1 Introduction
In this paper, we explore a new approach to understanding the molecular aspects of adhesion at polymerpolymer interfaces. Percolation theory is used as a means to parameterize the making and breaking of connectivity at polymerpolymer interfaces [1,2]. The strength G_{1c}, of polymer interfaces has been investigated by many [1–52], and different theories have been proposed to relate interface structure to strength. For welding symmetric A/A interfaces as a function of diffusion depth X, it was suggested that G_{1c} ~ X [4–7], or the number of bridges per unit area of interface, P, G_{1c} ~ P [9–14], or the crossing density ρ, G_{1c} ~ ρ [8], or the contour length L, G_{1c} ~ L [16–20]. For incompatible A/B interfaces as a function of equilibrium width w, it has been suggested that G_{1c} ~ w^{2} [21,22], or G_{1c}~ w [22]. For incompatible A/B interfaces as a function of the number of entanglements N, in the interface, the relation G_{1c} ~ N^{2} was proposed [23]. For an incompatible A/B interface reinforced with Σ compatibilizer chains per unit area, G_{1c} ~ Σ^{2}, has been suggested [24–31]. Other approaches to analyzing the strength of interfaces have involved simulations [32–34], theoretical modeling [34–41], experimental correlation between toughness and interfacial width [42–49], nonisothermal modeling of composite interfaces [50] and subT_{g} welding [51,52]. Theories relating structure to strength were proposed or utilized for each specific interface, and while all had a measure of success in describing the strength of a targeted interface, essentially none were readily transferable to describe other interfaces, or could be readily extended to provide an acceptable theory of strength for the bulk polymer. Thus, for example, no one has a theory that can simultaneously model strength development during welding of symmetric A/A interfaces and which can be readily extended to understanding the reinforcement of incompatible A/B interfaces by Σ compatibilizer chains, while at the same time predicting the molecular weight dependence of the virgin state. Even within a single interface type, such as welding of symmetric A/A polymer–polymer interfaces, there is little agreement between investigators. The Wool–O'Connor model [16] appears to have the correct form theoretically and experimentally [1] such that the fracture energy G as a function of time t and molecular weight M, behaves as G ~ L ~ (t/M)^{1/2} However, we can criticize the relation G ~ L since it does not have the number of chains per unit area Σ involved. Consequently, if we have just one chain of length L compared to 10^{14}chains, the average L can be the same in both cases but the strength can be radically different. Thus, we are unable to describe the data of Brown et al. [24–31] for reinforcement of A/B incompatible interfaces with Σ A–B compatibilizer chains, or derive the Brown Law [25], G ~ Σ^{2}. Therefore, the early Wool theory of welding cannot be readily extended to other interfaces, even though it appears to correctly predict the molecular weight dependence of the virgin fully healed state [1]. The purpose of this paper is to provide a theory of fracture, which can be applied universally to all interfaces and can be readily extended to understanding the bulk strength of polymers in terms of known microscopic parameters and material constants.
The general approach to evaluating the fracture energy G_{1c} of polymer interfaces is represented in Fig. 1 [1]. Material A is brought into contact with material B to form an A/B interface, the sample is fractured and the strength is related to the structure of the interface through microscopic deformation mechanisms. In the virgin state, or when welding or crack healing, A = B. For the incompatible A/B interface, we consider both the nonreinforced interface, and the interface reinforced with an areal density Σ of compatibilizer chains. Typically, 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 JIntegral method, as described by Hutchinson et al. [53–55], where G_{1c} is the integral of the traction stresses σ(δ) with crack opening displacements δ, in the cohesive zone, following yielding at a local yield or craze stress σ_{Y}. The cohesive zone at the crack tip breaks down by a vector percolation process, as described herein, at a maximum stress value, σ_{m} > σ_{Y}. Typical ratios of σ_{m} /σ_{Y} are about 4–10 [53]. Both σ_{m} and δ are rate dependent and in the simplest case, the fracture energy is determined by:
$${G}_{\text{1c}}={\sigma}_{\text{m}}\text{}{\delta}_{\text{m}}$$  (1.1) 
2 Rigidity percolation theory of fracture
The transmission of forces through a lattice as a function of the fraction p, of bonds in the lattice has been analyzed by Kantor and Webman [56], Feng and Sen [57,58], Thorpe et al. [58,59] and others [1,60,61]. De Gennes first suggested that conductivity or scalar percolation could be used to quantize the modulus of elasticity E, of randomly connected networks, such as gels [62]. Analyses based on the Born and Huang model of the microscopic elasticity of a lattice [63] gave results for the elasticity which resembled conductivity percolation when shear terms were neglected in the Hamiltonian for the elastic energy, as:
$${E~[p\u2013p}_{\text{c}}{]}^{t}$$  (2.1) 
The vector or rigidity percolation process addresses several important points. First, consider a 2D lattice near the percolation threshold p_{c}, as shown in Fig. 2. Due to the random fractal connectivity of the lattice, the stress distribution ϕ(σ), in the bonds becomes highly nonuniform such that some bonds are highly stressed, while others bear little stress. The existence of highly stressed bonds is a prelude to molecular fracture and parallels the ‘hot bonds’ in conductivity percolation, where hot bonds arise from high current density in some individual bonds near the percolation threshold. The hot bonds overheat like electrical fuses in the high current density and break. The concept of mechanical ‘hot bonds’ is relevant to fracture of polymers in general and is the basis for understanding why materials fracture at macroscopic stresses, which are orders of magnitude less than the molecular fracture stresses. When polymers such as polypropylene and polyethylene are subjected to uniform tensile stresses, it has been shown using infrared and Raman spectroscopy that the molecular stress distribution can be quite broad, even though the applied stress is well below the macroscopic fracture stress [64,65]. The development of the molecular stress distribution ϕ(σ) is due to the inherent sloppiness of the lattice. Thus, in the Jintegral fracture mechanics model, the maximum fracture stress near the crack tip σ_{m}, described in Fig. 1 and Eq. (1), remains closer to the yield stress than to the much higher molecular fracture stress.
Another point of interest is that only a fraction [p – p_{c}] of the bonds needs to be fractured before complete failure occurs in a 2D or 3D network. Thus, in a deformation zone at a crack tip, the crack advances through the zone by breaking a fraction [p – p_{c}] of bonds or fibrils in parts of a craze network. The broken bonds do not lie on the same plane, as is in the Nail Solution [40], and is often assumed intuitively, but are distributed over the deformation zone volume. The deformation zone near fracture is best described as a volume of material preceding the crack tip that contains a considerable number of defects.
