Kinetics of healing of asphalt mixtures

Journal Pre-proof Kinetics of healing of asphalt mixtures Xue Luo, Bjorn Birgisson, Robert L. Lytton PII:

S0959-6526(19)34660-8

DOI:

https://doi.org/10.1016/j.jclepro.2019.119790

Reference:

JCLP 119790

To appear in:

Journal of Cleaner Production

Received Date: 17 May 2019 Revised Date:

4 October 2019

Accepted Date: 17 December 2019

Please cite this article as: Luo X, Birgisson B, Lytton RL, Kinetics of healing of asphalt mixtures, Journal of Cleaner Production (2020), doi: https://doi.org/10.1016/j.jclepro.2019.119790. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Kinetics of Healing of Asphalt Mixtures Xue Luo, Ph.D. College of Civil Engineering and Architecture Zhejiang University 866 Yuhangtang Road Hangzhou 310058, China Office: +86 571-88206542 Email: [email protected]

Bjorn Birgisson, Ph.D., P.E. Zachry Department of Civil Engineering Texas A&M University 3136 TAMU College Station, Texas 77843 Phone: (979) 845-6039 Email: [email protected]

Robert L. Lytton, Ph.D., P.E. Zachry Department of Civil Engineering Texas A&M University 3136 TAMU College Station, Texas 77843 Phone: (979) 845-9964 Email: [email protected]

* Corresponding author: Bjorn Birgisson, [email protected]

ABSTRACT Promotion of healing of asphalt mixtures is significant to enhance the resiliency and extend the service life of asphalt pavements. Extensive laboratory measurements of effects of healing and field observations have proven its impact. Many approaches have been developed to explain and model healing of asphalt mixtures based on its mechanisms and/or mechanical features. However, due to the complexity caused by the co-existence of several coordinated rate processes during healing, there are still unclear aspects in terms of mechanisms and appropriate representation of each rate process. Target this issue, this study proposes to investigate healing of asphalt mixtures using kinetic and thermodynamic quantities and formulations, aiming at clarifying the relationships among the coordinated processes involved in healing and developing an approach to predict the rate of each process. Healing is found to contain two coordinated rate processes: free energy change and bond restoration, and healing rate is proportional to the rates of these two processes, and the healing activation energy is the product of the activation energy of the free energy change and that of the bond restoration. In order to calculate each activation energy and rate constant, the recoverable strain energy (RSE) and recoverable pseudo strain energy (RPSE) measured in the rest time of a creep and step-loading recovery (CSR) test are used. The RPSE in the intact material and the surface energy constitute the total free energy change for healing. A three-parameter model is proposed to simulate a rate process like the change of the RSE/free energy change with time, and a dimensionless logarithmic rate is defined to obtain a representative change rate. This representative rate is ideal to be used as the rate constant in the Arrhenius equation to compute the activation energy. The activation energies for viscoelastic recovery, healing, free energy change, and bond restoration are determined for two types of asphalt mixtures at unaged and aged states. The relations between different activation energies are clarified so as the relationships among different rate processes. Furthermore, the healing activation energy obtained from this kinetic approach is proven to be the same as that calculated using the damage density from a mechanistic approach. A method to generate a damage density-based healing curve at any rest time at any temperature using just energy measurements is proposed in this study.

Keywords: healing; asphalt mixture; kinetics; activation energy; free energy

2

1. Introduction Healing of asphalt mixtures occurs when a cracked asphalt pavement is in the rest period. Under repeated traffic loading, cracks in asphalt mixture will initiate and grow to form new crack surfaces. During the rest period of traffic loading, crack surfaces tend to gradually close to restore original material properties of asphalt mixtures. This process is often called self-healing, which means that it could happen without execution of any external assistance like mechanical force. There have been numerous facts about the benefits of healing from laboratory investigations, such as extension of fatigue life of asphalt mixtures (Lytton et al., 1993; Shen et al., 2010). Field measurements also indicated that minor fatigue cracking completely healed in in-service asphalt pavements with low traffic loading (Nishizawa et al., 1997). Since the maintenance/rehabilitation of asphalt pavements will consume fossil fuels and release chemical pollutants to the air, increasing the service life by healing cracks is definitely beneficial to the environment. In addition, from the perspective of life cycle assessment, it was found that enhanced asphalt healing rates would lead to a significant reduction in greenhouse gas emission and energy usage (Butt et al., 2012). Due to these favorable characteristics, investigations of healing in asphalt pavements have been presented in many studies, which nowadays have been classified within environmentally sustainable construction strategies (Ayar et al., 2016). A number of factors affect the healing speed of an asphalt pavement, like the temperature, aging time, binder type, air void content, etc. (Si et al., 2002; Luo et al., 2015a). For instance, increasing temperature or changing the composition of asphalt mixtures can greatly accelerate the healing process. Accordingly, several new technologies have been invented to increase healing capability and/or rate. Introducing metallic additives or microcapsules into asphalt materials is one of the most popular methods at this moment (Gallego et al., 2013; Dai et al., 2013; Chung et al., 2015; Norambuena-Contreras and Garcia, 2016). The metallic additives like steel wool fibers increase the conductivity of an asphalt mixture, which enables more efficient heating by electromagnetic induction or microwaves. When the temperature increases more rapidly, the asphalt binder flows more easily between crack surfaces to promote healing. The microcapsules work in a different way, which sequester the healing agent, or called rejuvenator. When an approaching crack triggers the release of rejuvenators, a chemical reaction occurs in the asphalt around the capsule and help bond crack surfaces. 3

