J. Phys. Chem. B 1997, 101, 4613-4619
4613
Connector Chain Aggregation Effects in Elastomer-Elastomer Adhesion Promotion Christian Ligoure* and James L. Harden Groupe de Dynamique des Phases Condense´ es, Unite´ Mixte de Recherche CNRS/UniVersite´ Montpellier II no. 5581, Case 026, UniVersite´ Montpellier II, F-34095 Montpellier Cedex 5, France ReceiVed: December 16, 1996; In Final Form: March 27, 1997X
We consider the effects of chain aggregation during pull out on the adhesion of elastomer-elastomer surfaces fortified with interfacial connector chains. For the case of monodisperse end-grafted A homopolymer chains at an A/B elastomer interface, we show that lateral aggregation of connector chains in the gap into tethered micelle-like structures occurs at a characteristic length ζ ∼ 1/σ set by the grafting density σ. This is a firstorder transition, characterized by a metastable single-fibril state separated from a state of arbitrarily large aggregates by a gap-size dependent energy barrier. As a result, the threshold fracture toughness G0 as a function of σ is predicted to have a plateau reflecting a regime of spontaneous connector chain extraction induced by aggregation. Experimental access to this plateau depends on the rate of A/B elastomer separation, due to the slow kinetics of the aggregation instability. For the case of polydisperse connector brushes, aggregation processes are predicted to occur on many length scales, leading to a possible enhancement of adhesion promotion compared with monodisperse brushes. We briefly highlight this effect with several examples of polydisperse connector brushes and pseudobrushes.
Introduction The adhesive properties of composite material interfaces has been a subject of rather extensive studies during the last several decades, especially for the case of polymer adhesion. This interest is due in part to the technical importance of adhesion in many materials applications. The failure of adhesion of a polymeric material to a surface (either an inorganic material or another polymeric material) generally involves the nucleation and propagation of cracks at the interface between the materials. This is a rather complicated process which depends both on the bulk viscoelastic properties of the materials and on the intrinsic adhesive properties of the interface in the cohesive zone near the crack tip. On a macroscopic level, the state of the polymeric material (rubbery vs glassy) and the particular mode of crack propagation are important issues. On a microscopic level, the molecular mechanisms of adhesion play a critical role. The basic principles of polymer adhesion are discussed in several texts.1-3 Many of the recent advances in the field have been reviewed by Brown.4,5 Here we mention only a few of the salient features. The adhesive properties of an interface are usually characterized by the fracture toughness G, the energy required to create a unit area of fracture at the interface. Owing to the importance of viscoelastic energy dissipation processes in crack propagation, G usually depends very strongly on the rate of propagation. For the case of elastomeric materials, there is evidence that G can be written as a product of a threshold value G0 and a rate and temperature dependent factor due to viscoelastic losses.6-9 Typically, G is several orders of magnitude larger than G0 for realistic propagation rates. Nevertheless, an understanding of the underlying mechanisms controlling G0 is important since it sets the scale for the initiation of adhesive failure. In recent years there has been made a substantial theoretical and experimental effort toward understanding the molecular level processes governing fracture toughness. In particular, the role of connector molecules, polymer chains which bridge the interface between materials in contact, as adhesion promoters X
Abstract published in AdVance ACS Abstracts, May 1, 1997.