An important corollary to the existence of the threshold p_{c} is that when p < p_{c}, the lattice connectivity is broken and no significant strength exists beyond that of nonbonded potentials and Van der Waals interactions. Thus, the molecular lengths (L ~ M) must be long enough, the areal density of chains Σ, at the interface must be great enough and the number of entanglements in the lattice N, at an interdiffusion distance X, or interface width w, has to exceed the percolation threshold before strength develops. This means that an initial investment (p_{c}) is needed before strength develops, such that when G_{1c} ~ [p – p_{c}], there exists corresponding critical parameters such as M_{c}, L_{c}, Σ_{c}, X_{c}, N_{c}, w_{c}, etc., which are all related to each other through the percolation parameter p.
To convert these percolation concepts into quantitative fracture terms, consider the vector percolation experiment shown in Fig. 2, applied to any 3D lattice in general with tensile modulus E. The Hamiltonian for the stored elastic energy can be formulated using the Born and Huang [63], or the Kantor and Webman approach for specific lattices [56], or using the more simple engineering strain energy density approach as follows. The stored elastic strain energy density U in the lattice due to an applied stress σ is determined in the uniaxial approximation by:
$$U={\sigma}^{\text{2}}\text{/2}\text{}E$$  (2.2) 
The stored strain energy can also be determined for the general case of multiaxial stresses [1] and lattices of varying crystal structure and anisotropy. The stored strain energy dissipation per unit volume U_{f}, to fracture a network consisting of a bond (or entanglement) density of ν bonds per unit volume is:
$${U}_{\text{f}}=v\text{}{D}_{\text{o}}\text{}[p\u2013{p}_{\text{c}}]$$  (2.3) 
$$\sigma \text{*}=\{2\text{}E\text{}v\text{}{D}_{\text{o}}\text{}[p\u2013{p}_{\text{c}}]{\}}^{\text{1/2}}$$  (2.4) 
This equation predicts that the fracture stress increases with the square root of the bond density. The percolation parameter p, is in effect, the normalized bond density such that for a perfect net without defects, p = 1, and for a net that is damaged or contains missing bonds, then p < 1. Obviously, as p approaches p_{c}, the fracture stress decreases towards zero and we have a very fragile material. This fracture relation could therefore be used to evaluate durability, or retention strength of a material by tracking damage accumulation through a single parameter p. Note that the Net solution refers to the stress required to fracture a unit volume of the net in uniaxial tension.
When applied to interfaces, we let the volume of material or Net, contain the interface such that we can calculate σ* with a knowledge of p based on a local normalized entanglement density. In all applications of the RP model, the stressed state is the reference state to assess percolation and connectivity. This will become more apparent when we examine disentanglement for example, where an unraveling or disentanglement process in the stretched state breaks the connectivity.
3 Fracture by disentanglement
The bridge percolation model of entanglements proposed by Wool [1,66] was recently supported by computer simulations of Theodorou et al. [67,68] and is the basis for the disentanglement model described herein. M_{c} represents a segment of an entangled chain that is long enough to form a bridge or loop of 3crossings (3P) through a plane in the melt. An entangled net forms when the number of chain (Σ ~ M^{–1/2}) intersecting the plane equals the number of bridges. Thus, when Σ = 3 P, M_{c} = 9 (Σ P)^{2} M. Computer simulations of polyethylene melts by Uhlherr et al. [67] showed this model to be accurate. By sampling the amorphous melt, they found that the average mesh segment which intersected the plane three times was equivalent to the critical entanglement molecular weight M_{c}. Thus, the bridge with three crossings is the basic mesh element of the entanglement network capable of transmitting vectors and defines precisely the number of bonds per chain which must be broken or disentangled to obtain a critically connected structure.
Disentanglement is considered to proceed by the mechanism shown in Fig. 3, where we depict the response of an (average) entangled chain to a constant (step function) draw ratio λ [1,3]. Fracture by disentanglement occurs by first straining the chains to a critical draw ratio λ_{c} and storing mechanical energy of order G ~ (λ_{c} – 1)^{2}. The stretched chains then relax by Rouselike retraction and disentangle, when the energy released is sufficient to relax them to the critically connected state corresponding to the percolation threshold, p_{c}. When this occurs, a chain, which initially had many bridges (~ M^{1/2}), is reduced to a single critically connected bridge by the applied strain. The percolation parameters [p – p_{c}] associated with the disentanglement process at an interface are derived as follows; p is the normalized entanglement density defined as:
$$p=g(\lambda {)\text{}N}_{\text{v}}\text{/}v$$  (3.1) 
$$g(\lambda )=[M\text{/}{M}_{\text{e}}(\lambda )]\u2013\text{1}$$  (3.2) 
The chain ends effectively contribute to the loss of one entanglement. Since N_{v} = ρ/M and ν = ρ/M_{e}, then we have:
$$p=\frac{\{[M\text{/}{M}_{\text{e}}(\lambda )]\u20131]\}\rho \text{/}M}{\rho \text{/}{M}_{\text{e}}\left(\lambda \right)}$$  (3.3) 
$$p=[1\u2013{M}_{\text{e}}(\lambda )/M]$$  (3.4) 
$${M}_{\text{e}}(\lambda )={\lambda}^{\text{2}}\text{}{M}_{\text{e}}$$  (3.5) 
M_{e}(λ) increases between entanglement points due to the retraction process at constant λ. A more detailed treatment of disentanglement would account for the orientation function of the entanglements and lateral contraction, as discussed elsewhere [1]. Eq. (3.4) becomes:
$$p=1\u2013{\lambda}^{\text{2}}\text{}{M}_{\text{e}}\text{/}M$$  (3.6) 
An important consequence of the latter equation is that when λ = 1, there exists a critical value of molecular weight M = M_{c} for which p = p_{c} and we obtain the relation between M_{e} and M_{c} as:
$${M}_{\text{c}}=\frac{{M}_{\text{e}}}{\text{1}\u2013{p}_{\text{c}}}$$  (3.7) 
Since p_{c} ≈ 1/2, we note that M_{c} ≈ 2 M_{e}, as commonly observed. M_{e} is determined from the onset of the rubbery plateau by dynamic mechanical spectroscopy and M_{c} is determined at the onset of the highly entangled zeroshear viscosity law, η ~ M^{3.4}. This provides a new interpretation of the critical entanglement molecular weight M_{c}, as the molecular weight at which entanglement percolation occurs with the onset of longrange connectivity. Concomitantly, the dynamics changes from single chain, Rouselike behavior, to that of chains significantly impeded by others, as in Reptation. It also represents the transition from the Nail (weak fracture) to the Net (strong fracture) solution and the onset of significant strength development via the formation of stable, strong, oriented fibrillar material in the deformation zones preceding the crack advance.