In order to quantify the effectiveness of these new technologies, mechanical tests are usually performed on asphalt mixtures with and without healing processes, such as the threepoint beam bending test and semi-circular bending test. A healing ratio of modulus, load, or strength is defined in order to examine the change due to healing (Liu et al., 2013; Al-Mansoori et al., 2017). This ratio graphically demonstrates the effect of healing on a selected material property or parameter, which could help guide the mixture design with incorporation of new additives. However, since healing of an asphalt mixture is closely related to its composition and internal structure as well as environmental conditions, a phenomenological index like the modulus ratio or load ratio is not enough to discern the mechanism of improvement during the healing process. A mechanism-based or mechanics-based study would be more appropriate to gain a better understanding regarding how to enhance healing capability and healing rate for asphalt pavements. The investigation of healing mechanism of viscoelastic materials like polymer solids or asphalt mixtures has been conducted extensively. Early studies discussed the physical mechanisms involved in a healing process, which include the contact and adhesion of crack surfaces, interchange of molecular segments across the interface, and formation of new entanglements at the interface (Kausch et al., 1987). Wool and O’Connor (1981) proposed a convolution integral function to describe healing as a result of wetting and intrinsic healing process. For materials like asphalt mixtures, the wetting process is dictated by material properties such as creep compliance, surface energy, and geometric fractures of the cracks; the intrinsic healing process is dictated by the self-diffusivity and mobility of the asphalt molecules (Little et al., 2015). Using the Fourier-transform infrared spectroscopy (FTIR), a good correlation was found between the mobility of asphalt molecules and healing rate for asphalt binders (Kim et al., 1990). This conclusion was further confirmed based on the computational modeling of asphalt binders using molecular dynamics, which showed that the self-diffusivity of asphalt molecules was closely related to healing properties of binders (Bhasin et al., 2010). The diffusion coefficient of asphalt binders calculated from molecular dynamic modeling was found to increase with temperature and change with the binder composition (Sun et al., 2016). However, due to the complexities involved in the diffusion process, this part of healing is usually described by a phenomenological model (Bommavaram et al., 2009) or by the molecular dynamic simulations as mentioned above. 4

In addition to the mechanism-based approaches, healing of viscoelastic materials has been studied using the mechanical analyses. Schapery (1989) used the bonding speed of a crack in linear viscoelastic materials to describe the crack closing process. This speed was driven by the interfacial forces of attraction, namely surface energy, so the adjacent separated crack surfaces tended to be closed gradually over the bonding zone. The bonding zone length and bonding speed were calculated through rigorous mechanical derivations. Besides the surface energy, the authors introduced the concept of the internal stress to study healing of asphalt mixtures (Luo et al., 2013; 2015b). Under the circumstance of loading, the internal stress is one of the two components that constitute the applied stress (Raghavan and Meshii, 1997). When the load is removed, the internal stress is the stress left in a viscoelastic material to drive the deformation to recover (Luo et al., 2013). Accordingly, the energy redistribution occurs under the action of the internal stress and surface energy during the rest period. There is redistributed energy associated with the internal stress and released energy related to the surface energy. The authors further established the energy balance equations using an Energy-Based Mechanistic (EBM) approach, and solved for the reduction of the cracked area in an asphalt mixture during the rest period (Luo et al., 2015a; b). The EBM approach provides a cause-and-effect description of the change of internal geometry of cracks due to healing. It avoids phenomenological representations and enables a direct measurement of healing from a simple mechanical test. According to the mechanism- and mechanics-related studies reviewed above, healing of asphalt materials involves several coordinated rate processes, which are driven by different driving forces and dependent on the material composition, temperature, aging, etc. An explicit and accurate representation of each rate process would provide more insights to understand the essence of healing of a viscoelastic material. With respect to the pavement infrastructure, more accurate computation of healing rates for asphalt mixtures would provide better guidance for accelerating the healing process. Therefore, this study aims at investigating the kinetics of healing of asphalt mixtures to clarify the coordinated rate processes involved in healing, and developing an approach to predict the rate of each process based on the kinetic and thermodynamic principles. To achieve this objective, the next section will first analyze healing of asphalt mixtures from the perspective of kinetics and thermodynamics, which will define healing from a new perspective and establish the characterization as well as modeling approaches. Then the 5

following three sections will introduce the laboratory investigations and analysis methods, including the background about viscoelastic recovery and healing of asphalt mixtures, development of an appropriate rate term for a rate process, and computation of kinetic and thermodynamic quantities. Finally, a summary section concludes this study with the main contributions and future work.

2. Thermodynamics and Kinetics of Healing of Asphalt Mixtures Healing is a counter process to cracking. There is a number of initial flaws, or air voids in an asphalt mixture, which lower the global strength through the local stress magnification and serve as initial cracks. In order for these initial cracks to propagate, sufficient stress and work are needed at the atomic level to break the bonds between atoms and/or molecules (Anderson, 2005), so two crack surfaces are formed and gradually open. On the contrary, healing is the process of crack closure and bond restoration between two crack surfaces. In the following two subsections, more details about healing of asphalt mixtures are discussed from the perspective of thermodynamics and kinetics, respectively.

2.1 Thermodynamics of healing From a thermodynamic point of view, the discussion of healing of an asphalt mixture involves a parameter called Gibbs free energy, or free energy. Free energy is a function of other thermodynamic quantities like the representation of the internal energy and measurement of the randomness or disorder of a system. With respect to a change process, the quantity that indicates how the process proceeds is the change of the free energy. When the change of the free energy has a negative value, the process occurs automatically (Bejan, 2016). During the process of healing, there are two contributions to the total free energy change in an asphalt mixture. The first is the free energy difference in the intact material between a nonhealed state and a healed state, which is designated volume free energy for healing. The zone where the volume free energy is redistributed is called the supply zone. The second is the free energy difference in the two crack surfaces between a non-healed state and a healed state, called the surface free energy. Figure 1 illustrates a crack with two contact area growth zones, where the closure of crack surfaces and restoration of bonds occur. In the vicinity of the crack, the force lines are not uniformly distributed showing the nonlinearity and stress concentration. There are 6

two supply zones for the volume free energy, above and below the crack respectively. For each crack, there is certain surface free energy stored on the crack surfaces. When the crack gradually heals, the surface free energy is released as the crack surfaces close and the volume free energy is redistributed to the intact material in the supply zone. The magnitude of the contribution of volume/surface free energy is the product of the volume/surface free energy density and the corresponding volume/area. Thus, the total free energy change is expressed as: ∆ G H = ∆ GvH VGvH − γ S γ

(1)

where ∆GH is the free energy change for healing; ∆GvH is the volume free energy density for healing; VGvH is the volume of the supply zone; γ is the surface energy density, −γ indicates that the surface free energy is released, or lost when a crack closes; and S γ is the surface area of the contact area growth zone. Force line

Volume free energy in the supply zone

Intact material

Healed crack

Contact area growth zone

Surface free energy at crack surfaces

Figure 1. Illustration of volume free energy and surface free energy for an asphalt mixture during healing Based on Equation 1, the rates of the change of the volume, surface, and total free energy can be obtained, which are plotted schematically against the crack radius in Figure 2. The rate of the change of the surface free energy must be larger than that of the volume free energy so that healing could occur. When the rates are equal, the value of the free energy change rate is zero. Beyond this point, the process of self-healing cannot happen unless an external energy is added to the system, such as applying a compressive load or heat the material. This critical point is the

7

threshold between healable and non-healable cracks, called critical crack radius for healing as shown in Figure 2.