S1089-5647(96)04088-6 CCC: $14.00
has been carefully studied.10-36 Examples include linear homopolymer and copolymer connectors, either self-assembled at the interface between materials or chemically bounded to the surface of one or both materials. In general, such connector molecules substantially increase fracture toughness, although there are counter examples.21 The precise mechanisms governing adhesion promotion depend on the properties of the materials in contact. The case of adhesion between glassy materials is rather complex and may involve several competing mechanisms.10-21 In this paper, we consider connector chain adhesion promotion for elastomeric materials.22-35 For the case of homopolymer connector molecules bridging an elastomer-elastomer or elastomer-solid interface, there are essentially two contributing factors to this adhesion promotion: a thermodynamic penalty for extracting connector molecules from the bulk elastomer and viscous energy dissipation due to friction between the connector molecules and the elastomer matrix at finite rates of extraction. The first contribution affects G0 and is the focus of this paper. A number of theoretical models for the effects of connector molecules on the adhesion of elastomers have been proposed.22-30 Within these models, the contribution of connector molecules to the threshold toughness G0 is assumed to come from the independent sum of the work of extraction of individual connector molecules. The model of Raphae¨l and de Gennes23 considers monodisperse A homopolymer chains grafted at uniform surface density to a cross-linked elastomer of incompatible B polymer and placed in contact with a cross-linked elastomer of A polymer. It is assumed that initially the A connectors completely penetrate the A elastomer material. In response to separation of the A and B elastomers, the connector chains are progressively extracted from the A elastomer, forming single-chain fibrils. There is a free energy cost involved in stretching these connector chains and exposing them to the air in the gap. The contribution of the connector molecules to G0 is given by the product of the work required to fully extract a connector from the elastomer and the number of connector molecules per unit area, giving a contribution proportional to the grafting density σ and the degree of polymerization N of the A connector chains. © 1997 American Chemical Society
4614 J. Phys. Chem. B, Vol. 101, No. 23, 1997 Experimental studies on model polystyrene-polyisoprene elastomer systems with tethered polystyrene-polyisoprene diblock copolymer connectors31-33 found that the threshold toughness G0 increases monotonically with the molecular weight of the polyisoprene connector blocks,33 in qualitative agreement with the model of ref 23. The dependence of G0 on the grafting density of connectors is less clear. Initially, G0 clearly increases with σ, as predicted. However, the growth of G0 appears to saturate at sufficiently large σ and perhaps even attain a maximum value. Other experimental studies on the adhesion of polydimethylsiloxane elastomer networks to silica surfaces with attached polydimethylsiloxane connector chains found clear evidence of a maximum value of the threshold fracture toughness G0(σ).34,35 Furthermore, these latter studies have shown that irreversibly adsorbed connector chains are somewhat better adhesion promoters than end-grafted chains. This observed maximum in the fracture toughness with connector chain density can be explained by models accounting for partial interdigitation of connector chains.27-30 Although partial interdigitation effects do depend on the polydispersity of the connectors,29 it is not clear whether partial interdigitation alone can account for the differences between end-grafted and irreversibly adsorbed connector chains as adhesion promoters. One possible shortcoming of current theoretical models is the assumption of independent, single-chain connector fibrils. The connector chains in the air gap between elastomers are reminiscent of grafted polymer brushes in poor solvent conditions. The grafted chains in a poor solvent brush are unstable toward lateral microphase separation at intermediate grafting densities.37-39 For marginally poor solvents, brushes of twicegrafted chains (chains grafted by each end to opposing surfaces) are predicted to aggregate into bundles.37 Such bundles resemble the crazes which form during the adhesive failure of glassy polymers20 although their origins are quite different. The possibility of bundle formation during connector chain pull out was anticipated by Raphae¨l and de Gennes23 but apparently not pursued. In the limit of very poor solvents, appropriate for chains exposed to air, tethered micelle-like structures have been predicted both for ordinary and twice-grafted brushes.38,39 These aggregates consist of regularly-spaced dense globules of collapsed polymer attached to the grafting surface(s) by extended single-chain tethers. In this paper we discuss the possible effects of such aggregation phenomena for connector chains in elastomer-elastomer adhesion promotion. We first discuss the case of monodisperse grafted chains in some detail. Afterward, we briefly highlight the implications of such aggregation phenomena for connector chains forming polydisperse brushes and irreversibly adsorbed polymer layers. Thermodynamic Considerations Following ref 23, we consider the case of monodisperse A homopolymer chains of degree of polymerization N grafted at average lateral chain separation d to the surface of a cross-linked elastomer of incompatible B polymer and placed in contact with a cross-linked elastomer of A polymer. Let σ ∼ (d/a)-2 denote the dimensionless grafting density, where a is the Kuhn length of the A chains. It is assumed that initially, when the bulk polymer surfaces are in contact, the A connectors completely penetrate the A elastomer (We will briefly address partial interdigitation effects below). In response to separation of the elastomer surfaces, which we assume are essentially rigid, the connector chains are progressively extracted from the A elastomer. We assume that the extracted portions of the connector chains within the air gap of thickness H will either form single-chain fibrils or laterally aggregate to form tethered
Ligoure and Harden
Figure 1. Schematic representation of aggregated and nonaggregated connector chains. We assume that A connector chains, which are grafted at lateral separation d to a B elastomer and freely interpenetrate an A elastomer, from either independent single-chain fibrils or tethered micelles with core radius Rc and corona radius R in an air gap of size H.
