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Keywords:

  • biomimetic;
  • crack;
  • glass transition temperature;
  • interfaces;
  • modeling;
  • reptation;
  • self-healing;
  • simulation

Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MOLECULAR MECHANISMS
  5. HEALING MODELING
  6. DESIGN OF SELF-HEALING COMPOSITES
  7. CONCLUDING REMARKS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. Biographical Information
  11. Biographical Information

Bioinspired self-healing polymers have attracted more and more interests. Imparting self-healing ability to existing polymers or developing new polymeric materials capable of self-healing is considered to be a solution for improving their long term stability and durability. This article reviews achievements in the field of theoretical researches on re-establishment of bonding between broken surfaces of self-healing polymers from microscopic and macroscopic point of view. Chains interaction, mechanical models related to healing procedures and effect of healing, design of novel self-healing composite systems, and so forth are summarized and analyzed in detail. Both thermoplastics and thermosets are included to offer a comprehensive knowledge framework of the smart function. The scientific challenges are also highlighted, which are related to the production of more advanced self-healing polymers. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011


INTRODUCTION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MOLECULAR MECHANISMS
  5. HEALING MODELING
  6. DESIGN OF SELF-HEALING COMPOSITES
  7. CONCLUDING REMARKS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. Biographical Information
  11. Biographical Information

To understand the phenomenon of self-healing and eventually to prepare self-healing polymeric materials or to modify healing strategy, fundamental knowledge dealing with molecular mechanisms is necessary. This is because rupture resulting from yielding, crazing, crack initiation, propagation, and coalescence are bound to be associated with chain slippage, interchain fracture, and chain scission. Re-establishment of adhesion between the broken parts through secondary forces is of great importance besides direct chemical bonding. Therefore, some theoretical considerations and simulation experiments on molecular interaction during healing, which have general meaning, are worth being reviewed. On the other hand, computational models related to design of self-healing materials, healing procedures, and the effect of healing will also be shown hereinafter. Although many of these works have not yet been applied in practice, we should keep them in mind because of their predictability and guidability.

In general, rebuilding of strength at polymer interfaces is a complex process and depends on the physical and chemical nature of the polymer surface. Chemical interactions and molecular connectivity achieved across the interfaces determine the ability of polymers and polymer composites to be repaired as well as the concrete healing technique. Raghavan and Wool1 identified 10 polymer–polymer interfaces of prime importance. As shown in Table 1, these interfaces can be represented by compact tension (CT) specimens with A- and B-halves. Either A- or B-half can be a liquid (L), a virgin solid with as-cast surface (SV), a solid with fractured surface (SF), or a solid with fractured and treated surface (SFC). When the two halves of a CT specimen are brought together, an interface can be formed at the A-B contact plane and ready for evaluation in terms of fracture mechanics.

Table 1. Possible Polymer–Polymer Interfaces Encountered During Strength Restoration
inline image

On the whole, the examples listed in Table 1 demonstrates a panoramic view of the possible contact modes, involved in healing action of polymeric materials and can be taken as the reference starting points for further discussion about molecular mechanisms.

MOLECULAR MECHANISMS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MOLECULAR MECHANISMS
  5. HEALING MODELING
  6. DESIGN OF SELF-HEALING COMPOSITES
  7. CONCLUDING REMARKS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. Biographical Information
  11. Biographical Information

Self-Healing Below Glass Transition Temperature

Interdiffusion and entanglements are necessary for strength development at polymer interface, while the two factors sometimes oppose each other.2 On one hand, the shorter the chains, the faster the diffusion. On the other hand, the presence of entanglements in long chains increases the strength at the interface. Therefore, molecular weight of the polymers to be healed is also important for rebinding. When these requirements are met, healing of cracks even at a temperature below the glass transition temperature (Tg) would be feasible. That is, heat treatment above Tg is not always the prerequisite as imagined. This seems to contradict the conventional knowledge of translational motion of polymer segments, which is generally believed to be nonexistent below Tg. Nevertheless, molecular mobility on polymer surface layer at temperature below the bulk Tg has been shown,3–7 which means a decrease in the effective Tg of the surface layer (on the scale of radius of gyration, Rg) as compared with Tg of the bulk (Table 2). In this context, self-healing below Tg becomes possible.

Table 2. Tg for Poly(styrene-block-methyl methacrylate) Diblock Copolymer (PS-b-PMMA) with Mn of 45,500 in Different Depth Ranges
inline image

The simplest case in this aspect lies in craze healing. Actually, the two parts of the interface in a craze are not completely separated. Wool and O'Connor8 reported that craze in polystyrene (PS) can be recovered below Tg. By means of dark field optical microscope, it was found to disappear at Tg −28 °C following line mode in which uniform healing occurred along the entire length of the craze. The lowest possible temperature for PS craze healing was considered to be 52 °C, that is, Tg −46 °C. As the molecular-scale process of craze formation includes molecular disentanglement, it might be reversed by chain diffusion as a result of suitable thermally activated molecular motion.9 In a latter report, McGarel and Wool10 studied the kinetics of craze healing in PS. Craze healing time was revealed to increase with molecular weight.

Boiko et al.11 evaluated self-healing ability of PS with T-peel tests below the Tg. The measured fracture energy of the high molecular weight PS/PS interface developed after healing below its bulk Tg by 43 °C is 1.9 J/m2 (Table 3), larger by one order of magnitude than the work of adhesion (0.084 J/m2) for a PS/PS interface. It proved that the force of interaction at the interface did not only result from the wetting contribution due to physical attraction (van der Waals forces) but also from segments penetration across the interface. Kajiyama and coworkers13–15 have worked on the same topic. They confirmed the presence of a mobility gradient in the surface region of PS on the basis of interdiffusion experiments. The thickness of the PS/PS bilayer interface was definitely evolved with time at a temperature below the Tg. The thickness of the surface mobile layer was of the order of nanometers depending on the temperature and the molecular weight. In the surface layer, there is a gradient of chain and/or segmental mobility. Besides, the surface layer is not apparently related to the chain dimension.