+

Volume free energy change rate

Critical crack radius for healing (threshold between healable and non-healable cracks)

Free energy change rate, Crack radius

∆G& H

Direction of healing

-

Surface free energy change rate

Figure 2. Schematic curves for the volume free energy rate and surface free energy rate contributions to the total free energy change rate during healing 2.2 Critical crack radius for healing According to the definition above, when the crack radius is less than the critical crack radius, the cracks are healable. This means that the cracks could return to their original status, namely the initial flaws, or air voids for asphalt mixtures. To obtain an explicit expression of the critical crack radius for healing, the cracks in asphalt mixtures are assumed to be penny-shaped. Thus, the three-dimensional (3D) view of the volume free energy supply zone resembles a cone, and the 3D view of the zone that surface free energy is released looks like a toroid (Luo et al., 2015d). Accordingly, the free energy change for healing is computed by:

∆GH = ∆GvH VGvH − γ Sγ  2 M I π 2 cI 3 2 M N π 2 cN 3  2 2 = ∆GvH  −  − γ 2 M I π cI − 2 M N π c N 3 3  

(

)

(2)

where c I is the initial average crack size before healing; cN is the new average crack size after healing; M I is the number of initial cracks before healing; and M N is the number of new cracks after healing.

8

To obtain the critical crack radius for healing, Equation 2 is differentiated with respect to the crack radius, and set the resulting expression equal to zero (note that the crack radius is decreasing so the crack increment is negative):

d ∆GH = ∆GvH 2 M N π 2 cN 2 − γ ( 4 M N π cN ) = 0 dc

(

)

(3)

which leads to the result

ccH =

2γ ∆GvH π

(4)

where ccH is the critical crack radius for healing. If the crack is further specified as cohesive cracking (fracture within asphalt mastic) or adhesive cracking (fracture at the asphalt-aggregate interface), Equation 4 should be expressed as:

ccH

 ∆G c   ∆GvH π = a  ∆G  ∆GvH π

cohesive cracking (5)

adhesive cracking

where ∆Gc is the bond energy of cohesion; and ∆G a is the bond energy of adhesion. 2.3 Kinetics of healing The discussions of healing based on the concept of free energy demonstrate that the rate of the free energy change is essential to understand the mechanism of healing. This rate process depends on the mixture composition, temperature, aging, etc. In addition, another important rate process involved in healing is the restoration of bonds when two crack surfaces are in contact, which are also influenced by these factors. In order to characterize these rate processes, the Arrhenius equation is employed to establish the formulation. The Arrhenius equation is a promising model for asphalt materials. The authors had successfully used it to characterize aging of in-service asphalt pavement by relating the modulus change rate to the field aging temperature, based on which the kinetics-based aging prediction models were developed (Luo et al., 2015c; 2017; 2018). In the theory of kinetics, the rate constant, usually designated by the symbol k, measures how fast a process researches equilibrium. There is a minimum requirement for input of energy to enable the process to proceed; this energy is called activation energy. Different materials have 9

different activation energies, which are affected by the material itself and conditions in which the process takes place. Accordingly, the value of the rate constant is determined by these factors. The rate constant is usually obtained experimentally, which can be formulated with the activation energy using the Arrhenius equation (Bamford and Tipper, 1969):

k = Ae

−

Ea RT

(6)

where k is the rate constant for a process; A is the pre-exponential factor; Ea is the activation energy; T is the absolute temperature at which the process happens; and R is the universal gas constant. Since healing of an asphalt mixture contains two rate processes, the healing rate can be formulated to be proportional to the rate of the free energy change and bond restoration rate, which is:

(

k H = f ∆G& H ∗ k B

)

(7)

in which ∆G& H = A∆GH e

k B = AB e

−

Ea∆GH RT

(8)

EaB RT

(9)

where k H is the healing rate; k B is the bond restoration rate; A∆G and AB are the preH

exponential factor for free energy change rate and bond restoration rate, respectively; and Ea ∆G

H

and E aB are the activation energy for free energy change and bond restoration, respectively. Note that Equations 8 and 9 have a different sign for the activation energy, which will be explained later in Section 4. According to Equations 7 to 9, there is: E aH = − E a ∆G H + E aB

(10)

where EaH is the activation energy for healing.

2.4 Relationships between healing, damage, recovery, and aging Since the healing rate of an asphalt mixture is influenced by the material itself and conditions in which healing occurs, the relationships between healing, damage, recovery, and aging are discussed in more details based on the concept of kinetics. Figure 3 illustrates different states of an asphalt mixture: UU stands for undamaged and unaged, DU for damaged and unaged, UA for 10

undamaged and aged, DA for damaged and aged, DPU for plastic damage and unaged, DCU for cracking damage and unaged, DPA for plastic damage and aged, and DCA for cracking damage and aged. The solid line in Figure 3 indicates a full process (e.g. undamaged to damaged); the dashed line indicates a partial process (e.g. a damaged material can only recover its viscoelastic deformation and heal its cracking damage; plastic damage is irreversible).