When M > M_{c}, we obtain the critical draw ratio for fracture λ_{c} from Eq. (3.6) as:
$${\lambda}_{\text{c}}\approx {({M\text{/}M}_{\text{c}})}^{\text{1/2}}$$  (3.8) 
The maximum molecular weight M* at which disentanglement can occur is determined when strain hardening occurs at λ_{c} ≈ 4 such that^{1}:
$$M\text{*}\approx {\text{8}M}_{\text{c}}$$  (3.9) 
Donald and Kramer [69] also found that the draw ratio of crazes in several polymers was of order λ ≈ 4 and varied in a range of 2–5. In 1981, A. Donald explained to R.P. Wool (private communication) the significance of straightening the slack between entanglements, which is key to understanding the disentanglement process described herein. At M = M* = 8 M_{c}, G* ~ 0.42 M* such that Eq. (3.8) gives the molecular weight dependence of the virginstate fracture energy as [1]:
$${G}_{\text{1c}}\text{/}G\text{*}={\text{0.3}M\text{/}{M}_{\text{c}}[1\u2013({M}_{\text{c}}\text{/}M)}^{\text{1/2}}{]}^{\text{2}}$$  (3.10) 
We have shown that the latter equation gives an excellent fit to the molecular weight dependence of fracture [3]. At high rates of strain compared to 1/τ, the inverse disentanglement time, or when disentanglement cannot readily occur (M > M*), bond rupture occurs randomly in the network and the percolation parameter p becomes dominated by chain ends. In this case, the entanglement molecular weight M_{e} does not depend on strain and Eq. (3.4) gives:
$$p=\text{1}\u2013{M}_{\text{e}}\text{/}M$$  (3.11) 
Since G_{1c} ~ [p – p_{c}], and p_{c} = 1 – M_{e}/M_{c}, we obtain:
$${G}_{\text{1c}}=G\text{*}[1\u2013{M}_{\text{c}}\text{/}M]$$  (3.12) 
In addition to glassy polymers, the RPfracture model also describes the fracture of soft lightly crosslinked rubber materials, as described in detail elsewhere [3]. Rubber is an interesting case since the modulus E and crosslink density ν in Eq. (2.4) are related via E ~ ν, such that the fracture stress σ ~ E, σ ~ ν and the fracture energy at low deformation rates behaves as G_{1c} ~ ν. It is also found that a perfect rubber network (p = 1) cannot break without first undergoing significant strain hardening at λ ≈ 4, which is the common experience [3].
3.1 Comment on percolation and polymer rheology
Entanglement percolation effects will have a significant effect on the manifestation of rheological functions associated with the dynamics of disentanglement via Reptation processes, such as the zero shear viscosity η_{o} and creep compliance J_{e}^{o}. For example, the zero shear viscosity is well approximated by:
$${\eta}_{\text{o}}={{G}_{\text{N}}}^{\text{o}}\tau $$  (3.13) 
$$p(t)=1\u2013\text{4/}{\pi}^{\text{3/2}}\text{}[{t\text{/}T}_{\text{r}}{]}^{\text{1/2}}\u2013{M}_{\text{e}}\text{/}M$$  (3.14) 
$${\tau}_{\text{RP}}{\text{/}T}_{\text{r}}={\text{1.94}[{M}_{\text{e}}\text{/}{M}_{\text{c}}\u2013{M}_{\text{e}}\text{/}M]}^{\text{2}}$$  (3.15) 
When M = M_{c}, τ_{RP} = 0 as expected, and when M = ∞, τ_{RP} ~ T_{r}. Thus, for typical experimental M values in the order of 5–10 M_{c}, the mechanical relaxation time τ_{RP} is less than the dynamics relaxation time T_{r} due to rigidity percolation of the entanglement network in the terminal zone. A fraction [1 – p_{c}] of the entanglements relaxes by the single chain Reptation process and a fraction p_{c} relaxes by a multichain intermolecular percolation process with a Rouselike character. This produces an apparent η_{o} ~ M^{3.4} law and the exponent of 3.4 is a consequence of the percolation process and has no particular scaling law relevance. In the absence of percolation in the terminal zone, since the dynamics relaxation time τ ~ R^{2}/D, one can also obtain a 3.4 power by requiring that the diffusion coefficient D ~ M^{–2.4} while the endtoend vector behaves as R^{2} ~ M. An analysis of diffusion data by Lodge et al. [71] suggested that D ~ M^{–2.3}. However, this will not produce mechanical relaxation times that are significantly less than the chain dynamics relaxation time as observed experimentally.
The creep compliance J_{e}^{o} as a function of molecular weigh is predicted by the Doi–Edwards theory [72] to be independent of molecular weight for M > M_{c}. However, it is found experimentally that J_{e}^{o} ~ M at values of M < M* and J_{e}^{o} ~ M^{o} when M > M*, where M* ≈ 5–8 times M_{c}. In addition to the 3.4 power for η_{o}, it can be readily shown^{2} that the percolation correction also predicts the correct creep behavior using the rheological functions:
$${\eta}_{\text{o}}={{G}_{\text{N}}}^{\text{o}}\underset{\text{o}}{\overset{\infty}{\int}}p(t)\text{d}t$$  (3.16) 
$${{J}_{\text{e}}}^{\text{o}}=\underset{\text{o}}{\overset{\infty}{\int}}t\text{}p(t)\text{d}t\text{/}{{G}_{\text{N}}}^{\text{o}}{[\underset{\text{o}}{\overset{\infty}{\int}}p(t)\text{d}t]}^{2}$$  (3.17) 
Entanglement percolation effects had not previously been considered by any investigator in addressing the above longstanding unresolved issues in polymer rheology.