Table 3. Some Molecular and Mechanical Properties for Symmetric PS/PS Interfaces after Healing for 24 h
inline image

Wool et al.16, 17 treated the surface layer softening as a gradient rigidity percolation issue. For free surfaces there is a gradient of p(x) near the surface, where x < ξ (cluster size correlation length) and hence a gradient in both Tg and modulus E. If the gradient of p is given by p(x) = (1−x/ξ) then the value of xc for which the gradient percolation threshold pc occurs, and which defines the thickness of the surface mobile layer, is given by:

  • equation image(1)

where b is the bond length and ν is the critical exponent for the cluster correlation length ξ ∼ (p−pc)ν. For PS, when T = Tg−10 K, Tg = 373 K and using b = 0.154 nm, pc = 0.4, ν = 0.82, then the thickness of the surface mobile layer xc = 3.8 nm. This could allow for healing to occur below Tg assuming that the dynamics are fast enough. In addition, the relation between fracture energy GIC and ΔT (i.e., TgT) can be deduced as:

  • equation image(2)

It receives support from the experimental data in ref.11. In addition, Yuan et al.18 used neutron reflectometry to study the relaxation of the interface between both unentangled and entangled deuterated PS (d-PS) and hydrogenated PS (h-PS) films in the glass state (8–18 °C lower than Tg). The initially sharp interface between the glassy polymer layers were found to broaden with time, while the average interfacial thickness saturated to a healing thickness in a range between 1 and 3 nm after a long annealing time (1 h) for the range of temperatures investigated.

In principle, self-healing below Tg based on intimate contact is rather attractive, but the strength developed in this way is far from satisfactory. The fact that the macromolecular coils are compressed at the film surface in a direction normal to the interface plane, which leads to insufficiently thick interdiffusion layer, might partially take the responsibility.19–21 It is worth noting that the available self-bonding experimental results mostly come from a few thermoplastics and the healing temperature that is still higher than room temperature despite it is lower than Tg. In fact, molecular mobility is also perceivable in thermosetting polymer below the calorimetric glass transition.22 With the aid of encapsulated solvent, fractured epoxy resin has been healed under ambient temperature,23, 24 suggesting that the mechanism of chain segments interdiffusion and entanglements might also work. Compared to the case of thermoplastics, of course, either molecular theories or measures of investigation available for thermosets are challenging. Great effort has to be made to explore the exact mechanisms and more importantly to develop new technologies of healing.

Self-Healing Above Tg

It is well known that the contact of two identical or compatible polymer surfaces above the Tg leads to diffusion of molecular segments across an interface.25–35 Strong links would develop between the surfaces as the segments establish entanglements on the opposite side of the interface. To unscramble the complexity of strength development at polymer interfaces, Wool and O'Connor36 proposed five stages of crack healing (Fig. 1), which helps to separate the multiconvoluted time dependencies of the different mechanisms controlling crack healing and molecular processes. They are:

  • surface rearrangement,

  • surface approaching,

  • wetting,

  • diffusion,

  • randomization.

thumbnail image

Figure 1. Schematic diagram showing two random-coil chains on opposite crack surface during the five stages of crack healing: (a) rearrangement, (b) surface approaching, (c) wetting, (d) diffusion to a distance χ, and (e) diffusion to an equilibrium distance χ and randomization. Only one chain is shown for clarity. The dashed line represents the original crack plane. (Reproduced from Ref.36, with permission from American Institute of Physics.)

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Surface rearrangement has to be considered for healing cracks because topographic features of a newly formed fracture surface might change with time, temperature, and pressure following contact with healing fluid.17 This would in turn affect the rate of crack healing. It is of particular importance for chain ends when they are supposed to be used to react with healant. They can be designed to preferentially migrate to the surface; otherwise, the molecules would diffuse back into the bulk.37

Surface approaching is critical as no healing would occur if the surfaces of broken portions are not brought together, or debris from the damage process is left, or the interstitial is not filled with sufficient amount of healing agent. In the case of intrinsic self-healing, where no healing agent is pre-embedded in the materials, surface approaching must be ensured as no additional substances would appear forcing the surfaces together.

The stages of wetting and diffusion are responsible for the majority of the mechanical property recovery or healing.37 No matter whether the healing process involves healing agent, wetting at solid–solid or solid–liquid contact is needed to build an interface before repair. As indicated by Wool,17 wetting can proceed in a time-dependent fashion at the interface. Typically, the wetted areas are randomly nucleated at different times and then propagated until coalescence and complete wetting are achieved.36 For simplification, the problem can be treated as a two-dimensional (2D) nucleation and growth in analogy to crystallization of spherulites.38 That is, the fractional wetting area, W(t), is written as:

  • equation image(3)

where k and m are constants depending on the nucleation function and radial-spreading rates. W(t) = 1, corresponding to complete wetting is not always the case depending on how good the contact is. For crack healing, the wetting distribution function, W(t), was found to convolute with interdiffusion function, H(t), and affect the fracture energy of the interface:17

  • equation image(4)

where τ is the dummy variable of the convolution integral.