Damage

DPU DU

UU

DCU

Viscoelastic recovery & healing Aging

Aging Damage

DPA DA

UA Viscoelastic recovery & healing

DCA

Figure 3. Relationship matrix of different states of asphalt mixtures

In Equation 7 above, the healing rate is regarded to be a composite value. Similarly, a rate process that contains several coordinated sub-rate processes could be formulated in this way. For instance, the change from an undamaged asphalt mixture to a damaged one involves the evolution of plastic deformation and cracking. This means that the rate of the damage change is the product of the plastic deformation rate and cracking rate. When an asphalt mixture is not damaged, the recovery is a process of viscoelastic deformation, which is called viscoelastic recovery herein. Once the material is cracked, the recovery actually involves two simultaneous processes: viscoelastic recovery of the intact material and healing of cracks. Thus, when a damaged asphalt mixture recovers and heals, the change rate is actually determined by the viscoelastic recovery rate and healing rate. When the material is aged, the aging rate is affected by the oxidation rate and the oxygen diffusion rate. Therefore, the damage rate, recovery-healing rate, and aging rate can all be expressed as the product of the rates of its sub-processes. Within the scope of this study, the recovery-healing rate and the effect of aging are presented herein. Similar to Equation 7, the recovery-healing rate can be expressed as: k RH = f ( k R ∗ k H

)

(11)

11

where k RH is the rate of the change of a recovery-healing process; and k R is the viscoelastic recovery rate. Thus, there is the following relationship in terms of the activation energy:

EaRH = EaR − EaH

(12)

where EaRH is the activation energy for a recovery-healing process; and EaR is the activation energy for viscoelastic recovery. The sign of the activation energy will be explained later in Section 4 with the experimental data. Substitute Equation 10 into Equation 12, which gives: E aRH = E aR + E a ∆ G H − E aB

(13)

When the material is aged, Equations 11 and 13 are still valid, but the values of the activation energy are different. In order to determine each activation energy in Equation 13, this study proposes a new method to measure and calculate the rate constant of a rate process for asphalt mixtures, which are introduced in the following sections.

3. Characterization of Viscoelastic Recovery and Healing of Asphalt Mixtures Healing of asphalt mixtures is always companied by a recovery process. In the previous studies, the authors had designed the creep and step-loading recovery (CSR) test to measure the internal stress that drives the viscoelastic recovery and healing of asphalt mixtures (Luo et al., 2013; 2015b), which is elaborated below with the materials used in this study.

3.1 Laboratory tests and materials The CSR test is designed especially for measuring the internal stress when an asphalt mixture is unloaded. Due to the viscoelastic nature of asphalt materials, there are residual stresses left that drive the material to deform when the external load is removed. Such a residual stress is called internal stress, which diminishes with time. As shown in Figure 4, during the rest period of a CSR test, there are several three-step loads at different locations to measure internal stresses at different stages. The step-loads introduce interruptions to the strain in three ways: (1) when the stress induced by the step-load is less than the internal stress, the strain rate is less than zero; (2) when the stress is equal to the internal stress, the strain rate is zero; and (3) when the stress is larger than the internal stress, the strain rate is positive, as demonstrated by the magnified strain in Figure 4. At the second condition, the internal stress is measured by the added step-load. 12

Based on the principle of measurement for the internal stress, the nondestructive and destructive CSR tests are designed to measure the internal stress in undamaged asphalt mixtures and damaged ones, respectively. More details about the design and validation of this testing method can be found in Luo et al. (2013). The nondestructive and destructive CSR tests are performed using the Material Test System (MTS). Two types of asphalt binders selected for this study are called “NuStar” from New Jersey and “Valero” from California. The aggregate is Hanson limestone from Texas. The asphalt mixtures are compacted by the superpave gyratory compactor to produce cylindrical specimens, which are 102mm in diameter and 152mm in height. A pair of end-caps are glued to the specimens so as to apply tensile loading. The deformation is measured as the average of the readings of three axial linear variable differential transformers (LVDTs). Half of the asphalt mixture specimens are unaged and the other half are aged in the environment room for 6 months. The CSR test is conducted on each specimen nondestructively at three temperatures: 10°C, 20°C, and 30°C and then destructively at the same three temperatures.

250

250

145 61

200 Axial stress (kPa)

155

66 Time (s)

71

200

Magnified

150

150 Axial strain

100 Axial stress

50

100

3-step load

Axial strain (με)

Strain (με)

165

50

0

0 0

50

100 150 Time (s)

200

250

Figure 4. Typical axial stress and strain of an asphalt mixture measured from a CSR test

13

3.2 Mechanistic modeling of healing From the CSR test, a series of internal stresses of asphalt mixtures are obtained as shown in Figure 5. In the legend, “R” stands for the viscoelastic recovery of an undamaged material; “RH” stands for the viscoelastic recovery and healing of a damaged specimen; “0 mo” and “6 mo” stand for an unaged specimen and aged one for 6 months, respectively; “M” and “m” stands for the measured value and modeled value, respectively. Figure 5 presents the typical features of the internal stress of asphalt mixtures. It decreases with the rest time till it vanishes. When the asphalt mixture is subjected to a higher creep stress in the destructive test than that in the nondestructive test, the corresponding internal stresses are larger. For both undamaged and damaged asphalt mixtures, the internal stress is sensitive to the temperature, which increases as the temperature decreases. In order to generate a continuous curve using the discrete data points of the internal stress, an exponential model was proposed by the authors earlier (Luo et al., 2013):

σ i (t ) = σ ae

−

t

η1

+ σ be

−

t

η2

(14)

where σ i ( t ) is the internal stress; σ a , σ b , η1 , and η 2 are fitting parameters for σ i ( t ) . The predicted internal stresses by Equation 14 are shown in Figure 5. The model matches well with the measured values, and agrees with the fact that the internal stress eventually vanishes at some point of the rest time.

Internal stress (kPa)

80 R_M RH_M R_m RH_m

60 40 20 0 50

100 Time (s)

150

200

Figure 5. Example of measured and modeled internal stresses of asphalt mixtures

14

Using the internal stress, the EBM approach was established in our previous studies to determine the evolution of healing of asphalt mixtures in a pure mechanistic manner (Luo et al., 2015b). The EBM approach includes a force equilibrium principle and an energy balance principle. For a CSR test, the mathematical forms of the approach are: •

•

In the loading phase of a CSR test:

σ A S = σ A ( S − Sc )

(15)

DSE A = DSET

(16)

DPSE A = DPSE T

(17)

In the rest period of a CSR test:

RSE A = RSET

(18)

 2 M I π 2cI 3 2 M N π 2 cN 3  RPSE AVm = RPSE T Vm + RPSE T  −  3 3  

(

−γ 2 M I π cI − 2 M N π cN 2

2

)

(19)

where σ A is the apparent creep stress; S is the cross sectional area; σ T is the true creep stress, which is the stress in the intact material; Sc is the area of all the cracks on the cross section; DSEA is the apparent dissipated strain energy (DSE); DSET is the true DSE of the intact material; DPSEA is the apparent dissipated pseudo strain energy (DPSE); DPSET is the true DPSE of the intact material; RSEA is the apparent recoverable strain energy (RSE); RSET is the true RSE of the intact material; RPSEA is the apparent recoverable pseudo strain energy (RPSE); RPSET is the true RPSE of the intact material; and V m is the volume of the asphalt mastic in one layer of the asphalt mixture, whose thickness equals to the mean film thickness. The application of Equations 15 to 19 to the CSR test data quantifies the reduction of the cracking damage in an asphalt mixture due to healing through the damage density and average crack size, as shown in Figure 6. The extent of cracking damage is computed by the damage density φ ( φ = S c S ), which increases under a destructive loading and decreases as healing proceeds.