micelles with core radius Rc and tether corona radius R, as shown in Figure 1. Note that the single-chain fibril configuration is a special case of tethered micelles in the limit R ) Rc ) 0. Hence we may treat both situations within the same formalism. We first determine, as a function of gap size and grafting density, the preferred connector molecule geometry by comparing the free energy per unit area of single-chain fibrils and tethered micelles. We assume initially that the connector chains within a micelle core adopt essentially unperturbed Gaussian configurations and that there is no difference in chemical potential between the A connector monomers in the core and in the A elastomer. The former assumption can be verified a postiori, while the latter is equivalent to the complete interdigitation ansatz. The free energy per micelle is then given by the sum of contributions from the monomer/air interaction energies of the core and tethers and the elastic stretching energy of the tethers. The surface energy of a core is simply 4πR2c γ, where γ is the A monomer/air surface tension. This gives a core contribution per unit area
Fc ) 4γ(Rc/R)2
(1)
The free energy per tether has the same form as that of ref 23 for single-chain fibrils. Consider a tether of length L(F) containing n(F) monomers that joins the core centered at F ) 0 to an elastomer surface at F e R. The monomer/air interaction energy of this tether scales as γa2n, while its elastic stretching energy scales as kBTL2/na2. We assume that the cores reside at the center of the gap and that tethers adopt the shortest path between the core and the elastomer surface, implying L(F) ) (F2 + H2/4)1/2 - Rc. The total tether contribution comes from the sum of the stretching and monomer/air interaction energy contributions of each tether, leading to a contribution per unit area Ft of the form
Ft ) 4
γσ R2
k Tσ
L(F)2
∫0R Fn(F)dF + 4 aB4R2 ∫0R F n(F) dF
(2)
The total tethered micelle free energy density F is given by the sum of eqs 1 and 2. The equilibrium tethered micelle configuration is obtained by minimizing F with respect to n(F), Rc, and R. Functional minimization of F with respect to n(F), δF/δn ) 0, yields n(F) ) K-1/2L(F)/a, where K ) γa2/kBT. (We note that K ≈ 1 for very poor solvents.) This result indicates that, as in the case of single-chain fibrils,23 the tethers are fully extended strings of poor solvent blobs,40 all under an equal tension ft = 2kBTK1/2/ a.41 Using this relation between L(F) and n(F) in eq 2 and
Elastomer-Elastomer Adhesion Promotion
J. Phys. Chem. B, Vol. 101, No. 23, 1997 4615 TABLE 1: Characteristic Gap Size h*a σ
h*
n*
10-1 10-2 10-3
1.11 1.38 1.46
∼101 ∼102 ∼103
a h* as a function of σ and the corresponding average number monomers per chain n* extracted from the elastomer during the fluctuation-induced formation of an aggregate from single-chain fibrils.
j kBT the probability of aggregation is significant. We define the characteristic gap size h* corresponding to the onset of aggregation by ∆F(h*)ch ) kBT. Using an expansion of ∆F(h)ch around h ) 3/2 to quadratic order and assuming σ , 1, this criterion for h* gives Figure 2. Plots of the dimensionless free energy f(r,h) ) F(r,h)/F(0,h) versus the reduced tether corona radius r ) R/ζ for several values of the dimensionless gap size h ) H/ζ. The upper, middle, and lower curves correspond respectively to h ) 1, h ) 3/2, and h ) 2.