The interface formed by random walk chains diffusing by reptation across a polymer–polymer weld line are illustrated in Figure 2.16, 17, 39 The structure of the diffuse weld interface resembles a box with fractal edges containing a gradient of interdiffused chains. Using Sapoval's gradient percolation theory,40 Wool and Long39 required that chains, which contribute to the interface strength straddle the interface plane during welding, such that chains in the concentration gradient that have diffused further than their radius of gyration cease to be involved in the load bearing process at the interface. For time, t, being shorter than the reptation time (the time required by the chains to escape from their initial tubes in a curvilinear snakelike manner according to the reptation model developed by de Gennes41), Tr, and diffusion distance, Ld, smaller than the radius of gyration, Rg, self-similarity of the interface is lost due to the correlated motion of the chains creating gaps. Only at t > Tr and Ld > Rg, the interface becomes fractal. When Ld >> Rg, the polymer diffusion front behaves as the monomer case. The fractal nature of diffuse interfaces plays an important role in controlling the physical properties of polymer–polymer and interfaces. When local stress exceeds the yield stress, the deformation zone forms and the oriented craze fibrils consist of mixtures of fully entangled matrix chains and partially interpenetrated minor chains (Note: here the so-called minor chains represent the portions of a chain that are no longer in the initial tube increase with time42). Fracture of the weld occurs by disentanglement of the minor chains or by bond rupture. If the stress rises to the point, where random bond rupture in the network begins to dominate the deformation mechanism instead of disentanglement, then the weld will appear to be fully healed regardless of the extent of interdiffusion. This can occur at high rates of testing when the minor chains cannot disentangle and bond rupture pervades the interface breaking both the minor chains and the matrix chains.17

thumbnail image

Figure 2. Fractal interface formed by interdiffusing polymer chains. Only one side is shown. (Green region) Chains connected with other side at bottom (Yellow chains). Those chains which have reached their equilibrium-diffused distance and continue to diffuse away but are no longer connected top the other side. (Red region) The fractal diffusion line separating the nonconnected from the connected chains which provide strength at the interface. (Reproduced from Ref.17, with permission from The Royal Society of Chemistry.)

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To correlate the microscopic description of motion of chains to the macroscopic measurements, Prager and Tirrell,43 and Kim and Wool42 showed that the stress at fracture is a function of the contact time, t, to the fourth power. The same scaling law for the development of strength was also derived by Jud et al.34 They used the Griffith theory of fracture according to which the crack will initiate when the release of stored elastic energy per unit increase in crack area matches the energy required per unit area of fresh crack surface exposed. The elastic energy release goes as the square of the applied stress (if the crack is propagating through virgin material which is linearly elastic). The energy of fresh crack surface would be expected to be proportional to the chain crossing density (the number of links per unit surface area, n(t)). Therefore, the measured stress at refracture should depend on n(t)1/2. As n(t) ∼ t1/2 for equilibrium interfaces, the refracture stress grows with t1/4, as experimentally observed (Fig. 3).

thumbnail image

Figure 3. Double logarithmic plot of fracture toughness KIC against healing time. Curves 1–4: healing of broken PMMA specimens immediately after fracture (the points are average values taken from 20 to 30 measurements). Curve 5: surfaces welded after vaccum drying and polishing (all data points are indicated individually). SAN: styrene-acrylonitrile copolymer. Both PMMA and SAN have the same glass transition temperature, Tg = 375–377 K. (Reproduced from Ref.34, with permission from Springer Science+Business Media.)

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In addition to time for interdiffusion and eventually randomization, molecular weight is also an important factor that influences the effect of crack healing. Prager and Tirrell's model predicts that GICt1/2M3/2 (where M denotes molecular weight).43 The model of Jud et al.34 predicts that GICt1/2M1. Kim and Wool's model predicts that GICt1/2M1/2.42

From the point of view of practical application, the models discussed in this subsection seem to be mostly suitable for intrinsic self-healing in thermoplastic rather than in thermosets,44 as supported by the experimental data for single crack healing and processing of pellet resin. The dependence of stress intensity factor on healing time to the power of 1/4,34 for example, is not applicable for healing cured epoxy materials either below or above the Tg,45, 46 despite that the two parameters are linearly correlated in double logarithmic coordinates. The situation has been changed since the invention of the approaches based on microencapsulation47 and microvascular networks,48 which showed prosperous healing capability in thermosetting material. In recent years, particular emphasis was placed on structure design and performance prediction of extrinsic self-healing composites containing healing agent through proper theoretical modeling and computational simulation (refer to the following subsections). In these cases, chemical reaction plays the leading role in re-establishing bonding across the fractured sites,49 so that chain mobility and physical interaction are no longer the decision factors.

HEALING MODELING

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MOLECULAR MECHANISMS
  5. HEALING MODELING
  6. DESIGN OF SELF-HEALING COMPOSITES
  7. CONCLUDING REMARKS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. Biographical Information
  11. Biographical Information

Percolation Modeling

Theoretical and numerical modelings of self-healing materials are still in the initiation stage, which roughly stays in step with the experimental attempts in the field. Privman et al.50 focused the modeling program on time dependence of a gradual formation of damage (fatigue) and its manifestation in material composition. They formulated continuum rate equation for such a process and then carried out Monte Carlo simulations. Supposing u(t) stands for the fraction of material that is undamaged, g(t) the fraction of material consisting of healant-carrying capsules, d(t) the fraction of material that is damaged, and b(t) the fraction of material with broken capsules, the following expression can be given as:

  • equation image(5)

Here, the regime of small degree of degradation of the material is considered. At least for small times, t, u(t) ≈ 1, whereas d(t), b(t), and g(t) are relatively small (b(0) = 0). For purposes of simple modeling, Privman et al.50 assumed that on average the capsules degrade with the rate, P, which is somewhat faster than the rate of degradation of the material itself due to its continuing use (fatigue), p, that is, P > p. The latter assumption was made to mimic the expected property that a significant amount of microcapsules embedded in the material may actually weaken its mechanical properties and, were it not for their healing effect, reduce its usable lifetime. Thus, by approximately taking

  • equation image(6)

g(t) can be yielded:

  • equation image(7)

For the fraction of the undamaged material, equation image is written as:

  • equation image(8)

where H(t) is the healing efficiency:

  • equation image(9)

The healing efficiency is proportional to the fraction of capsules, as well as to the fraction of the damaged material, because this is where the healing process is effective. The latter will be approximated by d(t) ≈ 1−u(t), which allows for obtaining a closed equation for u(t). Accordingly, H(t) is expressed by:

  • equation image(10)

The healing efficiency is controlled by the parameter

  • equation image(11)

Although the formulated model is simple, it has the advantage of offering an exact solution.

