Self-healing materials: a review

REVIEW www.rsc.org/softmatter | Soft Matter

Self-healing materials: a review

Richard P. Wool

Received 31st July 2007, Accepted 30th November 2007

First published as an Advance Article on the web 10th January 2008

DOI: 10.1039/b711716g

The ability of materials to self-heal from mechanical and thermally induced damage is explored in this

paper and has significance in the field of fracture and fatigue. The history and evolution of several

self-repair systems is examined including nano-beam healing elements, passive self-healing, autonomic

self-healing and ballistic self-repair. Self-healing mechanisms utilized in the design of these unusual

materials draw much information from the related field of polymer–polymer interfaces and crack

healing. The relationship of material damage to material healing is examined in a manner to provide an

understanding of the kinetics and damage reversal processes necessary to impart self-healing

characteristics. In self-healing systems, there are transitions from hard-to-soft matter in ballistic impact

and solvent bonding and conversely, soft-to-hard matter transitions in high rate yielding materials and

shear-thickening fluids. These transitions are examined in terms of a new theory of the glass transition

and yielding, viz., the twinkling fractal theory of the hard-to-soft matter transition. Success in the

design of self-healing materials has important consequences for material safety, product performance

and enhanced fatigue lifetime.

1.0 Introduction and overview

Self-healing materials are polymers, metals, ceramics and their

composites that when damaged through thermal, mechanical,

ballistic or other means have the ability to heal and restore the

material to its original set of properties. Few materials intrinsi-

cally possess this ability, and the main topic of this review is

the design for self-repair. This is a very valuable characteristic

to design into a material since it effectively expands the lifetime

use of the product and has desirable economic and human safety

attributes. In this review, the current status of self-healing mate-

rials is examined in Section 1, which explores the history and

Richard P: Wool

Dr Richard Wool is a Professor

of Chemical Engineering and

Director of the Affordable

Composites from Renewable

Resources (ACRES) Program

in the Center for Composite

Materials at the University of

Delaware. He is author of the

books ‘‘Bio-Based Polymers

and Composites’’ and ‘‘Polymer

Interfaces: Structure and

Strength’’. His research inter-

ests are in the fields of bio-based

polymers and composites, crack

healing, fracture, interfaces,

glassy state, polymer entangle-

ments and dynamics.

Department of Chemical Engineering, University of Delaware, Newark DE19716-3144, USA. E-mail: [email protected]

400 | Soft Matter, 2008, 4, 400–418

evolution of several self-repair systems including nanobeam-

healing elements, passive self-healing, autonomic self-healing

and ballistic self-repair. Section 2 examines self-healing mecha-

nisms, which could be deployed in the design of these unusual

materials and draws much information from the related field

of polymer–polymer interfaces and crack healing. The relation-

ship of material damage to material healing is examined in

Section 3 in a manner to provide an understanding of the kinetics

and damage-reversal processes necessary to impart self-healing

characteristics. In self-healing systems, there are transitions

from hard-to-soft matter in ballistic impact and solvent bonding

and conversely, soft-to-hard matter transitions in high rate

yielding materials and shear-thickening fluids used in liquid

armor. These transitions are examined in Section 4 in terms of

a new theory of the glass transition and yielding, viz., the twin-

kling fractal theory of the hard-to-soft matter transition. Section

5 gives an overview of the most recent advances in the self-heal-

ing field, including the biomimetic microfluidic healing skins,

and provides some prospective for the future design of self-heal-

ing materials. The biological analogy of self-healing materials

would be the modification of living tissue and organisms to

promote immortality, and many would agree that partial success

in the form of expanded lifetime would be acceptable. Hopefully,

the reader of this review is left with a sense of what-to-do and

what-not-to-do when designing self-healing materials, perhaps

not always as this author intended.

1.1 Observation of self-healing materials

Materials such as polymers and composites experience damage

and fatigue during their normal utilization and the concept of

eliminating this damage through a self-healing mechanism holds

the promise of enhanced lifetimes and enduring strength.1 This is

especially important in materials that are intended to perform in

a designed manner for significant times where repair is not

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possible. Our early attention to self-healing materials in the 1970s

arose through the need to understand the constitutive properties

of filled elastomers,2–4 such as those used in solid rocket propel-

lants for applications in space exploration (Apollo), Earth studies

(Shuttle) and more recently, ballistic missile interdiction. These

materials consisted of ammonium perchlorate-filled hydroxyl-

terminated polybutadiene cross-linked with diisocyanates. It

was clear that mechanical action caused microscopic damage at

the nanoscale, which could coalesce to form larger microscopic

cracks, which in turn could propagate as macroscopic cracks

and cause catastrophic loss of the material and payload.

However, it became readily apparent that much of this damage

could self-heal and measures of damage through modulus loss

or mechanical stress–strain hysteresis were developed to quantify

the damage and healing processes.1 The first self-healing mechan-

ical element is shown in Fig. 14.

This reversible self-healing crack element was used to examine

some of basic phenomena of an idealized self-healing element,

which could be used in series or parallel with elementary springs

and dashpots, similar to the Maxwell or Voigt models. It

consisted of two surfaces with an anharmonic potential function

V(l) which would permit yielding at a critical strain ec ¼ lc�1.

What was most interesting about this model was that at high

strain, e.g. l ¼ 10, during stress relaxation, the applied mechan-

ical work was stored reversibly in the new surfaces created, and

would cause the stress to eventually increase before relaxing

completely. This could result in materials with most peculiar

mechanical properties, such as negative creep compliances J(t)

and reactive stress relaxation moduli G(t) where the stress would

first decrease with time and then suddenly increase at constant

strain,4 which bemused unsuspecting visitors to our lab. Such

self-healing material elements can now be made with intercalated

nanoclays and carbon nanotube bundles.5–7 Elastomers based on

functionalized fatty acids were found to critically intercalate and

partially exfoliate nanoclay galleries. The nanoclay galleries were

Fig. 1 (a) The stress–strain response of a healing element is shown where th

function. (b) Viscoelastic elements made with the healing element in series an


expanded to near-exfoliation by the elastomer and could open or

close with mechanical action thereby producing a mechanical

self-healing process similar to the nanobeams in Fig. 1.

An explicit example of a self-healing material was demon-

strated with hard elastic polypropylene (HPP).8,9 HPP is a

stress-crystallized PP which has a morphology consisting of

stacked lamella perpendicular to the extrusion direction of the

fibers or films. When stretched along the fiber axis, the lamellae

would splay apart permitting large reversible strains up to 500%,

as shown in Fig. 2 and 3.

The initial high modulus of this polymer and its subsequent

elastomeric behavior earned it the name of hard-elastic polymers,

which are quite unique to row-nucleated semicrystalline poly-

mers. The resulting network fracture morphology was often

compared to a ‘‘Christmas paper bell’’. The mechanical proper-

ties shown in Fig. 3 show that the HPP when allowed to rest

at room temperature could completely recover its high elasticity

as the interlamellar microvoids healed. In detailed studies, we

found that the interlamellar fracture damage could be

completely healed at room temperature and accelerated with

higher temperature.1,9

We subsequently discovered that crack healing was fairly

ubiquitous in polymer materials and systematically studied

microvoid healing, craze healing and crack healing in a variety of

materials, such as crazes in polystyrene,10,11 microvoids in SBS

block copolymer elastomers12 and others which are reviewed in

ref. 1. In a seminal study, Kausch and co-workers13–15 showed

that single cracks that were formed at room temperature in frac-

ture mechanics specimens of PMMA could be completely

rehealed when the surfaces were rejoined and welded above the

glass transition temperature Tg. Much work has been done by

our group16–25 and others26–68 to understand the concept of

strength development at polymer–polymer interfaces and poly-

mer–solid interfaces69–75 in terms of the fundamental dynamics

of entangled polymers. Using DeGennes’ reptation dynamics

e two parallel surfaces interact by an anharmonic non-bonded potential

d parallel with a viscous dashpot [Wool 1978].

Soft Matter, 2008, 4, 400–418 | 401

Fig. 4 Schematic drawing of hollow fiber fluid-release design to repair

cracks in fiber-filled composites (courtesy of C. Dry, University of

Illinois).

Fig. 3 Cyclic stress–strain curves of a HPP fiber with intervening

healing times from 10 to 1000 min at room temperature.1,9

Fig. 2 SEM micrographs showing the surfaces of a 2.78 tex (25 denier,

hard elastic polypropylene after extensions of: (a) up to 50%; (b) up to

100% (note the appearance of the surface cracks; (c) up to 200% (note

the widening of the cracks); (d) the surface of (c) at higher magnification.

Final magnifications are (a–c) 1500x, (d) 3900x. (courtesy of Ludmila

Konapasek).

and the entanglement percolation model,17,76 we derived a unified

theory for the strength development of polymer–polymer inter-

faces.17 While these relations are useful for welding above Tg

using molecular mobility and interdiffusion processes, they

also provide the keys to the important molecular and micro-

scopic connectivity relations necessary to promote self-healing

in materials below their Tg values where molecular mobility is

more restricted, typically.

1.2 Design of self-repair materials

A major issue for rigid materials below their Tg value, e.g.,

composite matrices, is how to induce or design for self-healing.