minimizing the resulting tethered micelle free energy density F with respect to Rc at fixed R, gives a relation between Rc and R. A natural characteristic length, ζ ) 4/3K1/2σ-1a, appears as a result of this analysis, which we use to nondimensionalize all remaining length scales via h ) H/ζ, r ) R/ζ, and rc ) Rc/ζ. This minimization gives rc ) 2/3r2 and the resulting free energy density
F(r,h) ) 4/9γ[8(r2 + h2/4)3/2r-2 - 4r2 - h3r-2]
(3)
Note that F(0,h) ) 8γh/3 corresponds to the single-chain fibril free energy of Raphae¨l and de Gennes.23 To ascertain the preferred equilibrium geometry for a given value of h, one must analyze the behavior of F(r,h) as a function of r. We note first that F(r,h) always has an extremal value at r ) 0. For h < 3/2, this extremal value is a local minimum. Furthermore, there is also a global maximum of F(r,h) for h < 3/2 at r ) rs given by
rs )
1 [1 + (1 - h)(2h + 1)1/2]1/2 x2
(4)
The position of this maximum is a decreasing function of h, which vanishes at h ) 3/2. For h g 3/2, F(r,h) then takes its global maximum at r ) 0. Figure 2 gives plots of the scaled free energy f(r,h) ) F(r,h)/F(0,h) vs r for h ) 1, 3/2, and 2. This figure shows that single-chain fibrils are energetically preferred until h ) 3/2, where this fibril state becomes unstable toward the formation of aggregates. We note that, in principle, arbitrarily large aggregates are preferred for h > 3/2, although we are incapable of treating aggregates that are larger than rc ) h/2 due to geometrical constraints imposed by our model. Presumably, this aggregation process induces the total extraction of connector chains from the A elastomer, resulting in a dense, molten slab of A chains residing in the gap. Metastability and Threshold Toughness The thermodynamic analysis of the previous section showed that aggregation of connector chain molecules is expected for a reduced gap size h g 3/2. However, there is also a finite probability of chain aggregation for h < 3/2, due to thermal fluctuations. Let ∆F(h)ch ) [F(rs,h) - F(0,h)]/σ denote the activation energy per chain for the formation of an aggregate of critical size r ) rs(h) from the single-chain fibril state, r ) 0. Note that, at fixed h, this activation barrier decreases with increasing grafting density σ. If ∆F(h)ch . kBT, such a fluctuation-induced aggregation event is highly improbable and the single-chain fibril state is metastable. However, for ∆F(h)ch
3 3σ 1/2 h* = 2 2K
( )
(5)
Values of h* for K ) 1 and several values of σ are given in Table 1. Note that h* is a decreasing function of σ. We are now in a position to discuss the pull-out process and the connector chain contribution to the threshold toughness G0. Starting at contact, there is the Dupre´ work of adhesion W, associated with the bare elastomer-elastomer interface, required to open an infinitesimal gap. As the gap thickness is increased, the connector chains are progressively pulled out from the elastomer until all chains have been fully extracted at some gap size Hmax. Since the connector chain tension, ft = 2kBTK1/2/a, is independent of the geometry (fibrils vs aggregates), the form of the total work per unit area required for complete extraction is δW = σa-2ftHmax for both fibrils and tethered micelle aggregates. However, the value of Hmax depends on the geometry, which in turn depends on the characteristic length ζ and on N. The maximum gap size for single-chain fibrils is H(f) max = (f) 1/2 K Na. If Hmax e ζh*, fibrils are preferred throughout the pull-out process, giving the Raphae¨l and de Gennes23 result δG0 = 2σγN, where δG0 ) G0 - W. This regime is restricted to σ e σ*(N), where σ*(N) is the grafting density corresponding to H(f) max ) ζh* and is given by
σ*(N) )
2 (1 - 2/(3KN)1/2) N
(6)
The O(N-1/2) correction term in eq 6 is due to thermal fluctuation effects. The corresponding maximum contribution to the threshold toughness in the single-chain fibril regime, δG*0 ) 2σ*γN, is given by
δG*0(N) ) 4γ (1 - 2/(3KN)1/2)
(7)
For σ > σ*(N), connector chains are sufficiently long to eventually favor aggregation. In this case, pull-out proceeds in the fibril mode until H = ζh*. Once H J ζh*, the activation barrier is ineffective and spontaneous chain aggregation occurs. The thermodynamic preference for arbitrarily large aggregates once beyond the activation stage (see Figure 2) implies that the aggregation process induces the total extraction of tethers from the A elastomer matrix, resulting in adhesion failure. We now can summarize the effects of chain aggregation on δG0. Figure 3 shows a logarithmic plot of δG0/γ vs σ/σ* for K ) 1 and N ) 103, corresponding to σ* = 0.002. In the fibril regime, we have the result of Raphae¨l and de Gennes, δG0 = 2σγN for σ < σ*. Beyond σ* ∼ N-1, however, there is a plateau with δG* 0 ≈ 4γ, corresponding to complete chain pullout at h*. We should emphasize that the δG0(σ) presented in Figure 3 is
4616 J. Phys. Chem. B, Vol. 101, No. 23, 1997
Ligoure and Harden our estimates, we will take a simplified view which ignores the effects of chain tension on τ*. We characterize the degree of connector chain entanglement with the elastomer matrix by the average number Ne of connector chain monomers between entanglement points48 and estimate τ* as the characteristic relaxation time of a subchain of n* monomers in the elastomer. If n* . Ne, relaxation occurs through reptation of the connector chain in the elastomer matrix,48 with τ* ∼ n3*/Ne. On the other hand, for n* j Ne, relaxation occurs through a Rouse process48, for which τ* ∼ n2*. Thus we have
τ*(h,σ) )
Figure 3. Double logarithmic plot of the connector chain contribution δG0 to the threshold fracture toughness scaled by γ as a function of the reduced connector chain density σ/σ* for the case of K ) 1 and N ) 103, corresponding to σ* = 0.002.