  • equation image(12)

Equation 11 suggest that an important challenge in the design of self-healing materials will be to have the healing effect of most capsules cover volumes much larger than a capsule to compensate for a relatively small value of g(0), which is the fraction of the material volume initially occupied by the healant-filled capsules. As the healant cannot “decompress,” its healing action, after it spreads out and solidifies, should have a relatively long-range stress-relieving effect to prevent further crack growth over a large volume.

Considering that the above continuum modeling cannot address the details of morphological material properties and healant transport, Dementsov and Privman51 introduced conductance as a measure of self-healing material integrity. That is, conductivity is directly proportional to the local material “health.” By assuming square lattice (coordination number z = 4) bond percolation, the conductance, C(t), can be calculated through the bond-percolation mean-field formula:52

  • equation image(13)

The conductance is normalized to have C(0) = 1, and for simplicity the bond percolation probability is given by u(t), that is, the situation when the conductance of the healthy/healed material is maximal is considered, whereas, the other areas do not conduct at all. In the regime of a relatively low damage, which is likely the only one of practical interest, and also the one where the mean-filed expressions are accurate, we note that the conductance provides a convenient, proportional measure of the material degradation,

  • equation image(14)

where the constant K = z/(z−2) depends on the microscopic details of the material conductivity. Here K = 2, but in practical situations this parameter can be fitted from experimental data.

Results of Monte Carlo simulations51 have shown the general features of self-healing process. In particular, the onset of the material fatigue is delayed by developing a plateau-like time dependence of the material quality at initial times. In this low-damage regime, the changes in conductance, and likely in most other transport/response properties of the material that can be experimentally probed, measure the material quality degradation proportionally, whereas for larger damage at latter times, transport properties may undergo dramatic changes, such as the vanishing of the conductance, and they might not be good measure of the material integrity.

In a recent report, Dementsov and Privman53 extended the calculation of the conductance from 2D lattices to three-dimensional (3D). They demonstrated the competition between the healing capsules for three dimensions, where it is more profound than in their earlier-studied 2D systems. Even for low initial densities of the healing capsules, they interfere with each other and reduce each other's effective healing efficiency. Accordingly, it was suggested that not just the capsules density but their uniform dispersion in the medium (to avoid clumping) is of importance in designing self-healing materials.

Continuum and Molecular-Level Modeling of Fatigue Crack Retardation

To model the behavior of self-healing polymeric system under more realistic loading environments, Geubelle and coworkers54 constructed a numerical approach to describe the fatigue crack retardation in the self-healing polymer developed by White et al.,47 which contained microencapsulated dicyclopentadiene and Grubbs' catalyst. Before review of their model, experimental observation of fatigue crack propagation and healing induced-reduction in crack growth rate in the composites55 is summarized hereinafter.

It has been found that for successful in situ self-healing, the healing agent released into the crack plane must have enough time to polymerize. If the crack growth rate is too fast, little or no healing will occur. These observations can be explained in terms of a competition between two time scales. The development of healing efficiency in this material system is as follows.56 After an initial dwell period of about 30 min during which no appreciable healing is measured, the healing efficiency increases rapidly, tapering off to a maximum healing efficiency after about 10 h, showing the presence of a characteristic time for healing denoted hereafter by τheal. On the other hand, retardation and arrest of the propagating crack in the self-healing material is influenced by the healing kinetics, the cyclic loading applied on the structure and the fracture properties of the material determine the crack growth rate. This fact introduces in the self-healing system another time scale τcrack, the characteristic time for crack propagation. This time scale may increase or decrease with crack length depending on whether the crack growth is unstable or stable, respectively. For higher loading levels, no crack retardation is possible as τcrack is substantially less than τheal, thus allowing little or no healing. If the load level is decreased, the crack propagation rate in the material also reduces, thereby increasing τcrack and enabling the crack retardation to take place. The presence of a rest period during which the loading is not applied but healing takes place further facilitates crack healing. Finally, for much lower load levels, τheal becomes considerably smaller than τcrack, and the crack retards or gets completely arrested.

Crack retardation in self-healing polymers is mainly controlled by the following two effects: crack bridging (or adhesive) effect associated with the adhesion of the healing agent to the crack flanks, and crack closure effect associated with the solid wedge formed by the deposited polymer behind the crack. Geubelle and coworkers54 combined the cohesive modeling for fatigue crack propagation57 and a contact algorithm to enforce crack closure. The healing kinetics of the self-healing system was incorporated by introducing along the fracture plane, a state variable representing the evolving degree of cure of the healing agent. The atomic scale processes during the cure of the healing agent were modeled using a coarse-grain molecular dynamics model specifically developed for this purpose. The core part of their modeling is as follows.

The cohesive model of an inserted wedge in the wake of an advancing fatigue crack gives the rate of change of the cohesive stiffness, kcoh, which describes an irreversible unloading–reloading path that accounts for the fatigue-induced irreversibility:57

  • equation image(15)

where Nf denotes the number of loading cycle experienced by the material point as the onset of failure, ψ and β are material parameters describing the fatigue-induced degradation of cohesive properties, and equation image is the rate of change of the normal separation. When a wedge of thickness Δn* is inserted behind the crack tip, the crack faces experience a contact force whenever Δn − Δn* ≤ 0.