4 Polymer–polymer welding
Fig. 4 shows an interface formed by random walk chains diffusing by reptation across a polymerpolymer weld line [34]. The molecular aspects of interdiffusion of linear entangled polymers (M > M_{c}) 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 polymer–polymer interfaces using Dynamic Secondary Ion Mass Spectroscopy (DSIMS) and Neutron Reflectivity [73–78]. These experiments involved interfaces consisting of the following polymer pairs; HDH/DHD, HDH/HPS, HDH/DPS, DHD/DPS, DHD/HPS and DPS/HPS, where HDH, DHD, DPS and HPS were centrally deuterated (25%) PS chains, End deuterated (25% each end) PS chains, fully deuterated and fully protonated (normal) polystyrene chains, respectively. The HDH/DHD ‘ripple’ experiments clearly showed that DeGennes' reptation dyanmics model was an excellent model to describe the interdiffusion process during welding, the HDH/DPS and DHD/HPS showed the distinct motion of the chain ends and centers and the HPS/DPS demonstrated the overall concentration profiles. The scaling laws and the complete concentration profiles were calculated by Kim et al. [19] and Zhang et al. [20]. The important result for the contour length L ~ (t/M)^{1/2}, (which is the basis for the early Wool theory) was also supported by welding computer simulations of Windle et al. [32] Initially, as the symmetric (A=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 3 nm. This can also occur below T_{g} when the top surface layer becomes more mobile than the bulk and can be explained by finite size rigidity percolation theory [79]. However, at this point, the interface is very weak and fracture can be described by the Nail solution [40]. At the wetting stage, the frictional pullout of intermeshed chain segments, which have ‘elbowed’ their way across the interface, determines the fracture energy (ca 1 J/m^{2}). As welding proceeds, Σ 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 chains, which are fully entangled (ignoring surface reflection configuration effects on entanglement density).
Molecular aspects of interdiffusion at a polymer–polymer interface
Molecular Aspect  Symbol  Dynamic relation, t < T_{r}  Static relation H t = T_{r}  r, s 
General property  H(t)  t^{r/4} M ^{–s/4}  M ^{(3r–s)/4}  r,s 
Average contour length  l(t)  t^{1/2} M ^{–1/2}  M  2,2 
Number of chains  Σ(t)  t^{1/4} M ^{–5/4}  M ^{–1/2}  1,5 
Number of bridges  P(t)  t^{1/2} M ^{–3/2}  M ^{0}  2,6 
Average monomer diffusion depth  X(t)  t^{1/4} M ^{–1/4}  M ^{1/2}  1,1 
Total number of monomers diffused  N(t)  t^{3/4} M ^{–7/4}  M ^{1/2}  3,7 
 X_{cm}  t^{1/2} M ^{−1}  M ^{1/2}  2,4 
 N_{f}  t^{1/2} M ^{–3/2}  M ^{0}  2,6 
The structure of the diffuse weld interface in Fig. 4 resembles a box of width X, with fractal edges containing a gradient of interdiffused chains, as shown by Wool and Long [34]. Using Sapoval's gradient percolation theory [80], we require that chains, which contribute to the interface strength, straddle the interface plane during welding, such that chains in the concentration gradient which have diffused further than their radius of gyration 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 [1,19,20]. However, for broad molecular weight distributions, the fraction of nonconnected chains expressed through gradient percolation, can be significant and impacts on the observed time dependence of welding [1 (p. 76)].
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 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.
The percolation term [p – p_{c}] determines the number of bonds to be broken, or disentangled such that when Σ chains, each with L/L_{c} entanglements per chain, interdiffuse in an interface of width X, we obtain
$$[p\u2013{p}_{\text{c}}]~{\text{{}\Sigma \text{}L\text{/}X\u2013[\Sigma \text{}L\text{/}X]}_{\text{c}}\text{}}$$  (4.1) 
The interface of width X is composed of a fraction L/M of diffusing chains and the matrix chain fraction (1 – L/M), into which the chains are diffusing. The total stored strain energy in the interface U ~ X, is consumed in disentangling only the Σ minor chains of length L, from the matrix chains, which are being stretched also but cannot disentangle at the same rate as the minor chains, and we obtain, G_{1c} ~ p as:
$${G}_{\text{1c}}~\Sigma \text{}L\text{/}X$$  (4.2) 
When the matrix chains disentangle or break along with the interdiffused chains, then p = 1 and the virgin strength is reached. The number of diffusing chains per unit area Σ, contributing to the interface strength is governed by gradient percolation, such that only those chains which straddle the interface are counted. Thus, only a subset of the concentration depth profile is contributing to strength, namely, those chains which are simultaneously connected to the A and B side of the interface. Also, the length L implies the number of entanglements per minor chain (L/L_{c} – 1), which can decrease significantly, for example, if brushlike ordering occurs at the interface, or the entanglement topology changes such that L_{c} becomes very large as in a solvent where M_{c} depends on polymer concentration ϕ, as M_{c} = M_{c}(1)/ϕ.
Applying Eq. (3.1), p = g(λ) N_{v}/ν to the interface, we obtain the number of entanglements per contour chain length L as, g(λ) = (L/L_{e} – 1), the number of chains in the interface N_{v} = Σ/X ~ 1/L_{∞} (from Table 1), the crosslink density ν ~ 1/L_{e}, the stretchdependent entanglement length L_{e}(λ) = λ^{2} L_{e}, such that Eq. (3.1) gives the percolation parameter:
$$p=(L\u2013{\lambda}^{\text{2}}\text{}{L}_{\text{e}}){\text{/}L}_{\infty}$$  (4.3) 
$${p}_{\text{c}}=({L}_{\text{c}}\u2013{L}_{\text{e}}){\text{/}L}_{\infty}\approx {L}_{\text{e}}{\text{/}L}_{\infty}$$  (4.4) 
$${\lambda}_{\text{c}}=[{L\text{/}L}_{\text{e}}\u2013{\text{1}]}^{\text{1/2}}$$  (4.5) 
Note that when L = 2 L_{e} = L_{c}, then λ_{c} = 1 as required. Also, when strain hardening occurs at λ_{c} = 4, then L* ≈ 8 L_{c}, which is the transition from disentanglement to bond rupture as the maximum strength is obtained. Since G_{1c} ~ (λ_{c} – 1)^{2}, and the time evolution of the minor chain length L ~ (t/M)^{1/2} is orders of magnitude longer that that for the entanglement length L_{e}, the time dependence of welding is given by G_{1c} ~ L, as:
$${G}_{\text{1c}}{\text{/}G}_{\text{1c}}\text{*}=[t\text{/}\tau {\text{*}]}^{\text{1/2}}\text{}(t\le \tau \text{*})$$  (4.6) 
Here G_{1c}* is the maximum strength obtained at the welding time τ* ~ M. Note that when M < M*, L = L_{∞} (t/T_{r})^{1/2} such that G_{1c} ~ [t/M]^{1/2} and full interdiffusion of the contour length is required to L = L_{∞} and the welding time occurs at t = T_{r}. However, when M > M*, full interdiffusion is not required and full strength is achieved at L = L* = 8 L_{c} and L = L* (t/τ*)^{1/2}, where τ* ~ M. For all molecularweights, the molecularweight dependence of welding remains as G_{1c} ~ (t/M)^{1/2}. Experimental support for Eq. (4.64) was reported by O'Connor [16,17] and McGarel et al. [18] and reviewed in reference [1]. The applicability of the welding law G_{1c} ~ [t/M]^{1/2} has been demonstrated not only for glassy polymers but also for hottack experiments and rubbery polymers [1 (Chapter 8)].