somewhat schematic; the actual behavior of δG0 near σ* should of course be a smooth function of σ. Nevertheless, the predction that δG0 saturates at large σ is consistent with some of the experimental results of refs 32-34. Aggregation Kinetics The results presented above are subject to certain kinetic limitations, both below and above the instability threshold at h ) 3/2. In particular, the duration of an adhesion experiment must be long compared with the characteristic time required to form aggregates, in order for aggregation-related effects to be observable. The existence of an activation barrier below the instability threshold suggests that the formation of aggregates will follow an Arrhenius law in this regime. In this case, the characteristic time τa of aggregate formation for a fixed gap size takes the form43
τa = τ* exp[∆F(h)ch/kBT]
for
h e 3/ 2
1 n*(h,σ) = K1/2σ-2rs(h)4 2
for
h e 3/ 2
(9)
where rs(h) is given by eq 4.44,45 Note that n* is a rapidly decreasing function of h and σ. Table 1 gives values of n* evaluated at h ) h* for several values of σ. Due to the dependence of h* on σ (see eq 5), n*(h*,σ) ∼ σ-1 to leading order in σ. The characteristic time τ* depends on the degree of entanglement of the connector chains with the A elastomer matrix. Since these connector chains are under an applied tension, their extraction dynamics is rather complicated in general.46,47 In
(10)
where τ0 is a monomer relaxation time and n*(h,σ) is given by eq 9. Equations 9 and 10 imply that τ*(h*) ∼ σ-2 for n* j Ne, while τ*(h*) ∼ σ-3 for n* > Ne. For Ne = 102, Table 1 shows that the crossover from reptation to Rouse behavior occurs near σ ) 10-2. Finally, the characteristic aggregation time τa for h < 3/2 is given by eqs 8-10. We note that τa can be quite large even for h ) h*, since τ*(h*,σ) can be very large for sufficiently small σ. For h g 3/2, single-chain fibrils are mechanically unstable and spontaneous aggregation is favored. However, if the rate of separation V of the A and B elastomers is sufficiently high, these aggregates will not form during the course of an experiment and the connector chain extraction will occur solely in the single-chain fibril mode. For simplicity, we suppose that V is constant during an experiment. At the instability threshold, h ) 3/2, nf = 2/σ monomers have been extracted from the elastomer matrix in the fibril mode (ignoring activated events). Thus, there are on average N - nf monomers per connector chain remaining in the A elastomer at h ) 3/2. The time required to extract the remaining monomers in the fibril mode τf is given by incremental gap size, ∆H = (N - nf)aK1/2, occurring at the point of total fibril extraction divided by V,
N(1 - σ*/σ)K1/2a τf = V
(8)
where τ* is the characteristic time required for the partial extraction of a typical chain from the elastomer during the aggregation process and ∆F(h)ch ) [F(rs,h) - F(0,h)]/σ is the free energy barrier height. The characteristic time τ* is a function of the number of monomers per chain extracted from the matrix and may be rather large. We may estimate the average number of extracted monomers per chain n* from the difference between the average numbers of monomers per connector chain forming a tethered micelle and forming a singlechain fibril for a given gap size h. This difference is dominated by the average number of monomers per connector chain in a micelle core, 〈Nc〉 ∼ (Rc/a)3/[σ(R/a)2], giving
{
τ0n2* for n* j Ne 3 -1 τ0n*Ne for n* > Ne
(11)
where σ* is given by eq 6. On the other hand, the characteristic time τa required to extract the remaining monomers by spontaneous aggregation is approximately the relaxation time of a subchain of N - nf monomers in the elastomer. Following the arguments leading to Eq 10, we have
τa )
{
τ0(N - nf)2 = τ0N2(1 - σ*/σ)2 for N - nf j Ne 3 -1 3 -1 3 τ0(N - nf) Ne = τ0N Ne (1 - σ*/σ) for N - nf > Ne (12)
where Ne is the average number of connector chain monomers between entanglements in the elastomer matrix. For a given V, the experimentally observable behavior may be anticipated by comparing τf and τa. If τf . τa, there is sufficient time during an experiment for aggregates to form and the aggregationinduced spontaneous chain extraction and the associated plateau in G0 are expected. However, if τf , τa, connector chain extraction should occur solely in the single-chain fibril mode, in accordance with the model of Raphae¨l and de Gennes.23 By equating τf and τa, one may define a critical rate of elastomer separation Vc as
Elastomer-Elastomer Adhesion Promotion
Vc )
{
-1 τ-1 (1 - σ*/σ)-1 for N - nf j Ne 0 aN -1 -2 -2 τ0 aNe N (1 - σ*/σ) for N - nf J Ne