As soon as the healing reaction is triggered, the monomer released behind the crack tip polymerizes and creates a solid wedge in the wake of the crack. The wedge thickness, δ*, and the degree of cure, α, can be correlated through an explicit linear form based on the molecular-level simulations:

  • equation image(16)

where Δn* denotes the maximum thickness that can be achieved by the wedge, and αthreshold the threshold value of curing degree (below which the polymer acts like a liquid, and above which it responds stiffly).

As alluded to earlier, the key issue that needs to be captured when modeling the fatigue failure of the self-healing composite is the competition between how quickly the crack propagates and how fast it can be healed. The healing and failure processes are each characterized by their own time scales. For the cure kinetics process, the time scale is simply given by:

  • equation image(17)

where B is the rate constant depending on temperature and the catalyst concentration. The other time scale, τcrack, can be expressed as the time needed for the fatigue crack to propagate a distance equal to the active cohesive zone length, lc, located ahead of the crack front:

  • equation image(18)

where da/dN denotes the crack advance per cycle and ω the loading frequency. The cohesive zone length is related to the failure properties of the material.

Accordingly, Geubelle and coworkers54 conducted numerical simulations. The study of the effect of different loading and healing parameters shows a good qualitative agreement between experimental observations and simulation results.

Continuum Damage and Healing Mechanics

Barbero et al.58 studied damage and healing behavior of fiber reinforced polymer composites. The microcracks were taken as distributed damages, which proved to be able to initiate the release of healing agent from microcapsules embedded in composites' matrix. Similarly, healing processes can be considered opposite to damage. Therefore, continuum damage mechanics can be applied. Barbero et al. firstly generalized continuum damage mechanics including healing processes and then proposed continuum damage-healing mechanics. Furthermore, the theory was used to establish a specific model for fiber-reinforced polymer-matrix composites experiencing damage, plasticity, and healing. The mesoscale constitutive model was developed in a consistent thermodynamic framework that automatically satisfies the thermodynamic restrictions. The degradation and healing evolution variables were obtained by introducing proper dissipation potentials motivated by physically based assumptions.

Recently, Barbero et al.59 simplified their model. A damage/healing model where both effects were described in a single thermodynamic space was presented, which consists of three main ingredients the damage variable, the free-energy potential, and the damage evolution equations. Some of the main issues are reviewed as follows.

Damage represents distributed, irreversible phenomena that cause stiffness and strength reductions. When distributed damage controls the mechanical behavior, many materials, including polymer-matrix composites, usually exhibit a quasibrittle macroscopic behavior. For example, experimental observations on polymer-matrix composite before failure show a continuous distribution of microcracks in the matrix. During loading, the total energy of the system is dissipated mainly into new surface formation, whereas a minor fraction is used to nucleate existing microcracks. In this context, damage can be represented by a second-order tensor D. As healing is related to the extent of repair of distributed damage, it is also represented by a second-order diagonal tensor H with principal directions aligned with those of the damage tensor D. The principal values h1, h2,h3 represent the area recovery normal to the principal directions, which are aligned with the material directions x1, x2, x3 in the material coordinate system. It is postulated that healing tensor, hi, is proportional to the damage tensor, di:

  • equation image(19)

where the proportionality constant, ηi, is the efficiency of the healing system. The principal values of the healed-damage tensor are given by:

  • equation image(20)

Experimentally, it is possible to measure the following parameters via cyclic shear stress-strain tests: (i) the virgin moduli equation image from the initial slope of the first cycle of loading; (ii) the damaged moduli Gmath image from the unloading portion of the first cycle; and (iii) the healed moduli Gmath image from the loading portion of the second cycle (after healing). From these data, the efficiency is defined as:

  • equation image(21)

For fiber composites, the self-healing system is incapable of healing fiber damage, which results in η1 = 0. Once the healing agent is released, it travels by capillary action and penetrates all the microcracks regardless of orientation, and thus η2 = η3 = η.

For identification, all that is required is experimental determination of the healing efficiency as a function of damage. Under shear loading G12, the amount of damage d1 in the fiber direction is negligible when compared to the amount of damage d2 transverse to the fibers. Therefore, the change in the (unloading) shear modulus due to both damage and healing is given by

  • equation image(22)

and the healing efficiency can be calculated as

  • equation image(23)

Taking into account that the induced damage d2 is a function of the applied strain, it is possible to represent the efficiency as a function of damage with a polynomial as:

  • equation image(24)

Shear tests of composite laminates were performed. The results indicated that the computational model tracks the damaging stress-strain behavior very well. Additional tests on the samples not used in the parameter identification were also carried out to verify the predictive capabilities of the model. It was observed that it indeed possesses certain predictability as expected.

Discrete Element Modeling and Numerical Study

So far, most theoretical works dealing with self-healing materials are based on continuum approaches. They act on a coarse grained level and the materials must be (or are assumed to be) sufficiently homogeneous on that coarse grained level. In contrast, many-particle simulations like the discrete element model (DEM)60, 61 complement experiments on the scale of small “representative volume elements” (RVEs). They allow detailed insight into the kinematics and dynamics of the samples as all information about all particles and contacts is available at all times. DEM requires only the contact forces and torques as the basic input to solve the equations of motion for all particles in such systems. Furthermore, the macroscopic material properties, such as, among others, elastic moduli, cohesion, friction, yield strength, dilatancy, or anisotropy can be measured from such RVE tests.

Herbst and Luding62 presented a model for self-healing in particulate materials based on a piecewise linear contact model for elastoplastic, adhesive, viscous, and frictional particle–particle interactions. The contact model included a memory variable, that is, the contact laws were history dependent. The proposed self-healing model admitted to set the damage detection sensitivity and rate as well as the strength of the healed contacts. It could therefore be a model for a material including a healing agent in, for example, capsules.