402 | Soft Matter, 2008, 4, 400–418

This is very important for thermosetting composite matrices,

which are typically highly cross-linked, and for glassy amor-

phous polymers and semicrystalline materials. Carolyn Dry, an

architect at the University of Illinois, presented an interesting

solution to self-healing materials to me. She was interested in

self-healing of fiber filled composites,77–79 for both polymer and

concrete matrices, and had developed several patents based on

these novel materials. In collaboration with N. Sottos, she devel-

oped a system of hollow fibers filled with a reactive fluid, which

was released by mechanical trauma and subsequently reacted to

heal the local cracks, as shown in Fig. 4.79

C. Dry had analyzed passive self-healing materials in the

following 5 steps, as described in ref. 1 (chapter 12): (1) the mate-

rial must be subjected to gradual deterioration, for example by

dynamic loading that induces microcracks; (2) the fiber must

contain self-repair fluid; (3) the fibers require a stimulus to

release the repairing chemical; (4) a coating or fiber wall must

be removed in response to the stimulus; and (5) the fluid must

promote healing of the composite damage.

Several interesting self-repair tests were conducted by Dry and

co-workers on polymer matrices with different fibers. Experi-

ments were performed on polymer samples containing an

embedded continuous metal fiber with two self-repair fibers in

its vicinity, one containing a (epoxy) monomer and the other

containing the diamine cross-linking agent. They first mechani-

cally debonded the metal fiber in a fiber-pullout test without

disturbing the self-repair fibers. At this point, the metal fiber

had very little adhesion with the matrix and could be readily

removed with little stress. They then subjected the composite

to sub-critical loads, which released the fluids from the self-repair

fibers. The fluids mixed and diffused to the polymer–metal inter-

face and acted to repair the interface damage, which resulted in

a considerable gain of the fiber pullout stress.

This approach to self-repair is very flexible with regard to the

design of fiber, number of fibers, healing fluid, fiber construction,


fiber coating and so forth. The healing fluid could be a cross-

linking epoxy or polyester, a chemical agent that reacts with

the matrix, or one that reacts selectively with the damaged

surface. For example, one could use vinyl monomers in places

where microcracks in the matrix generate free radicals. The

free radicals would then polymerize the fluid and help promote

strength. Matrix solvents could also be used to promote solvent

bonding in microvoids. Thermally induced healing can also be

induced by exothermic reactions of the self-repair fluid, either

with itself or with components of the composite. We will return

to this passive self-healing mechanism later.

Fig. 6 (a) Healed fracture toughness KIC as a function of microcapsule

concentration. (b) Healing efficiency KIC(healed)/KIC(virgin) vs. concen-

tration for microcapsule diameters 180 mm and 50 mm. The healing time

was 24 h (courtesy of S. White et al.).

1.3 Autonomic self-repair design

An interesting variant of the passive self-healing method was

developed by White and Sottos in 2001,80–83 where instead of

using hollow fibers, they used spheres of healing fluid as shown

in Fig. 5. They describe the autonomic healing concept as

follows: a microencapsulated healing agent is embedded in

a structural epoxy matrix containing a catalyst (Grubb’s)

capable of polymerizing the healing fluid. (a) Cracks form in

the matrix wherever damage occurs. (b) The crack ruptures the

microcapsules releasing the healing agent into the crack plane

through capillary action. (c) The healing fluid contacts the

catalyst, which is randomly distributed in the matrix and triggers

the polymerization reaction that helps to heal the damage by

bonding the crack surfaces closed.

This approach was successfully demonstrated for both frac-

ture and fatigue healing in composite matrices80–84 and was

discussed by this author in a companion paper.85 Using a tapered

double cantilever beam (TDCB), which permits stable crack

propagation, they first propagated a crack with known length.

The propagating crack ruptured the urea formaldehyde spheres,

which released the dicyclopentadiene fluid. In the rest state, the

released fluid contacted the ruthenium (Grubb’s) catalyst

embedded at random in the epoxy matrix, which initiated the

polymerization of the fluid and repaired the crack. A second

loading of the TDCB gave a measure of the fracture energy of

the repaired original crack length. They report about 75%

recovery of the fracture toughness, or maximum load ratio at the

same crack length. Fig. 6 shows the fracture toughness healing

experiments as a function of microcapsule size and concentration

Fig. 5 The autonomic self-healing system developed at the University of

Illinois [Courtesy of S. White and N. Sottos].


(Brown et al., 2003). It is clear with these experiments that they

are capable of recovering about 80% of the virgin KIC values of

their material through this unique self-healing mechanism. In

terms of fracture energy GIC, since GIC � KIC2 in the linear elastic

fracture mechanics approximation, this corresponds to about

64% energy recovery.

Similar experiments in fatigue (sub-critical crack growth with

cyclic stress) showed that the initial fatigue crack propagation

rate (da/dN) could be recovered due to this healing process.82

Fig. 7 shows related work of Brown et al. (2005) on fatigue

healing.

The slowly propagating crack is attempting to heal due to

release of the healing fluid from the ruptured microcapsules.

Fig. 7 Crack extension in fatigue (5 Hz) shows that the precrack had

healed and required � 3000 cycles to reopen. Subsequently, the crack

extension per cycle shows regions of constant slope and retardation

due to healing during propagation (courtesy of S. White et al.).

Soft Matter, 2008, 4, 400–418 | 403

Fig. 8 Healed EMMA film following puncture (courtesy of S. Kalista).

They observe that if the fatigue is halted and the sample rested,

the cracks will heal up. These results are particularly impressive

since fatigue is quite insidious in that cracks which appear to

have healed in terms of strength can remain quite weak in

fatigue: for example, we have observed that for welding of poly-

mer–polymer interfaces that full strength could be obtained in

terms of KIC measurements after a certain weld time but the

weld remained very weak in terms of fatigue crack propagation

rates.21 Thus in Fig. 6 where they show 80% healing of KIC,

one might suspect that fatigue healing would be substantially

impaired but this is not the case, as shown in Fig. 7 and related

experiments of Brown et al. Several significant advances have

been made in both the chemistry and self-repair methodology

of this unique system81–85 and are discussed in Section 5.

The work by White and Sottos et al. is considered by many to

represent the current leading edge of the field of self-healing

materials. However the field has become very active and several

papers have been presented by various groups where other chem-

ical means were used to initiate healing of the fluid or matrix.86–90

Balazs et al.91 explored the use of nanoparticle migration to crack

tips in thin films by computer simulation and found that this

migration could lead to substantial healing processes. The nano-

particles become localized in nanoscale cracks, the precursors to

microcracks, and effectively form patches to repair the damaged

region. This approach assumes that the nanoparticles have suffi-

cient mobility in the polymer matrix where the test temperature

was assumed to be greater than Tg. The repaired composites were

expected to recover 75–100% of their mechanical strength. Yu

et al.92 (1995) examined the rate of crack closure during healing

of several materials. They found that the rate of crack closure

during the wetting stage was relatively constant in solvent treated

(CCl4) polycarbonate, ethanol treated PMMA, cracks in single

crystals of KCl under pressure and single crystals of LiF at

elevated temperatures. We note that crack surface wetting is

a necessary but not a sufficient condition formechanical recovery.

For example, in both crack and craze healing, the cracks can

disappear at the nanoscale during the wetting stage but the inter-

face remainsweak until considerable interdiffusion has occurred.1

1.4 Ballistic impact self-repair

Perhaps the most provocative self-repair experiment was the

observation by R. Fall et al.93 and Kalista et al.94,95 that bullet

holes in a plastic plate would heal up instantly. The high velocity

projectile could penetrate the polymer plate and the holes would

reseal faster than the eye could see. This phenomenon was

explored in some detail by Kalista et al.95 They examined self-

healing in several poly(ethylene-co-methacrylic acid) copolymers

(EMMA) following projectile puncture. Fig. 8 shows a typical

result where a 4.5 mm projectile of mass 0.51 g and velocity

196.6 m s�1 has penetrated an EMMA film of thickness 1 mm.

The polymer contained about 5.4 mol% methacrylic acid groups,

which had been partially (60%) neutralized by sodium and had

a melting point Tm ¼ 93 �C. One sees in Fig. 8 that the central

damage zone is about the size of the bullet cross-section but

has been healed. The healing was determined by a pressure burst

test. Interestingly, when the samples were tested at elevated

temperatures (60 �C), and some, but not all at low temperature

(�30 �C), no healing occurred. They concluded that the ionic

404 | Soft Matter, 2008, 4, 400–418

content of the polymer was not important for healing and that

self-repair occurred by a two-stage process: Stage 1 involved

melt elastic recovery followed by Stage 2 with sealing and poly-

mer-chain interdiffusion. Kalista et al. conclude that the impact

energy of the projectile was sufficient to melt the polymer in the

damage zone and melt recoil followed by interdiffusion

promoted self-repair.

A minimal energy balance analysis of this process would give

the temperature rise, DT ¼ T (impact) � T (sample), in the

damage zone of radius R and mass m as:

DT ¼ DU � m DHf

m Cp

(1)

where DU ¼ 1⁄2 m(V12 � V2

2) is the energy dissipated with impact

velocity V1 and exit velocity V2, DHf is the heat of fusion to melt

the polymer and the Cp is the heat capacity. The mass of the

damage zone is m ¼ rpR2h. The impact velocity in Fig. 8 gives

an energy of U1 ¼ 9.9 J (we do not know the exit velocity and

DU should be smaller), R ¼ 2 mm, h ¼ 1 mm, Cp z 2.7 J K�1

g�1, DHf ¼ 429 J g�1 (PE value) and r z 1 g cc�1. These values

give DT z 100 �C in the damage zone, which is sufficient for

melting and rehealing with Tm ¼ 92 �C and T (sample) ¼ 22 �C.