15

5.2% Damage density (%)

20C_cracking

20C_healing

5.1% 5.0% 10C_healing

4.9% 10C_cracking

4.8%

End of creep phase & start of healing

4.7% 0

30

60

90 Time (s)

120

150

180

Figure 6. Examples of damage density of asphalt mixtures in destructive CSR test 3.3 Characteristics of recovery strain energy and free energy for healing

In the EBM approach, two types of energy are associated with healing of asphalt mixtures: RSE and RPSE. The RSE contains the energy for viscoelastic recovery and healing, while the RPSE represents the energy that is only associated with healing. Therefore, the RSE can be used to study the combined recovery-healing process, and the RPSE can be used to compute the free energy for healing. The RSE is calculated by:

RSE = ∫ σ i ( t ) d ε r t

t0

(20)

where t0 is the start of the rest time; t is the rest time; and ε r is the residual strain in the rest period, which is modeled as follows:

ε r ( t ) = ε 0,r + ε1,r e

−

t

γr

(21)

in which ε 0,r , ε 1, r , and γ r are fitting parameters for ε r ( t ) . Then the RSE is calculated for each asphalt mixture specimen tested in the CSR test. A few examples of the computed RSE are shown in Figure 7. Similar to the internal stress, the RSE diminishes with the rest time. When the asphalt mixture specimen is aged or subjected to a higher load, the value of the RSE increases. The RSE also varies with the change of the temperature; at lower temperatures the RSE is larger.

16

RSE (J/m3)

1.2E-02 R_0 mo_20C RH_0 mo_20C R_6 mo_20C

8.0E-03

4.0E-03

0.0E+00 50

100

150

200

Time (s) (a) NuStar mixture (0 and 6 months)

RSE (J/m3)

2.8E-01 RH_10C RH_20C RH_30C

2.1E-01 1.4E-01 7.0E-02 0.0E+00 50

100 Time (s)

150

200

(b) Valero mixture (6 months) Figure 7. Examples of RSE of asphalt mixtures from the CSR test

On the other hand, based on the calculated average crack size from the RPSE balance equation (Equation 19), the free energy change for healing can be computed for each time increment i:  2 M Ii −1π 2 c Ii −13 2 M Niπ 2 c Ni 3  2 2 ∆ G Hi = RPSE iT  −  − γ 2 M Ii −1π c Ii −1 − 2 M Niπ c Ni 3 3  

(

)

(22)

 2 M Iiπ 2cIi 3 2 M Niπ 2 cNi 3  in which i = 1, 2, 3, …; RPSEiT  −  is the volume free energy for healing 3 3  

(

)

at time increment i; and γ 2M Iiπ cIi 2 − 2M Niπ cNi 2 is the surface free energy at time increment i. 17

An example plot of these two types of free energy is given in Figure 8. Both the surface and volume free energy change as the crack radius decreases. The absolute value of the surface free energy is much larger than the volume free energy. This is because the cracking damage in the material is very small (namely the crack radius is small), as indicated by the damage density in Figure 6, so the volume free energy associated with micro-cracks is relatively much lower than

1.147 0.0E+00

1.148

1.149

1.150 1.0E-12 Direction of healing

-2.0E-10

8.0E-13

-4.0E-10

6.0E-13

-6.0E-10

Surface free energy Volume free energy

4.0E-13

-8.0E-10

2.0E-13

-1.0E-09

0.0E+00

Volume free energy (J)

Surface free energy (J)

the surface free energy.

Crack radius (mm) Figure 8. Example of surface free energy and volume free energy of an asphalt mixture measured from the CSR test As a result, the total free energy change for healing at an arbitrary moment during the rest period is: ∆ G H = ∑ ∆ G Hi

(23)

Figure 9(a) presents an example of the values of ∆GH at three temperatures, showing its temperature sensitivity. At a lower temperature, the free energy change is larger and this is because the value of RPSET, i.e. volume free energy increases. In addition, the total free energy change varies with the binder type and aging time, as shown in Figure 9(b). Note that Figure 7 utilizes the time from the beginning of a CSR test; the rest time in Figure 9 starts from the beginning of the rest period of a CSR test.

18

0

30

60

90

Free Energy Change (J)

0.0E+00 -5.0E-08 10C 20C 30C

-1.0E-07 -1.5E-07 -2.0E-07

Rest Time (s)

(a) NuStar mixture at different temperatures 0

30

60

90

Free Energy Change (J)

0.0E+00

-5.0E-08

NuStar_0 mo_30C NuStar_6 mo_30C Valero_6 mo_30C

-1.0E-07

-1.5E-07

Rest Time (s)

(b) Unaged and aged NuStar and Valero mixtures Figure 9. Total free energy change for healing of an asphalt mixture measured from the CSR test 4. Kinetics-Based Modeling of Viscoelastic Recovery and Healing in Asphalt Mixtures The results of the RSE and ∆GH of asphalt mixtures above demonstrate that their rates keep changing as the rest time increases. In order to develop a kinetics-based relationship, a characteristic rate of the change of the RSE or ∆GH must be obtained first by properly modeling the energy evolution curve and formulating a representative energy change rate. Since the curves of the RSE and ∆GH are similar, only the RSE is taken as an example and ∆GH is analyzed in the same way.