J. Phys. Chem. B, Vol. 101, No. 23, 1997 4617 Polydispersity Effects
(13)
For V , Vc, aggregation-induced adhesion failure may occur, while for V . Vc adhesion failure should occur in the fibril extraction mode. In particular, this implies that the apparent threshold toughness δG0 for σ > σ* depends on V, with the limiting behavior
δG0 )
{
4γ [1 - 2/(3KN)1/2] for V , Vc 2σγN for V . Vc
(14)
It is instructive to estimate Vc under typical experimental conditions. Consider the case of well-entangled connector chains with a ) 5 Å, τ0 ) 10-9 s and Ne ) 102. We will assume that the chains are grafted at high density, σ . σ*, and that there are no significant partial interdigitation effects. Then, for N ) 103 we have Vc = 50 µm/s, while for N ) 104 we have Vc = 0.5 µm/s. In a typical JKR adhesion experiment, the velocity is in the range 0.01 < V < 1.0 µm/s. For the lower molecular weight example, Vc = 50 µm/s is quite large compared with experimental velocities and we would expect aggregationinduced adhesion failure to occur, as detailed above. On the other hand, for the higher molecular weight example, Vc = 0.5 µm/s lies within the range of V studied in typical JKR experiments. Experiments measuring G(V) in this case may be affected by the dependence of G0 on V: In a certain range of V, δG0 would shift from the low-velocity to the high-velocity limit of eq 14 with increasing V. As a result, the linear dependence of G ) G0 f(V,T) on V, predicted on the basis of a velocity independent threshold toughness G0 and a linear viscoelastic dissipation factor f(V,T),23,25 would not be observed. This scenario is in qualitative agreement with the experiments of ref 36, in which G(V) initially increased more rapidly than linearly with V at low rates of separation. Partial Interdigitation Effects Up to this point, we have ignored the effects of crosslinks in the A elastomer. However, in principle there should be a difference in the monomer chemical potential between the cores and the elastomer, due to the finite elasticity of the latter. These effects have been considered for the single-chain fibril mode of connector chain extraction,27-30 where they were shown to cause partial interdigitation of the connector chains with the A elastomer network. In the case of tethered micelles, this finite monomer exchange potential also results in partial interdigitation, as well as in a slightly renormalized effective surface tension of the micelle cores.44 This latter effect is rather small, however. As in the single-chain fibril case, and using the simplest model of interdigitation between a brush and an elastomer,28 partial interdigitation occurs for σ J PN-3/2 and interdigitation failure occurs for σ J P-1/2, where P is the average number of monomers between crosslinks in the elastomer. Depending on the value of P, the range of σ corresponding to partial interdigitation could lie either above or below σ*. Within the partial interdigitation regime, the effective degree of polymerization of the connector chains is renormalized to N ˜ ˜ on σ can lead ∼ (P/σ)2/3. Qualitatively, the dependence of N to a smoothing of the behavior of G0 near σ* and a shift of the plateau region. Furthermore, the renormalization N f N ˜ could strongly affect the kinetics of aggregation through the resulting renormalization of the characteristic chain relaxation times presented in eqs 10 and 12. A detailed discussion of these effects will be presented elsewhere.44
We now would like to briefly discuss the effects of connector chain aggregation for polydisperse brushes and pseudobrushes. For the case of monodisperse brushes presented in the previous sections, the aggregation instability of single-chain fibrils during pull out was associated with a characteristic length scale ζ(σ) set by the grafting density σ. A polydisperse brush usually consists of grafted chains with a distribution of different N and σ. Thus, there is in principle a cascade of characteristic lengths for brushes of polydisperse connector chains. The aggregation behavior of the connector chains and the implications for adhesion are in general very complex for such brushes, with many regimes to consider.44 We will present a detailed description of the polydisperse case elsewhere. In this paper, we only present several simple examples. Consider first a bimodal brush consisting of A chains with N1 and N2 monomers grafted at the dimensionless densities σ1 and σ2 to a B elastomer surface and placed in contact with an A elastomer. We will assume that the A connectors completely penetrate the A elastomer. Let N1 < N2 and let σ0 ) σ1 + σ2 be the initial density of bridging connector chains. We consider the limit of low grafting