However, at�30 �C, this is not sufficient energy formelting. Also,

if the mass of the damage zone increases with increasing temper-

ature, the numerator in the above relation goes rapidly to zero

and healing will not occur, as observed by Kalista et al. In related

experiments, Kalista et al. demonstrated that the same polymer

plates also self-repair when cut with a saw since the friction gener-

ated by the saw is sufficient to thermally weld the surfaces

together. However, the polymer plate when cut with a very sharp

razor blade did not self-heal due to insufficient mechanical energy

dissipation at the crack interface. Room temperature projectile

testing with LDPE films showed no tendency for self-repair.95

In general, one should be able to design most thermoplastics for

ballistic self-healing. These could find unique applications in

space-capsule protection against micro meteorites.

In the Kalista ballistic-healing experiments, self-healing is

obtained by transforming ‘‘hard matter’’ to ‘‘soft matter’’ due

to mechanical action. Decker et al.96 have shown that ballistic

impact resistance can be obtained by transforming a liquid to

a solid, through the use of shear-thickening fluids (STF). The

STF consists of a colloidal particle suspension that percolates

rigidity under high deformation rates and has become the basis

for the invention of ‘‘liquid armor’’ at the University of

Delaware. The advantage of this system is that after impact, the


instantly rigidized matter returns to the liquid state and rapidly

heals itself by restoring the local concentration of particles.

The STF suspensions when mixed with Kevlar fibers have shown

remarkable stab resistance when subjected to repeated high

velocity trauma from sharp objects such as knives and ice picks.96

We further explore this mechanism in Section 4.

In this review, we examine several relevant theories for self-

healing and compare with experimental results. It is intended

that this paper serve both as a review of work done in the field

of self-healing materials and act as a design tool for future gener-

ations of these interesting materials which proffer the unusual

promise of everlasting material life.

2.0 Self-healing mechanisms

2.1 Stages of passive self-healing

Discussions with C. Dry (�1993) resulted in the following anal-

ysis of self-healing systems using passive fluids, solvents, reacting

catalysts, etc. The results are applicable to both the passive heal-

ing models of Dry77–79 and White and Sottos,80–85 models using

nanoscale healing elements,4 other general models using chemi-

cal reactions,86–90 nanoscale segregation91 and ballistic impact

self-repair.95

Five stages of crack healing were developed by Wool and O’

Connor19 in an effort to unscramble the complexity of strength

development at polymer interfaces. That approach proved to

be successful in separating the multi-convoluted time dependen-

cies of the different mechanisms controlling crack healing and

their underlying molecular processes. Repair of cracks and

microscopic damage has been described in terms of the following

stages: (a) surface rearrangement; (b) surface approach; (c)

wetting; (d) diffusion; and (e) randomization. These are discussed

with respect to self-healing materials in the following sections.

2.2 Surface rearrangement

When the freshly damaged surfaces or microvoids are created in

fracture or fatigue, one should consider the roughness or topog-

raphy of the surface and how it changes with time, temperature

and pressure following contact with the healing fluid. In frac-

tured polymers, rearrangement of fibrillar morphology and other

factors affect the rate of crack healing. Chain-end distributions

near the surface can change as molecules diffuse back in to the

bulk.20 If the chain ends are needed for reaction with the fluid,

they could be designed to preferentially migrate to the surface

using lower surface tension moieties on the chain ends. Spatial

changes of the molecular weight distribution can also occur,

for example, where the low molecular weight species preferen-

tially migrate to the surface. Nanoparticles in the bulk could

preferentially migrate into nano voids.91 In time-release solvents

or adhesives, surface rearrangement is affected by the polymer–

solvent interaction. Chemical reactions, for example oxidation

and cross-linking can occur on the surface and complicate the

dynamics of diffusion. Solvents used in the passive healing exper-

iments could also cause additional damage for example, by

causing crazes and microvoids to swell and allow them to prop-

agate further.

Each material and self-repair release technique posses unique

surface-rearrangement processes that may need to be quantified.


The use of solvent systems that promote surface segregation of

chain ends for example, would be highly conducive to rapid

healing of damage. The relation between solvent concentration

and surface-rearrangement dynamics needs to be quantified,

primarily through the effect of solvent on both the glass transi-

tion temperature Tg and the relaxation times of the surface mole-

cules, as discussed in ref. 1 (chapter 7). The critical entanglement

molecular weight Mc will also be changed with surface polymer

concentration 4 in the good solvent as,76

Mc(4) ¼ Mc(1)4�5/4 (2)

Where Mc(1) is the unperturbed Mc value with 4¼ 1. This means

that when a compatible healing fluid interacts with the polymer

surfaces, the entanglement molecular weight increases. Conse-

quently, the entanglement density n(4) decreases via:

n(4) ¼ n(1)49/4 (3)

Consequently, the plateau elastic modulus GN0 behaves as76

GN0 ¼ GN

0(1)49/4 (4)

Thus, the surfaces are expected to become quite soft in the

presence of a healing fluid and this could be quite beneficial

to promote interdiffusion if necessary for healing. The local

reptation times t will be affected by solvents as,76

t ¼ t(1)45/4 (5)

in which t(1) � M3. The viscosity h of the surface layer will be

changed accordingly as

h ¼ h(1)43.5 (6)

Thus, a 50% decrease in polymer concentration on the surface as

it mixes with the healing fluid would reduce its local viscosity by

0.09 times its original value. Solvation to 10% causes a decrease

of the viscosity to 3� 10�4 times h(1). Since these effects basically

occur at room temperature, an important role of the healing

fluid could be to enhance molecular mobility of the surfaces

and facilitate the other stages of healing. The dynamics of the

surface-layer rearrangement is similar to that in some bulk

processes, but in general, the surface molecules should have

enhanced mobility due to higher degree of freedom and an

altered photon density of states reducing both the Tg and heat

capacity of the surface layers. Because surfaces have a lower

heat capacity, less energy is required to raise their temperature.

For polymer–solid interfaces as commonly encountered in

composites, surface restructuring dominates the mechanism of

adhesion between the polymer and the solid. When a mole

fraction of sticker groups f(X) is used to bond the polymer to

the surface, an optimal sticker group concentration f*(X) is

needed to maximize adhesion while minimizing cohesive failure

in the boundary layer adjacent to the solid.69–75 When f < f*,

adhesive failure dominates, the fracture energy G1c � f and the

solid separates cleanly from the polymer. When f > f*, cohesive

failure occurs in a polymer layer adjacent to the surface and G1c

� 1/f. What happens to this balance in the presence of a healing

fluid? This restructuring is quite delicate and important to soft

Soft Matter, 2008, 4, 400–418 | 405

Fig. 9 Schematic of a time-dependent wetting process W(t) in a portion

of a polymer–polymer interface. The wetted shaded pools of radius r are

nucleated at different times and propagate until coalescence is achieved,

and the fractional wetted area W ¼ 1.

materials. The f* value is determined from the entanglement

percolation theory to be75

f* ¼ 4j(Mj/Mc) (7)

where Mj is the molecular weight per bond of a random walk

chain (e.g., Mj ¼ 14 g mol�1 for PE). The critical entanglement

molecular weight Mc is determined by the random walk charac-

teristic ratio CN and Mj as76

Mc ¼ 62CNMj4�5/4 (8)

Combining the latter two relations, we obtain the f* value in the

presence of the solvent as

f* ¼ (0.0745/4)/Cf (9)

Since Cf is of order 7–20 for many polymers, then f* � 1%, as

observed.69–75 Thus, in the presence of a healing fluid, surface

restructuring would produce a new value of f* which is consider-

ably less than the original. The resulting restructuring could

result in a very weak polymer–solid interface since the weak cohe-

sive failure mode would be favored. The surface-restructuring

kinetics of polymer–solid interfaces are complex and have been

examined by Gong and co-workers and Lee and co-workers.69–75

2.3 Surface approach

In controlled laboratory healing experiments, this stage is

considered fairly trivial since surface preparation and suitably

applied pressure usually ensures that the surfaces are brought

together. However, this stage could be the most critical for

self-healing materials. Simply, no healing occurs if the surfaces

are not brought together or the gap is not filled with the healing

fluid. Thus, any debris left over from the damage process could

pry the surface apart to prevent surface approach and terminate

the self-healing process. Surface approach applies to crack

surfaces that are brought together to heal alone or in the pres-

ence of the healing fluid. This stage of healing considers the

time-dependent contact of the different parts of the surfaces to

create the interface. Surface approach may be especially impor-

tant in composites where the nature of the damage can involve

the polymer matrix, fibers and the matrix–fiber interface. If the

healing fluid only bonds to one surface, then little or no healing

will occur. The healing solvent may also force the surfaces

together by the pressure of swelling.

2.4 Wetting

When the damaged surfaces approach, they need to wet each

other and form an interface before the healing process can

continue. With self-repair fluids, the wetting and compatibility

of the damaged surface by the fluid must also be considered.