Thus, for all welds, there exists a critical interdiffusion distance X* to obtain the maximum strength G_{1c}* as:
$$X\text{*}={\text{0.8}R}_{\text{g}}\text{*}$$  (4.7) 
$$\tau \text{*}={\text{64}(M}_{\text{c}}{\text{/}M)}^{\text{2}}\text{}{T}_{\text{r}}$$  (4.8) 
4.1 Fracture vs. fatigue
The full interpenetration of chains (X approaches R_{g}) is not necessary to achieve complete strength, when M > M* and τ* < T_{r}. However, a cautionary note: while complete strength may be obtained in terms of critical fracture measures, such as G_{1c} and K_{1c}, the durability, measured in subcritical fracture terms, such as the fatigue crack propagation rate da/dN, may be very far from its fully healed state at τ*. We have shown that while the weld toughness K_{1c} increases linearly with interdiffusion depth X as K_{1c} ~ X, the fatigue crack propagation behavior of partially healed welds behaves as [1,18]
$$\text{d}a\text{/d}N~{X}^{\text{\u20135}}$$  (4.9) 
This fatigue behavior is a very strong function of interdiffusion depth and underscores the penalty to pay for partial welding. Thus, the weld strength may be deceptively close to the virgin strength, but the fatigue strength may be dramatically reduced below its maximum value. Thus, one should always design a welding temperaturetime process window with respect to T_{r} to achieve maximum durability of welds and interfaces.
The welding times can be readily calculated. The reptation time T_{r} is determined from the selfdiffusion coefficient D and the endtoend vector R, by [81]:
$${T}_{\text{r}}={R}^{\text{2}}\text{/}(\text{3}\text{}{\text{\pi}}^{\text{2}}\text{}D\text{)}$$  (4.10) 
For example, when welding polystyrene at 125 °C, D = 4 × 10^{−6}/M^{2} (cm^{2}/s) [77,84], R^{2} = 0.45 × 10^{−16} M (cm^{2}) such that T_{r} = 4 × 10^{−13} M^{3} (s) and τ* = 0.0234 M (s). For the case where M = 400 000 g/mol and M_{c} = 30 000 g/mol, we have τ*/T_{r} = 0.36, where T_{r} = 435 min and τ* = 156 min. In this example, if the maximum weld strength were obtained at an allowed welding time of 156 min, the durability as measured by da/dN, would only be about 1/5 of its virgin value compared to complete welding at T_{r} = 435 min. When plastic parts are being injection molded, laminated, sintered or coextruded, many internal weld lines are encountered and this aspect of welding needs to be considered in designing materials with optimal durability [1].
Recent studies [83] have suggested that while chains diffuse in a reptationlike mode, the monomer friction coefficient (assumed constant for reptation) may have a weak molecular weight dependence, in the order of M^{0.3}, resulting in an exponent of 3.3, instead of 3.0 for the molecular weight dependence of the relaxation time. If true, this would cause a small change in the exponents for the molecular weight dependence of welding, but would not affect the time exponents. For example, the minor chain length L, which from Table 1 behaves as L ~ t^{1/2}M^{–0.5} with τ ~ M^{3}, would become L ~ t^{1/2} M^{–0.65}, when τ ~ M^{3.3}.
4.2 Chainend segregation
In the case of chainend segregation to the surfaces, as can occur in crack healing and some latex particle coalescence during film formation, the number of chains Σ is constant and the percolation term becomes p ~ L/X, or p ~X, since X ~ L^{1/2}. Thus, from Table 1, the strength development would be G_{1c} ~ (t/M)^{1/4}, rather than the usual t^{1/2} dependence. This t^{1/4} result was also predicted by Prager and Tirrell, using a crossing density analysis [8], but with a different molecular weight dependence for both the welding and virgin state.
4.3 Welding below T_{g}
Welding below T_{g}, as recently demonstrated by Boiko et al. [51,52] can occur due to softening of the surface layer. We have treated the surface layer softening as a gradient rigidity percolation issue [79]. The surface melting and glass transition temperature of thin films is an important issue for nanomaterials, thin film coating processes, sealing and welding of polymer materials. A significant number of papers have been published in this field dealing with the dynamics of heterogeneous media near T_{g}, confinement effects, surface effects, measurement methodology, thin film melting, thermal and mechanical properties. We have treated this thin film and surface mobile layer problem as a finite size vector percolation problem. The percolation threshold is reduced by the thickness of the film due to finite size clusters spanning the film.
The intermolecular bonding between atoms is anharmonic and an atom no longer transmits rigidity when it has thermally expanded beyond a critical distance, ca 0.22 bond strain, which is related to the position of the first derivative (force) maximum in the intermolecular potential energy function. Lindemann, ca 1910, proposed this as a mechanism for melting due to the onset of vibrational instability in the lattice with a sufficient number of LA atoms. This concept was later expanded upon by Born (1939) as the Shear Rigidity Catastrophe theory. We have elaborated further on the Born criterion using finite size vector percolation theory. During thermal expansion, we assume that the number of LA is proportional to temperature, and is in dynamic equilibrium such that their fraction p ~ T, and p_{c} ~ T_{g}^{∞}, where the latter is the T_{g} of the bulk glass at infinite thickness. Since the elastic modulus E ~ [p – p_{c}]^{v}, where the exponent v ≈ 1, the glass to rubber transition occurs when there are sufficient connected clusters of LA atoms at p_{c} and the high glass modulus decreases towards zero: E does not actually go to zero experimentally since the rubbery modulus is finite.