19

4.1 Modeling a rate process At the beginning of the rest period, there is a maximum RSE stored in an asphalt mixture, so the change of the RSE can be calculated by:

∆RSE = RSEmax − ∫ σ i ( t ) d ε r t

t0

(24)

where ∆RSE is the change of the RSE of an asphalt mixture; and RSE max is the maximum RSE at the beginning of the rest period. Fit the curve of the ∆RSE versus the rest time using the following model:

∆RSE = ∆RSE∞ e

ρ −  t 

β

(25)

where ∆ RSE ∞ , ρ , and β are fitting parameters for ∆RSE . The advantage of this model is that the three parameters have their own mathematical meanings, which are illustrated in Figure 10. When the rest time t increases toward infinity, the value of ∆RSE approaches ∆RSE∞ as shown in Equation 26 and Figure 10. In other words, ∆RSE∞ is the maximum amount of the RSE change that an asphalt mixture can possess, so ∆RSE∞ is equal to RSEmax.

limt →∞ ∆RSE∞ e

ρ −  t 

β

= ∆RSE∞

(26)

In Equation 25, ρ is a scaling factor along the horizontal axis. The curve of ρ = 1 (red dashed one) is sketched in Figure 10. When ρ > 1, the curve is stretched horizontally (blue solid one). Thus, a larger value of ρ means that it requires more rest time for the material to make the same change of the RSE. A characteristic time point is t = ρ, at which the value of ∆RSE is:

∆RSEt = ρ =

∆RSE∞ e

(27)

When ρ > 1 and β > 1, the curve exhibits a clear S-shape (green dash-dotted one) with an inflection point at which the sign of the curvature changes from positive to negative. The mathematical expression for this point is given by: 1

 β β tc = ρ    β +1 

(28)

where tc is the time of the inflection point.

20

ρ = 1, β<1

∆RSE ∆RSE∞

ρ > 1, β<1

ρ > 1, β>1

Inflection Point

t

Rest Time ρ ρt tc

Figure 10. General shape of the proposed model for the change of RSE

Using the proposed model in Equation 25, plots of measured and predicted ∆RSE versus the rest time of undamaged and damaged asphalt mixtures are generated and given in Figure 11. For undamaged asphalt mixtures, there is only a viscoelastic recovery process during the rest period; for damaged asphalt mixtures, there are the closure of crack surfaces and restoration of bonds along with the viscoelastic recovery of the material. The observations reveal that the ∆RSE measured from the CSR test is a good indicator of what actually happens in the material during the rest period. The proposed model in Equation 25 can accurately describe (1) the viscoelastic recovery process; and (2) the combined viscoelastic recovery and healing process. Similarly, for the curve of ∆GH versus the rest time like those in Figure 9, Equations 24 and 25 are applied for each tested asphalt mixture to obtain the three model parameters.

21

∆RSE (J/m3)

4.0E-03 3.0E-03 2.0E-03 1.0E-03 0.0E+00 0

30 60 90 Rest Time (s) R_0 mo_10C_M R_0 mo_10C_m R_0 mo_20C_M R_0 mo_20C_m R_6 mo_30C_M R_6 mo_30C_m (a) Undamaged NuStar mixture 4.5E-02 ∆RSE (J/m3)

3.6E-02 2.7E-02 1.8E-02 9.0E-03 0.0E+00 0

30 60 Rest Time (s)

90

RH_0 mo_10C_M RH_0 mo_10C_m (b) Damaged NuStar mixture Figure 11. Examples of measured and modeled change of the RSE of asphalt mixtures

4.2 Formulation of representative energy change rate After defining the model for the change of the RSE and ∆GH , the next step is to formulate a representative energy change rate to describe the speed at which the viscoelastic recovery and healing of asphalt mixtures proceed during rest period. The procedure of the derivation is listed as follows using ∆RSE as an example. 22

First, calculate the rate of the change of ∆RSE:

∂∆RSE = ∆RSE βρ β t − β −1 ∂t

(29)

The rate in Equation 29 is then transformed to a dimensionless logarithmic rate by performing the following mathematical manipulation:

 ∂∆RSE     ∆RSE  = ∂ ln ( ∆RSE ) ∂ ln ( t )  ∂t     t  in which

∂ ln ( ∆RSE ) ∂ ln ( t )

(30)

is the dimensionless logarithmic rate of the change of ∆RSE. Combining

Equations 29 and 30 gives:

∂ ln ( ∆RSE ) ∂ ln ( t )

=βρ β t − β

(31)

According to Equation 31, the dimensionless logarithmic rate of the change of ∆RSE varies with the time. Take the logarithm of both sides of Equation 31, which yields:

 ∂ ln ( ∆RSE )  β ln   = ln βρ − β ln t  ∂ ln ( t ) 

(

)

(32)

It means that in the log-log space this dimensionless logarithmic rate has a linear relationship with the time, which is defined by the slope of −β . As a result, β can be regarded as a representative dimensionless rate that describes how ∆RSE changes in relation to the time. Since the maximum amount of change that ∆RSE can have is ∆RSE∞, a representative energy change rate is defined as follows: k RSE = ∆ RSE ∞ ⋅ β

(33)

where kRSE is the representative energy change rate for ∆RSE. Similarly, a representative energy change rate can be calculated for the change of ∆GH : k ∆G H = ∆ G H ∞ ⋅ β ∆GH

(34)

where k ∆G is the representative energy rate for the change of ∆GH ; ∆GH ∞ and β ∆G are the H

H

model parameters of the change of ∆GH , similar to those in Equation 25.

23

4.3 Determination of kinetic quantities in the Arrhenius equation

The representative energy change rates defined above are used to formulate the Arrhenius equation for three processes: 1) Viscoelastic recovery of undamaged asphalt mixtures; 2) Viscoelastic recovery plus healing for damaged asphalt mixtures; and 3) Free energy change for healing for damaged asphalt mixtures. The corresponding Arrhenius equation is: E aR

k RSER = AR e RT

k RSE RH = ARH e

k∆GH = A∆GH e

E aRH RT

(36)

Ea∆GH

where k RSE and k RSE R

(35)

RT

RH

(37) are the representative energy change rate for ∆RSE in the viscoelastic

recovery process and viscoelastic recovery plus healing, respectively; AR and ARH are the preexponential factor for viscoelastic recovery and viscoelastic recovery plus healing, respectively; Rewrite Equations 35 to 37 as

ln kRSER = ln AR +

EaR RT

(38)

ln kRSERH = ln ARH +

EaRH RT

ln k ∆GH = ln A∆GH +

E a ∆ GH

(39) (40)

RT

1 gives a straight line with a slope of E a R RT ) and an intercept of ln AR (or ln ARH , or ln A∆G ).