The topic of wetting and spreading of a fluid on a surface has

been treated by Brochard.97 In the capillary action required by

the Sottos and White model to fill the voids, the wettability of

the surfaces by the fluid can be determined as part of the self-

healing design; some fluids may be better than others. One could

also inquire if the fracture surfaces have been altered by oxida-

tion due to chain fracture and are no longer wettable by the fluid.

406 | Soft Matter, 2008, 4, 400–418

Wetting can occur in a time-dependent fashion at the inter-

face. For our purposes we provide a brief phenomenological

description of wetting to illustrate potential problems in evalu-

ating the time-dependence of healing. Fig. 9 shows a schematic

region of the plane of contact of a polymer interface.19 Due to

surface roughness, etc., good contact and wetting are not

achieved instantaneously at all locations. Typically, wetted

‘‘pools’’ are nucleated at random locations at the interface and

propagate radially until coalescence and complete wetting are

obtained. This problem has been treated as a two-dimensional

nucleation and growth process such that the fractional wetted

area, W(t), is given empirically as:19

W(t) ¼ 1 � exp(�ktm) (10)

where k and m are constants depending on the nucleation func-

tion and radial-spreading rates. This function predicts that 100%

wetting will occur eventually, but this may not be the case

depending on how the surfaces are permitted to approach or

be filled in with the healing fluidFor crack healing, we have

observed that the wetting function W(t) convolutes with the

interdiffusion function H(t) and affects the fracture energy of

the interface via:

G1c ¼ðt

O

Hðt � tÞdW=dt dt (11)

in which t is the dummy variable of the convolution integral.

For polymer welding, H(t) is controlled by the average interpen-

etration contour length L, which behaves as,22

L(t) � (t/M)1/2 (12)

Under high pressure surface contact, if the wetting rate is

constant dW(t)/dt � W0, then the convolution process predicts

that the fracture energy should increase as


Fig. 10 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 equilib-

rium-diffused distance and continue to diffuse away but are no longer

connected top the other side. (Red region) The fractal diffusion line sepa-

rating the non-connected from the connected chains which provide

strength at the interface. [Wool and Long]

G1c � W0

ðt

0

LðtÞdt � t3=2 (13)

For self-healing reacting materials, the wetting function will

convolute with the reaction kinetics relation for strength devel-

opment. In the absence of diffusion at polymer–polymer inter-

faces, the wetting process delivers very little strength. However,

for rebonding the polymer matrix to a solid substrate such as

a composite fiber, the wetting stage should be rate determining,

coupled with surface rearrangement of sticker groups.

2.5 Diffusion stage

The diffusion stage is the most critical stage of strength develop-

ment for both crack healing and self-healing systems at polymer–

polymer interfaces. It is a dominant stage in the ballistic

self-healing experiments when the elastically recoiling melt from

the penetration hole rapidly wets and interdiffuses to make

a good seal across the bullet hole.

The molecular aspects of interdiffusion of linear entangled

polymers (M > Mc) during welding of polymer interfaces are

summarized in Table 1.1

The reptation dynamics and the interface structure relations in

Table 1 have been demonstrated experimentally by a series of

interdiffusion experiments with selectively deuterated polymers

using dynamic secondary ion mass spectroscopy (DSIMS) and

neutron reflectivity.98–103 The scaling laws and the complete

concentration profiles have been calculated by Kim and Wool22

and Zhang and Wool23 The important result for the contour

length L � (t/M)1/2, was also supported by welding computer

simulations of Windle et al.49 Initially, as the symmetric (A ¼ B)

interface wets by local rouse segmental dynamics, we find that

rapid interdiffusion occurs to distances of the order of the radius

of gyration of the entanglement molecular weight, ca. 30 A. This

can also occur below Tg when the top surface layer becomes more

mobile than the bulk and can be explained by finite size rigidity

percolation theory17 (Section 4). However, at this point, the

interface is very weak and fracture can be described by the nail

solution.57 At the wetting stage, the frictional pullout of

intermeshed chain segments, which have ‘‘elbowed’’ their way

across the interface, determines the fracture energy. As welding

proceeds, S minor chains of length L diffuse into an interface

of width X and considerable strength develops. The diffusing

chains are fractal random walks and interpenetrate with the

matrix chains, which are fully entangled (ignoring surface

reflection configuration effects on entanglement density).

The structure of the diffuse weld interface in Fig. 10 resembles

a green box of average width X, with fractal edges containing a

Table 1 Molecular aspects of interdiffusion at a polymer–polymer interface

Molecular aspect Symbol Dynam

General property H(t) tr/4M�s

Average contour length l(t) t1/2M�

Number of chains S(t) t1/4M�

Number of bridges P(t) t1/2M�

Average monomer diffusion depth X(t) t1/4M�

Total number of monomers diffused N(t) t3/4M�

Center of mass diffusion Xcm t1/2M�


gradient of interdiffused chains, as shown by Wool and Long.51

Gradient percolation theory104 requires that chains, which

contribute to the interface strength (green), straddle the interface

plane during welding, such that chains in the concentration

gradient that have diffused further than their radius of gyration

(yellow) cease to be involved in the load-bearing process at the

interface. We have shown that this amounts to a very small

number and for narrow molecular weight distributions, can be

ignored.1,51 However, for broad molecular weight distributions,

the fraction of non-connected chains expressed through gradient

percolation, can be significant.105 When the 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. Fracture of the weld

occurs by disentanglement of the minor chains, or by bond

rupture. It is interesting to note that if the stress rises to the point

where random bond rupture in the network begins to dominate

the deformationmechanism, instead of disentanglement, then the

weldwill appear tobe fully healed, regardless of the extent of inter-

diffusion. This can occur at high rates of testing when the minor

chains cannot disentangle and bond rupture pervades the inter-

face, breaking both the minor chains and the matrix chains.

3. Damage and healing theories

3.1 Percolation theory of damage and healing

We can learn about healing processes in materials by studying

their damage mechanisms. A useful approach to evaluating the

ic relation, t < Tr Static relation, Ht ¼ Tr r, s

/4 M(3r�s)/4 r, s1/2 M 2, 25/4 M�1/2 1, 53/2 M0 2, 61/4 M1/2 1, 17/4 M1/2 3, 71 M1/2 2, 4

Soft Matter, 2008, 4, 400–418 | 407

Fig. 11 The microscopic entanglement structure, e.g. at an interface or

in the bulk, is related to the measured macroscopic fracture energy G1c

via the RP theory of breaking connectivity in the embedded plastic

zone (EPZ) at the crack tip. The RP theory determines smax in the

EPZ, which is related to G1c via Hutchinson’s J-integral theory. The

percolation parameter p, which is a measure of damage and healing, is

related to the interface molecular structure via p �SL/X, where S is

the number of chains of length L in an interface of width X.1Fig. 12 The role of percolation in the random fracture of bonds in

a model net at constant strain is shown. (a) The net of initial modulus

E, is stressed in uniaxial tension to a stress s. (b) Release of stored strain

energy by random bond fracture in the net results in a percolating system

near the fracture threshold and a very broad distribution of stress on the

bonds.1

fracture energy G1c, of a healing interface of A–B polymers, is

represented in Fig. 11.1 In this double cantilever beam (DCB)

system, a crack propagates through the interface region preceded

by a deformation zone at the crack tip. For cohesive failure, the

fracture energy can be determined by the J-integral method, as

described by Hutchinson and Tvergaard,106–108 where G1c is the

integral of the traction stresses with crack opening displacements

d, in the cohesive zone, following yielding at a local yield or craze

stress sY. The cohesive zone at the crack tip breaks down by

a percolation process,17,18 as described herein, at a maximum

stress value, sm > sY.

Typical ratios of sm/sY are about 4–5.106 The yield stress domi-

nates the fracture process for non-crazing matrices such as ther-

mosets and this is determined by the twinkling fractal theory

(TFT)109 described in Section 4. Both sm and d are rate dependent

and in the simplest case, the fracture energy is determined by:

G1c ¼ smdm (14)

where dm is the critical crack opening displacement. Both smand

dm depend on the damage-zone structure and the microscopic

deformation mechanisms controlling the percolation fracture

process via disentanglement and bond rupture. To convert these

percolation concepts into quantitative fracture terms for healing

processes, consider the experiment shown in Fig. 12.1

This experiment can be used to interpret fracture and healing

on any 2d or 3d lattice with initial tensile modulus E and subject

to random bond fracture. Random bond scission causes the

formation of microvoids, which coalesce into larger voids and

facilitate a macroscopic crack propagating through the net at

the percolation threshold. The Hamiltonian for the stored elastic

energy can be formulated using the Kantor and Webman110

approach for specific lattices,111–115 the Born and Huang

method,116 or using the simple engineering strain energy density

approach as follows. The stored elastic strain energy density U,

(energy per unit volume), in the lattice due to an applied uniaxial

stress s is determined by,

U ¼ s2/2E (15)

408 | Soft Matter, 2008, 4, 400–418

The stored strain energy dissipation per unit volume Uf, to

fracture a network consisting of n bonds per unit volume is,

Uf ¼ nD0[p � pc] (16)

where Do is the bond fracture energy, and [p � pc] is the percola-

tion fraction of bonds which must be broken to cause fracture in

the network. In this approach, the strain energy U, is first stored

in the net and we inquire if this energy is sufficient to break

n[p�pc] bonds per unit volume when it releases at a critical strain

energy density U*¼ s*2/2E, such that at the critical condition,

U* $ Uf (17)

Substituting for U* and Uf in eqn (17) and solving for the critical

stress s*, we obtain the ‘‘net solution’’ for the critical fracture

stress as

s* ¼ {2EnD0[p�pc]}1/2 (18)

This relation was found to be applicable to damage and heal-

ing events in carbon nanotubes.18 Applications of eqn (18) to

a range of polymer materials are shown in Table 2 where the

entanglement density is v ¼ r/Mc.