For thin polymer films containing a fraction p of LA atoms at p < p_{c}, clusters of LA can be accessed and connected from the surface, as shown in Fig. 5. These fractal clusters are dynamic and if the LA were lights turning on and off as bonds are broken and reformed, the clusters would be blinking and dancing with interesting frequencies. We have shown that the accessed fraction f (dark clusters in Fig. 5), can be described by the finite lattice size percolation relation [1],
$${f=S(b\text{/}h)\text{}[\text{1}\u2013p\text{/}p}_{\text{c}}{]}^{\u2013\alpha}{(pp}_{\text{c}})$$  (4.11) 
$$\alpha =\upsilon \text{}(D\u2013d+\text{1})$$  (4.12) 
$$\xi =b\text{}{p\u2013p}_{\text{c}}{\text{}}^{\text{\u2013}\upsilon}$$  (4.13) 
When heat is applied to the thin film, as implied in Fig. 5, the free surfaces effectively have a monolayer of liquid atoms, which enhance the connectivity of the LA clusters at the surface. Thermal energy invades from the surface as vibrational waves with random amplitude causing intermolecular dissociation events on the amorphous ‘lattice’ of anharmonically bonded atoms on the polymer chains. Using Equation (4.11), and substituting for p/p_{c} = T/T_{g}^{∞}, the finite size percolation threshold f(h) = p*, such that we obtain the thickness dependence of T_{g}(h) as:
$${T}_{\text{g}}(t)={{T}_{\text{g}}}^{\text{\u221e}}\text{}{[1\u2013(B\text{/}h)}^{\text{\gamma}}]$$  (4.14) 
$$B=S\text{}b\text{/}p\text{*}$$  (4.15) 
$$\gamma =1\text{/}[\upsilon \text{}(D\u2013d+1)]$$  (4.16) 
The parameter S can have values of 0, 1, 2, or 3. For two free surfaces, S = 2, and the value of B ≈ 0.8 is determined using b = 0.154 nm for a C–C bond, and a percolation threshold p* = 0.4. For one free surface, e.g., a bulk surface or a thin film deposited on a neutral substrate, S = 1, and B = 0.4; for a thin film in contact with two neutral surfaces, S = 0 and B = 0, such that the thin film properties are the same as the bulk; for S = 3, e.g., with 3D nanoparticles of volume V ~ h^{3}, then B ≈ 1.16, which shows the greatest effect of T_{g} reduction with h. For strongly adsorbing thin films, the mobility of the surface layer is suppressed and T_{g} and T_{m} will actually increase relative to the bulk value. Thin films with one side free and the other side strongly adsorbed could provide some interesting local mobility battles. The value of γ is determined by the vector percolation values of ν and D, and is of order unity. For example with d = 3, ν = 0.82 and D = 2.85, Eq. (4.16) gives γ = 1.44. This relation for T_{g}(h) is in accord with data recently obtained by several investigators [85,86].
The surface rubbery layer conceptcontroversy in thick films is interesting and this percolation theory suggests that for free welding surfaces with S = 1, it exists, but there is a gradient of p(x) near the surface, where x < ξ as implied in Fig. 5, and hence a gradient in both T_{g} and modulus E. If the gradient of p is given by p(x) = (1 – x/ξ), then the value of X_{c} for which the gradient percolation threshold p_{c} occurs, and which defines the thickness of the surface mobile layer, is given by the percolation theory as:
$${X}_{\text{c}}=b\text{}(\text{1}\u2013{p}_{\text{c}})\text{/}\{{p}_{v}^{\text{c}}[1\u2013T\text{/}{T}_{\text{g}}]v\}$$  (4.17) 
$${G}_{\text{1c}}\text{~}\text{\Delta}{T}^{\text{\u20132}\upsilon}$$  (4.18) 
This appears to be in qualitative agreement with Boiko's data [52], who examined the fracture energy of polystyrene interfaces during welding at temperatures up to 80°K below T_{g}.
4.4 Summary comment on welding
In summary, the strength development during welding of polymers is well described by the relation:
$${G}_{\text{1c}}={G}_{\text{1c}}\text{*}\text{}(t\text{/}\tau {)}^{\text{1/2}}$$  (4.19) 
5 Fracture of incompatible interfaces
Consider the incompatible A/B polymer interface shown in Fig. 6. The equilibrium interface width d, which is typically much less than R_{g} of either the A or B chains, can be described by the Helfand relation [35]:
$$d=\text{2}\text{}b\text{/}(\text{6}\chi {)}^{\text{1/2}}$$  (5.1) 
$$L~k\text{}T\text{/}\chi $$  (5.2) 
Since the interface width d ~ L^{1/2}, the equilibrium incompatible interface thickness is derived as d ~ 1/χ^{1/2}, as expressed by Helfand in Eq 5.1. With increasing compatibility, or as χ approaches zero, d approaches the normal interface width X ~ R_{g} and the intermeshing segments becomes highly entangled, thereby producing much higher fracture energy comparable to the virgin state [1,21].
To understand the strength G, of incompatible interfaces as a function of their width d, we first consider the random walk of length L, shown in Fig. 6. This length L is part of a much larger random walk chain, and is a segment which begins on the Bside and traverses into the Aside, and returns to the Bside. In this respect, it is a bridge segment (of a larger chain) of length L_{p}, rather than a free chain of length L, such that the equilibrium interface width is properly described by:
$$d~{L}^{\text{1/2}}~{L}_{\text{p}}$$  (5.3) 
The number of bridges per unit area crossing the A/B interface is Σ_{p}, which is independent of molecular weight. As L_{p} increases, entanglements develop, crazes form and the percolation relation G ~ [p – p_{c}] applies. Here, the percolation parameter p = Σ L/X is now defined by:
$${p~\Sigma}_{\text{p}}{(L}_{p}{\text{/}L}_{\text{e}})\text{/}d$$  (5.4) 
$${G~[d\u2013d}_{\text{c}}]$$  (5.5) 
Here d_{c} is the critical interface width corresponding to p_{c}, which will be in the order of R_{ge}, and below which no strength exists, other than that of simple pullout and surface energy terms, as described by the nail solution. Letting the normalized width w = d/d_{c}, Eq. (5.5) becomes:
$$G~[w\u20131]$$  (5.6) 
The maximum strength G* is determined by:
$$G\text{*~[}w\text{*\u20131]}$$  (5.7) 
$$G\text{/}G\text{*}=(w\u20131)\text{/}(w\text{*\u20131})$$  (5.8) 
To investigate the latter relation, a plot of G/G* vs. w, should have a slope of 1/(w* – 1) ≈1, an intercept on the w axis at w_{c} = 1, and maximum strength attained (G/G* = 1) at w* ≈ 2, or the value of w* corresponding to w* ≈ 2 w_{c}.