A typical self-healing process consists of locally detection of damage and initiation of local healing when damage is detected. Accordingly, the following self-healing scheme that is compatible with (but not limited to) the contact model was introduced: (i) detect damage, (ii) heal detected damage, (iii) run simulation for a time τD, and (iv) go to (i). This procedure allowed to set the damage detection rate 1/τD at which damage is detected (and subsequent self-healing is activated), and to specify the healing sensitivity and the healing strength separately.

The model parameters have been systematically studied most prominently by examining the stress–strain response of the samples and additionally by monitoring the fraction of healed contacts. It was found that low damage detection sensitivity led to unsatisfactory self-healing as too few contacts were healed. On the other hand, very high damage detection sensitivity resulted in healing of contacts that were not critical. Healed contacts within the framework of this model, once healed cannot be healed again when they became critical again in the future. The healing intensity was a nonlinear function of the adhesion parameter kSH. It saturated for large kSH and set the strength of the healed contacts. With increasing healing intensity, the material can sustain larger stresses and failed at higher strains. As a function of kSH the material strength showed strong fluctuations for large kSH, that is, in the regime where the healing intensity was almost constant. These fluctuations were due to local contact instabilities. For very large healing intensities, that is, for very large adhesive strength of the healed contacts, a contact rapidly failed when the tensile limit was reached, resembling a local, brittle failure at the contact level. Thus, moderate values for the adhesive constant after healing, kSH, led to the best healing results. The fraction of healed contacts increased a little with the healing intensity, which corresponded to the final strength of the (solidified) healing agent, that is, the strength of the final bonding.

Based on a RVE, in which either spherical or elongated capsules were uniformly dispersed, Mookhoek et al.63 conducted a numerical study of the release of healing agent for liquid-based self-healing systems. In their model, the capsules did not overlap nor were in direct contact with each other. In the case of elongated capsules, a cylindrical shape terminated by a half sphere at either end was assumed. A cubic RVE was defined using periodic boundary conditions. To obtain a RVE with a desired capsule volume, capsules were sequentially positioned in this RVE using a random number generator for the position of centre of gravity of each capsule. Accordingly, the released healing agent volume, Vreleased, per crack area was given by:

  • equation image(25)
  • equation image(26)

where A denotes the cross-sectional area of the RVE with volume V, φ is the capsules volume fraction, r is the spherical capsule radius, Vcap is the single capsule volume, and AR is the aspect ratio of cylindrical microcapsules.

Equation 25 shows that with decreasing microcapsule size the total amount of liquid released decreases rapidly, which agrees with the estimation of Rule et al.64 Small, for example, nanosized, spherical microcapsules were suggested only to be used for healing of small scale damage (such as interfacial debonding) and not for enhancing the healing efficiency in general. Comparison between the results from eq 25 and eq 26 indicated that for a given capsule volume and a given capsule volume fraction, more healing agent can be delivered to a crack surface by the cylindrical capsules. Therefore, a strong improvement in the healing potential was predicted for elongated capsules. The spatial orientation of such elongated microcapsules also had a significant effect on the healing potential. Considerable reductions of capsule loading concentrations for a particular healing efficiency should be considered to preserve the material intrinsic properties to a larger extend.

DESIGN OF SELF-HEALING COMPOSITES

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MOLECULAR MECHANISMS
  5. HEALING MODELING
  6. DESIGN OF SELF-HEALING COMPOSITES
  7. CONCLUDING REMARKS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. Biographical Information
  11. Biographical Information

Entropy Driven Self-Assembly of Nanoparticles

Besides understanding healing processes, computational research into self-healing materials can ultimately facilitate the fabrication of the next generation of adaptive materials that both monitor their structural integrity and mend themselves before any catastrophic failure can occur.65 Balazs and coworkers66 used computer simulation to show the possibility of creating self-healing composites with nanoparticles. Their work is focused on multilayered composites that are comprised of alternating ductile polymers and brittle films of ceramics or metals. These composites are essential for optical and anticorrosion coatings, microelectronics packaging, solid-state devices, biomedical applications, and so forth.67, 68 Under large temperature gradient or external load, the brittle layer would form cracks that lie transverse to the polymer layers.69, 70 To utilize these composites in various circumstances, self-healing mechanisms that will repair cracks in the brittle layers must be introduced. The strategy is different from that based on microencapsulation,47 which heals cracked polymer layers.

The model of Balazs and coworkers shows that the addition of nanoparticles to multilayer composites yields a self-healing system, which actively responds to the damage and can potentially heal itself for multiple times. Figure 4 illustrates the geometry of the composite: a nanocrack is located in the solid film, which is sandwiched between two polymer layers. The Tg for the polymer is assumed to lie below room temperature, so that locally, the chains within the solid (rubbery) polymer layers are relatively mobile. At the onset of the calculations, nanoparticles are randomly distributed within the polymer films. They used Monte Carlo simulations and a hybrid self-consistent filed/density functional theory to determine the equilibrium distribution of the polymers and particles in the system. The output of the morphological studies served as the input to the lattice spring model enabling mechanical investigation. In this way, the effects of the observed particle distributions on the macroscopic behavior of the composites can be captured.

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Figure 4. 3D Monte Carlo simulation of a self-healing multilayer composite. The red, solid layer contains a crack and represents a fractured brittle layer. The blue region marks the polymer chains that lie above and below this brittle layer. The black cubes are nanoparticles. (Reproduced from Ref.65, with permission from Elsevier.)

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Through a variety of computer simulations, the researchers found that for sufficiently large nanoparticles that are compatible with the host matrix, the mobile chains expel the nanoparticles from the polymer film and into the cracks. In particular, the radius of the nanoparticles should be comparable to the radius of gyration for the chains. Entropy provides the driving force for this behavior: the chains gain greater conformational degrees of freedom by driving the particles to the surface and into the cracks. The system behaves in an autonomous manner; no intervention is necessary to localize the particles at the damaged or defective sites. Moreover, if new cracks appear in the confining surfaces, the entropy driving force again expels nanoparticles within the polymer film to these fissures. As the nanoparticles are made of the same substance (e.g., glass beads or metallic particles) as the brittle layer, their self-assembly would thereby effectively mend the material.