Eqn (18) predicts that the fracture stress increases with the

square root of the bond density n � 1/Mc. The percolation

parameter p, is in effect, the normalized bond density such that

for a perfect net without defects, p ¼ 1, and for a net that is

damaged or contains missing bonds, then p < 1. Obviously, as

p approaches pc, the fracture stress decreases towards zero and

we have a very fragile material. The fatigue lifetime is also calcu-

lated using this concept and is presented in a later section. This

fracture relation could therefore be used to evaluate durability,

fatigue damage accumulation, healing processes, or retention


Table 2 Comparison of RP theory and experimental fracture stress18

Polymer Mc/g mol�1

s (theory)/104MPa Mc

�1/2

s (expt)/MPa T/�C Ref.

PE 4000 158 160 �196 118PP 7000 119 98 �120 118PVC 11 000 95 142 �180 118PMMA 18 400 74 68 �60 118

PC 4800 144145 �140 118120 20 117

PS 31 000 57 56 20 118PTFE 13 200 87 117 �196 118PLA 12 000 65 (E ¼ 1 GPa) 64 20 5Starch 200 000 22 — 20 5

strength of a material by tracking damage through a single

parameter p. For thermosets, p is related to the extent of reaction

of the cross-link groups and this could be critical in the fiber–

matrix interface of composites.120

The time dependence of healing R(t) can be described from

eqn (18) as:

R(t) ¼ s(t)/sN ¼ [(p(t) � pc)/(1 � pc)]1/2 (19)

in which sN is the virgin strength with p ¼ 1 and p(t) is the time-

dependent bond fraction recovery. For example, if a diffusion

process controls p � t1/2, then R � t1/4, as observed for crack heal-

ing.13–15,119 The latter equation is similar in some respects to the

generic kinetic healing rate equation we proposed for mechanical

recovery in solid rocket propellants, filled elastomers, block

copolymers and hard elastic polypropylene:1

R(t) ¼ 1 � [1 � R0]/[1 + Kt]a (20)

Here R0 is an instantaneous recovery fraction, which is depen-

dent on the initial extent of the damage, K is a temperature-

dependent parameter and a is a rate constant. This empirical

relation gives R ¼ R0 at t ¼ 0 and R ¼ 1 as t goes to infinity

and was found to adequately describe many healing processes.1

3.2 Fracture and healing by bond rupture and repair

The percolation theory predicts that fracture by bond rupture of

linear polymers is in accord with Flory’s suggestion of the chain-

end effect, via p ¼ 1 � Mc/M and pc ¼ 1 � Me/Mc, such that:18

Table 3 Table of interfaces encountered during repair, recycling, joining and

Liquid B (L)Solid with Acurface (SV

Liquid A (L) L–L virgin (control)L–SV repair

recyclingSolid with as-cast surface (SV) SV–SV wieldSolid with fractured surface (SF)Solid with fractured and treated

surface (SFC)


G/GN ¼ [1 � Mc/M(t)]/[1 � Mc/Mf]. (21)

Thus healing in this system involves repairing the molecular

weight from Mc back to its original value Mf. This could be

done using a liquid monomer, which reacts with catalyst to

make new chains in a crack zone, as suggested by White

et al.80 and Wudl et al.88

3.3 Fracture and healing of an ideal rubber

For an ideal rubber, p ¼ 1, the modulus E ¼ vkT where v is the

cross-link density. Thus, eqn (18) predicts:

s(t)/sN ¼ E(t)/EN (22)

G(t)/GN ¼ v(t)/vN (23)

An ideal rubber with a perfect lattice was essentially unbreakable

at low strains (l < 4) since insufficient strain energy could be

stored to break the requisite percolation number of bonds, but

fracture could occur by strain hardening to produce a ten-fold

increase in modulus E, by introducing defects (p z pc) via fatigue

or radiation, by decreasing the bond energy D0 an order of

magnitude, or by raising the temperature to 3000 K. Alterna-

tively, by making the cross-links through the use of partially

intercalated nanoclays, damage occurs by opening the

nanobeams of the clay (similar to Fig. 1) and reducing the

cross-link density. However, this system can self-repair through

diffusion and intercalation to restore the original cross-link

density, as noted by Zhu and Wool7.

3.4 Fracture and healing of thermosets

With highly cross-linked materials such as typical unsaturated

polyesters and epoxies used in composite materials, the percola-

tion theory predicts:

s(t)/sN � [v(t)/vN]1/2 (24)

The repair of the cross-link density v(t) can be done by chemical

means, as suggested by White et al.,80 Dry and Sottos79 and Chen

et al.88 for thermally re-mendable cross-linked materials.

Raghavan and Wool90 examined a matrix of possibilities for

healing and repair in thermoset polymers as shown in Table 3.

The results are summarized in Fig. 13.

manufacturing of polymer composites

s-cast)

Solid with fracturedsurface (SF)

Solid with fracturedand treated surface (SFC)

andL–SF repair and recycling L–SFC repair and recycling

ing SV–SF welding and repair SV–SFC welding and repairSF–SF crack healing SF–SFC wear and repair

SFC–SFC wear and repair

Soft Matter, 2008, 4, 400–418 | 409

Fig. 13 Results of fracturing the A–B thermoset interfaces listed in

Table 3.90

Here we examined a compact tension (CT) fracture mechanics

specimen made with bottom part A and top part B (see inset in

Table 3) with the A–B interface along the crack plane. The 10

A–B interfaces considered were as follows:

1. Liquid–liquid thermoset (L–L). This is the virgin state for

reference after curing the vinyl ester (Dow Derakane 411-C50)

by free-radical polymerization using an organic peroxide initi-

ator (USP 245). For reference, the virgin L–L thermoset had

a fracture energy G1c z 500 J M�2.

2. Liquid–solid virgin interface (L–Sv). The liquid was cured

on a virgin solid made from the same liquid. This is important

in composite repair and recycling processes. This interface

surprisingly only achieved values of G1c z 15–200 J M�2, which

is substantially less than the virgin state. Many different forms of

cure were attempted but the result was always the same, a pre-

cured solid sample does not bond with 100% strength to its

own liquid, which could be a major problem in the design of

autonomic self-healing materials. Perhaps a chain-transfer agent

would have helped to promote additional connectivity between

the polymerizing liquid and the solid cross-linked material.

3. Solid virgin–solid virgin interface (Sv–Sv). This was a welding

experiment with the two solid virgin surfaces. This interface

showed some strength development when welded above Tg and

G1c ¼ 66–107 J m�2, The best results were obtained with the

lowest cross-link density. This interface clearly can benefit

from the release of self-healing fluids.

4. Solid virgin–solid fracture interface (Sv–SF). The fractured

surface was welded with a virgin surface. No strength was devel-

oped after welding above Tg, as might be expected from the very

difficult wetting stage involving the rough fracture surface and

the smooth virgin cured surface.

5. Solid fracture–solid fracture interface (SF–SF). This is

a crack healing experiment and little strength was developed at

healing temperatures above Tg with G1c ¼ 8 J m2�. Below Tg,

healing fluids would have to be released into the interface.

6. Liquid–solid fracture interface (L–SF). The fractured solid is

cured in contact with its own liquid. This is important for auto-

nomic healing processes where the healing liquid in the hollow

fibers or UF spheres is similar to the matrix. This interface

behaved like the L–Sv interface and only a small fraction of

the strength was recovered, again raising a warning signal for

autonomic self-healing design in highly cross-linked thermosets.

410 | Soft Matter, 2008, 4, 400–418

Again, it should be possible to improve on this healing mecha-

nism with free-radical transfer agents.

7. Liquid–solid fracture, chemically treated interface (L–SFC).

This interface involves first treating the fracture surface with

a chemical treatment, such as styrene monomer which would

polymerize along with the liquid. This is important in repair

and recycling. The in situ polymerization of styrene gave the

best results with G1c ¼ 320 J m�2, which is still considerably

less than the virgin state (500 J m�2). As shown in Fig. 13, with

increasing molecular weight of the PS samples, the repair

strength decreased.

8. Solid virgin with solid fractured chemically treated interface

(Sv–SFC). This interface is relevant to welding and repair. Super-

glue gave a repair strength of 107 J m�2, which is 22% of the

virgin state.

9. Solid fracture–solid fracture chemically treated interface

(SF–SFC). This interface is important in wear and repair. Typical

values with adhesives (Superglue) gave strengths z 100 J m�2.

This interface resembles that used in the autonomic healing

experiments of White and Sottos where the healing fluids are

contained in the UF spheres and released to the interface when

damage occurs.

10. Solid fracture chemically treated–solid fracture chemically

treated interface (SFC–SFC). The results were similar with the

chemical treatments giving some of the best results with fracture

energies near 300 J m�2. One of the chemical treatments consisted

of using 2 wt% solutions of polystyrene in toluene. The PS

molecular weights were 220, 596 and 4340 kDa. The idea was

that connectivity between the solid interfaces could be achieved

by diffusion of the PS chains into the cross-linked network,

which would have been swollen by the toluene solvent. This

result is encouraging for the design of solvent-healing systems.