A plot of ln k RSE (or ln k RSE , or ln k ∆G ) versus R

(or EaRH , or E a ∆ G

RH

H

H

H

Based on Equations 38 to 40, the activation energies and pre-exponential factors can be determined for viscoelastic recovery of undamaged specimens and that for viscoelastic recovery plus healing as well as the free energy change of damaged specimens tested at 10˚C, 20˚C, and 30˚C. Figure 12(a) and (b) present some examples of the plots of Equations 38 and 39 using the representative energy change rate computed by Equation 33; Figure 12(c) shows that for Equation 40 using the representative energy change rate computed by Equation 34. At lower

24

temperatures, the values of kR and kRH are larger, which indicates that the RSE changes faster during the rest period. This is because asphalt mixtures behave in a more elastic way at low temperatures so that the energy stored in the material recovers at a higher rate. Thus, E a R and EaRH are positive. The same finding of a positive E a ∆G H is observed from Figure 12(c), which is

due to the fact that the RPSET, i.e. volume free energy density becomes lager and changes faster at a lower temperature.

0.39

0.40

0.41

0.42

0.43

0 Ln(kRSE) (J/m3)

Activation energy -2 -4

y (RH) = 94.189x - 43.088 R² = 0.9886

-6 -8

y (R) = 129.17x - 60.59 R² = 0.9829

-10

1/RT (mol/kJ) R Linear (R)

RH Linear (RH)

(a) NuStar mixture (0 month)

0.39

0.40

0.41

0.42

0.43

Ln (kRSE) (J/m3)

0 -2 -4

y (RH) = 75.861x - 35.08 R² = 0.9998

-6 -8

y (R) = 93.04x - 45.318 R² = 0.9936

-10

1/RT (mol/kJ) R Linear (R)

RH Linear (RH)

(b) Valero mixture (0 month)

25

7 Ln (kΔGH) (J)

6

y (6 mo) = 59.304x - 19.17 R² = 0.8713

5 4 y (0 mo) = 61.53x - 21.183 R² = 0.9379

3 2 0.39

0.40

0.41 0.42 0.43 1/RT (mol/kJ) GH_0 mo GH_6 mo Linear (GH_0 mo) Linear (GH_6 mo) (c) NuStar mixture

Figure 12. Examples of Arrhenius plot of representative energy change rates

The activation energies and pre-exponential factors are calculated for each tested asphalt mixture specimen, and the results of activation energy for viscoelastic recovery, viscoelastic recovery plus healing, and free energy change for healing are given in Figure 13. The activation energy of viscoelastic recovery represents the rate of recovery of an undamaged asphalt mixture, a higher value indicting a slower recovery rate. When an asphalt mixture is aged, it becomes stiffer and phase angle deceases; then the material becomes more elastic to recover part of the deformation. Accordingly, the rates of recovery of both NuStar and Valero mixtures increase after 6-month aging, and the activation energy for viscoelastic recovery is smaller than that of unaged mixtures. 150 EaR

Ea (kJ/mol)

120

EaRH

EaGH

90 60 30 0 0 month

6 months

NuStar

0 month

6 months

Valero

Figure 13. Calculated activation energy of asphalt mixtures from the CSR test 26

From the results in Figure 13, it is clear that the activation energy of viscoelastic recovery plus healing becomes smaller when healing occurs in a damaged mixture during the rest period. The difference between the activation energy for viscoelastic recovery and that for viscoelastic recovery plus healing is caused by the energy requirement for healing, which changes the value of the activation energy. Mathematically, the activation energy for healing is computed by Equation 12 as mentioned earlier. The difference between the two intercepts is the intercept for healing: ln AH = ln ARH − ln AR

(41)

where AH is the pre-exponential factor for healing, which is associated with the RSE. Thus, the representative energy change rate for ∆RSE due to healing can be expressed as: k H = AH e

−

EaH RT

(42)

where k H is the representative energy change rate for ∆RSE in the healing process. Plots of ln k H versus

1 are given in Figure 14. A negative value of the slope means RT

that the energy rate associated with healing increases as the temperature increases, which is in accordance with the fact that healing occurs faster at higher temperatures. Figure 15 presents the values of E aH for all of the tested specimens. The NuStar mixtures have higher healing activation energy than the Valero mixtures, which indicates that the Valero mixtures heal faster. Similar observations of the healing activation energy for asphalt mastic are also reported in the literature. The changes of the shear moduli of asphalt mastic measured from the dynamic shear rheometer were used to determine the healing activation energy, which is in the range of 16.05 to 84.53 kJ/mol (Sun et al., 2015). After obtaining the healing activation energy, the activation energy for bond restoration can be calculated by Equation 10, which is shown in Figure 15. The difference in the bonding rate between the NuStar and Valero mixtures is obvious. Moreover, the bond restoration rate slows down a little when the asphalt mixture is aged as the activation energy becomes higher.

27

Ln (kH) (J/m3)

4.0 3.5 3.0 2.5 0.39

0.40

0.41 1/RT (mol/kJ)

NuStar_0 mo Valero_0 mo Linear (NuStar_0 mo) Linear (Valero_0 mo)

0.42

0.43

NuStar_6 mo Valero_6 mo Linear (NuStar_6 mo) Linear (Valero_6 mo)

Figure 14. Arrhenius plots of representative ∆RSE change rate for healing

Ea (kJ/mol)

120 EaH

100

EaB

80 60 40 20 0 0 month

6 months

NuStar

0 month

6 months

Valero

Figure 15. Calculated healing activation energy and bond restoration activation energy of asphalt mixtures from the CSR test 4.4 Comparison of healing activation energy from kinetic and mechanistic approaches