On the other hand, Young's moduli of the damaged (crack, no particles) and repaired (crack with particles) systems were compared with that of original, undamaged composite (no crack, no particles) in terms of micromechanics simulations. The results revealed that the mechanical property can be potentially restored to 75–100%.

Tyagi et al.71 did a similar work by considering a nanocomposite coating applied to a damaged surface, which contained a nanoscale notch. The integrated molecular dynamics and lattice spring model simulations showed that the nanoparticles were driven to localize in the notch because of polymer-induced depletion attraction. The depletion attraction between the particles and the notch was more pronounced for the larger particles. The calculations revealed that the stress concentration at the notch tip was greatly reduced due to the presence of the nanoparticles, relative to the case where the notched surface is just coated with a pure polymer layer. Consequently, the system would be inhibited from undergoing further damage when an external load was applied.

Gupta et al.72 experimentally verified the aforesaid theoretical predictions. They showed that the poly(ethylene oxide)-covered spherical nanoparticles (5.2 nm) in a poly(methyl methacrylate) (PMMA) matrix diffused to cracks in the adjoining silicon oxide layer. The findings confirm the importance of entropic interactions in segregating the nanoparticles to the crack.

The latest development in this aspect is the “repair-and-go” system.73, 74 Balazs and coworkers performed the 3D computer simulations using a hybrid computational approach.73 Amphiphilic capsules containing hydrophobic nanoparticles were assumed to be released in a flow field. The capsules could recognize cracks, which have hydrophobic interiors, in the hydrophilic surface of the material. Entropic forces caused the nanoparticles to be released into the crack. Once the crack was repaired, or the flow conditions changed, the capsules were released and went with the flow again. The most interesting issue of the work by Balazs and coworkers, as pointed out by White and Geubelle,74 is that it seems to capture the essential nature of wound healing by white blood cells—a regulated delivery of healing components to specific sites of damage. Putting this idea into practice will require experimentalists to develop capsules that are both robust and deformable, possess good adhesion characteristics, and able to release nanoparticles through their walls under appropriate conditions.

Optimization of Microvascular Networks

The self-healing composite containing microvascular networks48 has been regarded as the second generation of the biomimetic materials.75 Compared to the first generation of these materials with embedded microcapsules47 that can only heal a single fracture event, the newly developed system is capable of repeating the healing action so long as flow of the healing agent to the damaged areas via the microchannels is ensured. The demand originates from the fact that the crack position is not known in advance and several cracks may form at different sites simultaneously and repeatedly. One-time healing is invalid in this case.

Intuitively, the microvascular network needs to be dense enough so that cracks can be arrested before they reach a critical size. Therefore, microvascular networks embedded in a polymer must cover as much space as possible. However, the void space introduced by the presence of the microchannels tends to reduce the stiffness and the strength of the polymer. Furthermore, the amount of energy needed to drive the healing agent through the network is related to the pressure drop between the inlet and outlet locations of the fluid. A sound design would maintain this energy to a minimum. In this context, optimization of the microvascular network in consideration of the trade-off among different objectives is necessary.

Aragón et al.75 adopted a multiobjective genetic algorithms scheme to optimize the network topology against conflicting objectives, which include (i) optimizing the flow properties of the network (i.e., reducing the flow resistance of the network to prescribed mass flow rate) and (ii) minimizing the impact of the network on the stiffness and strength of the resulting composite in terms of void volume fraction associated with the presence of the microvascular network. They supposed that the network covers an entire 2D square domain with a network of equally spaced vertical and horizontal microchannels. The optimization was carried out on the diameters of the microchannels to capture the trade-off between the void volume fraction left by the presence of the network and the pressure drop between the source and target locations of the healing fluid. The flow analysis of the network was performed on the assumption of fully established Poisenille flow in all segments of the network, leading to the classical proportionality relation between the pressure drop along a segment and the network flow rate. It was expected that the optimized structures resulting from the optimization can then be manufactured using an automated process (“robotic deposition”) that involves the extrusion of a fugitive wax to define the network.76

Subsequently, Aragón et al.77 published a more detailed report on the same topic. In contrast to the preliminary study on orthogonal networks,75 this one also allows for diagonal microchannels, which are permitted to intersect only at locations determined by the spatial point lattice in the optimization process. In this context, a network containing microchannels intersecting outside the point lattice is marked as unfeasible. The effect of network redundancy, template geometry and microchannel diameters on the Pareto-optimal fronts generated by the genetic algorithm was investigated.

Bejan and coworkers78, 79 studied the network configuration that is capable of delivering fluid to all the cracks the fastest. The approach that they have chosen is based on constructal theory, which regards the generation of flow configuration as a natural phenomenon. All flow systems in nature have configuration (i.e., geometry, architecture, and drawing). Constructal theory places the occurrence of flow configuration on the basis of a physics principle (the constructal law): “For a flow system to persist in time (to survive) it must evolve in such a way that it provides easier and easier access to the currents that flow through it.”

Two classes of grids of interconnected channels filled with pressurized healing fluid were considered and optimized:78 (i) grids with one channel diameter and regular polygonal loops (square, triangle, and hexagon) and (ii) grids with two channel sizes. The best architecture of type (i) was found to be the grid with triangular loops. Although square grids and hexagonal grids are equally permeable, but that triangular grids are two times more permeable. Besides, the best architecture of type (ii) showed a particular (optimal) ratio of diameters that departs from 1 as the crack length scale becomes smaller than the global scale of the vascularized structure from which the crack draws its healing fluid. The optimization of the ratio of channel diameters cuts in half the time of fluid delivery to the crack.