The results of this work are summarized in Fig. 13.

3.5 Healing of polymer–polymer interfaces

Self-healing of thermoplastic polymer–polymer interfaces occurs

during ballistic impact, solvent bonding and sub-Tg surface

welding. For healing of symmetric A–A interfaces, the gradient

percolation width X is the average monomer interdiffusion

distance (see Fig. 10), the percolation parameter p ¼ SL/X,

where S is the areal density of minor chains of interdiffused

contour length L, and the percolation threshold pc z Lc/M z 0.

In the general healing case where the chain ends are randomly

distributed in space near the interface, then S/X �1/M (see Table

1) L � (t/M)1/2, then the fracture energy increases as G z L,

G(t)/GN ¼ [t/t*]1/2 (25)

where the weld time t* � M, when M > M*, and t � M3, when

M < M*. Typically, M* z 8Mc.

When the chain ends are segregated to the surface, which

could occur due to fracture or the presence of low surface energy

chain ends, S is constant with time since all the chains begin

diffusing at the same time and we obtain:

G(t)/GN ¼ [t/t*]1/4. (26)

where again, t* � M. This allows healing to occur faster

compared to eqn (25). In ballistic self-repair experiments, the


time t* is temperature dependent and the needed time for healing

competes with the cool-down process.

When using healing fluids with monomers that are different

than the matrix monomers, one can encounter an asymmetric,

potentially incompatible interface. For asymmetric incompatible

A–B interfaces of width d � c�1/2, where c is the Flory–Huggins

interaction parameter, we have again the percolation parameter

for the diffuse interface as p ¼ SL/X. In this case, X � d, L � d2,

and S is constant such that p � d. Since the fracture energy G �[d � dc], where dc is the tube diameter, then the fracture energy

depends on the normalized width w ¼ d/dc as,17

G(t)/GN ¼ [w(t) � 1]/[wN � 1] (27)

The latter relation is supported by experimental data on a wide

range of A–B interfaces, which were analyzed by Benkoski

et al.39 and Cole et al.40

Incompatible A–B interfaces are typically quite weak

compared to welded homopolymers. To make such interfaces

stronger, they can be reinforced with A–B co-polymer compa-

tiblizer chains, as demonstrated by Brown and co-workers41–48

Thus, one could incorporate compatibilizers into the healing

fluid, which when released would have sufficient mobility, e.g.

in a solvent, to heal the interface. For incompatible A–B inter-

faces reinforced by an areal density S of A–B compatibilizer

chains, an equilibrium diffuse interface is formed in which L

and X are constant, such that the percolation parameter p � S

and pc � Sc. Thus, the fracture energy as a function of areal

chain density becomes:17

G(t)/GN ¼ [S(t) � Sc]/[SN � Sc] (28)

The above connectivity relations for polymer interfaces are in

accord with much data obtained by several groups39–48 and are

reviewed in ref. 1 and 17. They provide the keys to understanding

damage evolution in fracture and its converse, healing.

3.6 Fatigue healing

For healing of thermoplastic interfaces the total interpenetration

of chains (X approaches Rg) is not necessary to achieve complete

strength when M > M* and t* < Tr. It is only necessary to diffuse

a distance equivalent to the radius of gyration of M*. However,

a word of caution: while complete strength may be obtained in

terms of critical fracture measures such as G1c and K1c , the dura-

bility, measured in sub-critical fracture terms, such as the fatigue

crack propagation rate da/dN, may be very far from its fully

healed state at t*. We have shown that while the weld toughness

K1c increases linearly with interdiffusion depth X, as K1c � X, the

fatigue crack propagation behavior of partially healed welds

behaves as:21

da/dN � X�5 (29)

which is a very strong function of interdiffusion and underscores

the penalty to pay for partial welding. Thus, the weld strength

may be near, or at the virgin strength, but the fatigue strength

may be dramatically reduced below its maximum value. Thus,

one should always design a healing time with respect to Tr to

achieve maximum durability of welds and interfaces. This is


a rather subtle processing point, which is often not appreciated

by the manufacturing industry and is important for self-healing

design, namely that fatigue and strength are related but not

similar in terms of healing parameters.

In fatigue of materials in general with applied stress sapp < sc,

the percolation theory suggests a new approach as follows: the

lifetime t occurs when the initial fraction of bonds pi is reduced

to pf, such that fracture occurs at the applied stress in accord

with eqn (18) as:

sapp ¼ {2ED0v[pf � pc]}1/2 (30)

This gives the critical bond fraction pf for the applied stress as

pf ¼ pc + sapp2/{2ED0v] (31)

The time dependence of p can be deduced from a steady-state

bond-fracture concept via

Pf ¼ pi � t(dp/dt) (32)

such that the failure time t is the time required to reduce pi to pfat the prevailing breakage rate dp/dt. The rate of bond rupture

dp/dt can be given by a thermally activated state theory

(reviewed in ref. 119) as:

dp/dt ¼ (1/t0)exp(�Do[1 � sapp/sc]/kT) (33)

in which the energy for bond fracture D0 is linearly reduced by

the applied stress. Substituting for dp/dt and pf and solving for

t, we obtain the lifetime of the material as:

t ¼ t0{[pi � pc] � sapp2/(2ED0v)}exp(D0[1 � sapp/sc]/kT) (34)

where D0/kT z 133 at room temperature and t0 z 10�12 s is the

vibrational period for the bonds being broken. For a material

without defects, pi ¼ 1 and sc is determined from eqn (18) using

pf¼ 1. Note that the applied stress enters in both the front factor

and exponential factor for t. When the applied stresses are small,

the exponential factors dominate and the ratio of two lifetimes at

applied stresses s1 and s2 would be:

t1/t2 ¼ exp[(s2 � s1)/sc]D0/kT (35)

When the damaged bond fraction is restored by self-repair

processes such that pi/ 1 in eqn (34), the material is rejuvenated

and its lifetime is considerably extended. Thus, in Fig. 7, we see

that the healing fluids allowed the initial pre-crack to be

completely healed (pi / 1) and that during fatigue, the self-

healing fluids retarded the crack advance compared to the

control as the rate of bond rupture dp/dt was reduced.

4 The hard-to-soft matter transition

4.1 Twinkling fractal theory of Tg

A topic that is most relevant to this discussion on self-healing is

the fundamental understanding of the hard-to-soft and soft-to-

hard matter transition, typically called the glass transition. The

glass transition temperature Tg is arrived at by heating a hard

cold glassy material to a suitable temperature Tg whereby it

Soft Matter, 2008, 4, 400–418 | 411

becomes soft; or by providing sufficient mechanical energy to

make the material flow by yielding; or by making the sample

dimensions sufficiently small, typically nanoscale where the Tg

drops considerably; or by examining the top layer of a solid

sample at T < Tg, where it appears to have a mobile layer; or

by increasing the cross-link density; or by changing molecular

weight or by removing a solvent, etc. We could ask whether it

is possible to get thermally induced self-healing below Tg, or

how does Tg affect the design of our chemically cross-linking

system? Can we change the nature of the material by the rate

at which we explore it, such as in liquid armor114 where a very

soft cloth-like material turns to apparent steel at high rates of

impact. Conversely, how do we utilize impact energy to change

a hard material to a soft material, which can self-heal, as in

the ballistic healing experiments?

An understanding of the glass transition remains as one of the

unsolved problems in the physics community, as discussed by

Angell,121 and is essential for soft matter. The classical onset of

the glass transition temperature Tg is shown in Fig. 14 where

the slope of volume V vs. T changes at Tg. The slopes are the

thermal expansion coefficients ag and aL for the glass and liquid,

respectively. We consider the anharmonic interaction potential

U(x) of quasiharmonic diatomic oscillators (Fig 15) in the liquid

state as a function of temperature. The anharmonicity controls

the vibrational frequencies u, with temperature T, pressure P

and stress s,122–124 and the extent of the anharmonicity controls

the magnitude of the thermal expansion coefficient a (Fig 15).

The vibrational frequencies, degree of freedom (d ¼ 1, 2 or 3)

and their density of states G(u) determines the thermal proper-

ties such as the specific heat capacity Cp and thermal conduc-

tivity K. As Tg is approached from above, the average bond

Fig. 14 Volume–temperature curve for a glass former. Tg occurs at Ps ¼Pc when the fractal structure form.

Fig. 15 Diatomic anharmonic Morse oscillator U(x) ¼ D0[1 � e�ax]2

with Boltzmann energy levels and populations 4(x) � exp[�U(x)/kT]

shows the liquid (X > Xc) and solid (X < Xc) phase diagram.

412 | Soft Matter, 2008, 4, 400–418

distance between the oscillators contracts, the volume V

decreases and the bond expansion factor x approaches its critical

value xc, such that when x > xc, the molecules are in the ergodic

liquid state and when x < xc, the molecules on average exist

in the non-ergodic solid state. The latter idea parallels the

Lindemann (1910) theory of melting, which states that when the

vibrational amplitudes exceed a critical value of the bond length

(x z 0.11), melting occurs. In our case we have a Boltzmann

distribution of oscillators at various quantized energy levels in

the anharmonic potential energy function of the atoms such

that at any temperature T, there exists a fraction Ps of solid

atoms (x< xc) and a fraction PL of liquid atoms.

