phases, the healing propensity could

The material properties of the individual phases are of paramount importance for the mechanical properties and the healing capacity of the entire bitumen. In the previous section the phase scans of bitumen under the AFM were discussed. The AFM can also be utilized to determine the material properties as well as the scanned phases simultaneously. The difficulty with observing bitumen using AFM is its viscoelastic nature and its highly temperature dependent flow behavior. In the following section, the utilization of AFM to

determine material properties of bituminous phases is discussed in more details.

loading cycle N

Fig. 18. Simulation of damage and healing development (Kringos et al., 2012)

*ps ve* 

*healing*

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

time [s]

0.0005

0.0010

0.0015

0.0020

0.0025

damage KSI

0.0030

No rest period Rest period VE Rest period VE & PS

*Phase mobility* 

equivalent stiffness in comparison with case A and B.

01234567

propensity of asphalt pavements in the field.

optimize the healing potential of an asphaltic mixture.

0.0015

be optimized.

0.0017

0.0019

0.0021

0.0023

0.0025

damage KSI [-]

0.0027

No restperiod Restperiod VE Restperiod VE & PS

Fig. 16. Stress development during fatigue simulation (Kringos et al., 2012)

Fig. 17. Damage development during fatigue simulation (Kringos et al., 2012)

!

*-0.15*

yy[MPa]!

*0.18*

0.0

max

t1 t2 t3 t4 t5 t6

t7 t8 t9 t10 t11 t12

t13 t14 t15 t16 t17 t18

t1 t2 t3 t4 t5 t6

t7 t8 t9 t10 t11 t12

t13 t14 t15 t16 t17 t18

Fig. 17. Damage development during fatigue simulation (Kringos et al., 2012)

Fig. 16. Stress development during fatigue simulation (Kringos et al., 2012)

Keeping in mind that these plots are made for a given location in the material, the equivalent stiffness of the overall mesh was also determined, Fig. 18. From these graphs it can be seen that case B again showed a lesser reduction of the intial stiffness after the six loading cycles as compared with case A. Case C showed even more restoration of the equivalent stiffness in comparison with case A and B.

Fig. 18. Simulation of damage and healing development (Kringos et al., 2012)

From these analyses is can be concluded that the 'healing' propensity of bitumen which is generally quantified by a fatigue test, could be partly due to visco-elastic loading and partly due to the healing mechanism that was presented in this chapter. If this is indeed the case, this would have important implications for the in time development of the healing propensity of asphalt pavements in the field.

Moreover, the two contributions (visco-elasticity and phase rearrangements) are controlled by completely different parameters. Which is important to understand when one wants to optimize the healing potential of an asphaltic mixture.

For instance it can be seen from Fig. 18 (b) that by varying the phase mobility, the rate of healing can be changed. This would mean that the applied rest periods in laboratory healing tests or the breaks between traffic loading on asphalt pavements would have important effects on the healing of generated damage. It is very well possible that some bitumen have intrinsic healing capacity, but due to a relatively low mobility of the phases, this is never maximized in practice. This would mean that if bitumen producers could change something in the bitumen blend to increase the mobility of certain phases, the healing propensity could be optimized.

The material properties of the individual phases are of paramount importance for the mechanical properties and the healing capacity of the entire bitumen. In the previous section the phase scans of bitumen under the AFM were discussed. The AFM can also be utilized to determine the material properties as well as the scanned phases simultaneously. The difficulty with observing bitumen using AFM is its viscoelastic nature and its highly temperature dependent flow behavior. In the following section, the utilization of AFM to determine material properties of bituminous phases is discussed in more details.

Atomic Force Microscopy to Characterize the Healing Potential of Asphaltic Materials 225

indentation of the Berkovich indenter. In AFM, a probe consisting of a cone with a nominal tip radius on the order of 10 nm or higher is typically used. The accurate determination of the AFM tip shape is in fact one of the major sources of uncertainty, when performing nanoindentation testing with AFM, particularly at lower load levels. As it has been argued by many investigators, in indentation testing at micro- and nanoscales it is very difficult to make a valid assumption concerning the indenter geometry, cf. e.g. Korsunsky (2001) and Giannakopoulos (2006). Even the best attempts at preparing a perfectly round spherical or perfectly sharp conical shapes inevitably produce flattened imperfect shapes. These deviations of the indenter shapes from the assumed ideal shape will affect measurements performed at small scales. The AFM tip is normally considered to be of conical shape with round spherical tip. The contact geometry is thus dominantly controlled by the spherical indenter tip at low load levels and is switching to conical geometry at higher loads. The procedure to extract viscoelastic properties with spherical indentation is illustrated below

Using ordinary notation the stress-strain relations for linear viscoelastic solids can be

<sup>1</sup>

<sup>2</sup>

1 1 , 3 3 *ij ij ij kk ij ij ij kk s e*

are the deviatoric components of stress and strain. In equations (4a) and (4b), *G*1 and *G*<sup>2</sup> are the so-called relaxation functions in shear and dilation, respectively. The viscoelastic

> () () <sup>1</sup> ( ) 2 () () *Gs Gs*

 0 *st f s f t e dt* 

In principle, in order to completely characterize the viscoelastic material one needs to determine two independent viscoelastic functions <sup>1</sup> *G t*( ) and <sup>2</sup> *G t*( ) . However, these two functions cannot be determined uniquely from experimental force-displacement data. Thus in most conventional testing techniques a constant viscoelastic Poisson's ratio is assumed,

t , is related to the relaxation functions in equations (4a) and (4b) as:

 2 1 2 1

*s Gs Gs*

 

*ii ii <sup>d</sup> t Gt d d*

*d* 

 

 

(6)

(7)

(4a)

(4b)

(5)

*ij ij <sup>d</sup> st Gt e d*

0

0 3 *t*

 

*s*

using Laplace transformed quantities according to

*t*

(Larsson & Carlsson, 1998)

where

Poisson's ratio,

formulated in relaxation form as:

Fig. 19. Calculation of equivalent stiffness (Kringos et al., 2012)