Fig. 7 shows data obtained by several investigators and analyzed by Benkoski, Fredrickson and Kramer [22] for several asymmetric interface pairs. Here, G/G* is plotted versus the normalized interface width w = d/d_{t}, where d_{t} is the reptation tube diameter, calculated as d_{t} = b (4/5 N_{e})^{1/2}. Significantly, no strength develops below some critical value w_{c}. The magnitude of w_{c} is of order unity, but varies for each polymer pair due to the slight differences in their normalization procedure (w = d/d_{t}) compared to the above analysis (w = d/d_{c}). However, the slopes are of order unity, as predicted herein, and the maximum strength occurs at w* ≈ 2, when w_{c} ≈ 1, or at w* = 2 w_{c}. The data in Fig. 7 could be readily normalized to w_{c} = 1 to form a master curve consistent with the very simple relation:
$$G\text{/}G\text{*}=w\text{\u20131}$$  (5.9) 
When w < w_{c}, or p < p_{c}, the Nail solution, G ~ Σ L^{2}, applies as the Σ nonentangled chain segments of length L pullout in simple friction. However, the chain segments do not pullout as linear strings of length L, as can be deduced in Fig. 6, but rather as intermeshed random walks of length L^{1/2}; the chain segment is attached to a very long chain, which is itself entangled, and hence, will not allow the segment L to pullout as a string. Thus, the critical stress behaves as σ ~ Σ μ L^{1/2}, where μ is the friction coefficient. The critical crack opening displacement behaves as δ ~ L^{1/2}, such that the fracture energy for pullout is:
$$G\text{~}\mu \Sigma L$$  (5.10) 
Since Σ is constant and L ~ d^{2}, it follows that in simple pullout at w < w_{c}:
$${G\text{~}d}^{\text{2}}$$  (5.11) 
However, this fracture energy is very low and orders of magnitude lower than that obtained at w > w_{c}. Both theories based on the friction contribution agree with the quadratic dependence G ~ d^{2}, as first proposed by Willett and Wool [21].
The adhesion between immiscible polymers as a function of interfacial width was also analyzed by Cole, Cook and Macosko [23] in terms of the number of entanglements N_{ent} in the interface. They define N_{ent} in the incompatible interface of width d as:
$${N}_{\text{ent}}={d\text{/}L}_{\text{e}}$$  (5.12) 
$$G~{N}_{\text{ent}}^{2}$$  (5.13) 
Their data is shown in Fig. 8 (Fig. 7 in Cole et al. [23]), where the slope of 2 from a plot of log G vs. log N_{ent} suggests support for the quadratic dependence in Eq. 5.13. The circles in Fig. 8 represent data obtained from interface pairs consisting of the following; PP/aPA, PS/aPA, PS/PP, PS/PEO, PS/PC, PS/PVC, PS/PE, PS/PMMA, PET/PC, using both melt and solvent lamination. The triangles in Fig. 8 represent literature values for PSrPMMA by Brown et al. [25], and the squares represent PC/SAN data obtained by Janarthan et al. [87]. Alternatively, using the percolation model, since p ~ d/L_{e}, then from Eqs. (5.5) and (5.12), we obtain:
$${G\text{~}[N}_{\text{ent}}{\u2013N}_{\text{c}}]$$  (5.14) 
$$G\text{/}G\text{*}=[{N}_{\text{ent}}\u2013{N}_{\text{c}}]\text{/}[{N\text{*}}_{\text{c}}\u2013{N}_{\text{c}}]$$  (5.16) 
Accordingly, a plot of G vs. N_{ent} should give a linear plot with intercept N_{c} as shown in Fig. 9, using data from Cole et al., (Table 3 in ref [23]). The linear fit correlation coefficient was R^{2} = 0.95 (neglecting G = 0 points) with intercept N_{c} = 0.7, and slope of 11 J/m^{2}. Cole et al. observed at least three G = 0 values in the vicinity of N_{c}, supporting the concept that little, or no strength exists below the percolation threshold
Thus, the data in Fig. 9 is linear with a nonzero intercept as expected, which meaningfully divides the data into two regions, N_{ent} < N_{c} for which G ≈ 0, consistent with very weak interfaces, and N_{ent} > N_{c}, which describes the strong interfaces. However, a power law fit with zero intercept, as required by the homogeneous function G ~ N_{ent}^{β}, will suggest an exponent of β ≈ 2, and also describes both weak and strong regions with the same function. Clearly, a plot of log G vs. log [N_{ent} – N_{c}] would give an exponent of β ≈ 1, consistent with the percolation theory. During welding, N_{ent} behaves as N_{ent} ~ t^{3/4} M^{–7/4} (Table 1), which we observe experimentally [1]. However, if one were to use the strength relation G_{1c} ~ N_{ent}^{2}, one would predict that G_{1c} ~ t^{3/2} M^{7/2}, and G* ~ M^{−2}, which is universally inconsistent with all welding and virgin–state data.