The healing activation energy determined above is based on the kinetic formulation of the change rate of a combined recovery and healing process in Equations 11 and 12. On the other hand, the healing activation energy can also be obtained from a mechanistic approach, like the EBM approach introduced in Section 3. Using the damage density evolution curve shown in Figure 6, the normalized extent of healing is defined in the following way:

h=

φ fc − φi , h ∈ [0,100%] φ fc − φ0

(43) 28

where h is the normalized extent of healing; φ fc is the initial damage density before healing starts; φi is the damage density at any moment i during the rest period; φ 0 is the initial damage density, which is equal to the air void content. The normalized extent of healing is used to generate a healing curve. Using the same approach in Section 4.1 and 4.2 to simulate the healing curve by the three-parameter model and compute the representative change rate for h: h = h∞ e

ρ  − h   t 

βh

(44)

k h = h∞ ⋅ β h

(45)

where kh is the representative change rate for the normalized extent of healing; h∞ , ρ h , β h are the model parameters for h. Then the Arrhenius equation for kh is expressed as: k h = Ah e

−

Eah RT

(46)

where Ah and E ah are the pre-exponential factor and healing activation energy associated with the normalized extent of healing, respectively. Based on Equations 43 to 46, the healing activation energy associated with h (h is computed by the EBM approach) is calculated for each asphalt mixture from the CSR test, as shown in Figure 16. The healing activation energies computed by Equation 12 using the representative energy change rate associated with the RSE are also presented in Figure 16. The values of the healing activation energy are almost the same. In other words, the kinetic approach and mechanistic approach yield the same healing activation energy.

Ea (kJ/mol)

60 EaH Eah

40 20 0 0 month

6 months

NuStar

0 month

6 months

Valero

Figure 16. Comparison of healing activation energy of asphalt mixtures from different methods

29

In addition to the healing activation energy, the relationships between other model parameters are explored using the analysis results from the CSR test. Figure 17(a) presents the relationship between the pre-exponential factors defined in Equations 42 and 46. Figure 17(b) shows that ρ h and β h are closely related to each other. Figure 17(c) gives the measured values of β h ( β h _ M ) and predicted β h ( β h _ P ). A multiple regression analysis is conducted to identify the relationship between β h and other model parameters. The final version of the regression function is given in Equation 47. It indicates that the value of β h can be predicted using the model parameters for ∆RSE. β h = 1.081 − 6.843 ∆ RSE ∞ RH + 0.003 ρ RH − 0.102 β RH − 36.598 ∆ RSE ∞ R

(47)

where ∆ RSE ∞ RH , ρ RH , and β RH are model parameters for ∆RSE of viscoelastic recovery plus healing; and ∆RSE∞R is the model parameter for ∆RSE of viscoelastic recovery. 25 Ln (AH) (J/m3)

20 15 10

y = 93.04x - 45.318 R² = 0.9936

5 0 0

5

10 Ln (Ah)

15

20

(a) Pre-exponential factors 1.0

ρh

0.8 0.6 0.4 y = -0.0406x + 0.9102 R² = 0.9522

0.2 0.0 0

3

6

9

12

βh (b) ρh and βh 30

1.2 y = 1x R² = 0.9473

βh_P

0.9 0.6 0.3 0.0 0.0

0.3

0.6 βh_M

0.9

1.2

(c) Measured and predicted βh Figure 17. Relationships between model parameters

Based on the relationships in Figures 16 and 17, a damage density-based healing curve versus the rest time can be easily generated from the CSR test just using the energy terms. The procedure is listed as follows: 1) Perform the nondestructive and destructive CSR tests on one asphalt mixture at three different temperatures, and measure the internal stresses; 2) Calculate the RSE for each test and determine the model parameters for the model in Equation 25; 3) Calculate the representative energy change rate using Equation 33 and determine the healing activation energy and pre-exponential factor AH using Equation 42; 4) Calculate Ah using the relationship in Figure 17(a), and then determine kh with the healing activation energy and Ah ; 5) Calculate β h using Equation 47, and then compute h∞ by Equation 45 and calculate ρ h based on Figure 17(b); 6) With determined h∞ , ρ h , and β h , the healing curve is produced by Equation 44. In this way, using the RSE measured from the CSR test, the amount of healing in terms of the reduction of damage density can be determined at any time at any temperature.

31

5. Conclusions and Future Work

This paper studies healing of asphalt mixtures from the perspective of kinetics and thermodynamics, and develops a new approach to determine the kinetic and thermodynamic quantities during the rest period. In this way, each coordinated rate process involved in healing are clarified and the associated change rate is computed to gain a better understanding of the essence of healing of a viscoelastic material. The main findings of this study are listed as follows.

•

Healing of an asphalt mixture contains two coordinated rate processes: free energy change and bond restoration. Thus, the healing rate is proportional to the rates of these two processes, and the healing activation energy is the product of the activation energy of the free energy change and that of the bond restoration.

•

A state change between undamaged, damaged, and aged asphalt mixtures usually contains several coordinated rate processes. The activation energy of a composite process is the product of the activation energies of the coordinated processes.

•

A rate process in which the rate keeps changing with time can be simulated by a threeparameter model, the parameters of which have clear mathematical meanings. According to this model, the dimensionless logarithmic rate has a linear relationship with time in the log-log space, based on which a representative change rate can be defined.

•

The representative energy change rate for the change of the RSE and that associated with the free energy change for healing are suitable to formulate the Arrhenius equation and determine the activation energy.

•

The difference between the activation energy for viscoelastic recovery of undamaged asphalt mixtures and that for viscoelastic recovery plus healing of damaged ones is the healing activation energy. The difference between the healing activation energy and that for the free energy change is the activation energy for bond restoration.

•

The activation energies obtained in this study, including those for viscoelastic recovery, healing, free energy change, and bond restoration, change with the mixture composition and aging time.

•

The healing activation energy determined based on the kinetic relationships is the same as that computed using the damage density from a mechanistic approach. This enables to generate a damage density-based healing curve at any rest time at any temperature using just energy measurements. 32

The concepts and methods proposed in this study could be applied to other state changes of asphalt mixtures, like the change from an undamaged state to damaged state due to plastic deformation and crack growth, or from an unaged state to aged state. Future work will be carried out in these aspects to clarify the relationships between each coordinated rate processes and predict the rate of each process.

Declarations of Interest

None References

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Declaration of Interest

The authors declare that they have no conflict of interest.