densities, for which both chain types are extracted entirely in the single-chain fibril mode. The process starts with connector chains at total density σ ) σ0. With increasing gap size, both populations of chains are progressively extracted until the shorter chains are completely pulled out of the A elastomer. This first stage proceeds in the fibril mode provided that σ0 e σ*(N1) = 2N-1 1 , for which Hmax (N1) = K1/2N1a. Thus, the contribution of the first chain population to the threshold fracture toughness is δG(1) ) 0 σ0a-2ftHmax(N1) = 2γσ0N1. In the second stage, σ ) σ2 and the longer chains are progressively extracted. This stage proceeds in the fibril mode provided that σ2 e σ*(N2) = 1/2 2N-1 2 , for which Hmax(N2) = K N2a. The contribution of the -2 second chain population is δG(2) 0 ) σ2a ft[Hmax(N2) - Hmax(N1)] = 2γσ2(N2 - N1). Thus the total connector chain contribution is
δG0 = 2γ(σ1N1 + σ2N2) e 4γ(2 - N1/N2)
(15)
where the upper bound comes from setting σ1 + σ2 ) σ*(N1) -1 = 2N-1 1 and σ2 ) σ*(N2) = 2N2 . Recall that the maximum contribution to δG0 from a monodisperse brush extracted in the fibril regime is δG0 = 4γ. Hence, the maximum bimodal brush contribution is approximately a factor of 2 - N1/N2 larger than the corresponding monodisperse value. This argument can be extended to arbitrary polydispersity, as follows. Let S(n) be the number of grafted connector chains per unit area having at least n monomers.49 If S(n) satisfies S(n) e 2a-2n-1, one can easily show that all connector chains will be extracted in the fibril regime. Following the same approach used in the bimodal brush example above, one finds a corresponding maximum contribution to δG0 of the form44
δG0 = 4γ log(N)
(16)
We note, however, that there will be corrections to eq 16 due to finite interdigitation effects.29 Consider next the case of A homopolymer connector chains that are irreversibly adsorbed to a B surface from the melt or from a solution with concentration Φ0. Layers of such adsorbed polymer have a wide distribution of loop and tail sizes. As such, they may be treated as polydisperse brushes with S(n) = -2 -1/2 pseudotails per unit area having at least n monoΦ7/8 0 a n mers each.50 Such pseudobrushes have been studied extensively
4618 J. Phys. Chem. B, Vol. 101, No. 23, 1997 as adhesion promoters. Experiments suggest that the threshold fracture toughness is larger for pseudobrushes than for endgrafted monodisperse brushes at the same total surface coverage.34,35 The effects of partial interpenetration of such pseudobrushes into elastomer matrices was considered in ref 29. As a general rule, interdigitation restrictions increase with the surface density of bridging chains. Thus, in the case of a pseudobrush, partial interdigitation effects are stronger for shorter tails than for longer ones. There is a minimum pseudotail length n0, depending on adsorption conditions and the elastomer properties, below which the penetration of the tails into the elastomer is hindered.29 Hence, before the onset of separation, only those pseudotails with n > n0 monomers will bridge the A/B interface, implying an initial effective -1/2 surface density σ0 = Φ7/8 of fibrils. Note that σ0 > σ*0 n0 (n0) initially and hence aggregation of pseudotails should occur immediately upon creation of a finite gap. This implies that, in contrast with monodisperse end-grafted brush connectors, connector chain pull out should in principle occur solely in the aggregation instability mode for the case of pseudobrushes. However, the aggregation process in this case is a bit different from the monodisperse brush case presented in the previous sections. In particular, aggregates of a finite preferred size may appear, due to the variation of ζ that occurs during the spontaneous extraction of polydisperse connector chains from the matrix.44 Discussion We have investigated the effects of lateral chain aggregation on the adhesion of elastomers to surfaces with irreversibly attached connector chains. Our principle result is that aggregation of connector chains in the gap may arrest the usual singlechain fibril extraction process. In the case of monodisperse endgrafted chains, aggregation occurs near a characteristic gap size ζ ∼ 1/σ. For sufficiently long connector chains, this fibril aggregation process occurs before individual fibrils may be fully extracted from the matrix and results in spontaneous connector chain extraction from the elastomer. This aggregation mechanism leads to a plateau in the threshold fracture toughness G0 as a function of σ. This plateau in G0(σ) may explain in part the apparent saturation of G with σ seen in some experiments32-36 and