1 Introduction

In this paper, we explore a new approach to understanding the molecular aspects of adhesion at polymer-polymer interfaces. Percolation theory is used as a means to parameterize the making and breaking of connectivity at polymer-polymer 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 ~ w² [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 ~ N² was proposed [23]. For an incompatible A/B interface reinforced with Σ compatibilizer chains per unit area, G_1c ~ Σ², 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], non-isothermal modeling of composite interfaces [50] and sub-T_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 10¹⁴chains, 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 ~ Σ². 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 non-reinforced 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 J-Integral 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:

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:

The vector or rigidity percolation process addresses several important points. First, consider a 2-D 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 non-uniform 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 J-integral 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 2-D or 3-D 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 non-bonded 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 3-D 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:

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:

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 3-crossings (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 Rouse-like 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:

The chain ends effectively contribute to the loss of one entanglement. Since N_v = ρ/M and ν = ρ/M_e, then we have:

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:

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:

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 zero-shear 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 long-range connectivity. Concomitantly, the dynamics changes from single chain, Rouse-like 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:

The maximum molecular weight M* at which disentanglement can occur is determined when strain hardening occurs at λ_c ≈ 4 such that¹:

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 virgin-state fracture energy as [1]:

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:

Since G_1c ~ [p – p_c], and p_c = 1 – M_e/M_c, we obtain:

In addition to glassy polymers, the RP-fracture 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].

4 Polymer–polymer welding

Fig. 4 shows an interface formed by random walk chains diffusing by reptation across a polymer-polymer 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/m²). 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).

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 non-connected 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

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:

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 brush-like 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 stretch-dependent entanglement length L_e(λ) = λ² L_e, such that Eq. (3.1) gives the percolation parameter:

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:

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 molecular-weight 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 hot-tack 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:

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 nano-materials, 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/h)[1–p/pc]–α(p<pc)

(4.11)

where b is the particle diameter, h is the film thickness, S is the number of free surfaces (S = 3, 2, 1, or 0) and α is determined by:

α=υ(D–d+1)

(4.12)

in which D is the fractal dimension of the clusters, d is the dimension of the sample (typically d = 2 for simulations (Fig. 5) and d = 3 for films and surfaces) and ν is the cluster correlation exponent, which gives the average size of the cluster as

ξ=b|p–pc|–υ

(4.13)

In 3-D, ν = 0.8, D ≈ 2.5, and α ≈ 0.4; in 2-D, ν = 4/3, D ~ 7/4 and α ≈ 1. The S-factor in Eq. (4.11) refers to the number of free surfaces of the thin material and this becomes unity if only one surface is free or the film is adhered to a substrate. In Fig. 5, the surface fraction accessed at p < p_c is determined using d = 2, p = 0.58, p_c = 0.5927, b = 1, h = 512 (lattice size), D = 7/4, and α = 1, such that Eq. (4.11) predicts that f ≈ 18%, which is in close agreement with the computer simulations (17%) at p < p_c. The cluster size correlation length in this case was ξ = 79 and complete connectivity would occur (f = 100%) if the thickness h were reduced from 512 to 93 lattice units.

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:

Tg(t)=Tg∞[1–(B/h)γ]

(4.14)

in which B and γ are described by:

B=Sb/p*

(4.15)

γ=1/[υ(D–d+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 3-D nano-particles of volume V ~ h³, 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 concept-controversy 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:

Xc=b(1–pc)/{pvc[1–T/Tg]v}

(4.17)

such that X_c ~ 1/ΔT^ν. For example, if T = (T_g – 10), b = 0.154 nm, p_c = 0.4, v = 0.82, then the thickness of the mobile layer X* = 3.8 nm. This could allow for healing to occur below T_g assuming that the dynamics are fast enough, since the mobile layers on both surfaces effectively disappear when the interface is formed. If G_1c ~ X² for entangled polymers, then we could deduce from Eq. (4.17) that for sub-T_g healing at ΔT = T_g – T:

G1c~ΔT–2υ

(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.

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]:

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 B-side and traverses into the A-side, and returns to the B-side. 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:

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:

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:

The maximum strength G* is determined by:

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:

When w < w_c, or p < p_c, the Nail solution, G ~ Σ L², applies as the Σ non-entangled 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:

Since Σ is constant and L ~ d², it follows that in simple pullout at w < w_c:

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 ~ d², 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:

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 PS-r-PMMA 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:

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 R² = 0.95 (neglecting G = 0 points) with intercept N_c = 0.7, and slope of 11 J/m². 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 non-zero 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_ent², 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 ~ Σ². Brown analyzed this and other similar data and derived a theory of fracture, which is referred as the Σ² law [25,26]:

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_gA of the diblock ends, the percolation model predicts the linear relation:

Normalizing this relation by the maximum strength G* at Σ*, we have:

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/m², 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 Σ-axis of Σ_c = 0.1/nm². The slope of this line is 11.1/nm². 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:

The radius of gyration of the PS ends with M_n = 83 200 g/mol is R_gA² = 63.2 nm², such that Σ_c ≈ 0.016 nm², which is in reasonable accord with the experimental value Σ_c = 0.01/nm² in Fig. 11. The maximum value of Σ* at G* can be determined from the entanglement bridge theory [66] by:

Examining both theories, G_1c ~ Σ² 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 first-principles 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 ~ M³, 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].

Acknowledgments

The author is grateful to the National Science Foundation and the Environmental Protection Agency for financial support of this work.