In another paper by Bejan and coworkers,80 the flow architectures were configured as two trees matched canopy to canopy (Fig. 5). A single stream flows through both trees and bathes every subvolume (crack site) of the material. Several types of tree–tree configurations are optimized. Trees that have only one level of branching and bathe a rectangular domain have optimal external shapes that are nearly square. They also have optimal ratios of channel sizes before and after branching. Trees optimized on square domains perform nearly as well as trees on freely morphing rectangular domains. The minimized global flow resistance decreases slowly as the number of subvolumes increases. It is more beneficial to bathe the entire volume with single (optimized) one-stream architecture than to bathe it with several streams that serve small clusters of volume elements. These conclusions were reinforced by an analytical optimization of the same class of architectures in the limit of a large number of assembled subvolumes. It was also revealed that the freedom to morph the design and to increase its performance can be enhanced by using tree–tree architectures with more than one level of branching.

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Figure 5. Rectangular domain with square elements bathed by two trees matched canopy to canopy. The self-healing composite is treated as a slab with rectangular face (H × L), and the flow layout is 2D. The stream enters the H × L domain through a trunk of hydraulic diameter D2, and splits into several thinner channels of diameter D1. These thinner channels form the canopy of the first tree. Further downstream, the flow from the D1 channels is reconstituted into a single stream that exits the domain through a channel of diameter D2, which is the trunk of the second tree. The canopy (D2) shared by the two trees bathes every area element of the H × L domain. (Reproduced from Ref.80, with permission from American Institute of Physics.)

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CONCLUDING REMARKS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MOLECULAR MECHANISMS
  5. HEALING MODELING
  6. DESIGN OF SELF-HEALING COMPOSITES
  7. CONCLUDING REMARKS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. Biographical Information
  11. Biographical Information

As viewed from the development course of science and technology of self-healing polymeric materials, it is seen that the theoretical analysis was ahead of experimental explorations in the early stage. The situation has been changed since the breakthrough at the route based on embedment of encapsulation of healing agent.47 The mutual promotion follows the general law of growth of a new discipline.

Comparatively, molecular mechanisms involved in self-healing of thermosets (including rubbers) are less investigated, while there are not many methods enabling successful self-healing of thermoplastics having been proposed. The former should be related to the lack of basic understanding of physics of the related materials. Knowledge of chain motion restricted by the crosslinked networks has not yet been well established. With respect to the latter, the difficulty of introducing appropriate healing strategy still needs to be overcome.

On the whole, explanation of observed phenomena of crack healing and prediction of novel healing techniques are of equal importance. Both would be very helpful for experimentalists to either refine the existing measures or work out practical solutions. Therefore, substantial achievements in theoretical aspects of self-healing are highly desired.

Acknowledgements

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MOLECULAR MECHANISMS
  5. HEALING MODELING
  6. DESIGN OF SELF-HEALING COMPOSITES
  7. CONCLUDING REMARKS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. Biographical Information
  11. Biographical Information

The authors thank the support of the Natural Science Foundation of China (Grants: 20874117, 50903095, 51073176, and U0634001), Doctoral Fund of Ministry of Education of China (Grant: 20090171110026), and the Science and Technology Program of Guangdong Province (Grant: 2010B010800021).

Note: The chapter from which this Review was adapted appeared in the book “Self-Healing Polymers and Polymer Composites,” written by Ming Qiu Zhang and Min Zhi Rong and published by John Wiley and Sons (ISBN: 978-0-470-49712-8). Print and online versions are available respectively at http://bit.ly/selfhealing_print and http://bit.ly/selfhealing_online.

REFERENCES AND NOTES

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MOLECULAR MECHANISMS
  5. HEALING MODELING
  6. DESIGN OF SELF-HEALING COMPOSITES
  7. CONCLUDING REMARKS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. Biographical Information
  11. Biographical Information

Biographical Information

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MOLECULAR MECHANISMS
  5. HEALING MODELING
  6. DESIGN OF SELF-HEALING COMPOSITES
  7. CONCLUDING REMARKS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. Biographical Information
  11. Biographical Information
Thumbnail image of

Ming Qiu Zhang has over 28 years of systematic experience in polymers, polymer blends, and polymer composites. He serves as a member of Asian-Australasian Association for Composite Materials (AACM) Council and the standing council of Chinese Society for Composites, as well as president of Guangdong Society for Composites. In 1997, he received the prestigious fellowship from the Natural Science Foundation of China for Outstanding Young Scientists; and in 2005, the Li Ka Shing Foundation and the Ministry of Education of China selected him as a Cheung Kong Scholar. In addition, Professor Zhang is on the editorial board of seven scientific journals and holds 28 patents.

Biographical Information

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MOLECULAR MECHANISMS
  5. HEALING MODELING
  6. DESIGN OF SELF-HEALING COMPOSITES
  7. CONCLUDING REMARKS
  8. Acknowledgements
  9. REFERENCES AND NOTES
  10. Biographical Information
  11. Biographical Information
Thumbnail image of

Min Zhi Rong obtained his PhD degree in polymer chemistry and physics in 1994 in Zhongshan University. Before that, he is a researcher and lecturer in Department of Materials Science and Engineering, Tianjin University. His main interests are focused on thermosetting/thermoplastic blends, polymeric functional materials, structure of polymer networks, polymeric nanocomposites, natural fiber composites, and self-healing of polymeric materials. Among his many professional accolades, Professor Rong won the 2007 Prize for Achievements in Natural Science Research for his work on polymer nanocomposites awarded by the Ministry of Education of China. Along with having been published in about 180 journal papers and book chapters, Professor Rong also holds 35 patents.