6 Fracture of reinforced incompatible interfaces
The role of A–B diblock compatibilizers or random A–B copolymers of aerial density Σ at incompatible A/B interfaces was investigated by Creton, Brown, Char, Deline and Kramer et al. [24–31]. Fig. 10 shows results of G_{1c} vs. Σ for PS/PMMA interfaces reinforced by PS(800)–PVP(870) diblocks. Most of the data are reasonably well described by a line with a slope of 2 on this log–log plot, suggestive of G ~ Σ^{2}. Brown analyzed this and other similar data and derived a theory of fracture, which is referred as the Σ^{2} law [25,26]:
$${G}_{\text{1c}}~{\Sigma}^{2}\text{/}{\sigma}_{\text{cr}}$$  (6.1) 
$${G}_{\text{1c}}~[(\sum \text{}L\text{/}X)\u2013(\sum \text{}L\text{/}X{)}_{\text{c}}]$$  (6.2) 
Since L and X are constant, then p_{c} ~ Σ_{c}, which represents a critical number of chains required to build up the network above the percolation level. Letting L/X ~ R_{g}_{A} of the diblock ends, the percolation model predicts the linear relation:
$${G}_{\text{1c}}~{R}_{\text{gA}}\text{}[N\u2013{N}_{\text{c}}]$$  (6.3) 
Normalizing this relation by the maximum strength G* at Σ*, we have:
$$G\text{/}G\text{*}=[\Sigma \u2013{\Sigma}_{\text{c}}]\text{/}[\Sigma \text{*}\u2013{\Sigma}_{\text{c}}]$$  (6.4) 
Fig. 11 shows a plot of G/G* vs. Σ, using Creton's data from Fig. 10. The fracture data was normalized by G* ≈ 110 J/m^{2}, which is the upper range of the data presented in Fig. 10. The linear relation for G/G* vs. Σ had a correlation coefficient of 0.9 and produced an intercept on the Σaxis of Σ_{c} = 0.1/nm^{2}. The slope of this line is 11.1/nm^{2}. The transition from Nails to Nets, or weak to strong interfaces, is demarcated by the threshold value Σ_{c}, which, as discussed by Creton et al., should occur near the overlap of the diblock random coils in the interface, such that:
$${\Sigma}_{\text{c}}\approx 1\text{/}{R}_{\text{gA}}^{2}$$  (6.5) 
The radius of gyration of the PS ends with M_{n} = 83 200 g/mol is R_{gA}^{2} = 63.2 nm^{2}, such that Σ_{c} ≈ 0.016 nm^{2}, which is in reasonable accord with the experimental value Σ_{c} = 0.01/nm^{2} in Fig. 11. The maximum value of Σ* at G* can be determined from the entanglement bridge theory [66] by:
$$\Sigma \text{*}=[{M}_{\text{c}}\text{/}M\text{*}{]}^{\text{1/2}}\text{/2}a$$  (6.6) 
Examining both theories, G_{1c} ~ Σ^{2} and G_{1c} ~ [Σ–Σ_{c}], as plotted in Figs. 10 and 11, respectively, there is sufficient data scatter in both plots such that one could not judge, based on this data alone, as to which theory was more valid. However, the percolation model, in addition to describing the A/B reinforced interface above, is universally consistent with welding data, virgin–state strength and the transition from weak to strong interfaces. It can be deduced that the exponent of 2, reported in several instances, is an accidental consequence of inhomogeneous functions for G_{1c} vs. N_{ent} with incompatible A/B interfaces, G_{1c} vs. Σ data for reinforced A/B interfaces and G_{1c} vs. M for virgin strength data. The percolation theory of incompatible interfaces is significantly different and in contradiction with theories proposed by Benkoski et al., Cole et al. and Brown et al.
7 Conclusion
A theory of fracture of entangled polymers was developed which was based on the vector percolation model of Kantor and Webman, in which the modulus E is related to the lattice bond fraction p, via E ~ [p – p_{c}]^{τ}. The polymer fractured critically when p approached the percolation threshold p_{c}, which was accomplished by utilizing the stored strain energy in the network to randomly fracture [p – p_{c}] bonds. The fracture energy was found to be G_{1c} ~ [p – p_{c}]. When applied to interfaces of width X, containing an areal density Σ of chains, each contributing L entanglements, the percolation term p ~ Σ L/X, and the percolation threshold was related to Σ_{c}, L_{c}, or X_{c}. This gave a unified theory of fracture for the virgin state of polymers in the bulk and a variety of polymer interfaces. The percolation theory has also been applied successfully to fracture of carbon nanotubes [3,88,89 and polymer–solid interfaces [90–92].
Several important results are summarized in Table 2 and include the following:
 (1) the fracture strength σ of amorphous and semicrystalline polymers in the bulk could be well described by the net solution, σ = [E D_{o} ρ/16 M_{e}]^{1/2}, and found to be in excellent agreement with a large body of data. This is a firstprinciples approach to fracture and requires no fitting parameters;
 (2) for welding of A/A symmetric interfaces, p = Σ L/X, and p_{c} ≈ L_{c}/M ≈ 0, such that when Σ/X ~ 1/M for randomly distributed chain ends, G/G* = (t/τ*)^{1/2}, where τ* ~ M, when M > M*, and τ ~ M^{3}, when M < M*. When the chain ends are segregated to the surface, Σ is constant with time and G/G* = [t/τ*]^{1/4};
 (3) for incompatible A/B interfaces of width d, normalized width w, and entanglement density N_{ent} ~ d/L_{e}, p ~ d such that G ~ [d – d_{c}], G ~ [w – 1], and G ~ [N_{ent} – N_{c}];
 (4) for incompatible A/B interfaces reinforced by an areal density Σ of compatibilizer chains, L and X are constant, p ~ Σ, p_{c} ~ Σ_{c}, such that G ~ [Σ – Σ_{c}].
Interface and bulk properties
Polymer system  Property  Relation  Comment 
Symmetric A/A  Welding fracture energy G_{1c}  G_{1c} = G* (t/τ)^{1/2} 

A/A  Toughness K_{1c}  K_{1c} ~ t^{1/4} M^{–1/4}  G_{1c} = K_{1c}^{2}/E 
A/A  Welding below T_{g} 


A/A  Chainend segregation  G_{1c} = G* (t/τ)^{1/4}  p ~ X 
A Thin film  T_{g} vs h  T_{g}(h) = T_{g}^{∞} [1 – (B/h)^{γ}] 

Virgin state  Fracture energy G_{1c}  G_{1c}/ G* = 0.3 M/M_{c} [1 – (M_{c}/M)^{1/2}]^{2} 

Virgin state  G_{1c}  G_{1c} = G* [1 – M_{c}/M] 

Virgin state  G_{1c}  G_{1c} = G_{o} + k M 

Fatigue  da/dN  da/dN ~ X^{−5}  T_{r} ~ M^{3} necessary 
Net solution  Fracture stress σ  σ* = {2 E ν D_{o} [p – p_{c}]}^{1/2}  D_{o} = C–C bond energy 
A/B Incompatible interface  Fracture energy G 


Incompatible with Σ(AB)  Fracture energy G  G_{1c} ~ R_{gA} [N – N_{c}]  Σ_{c} ≈ 1/R_{gA}^{2} 
Rubber  Fracture stress σ and energy G 


Thermosets  Fracture stress σ, energy G 
 ν = crosslink density 
Acknowledgments
The author is grateful to the National Science Foundation and the Environmental Protection Agency for financial support of this work.
^{1} The factor of 8 rather than 16 occurs due to an orientation correction [1].
^{2} R.P. Wool, American Physical Society Meeting (2000).