The solid fraction Ps at any temperature T is determined from

the integral (0 to xc) of the Boltzmann energy populations 4(x):

PsðTÞ ¼ðxc

0

fðxÞ dx (36)

in which 4(x) � exp[�U(x)/kT]. As Tg is approached from

above, as shown for example in Fig. 16, the fraction of solid

atoms Ps grows with a cluster size distribution function deter-

mined by percolation theory and eventually percolates at the

critical percolation threshold Pc.

The rigid (black) percolation cluster has a fractal dimension Df

and is not stationary but is quite dynamic since locally the solid

and liquid atoms are in dynamic equilibrium. Thus at Tg, to the

observer, there appears to exist a fractal structure which ‘‘twin-

kles’’ with a frequency spectrum F(u) as liquid and solid atoms

exchange. This ‘‘twinkling’’ fractal structure is invisible to the

usual scattering experiments since the liquid and solid are essen-

tially indistinguishable to such scattering experiments.

The existence of the twinkling fractal near Tg has a profound

effect on many properties, such as the mechanical loss peak. It

suggests that the onset of Tg is purely kinetic rather than thermo-

dynamic such that at very slow rates, the fractal appears liquid

Fig. 16 The twinkling fractal percolation cluster at Tg. The liquid

(white) regions are in dynamic equilibrium with the solid (black) clusters.

The twinkling fractal frequency spectrum mirrors the vibrational density

of states F(u) � uDf�1exp(�[|U(T) � Uc|])/kT such that at temperatures

much greater or lower than Tg, the twinkling rate slows down in the

pure liquid or very cold solid. Since the fractal is embedded in its liquid,

it is invisible to the usual scattering experiments.


Fig. 17 Finite size percolation effects are shown by the dark clusters for

one side of a thin film where p < pc. This also describes the surface of

a thick film at T < Tg.126–129

and at very high rates approaching u0, the fractal appears quite

rigid. This effect dominates the physics of the rate dependence of

yield in amorphous thermoplastic polymers and the physics of

liquid armor. The change in heat capacity DCp which appears

as a pseudo second order phase transition near Tg should be

determined predictably by the changes in the degree of freedom

of the oscillators from d ¼3 to d ¼ Df as well as the density of

states G(u)� udf, where df is the fracton dimension. The Tg value

of thin films will be affected by finite size percolation effects. The

existence of the twinkling fractal will also allow welding of glassy

polymers below their Tg via gradient percolation effects, where

the fractal danceswhile twinkling.

When the temperature drops below Tg, then the fractal has its

greatest manifestation on the evolution of the glassy structure.

With decreasing temperature, the low frequency components of

the twinkling fractal spectrum slow down or become negligible

and the fractal becomes quite rigid such that the normal volu-

metric contraction experienced in the liquid state deviates from

the extrapolated liquidus line VN and a non equilibrium

fractal-cavitation process commences resulting in the usual DV

noted between Vg and VN(Fig. 14) Thus, the thermal expansion

coefficient in the glass ag is less than in the liquid state aLapproximately as ag z pcaL. For the Morse oscillator of equilib-

rium interatomic distance R0, anharmonicity factor a, bond

energy D0, the critical distance xc z 1/3a and Tg is determined

from eqn (36), at pc ¼ ps, by:

Tg z (1 � pc)4D0/9k z 2D0/9k (37)

The (linear) thermal expansion coefficient in the liquid is

obtained from the average position of the bond length using

the integral of the Boltzmann populations (see Fig. 15) and is

obtained in the quasi-anharmonic approximation as:

aL ¼ 3k/[4D0R0a] (38)

which is related to Tg via eqn (37). For example, if D0 ¼ 3.5 kcal

mol�1, eqn (37) gives Tg¼ 391 K (118 �C); using R0¼ 3A and a ¼2/A, then eqn (38) predicts that aL z 70 ppm K�1 and ag z 35

ppm K�1, which are typical values for engineering thermoplas-

tics. It is interesting to note that the relation aLTg ¼ 1/(6aR0)

z 0.03, which was found to be true for many polymers and

composite resins. Since the modulus E � D0, one also expects

that E � 1/aL.

Physical aging occurs below Tg through the relaxation of DV

via the twinkling frequencies and is complex. Even well below

Tg, there is a non-zero predictable fraction of liquid atoms

remaining, which will cause the twinkling process to continue

at an ever-slowing pace but allow the non-equilibrium structure

to eventually approach a new equilibrium value near VN. A

near-equilibrium glass can be made by removing the fractal

constraints in 3d at Tg by forming the material using 2d vapor

deposition, as recently observed by Ediger et al.125 They used

vapor deposition of indometracin to make films, which were

reported to have maximum density and possess exceptional

kinetic and thermodynamic stability, consistent with the liquid

structure extrapolated into the glass. We have suggested that in

this case with DV z 0, a critical observation would be that

ag ¼ aL. The latter experiment is currently being done (private

communication with Mark Ediger).


4.2 Healing below the glass transition temperature

Healing of polymer–polymer interfaces below Tg, as demon-

strated by Boiko and co-workers67,68 can occur due to softening

of the surface layer as shown in Fig. 17. We have treated the

surface layer softening as a gradient rigidity percolation issue.17

The surface rubbery layer concept in thick films is interesting

and this percolation theory suggests that for free surfaces there

is a gradient of p(x) near the surface, where x < x (cluster size

correlation length) and hence a gradient in both Tg and modulus

E. If the gradient of p is given by p(x) ¼ (1 � x/x) 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 the percolation theory as5,17

xc ¼ b(1 � pc)/{pcv[1 � T/Tg]

v} (39)

Here b is the bond length and n is the critical exponent for the

cluster correlation length x � [p � pc]�n. For example with poly-

styrene, when T ¼ Tg �10 K, Tg ¼ 373 K and using b ¼ 0.154

nm, pc ¼ 0.4, n ¼ 0.82, then the thickness of the surface mobile

layer xc ¼ 3.8 nm. This could allow for healing to occur below

Tg assuming that the dynamics are fast enough.

If G1c � X2 for entangled polymers, then we could deduce that

for sub Tg healing at DT ¼ Tg � T, as

G1c � [1/DT]2n (40)

This appears to be in qualitative agreement with Boiko and

co-workers’ data68 who examined the fracture energy of polysty-

rene interfaces during welding at temperatures up to 40 K below

Tg. The TFT also suggests the presence of some interesting

dynamics in the mobile layer since the fractal clusters are essen-

tially ‘‘dancing’’ with a spectrum of frequencies related to the

density of states G(u) � udf, where the Orbach fracton dimension

df ¼ 1.33.

4.2 Twinkling fractal theory of yield stress

The following analysis is important for the design of self-repair

systems subject to ballistic impact and for the basic under-

standing of rate effects on yield stress of solids and shear thick-

ening fluids. For an amorphous solid below Tg, subjected to

triaxial stresses s1, s2, s3, the distortional strain energy function

Ud is determined by (Von Mises) as:

Ud ¼ (1 + n)[(s1 � s2)2 + (s2 � s3)

2 + (s2 � s3)2]/6E (41)

Soft Matter, 2008, 4, 400–418 | 413

in which n is the Poisson ratio and E is the isotropic elastic

Young’s modulus. In the simple uniaxial case, we obtain the

stored strain energy from eqn (41) with n ¼ 0.5 as U ¼ s2/2E.

This stored energy must be utilized to overcome a percolation

number of interatomic anharmonic oscillator bonds of energy

U(xc). For a Morse oscillator U(xc) ¼ 0.08 D0, which is the

energy necessary to reach the Lindemann bond expansion at

xc. The number of such oscillators per unit volume is 1/Vm,

where Vm is the molar volume. At the critical stored energy,

we have yielding when the amorphous solid is raised to the twin-

kling fractal state by the release of the mechanical energy and

flow begins in accord with:

sy2/2E $ 0.08 D0 [ps � pc]/Vm (42)

in which the solid fraction ps is given by eqn (36) and is both

temperature and rate dependent. Thus, the yield stress is

obtained as:123,124

sy ¼ {0.16E[ps � pc]D0/Vm}1/2 (43)

Note that the term D0/Vm corresponds to the traditional cohesive

energy density. For example, if E ¼ 1 GPa, p ¼ 1 (high rate of

deformation), pc ¼ 1/2, D0 ¼ 3 kcal mol�1, Vm ¼ 2M0/r, r ¼ 1

g cc�1, the monomer molecular weight M0 ¼ 100 g mol�1, then

eqn (43) predicts that sY ¼ 71 MPa. This value is typical for

high performance amorphous polymers with Tg z 63 �C.

The magnitude of ps is also rate and temperature dependent in

a manner determined by the twinkling fractal density of states.

As T approaches Tg from below, ps decreases towards pc and sydecreases accordingly. At high rates of deformation, p increases

and sy increases. At low rates of deformation, p decreases

towards pc and this is the basis for designing liquid armor with

shear-thickening fluids, which are liquid layers with particles at

p > pc that turn into solids at high rates. Thus, ps(g) � [g/u0]df

such that we can express eqn (43) as

sY/sN ¼ {[(g/u0)df � pc]/[1 � pc]}

1/2 (44)

in which sN is the yield stress at g ¼ u0. There exists a critical

deformation rate given by

gc ¼ u0pc (45)

such that when g < gc, then the material is liquid-like and when g

> gc, then the yield stress increases from zero to its maximum

value at u0. Thus, the yield stress increases with rate of testing

and the higher the rate, the more stored energy is required, which

facilitates ballistic healing at high rates.

The TFT concept of Tg suggests that twinkling fractal cluster

at Tg is quite soft, especially when sensed at low rates of

deformation. One can immediately understand why the loss

tangent tand ¼ E’’/E’(loss/storage modulus) reaches its

maximum value near Tg. The twinkling fractal frequencies are

given by

F(u) � udfexp�[|U(T) �Uc|]/kT (46)

Here the first term udf is the vibrational density of states for

a cluster of frequency u and the second exponential term is the

414 | Soft Matter, 2008, 4, 400–418

probability that the vibration will cause a ‘‘twinkle’’ or change

from solid to liquid or visa versa. Note that this energy difference

is always positive and is behaves as |U(T) � Uc| � [T2 � Tc2]. In

a single mechanical cycle, the stored energy is released or dissi-

pated by the twinkling process as liquid and solid clusters of

frequency u exchange at the fastest rates in accord with the

density of states G(u) � udf. As the temperature drops below

Tg, the percolating cluster increases in mass and the stored

energy increases. However, the twinkling frequencies decrease

their rate due to the increased energy barrier DE ¼ |U(T) � Uc|

for the solid-to-liquid transition and the energy dissipation

decreases. Near Tg, we can approximate the temperature depen-

dence of U(x) for a Morse oscillator as U(T) z a2D0a2T2, such

that the temperature dependence of F(u) is:

F(u,T) ¼ udfexp�[b(T2 � Tg2)/kT] (47)

in which the constant b ¼ a2D0a2. Thus the activation energy DE

is temperature dependent approximately as DE � [(T/Tg)2 � 1]

and changes rapidly with T. When T > Tg, the rigid clusters

decrease, the stored energy decreases and the rate of liquid-to-

solid transitions decreases. Thus, tand will reach a maximum

value near Tg typically. The twinkling fractal, though invisible

to scattering experiments could be the ‘‘dynamics engine’’ of

the amorphous state and plays an important role in complex

processes such as physical aging and self-healing.

5. Present and future advances in self-healingmaterials

A new self-healing system that mimics a vascular healing system

in skin was developed by the Sottos–White group at the Univer-

sity of Illinois, using a micro-fluidics variant on their earlier

liquid sphere system as shown in Fig. 18.84 The advantage of

this system over their earlier self-healing models, is that the

embedded vascular system would allow multiple healing events

to occur. The encapsulated spheres were limited to one healing

event per crack. In the above case, cracks initiated in bending

cause damage to the top coating (skin) which propagate and

arrest at the coating–substrate interface (Fig 18c). The healing

fluid in the micro fluidic channels then wicks up the crack to

the surface through capillary action. The fluid makes contact

with the epoxy skin, which contains the polymerizing agent

and the liquid-to-solid polymerization reaction is initiated.

They found that the healing fluid dicyclopentadiene (DCPD)

worked well with a Grubb’s catalyst, benzylidene bis(tricyclo-

hexylphosphine)dichloro ruthenium. This system has sufficiently

low viscosity to allow facile flow to the skin surface and the cata-

lyst remains active both during and after the reaction. The

authors successfully demonstrated that they could reheal the

skin layer seven different times.

In related experiments, White and Sottos et al. (private

communication) have found that in epoxy matrices, the DCPD

healing fluids along with the Grubb’s catalyst could be replaced

with a simple non-reactive solvent that was compatible with the

epoxy. This is essentially a solvent healing system using the

autonomic or microfluidic delivery system. It is remarkable

that high levels of healing can be obtained in highly cross-linked

polymers using just solvents alone. The new solvent system will


Fig. 18 (a). Schematic diagram of a capillary network in the dermis layer of skin with a cut in the epidermis layer. (b) Schematic diagram of the

self-healing structure composed of a micro vascular substrate and a brittle epoxy coating containing an embedded catalyst. The sample is in a 4-point

bending fracture configuration monitored with an acoustic emission sensor. (c) High magnification cross-sectional image of the coating showing that

cracks which initiate at the surface propagate towards the micro channel opening at the interface (scale bar represents 0.5 mm). (d) Optical image of

self-healing structure after cracks are formed in the coating (with 2.5 wt% catalyst) revealing the presence of excess healing fluid on the coating surface

(scale bar represents 5 mm) (courtesy of S. White).

considerably lower the cost of autonomic self-healing systems

that utilize both the urea formaldehyde spheres of healing fluid

and the biomimetic design discussed above.

Several other significant contributions to the design of self-

healing materials have been made. Yamaguchi et al.86 utilized

strong topological interactions of dangling chains in a polyure-

thane network polymer, which was obtained by eliminating the

sol fraction of the weak gel just beyond the transition point.

When cleaved with a razor blade at room temperature, complete

healing was observed. This was attributed to the presence of the

highly mobile long dangling chains near the gel threshold.

O’Conner20 had also observed healing of weakly cross-linked

polyurethanes which were used as solid rocket propellant

binders. When cut with a razor, the cracks would slowly heal,

as measured by fracture mechanics. In Mazayuki’s material,

the healing was very rapid by comparison, as if the weak gel

behaves as a fluid.

Yin et al.87 prepared a system similar to that of the Sottos–

White group. They used a two-component healing system

whereby UF capsules (30–70 mm diameter) filled with an epoxy

were distributed in a matrix that contained a latent hardener.

The latent hardener consisted of a complex of CuBr2 with

2-methylimidazole, which was quite soluble and well dispersed

in the epoxy composite matrix. When subjected to mechanical

stress, microcracks would form and break the UF spheres,

releasing the epoxy, which would then cure by contact with the

embedded hardener. They observed that the self-healing epoxy

with 10 wt% spheres and 2 wt% hardener achieved 111% of its


original fracture toughness, and a composite with woven glass

fibers achieved a healing efficiency of 68%.

A thermally re-mendable cross-linked polymer was developed

by Chen et al.88 A highly cross-linked polymer was made using

furan groups and maleimide, which would undergo a reversible

Diels–Alder reaction. The tough material when damaged was

heated above 120 �C and the molecular linkages would discon-

nect and then reconnect to reconstitute the network upon cool-

ing. The process is completely reversible and was used to

restore the strength of fracture samples several times. Perhaps

this type of chemistry could also be used to develop recyclable

thermosets, in addition to re-mendable materials.

Zhu and Wool explored the role of nanoclay cross-linked elas-

tomers on their self-healing ability as shown schematically in

Fig. 19.6,7 Linear polymer chains made from acrylated oleic fatty

acids methyl ester (AOME) were cross-linked through intercala-

tion with nanoclay (Cloisite 30B). The mechanical properties

were shown to depend on the nanoclay concentration 4 as

follows: Modulus E � 4, cross-link density v � 4, fracture stress

s� 4. The mechanical properties are interrelated through rubber

elasticity theory E � v and percolation theory for fracture stress

as s � [Ev]1/2. When subjected to mechanical loading, the elasto-

meric network could absorb mechanical energy by opening the

clay nanobeams, which would reduce the cross-link density.

Upon resting it was proposed that the nanoclay beams would

reheal and reconstitute the original mechanical properties of

the elastomer. This elastomer, which is both biodegradable as

well as biocompatible with human tissue, can be optimized

Soft Matter, 2008, 4, 400–418 | 415

Fig. 19 Schematic of self-healing mechanism in Nanoclay-filled elas-

tomer composites [Zhu and Wool].

in the future to provide toughening as well as self-healing

properties.

Future directions in the field of self-healing will most likely

reflect on the biomimetic materials, particularly those with

complex molecular aggregates derived from amino acid

sequences. These supramolecular protein-like structures interact-

ing with each other in large ensembles will have the ability to

react to external stress in a way to minimize the trauma and be

able to self-assemble back to the original structure. As in the

case of self-healing nano beams, it should be possible to work

with new materials composed of folded proteins for example,

which would partially unfold as a mechanism of absorbing

mechanical energy and then re-fold and self-heal. Such materials

would be both very tough and self-healing. Steps in this direction

are being explored by K. Kiick and D. Pochan at the University

of Delaware.130–132 Kiick is examining polypeptide-based macro-

molecules that can mediate the formation of inorganic materials,

some of which could even potentially heal composite fibers in

addition to the matrix. Pochan is examining responsive materials

constructed via peptide folding and consequent self-assembly.

Many advanced materials systems are designed to be either

tough or self-healing but typically not both. For example, rubber

particles toughen glassy plastics by promoting energy dissipation

in the form of local plastic deformation near the rubber particles

throughout the body of the material. However, the damage is

permanent and a new system which would self-heal would be

a significant advance in this field. Perhaps a simple mixture of

rubber toughening and self-healing solvent particles would

suffice?

Judging by the recent interest and advances in the field of self-

healing materials,133–141 it is safe to assume that the quest for

materials with eternal life mirrors that of its researchers.

Acknowledgements

The author is grateful to several funding agencies for supporting

this research over the years, including CCR-ARL, USDA-

416 | Soft Matter, 2008, 4, 400–418

CREE-NRI, NSF, EPA, DOE and ARO, and his many

colleagues and fellow researchers who provided the grist for

expansion of his imagination.

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Self-healing materials: a review

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