suggests that monodisperse grafted chains may not be effective adhesion promoters. For the more complex cases of polydisperse brushes and pseudobrushes, aggregation processes may occur on many length scales and can lead to situations for which there is an enhancement of adhesion promotion compared with monodisperse brushes [c.f. our bimodal brush example]. Although not fully explored in this communication, our results may explain some of the intrinsic differences seen in experiments between end-grafted and irreversibly adsorbed connector chains as adhesion promoters.34,35 In any event, we have shown that there are important differences in the underlying mechanisms of connector chain extraction between brushes and adsorbed layers which deserve future attention. The dynamics of chain extraction from an elastomer matrix is quite a complex process, even for monodisperse connector chains in the fibril mode of extraction.46,47 The precise nature of the extraction dynamics determines the energy dissipated during chain extraction. Since G . G0 under typical conditions, this is a very important issue. Although we have not considered the detailed dependence of the fracture toughness G on the rate of connector chain extraction, we have presented a simplified analysis of the aggregation kinetics and its implications to experimental measurements of the low-velocity adhesion strength G(V). We have argued that the aggregation instability that is
Ligoure and Harden predicted on thermodynamic grounds may not be observed in adhesion experiments conducted at finite V, due to the slow kinetics of connector chain aggregation. In particular, we have identified a characteristic rate Vc above which aggregation is not expected. One important consequence of this Vc is that adhesion experiments conducted in a range of V near V must be interpreted in terms of an apparent threshold toughness δG0 that is velocity dependent. Our model for the effects of aggregation on connector chain adhesion promotion has certain potential limitations. We have assumed an idealized spherical tethered-micelle aggregate structure. There are certainly many other candidate structures. In particular, we have not addressed the effects of preferential wetting of connector chains to an elastomer surface. Such a scenario has recently been studied for equilibrium tethered micelles formed from twice-grafted polymer chains in poor solvent conditions.51 In cases for which such wetting is relevant, one would expect pancake-like aggregates to form at an elastomer surface, rather than spherical aggregates in the gap. The precise quantitative adhesion behavior will depend on the aggregate geometry and perhaps also on the way the aggregates form. However, we believe that the qualitative features of our results are quite general and do not depend on the details of the aggregation process or the preferred aggregate configuration. We have not discussed in detail the effects of partial interdigitation of the A connectors into the A elastomer. The interplay between aggregation and partial interdigitation effects can give rise to non-monotonic dependence of the threshold fracture toughness G0 on the connector chain density σ. Furthermore, partial interdigitation effects can also potentially modify the kinetics of aggregation. In spite of its simplicity, our model exhibits very rich behavior, especially for the case of polydisperse connector chains. We will present a detailed analysis of this behavior for the cases of polydisperse end-grafted connector chains and irreversibly adsorbed connector chains in a future publication. Acknowledgment. We thank L. Le´ger and G. Porte for useful correspondence and discussions. One of us (J.L.H.) was supported by the CNRS Poste Rouge program. References and Notes (1) Williams, J. G. Adhesion and AdhesiVes, Science and Technology; Ellis Horwood: Chichester, U.K., 1984. (2) Kinloch, A. J. Fracture Mechanics of Polymers; Chapman and Hall: London, 1987. (3) Vakula, V. L.; Pritykin, L. M. Polymer Adhesion, Physico-Chemical Principles; Ellis Horwood: Chichester, U.K., 1991. (4) Brown, H. R. Annu. ReV. Mater. Sci. 1991, 21, 463. (5) Brown, H. R. IBM J. Res. DeV. 1994, 38, 379. (6) Gent, A. N.; Schultz, J. J. Adhes. 1972, 3, 281. (7) Andrews, E. H.; Kinloch, A. J. J. Proc. R. Soc. London, A 1973, 332, 385. Ibid. 1973, 332, 401. (8) Carre, A.; Schultz, J. J. Adhes. 1984, 18, 171. (9) Gent, A. N. Langmuir 1996, 12, 4492. (10) Brown, H. R.; Deline, V. R.; Green, P. F. Nature 1989, 341, 221. (11) Creton, C.; Kramer, E. J.; Hadziioannou, G. Macromolecules 1991, 24, 1846. (12) Brown, H. R. Macromolecules 1991, 24, 2752. (13) Xu, D. B.; Hui, C. Y.; Kramer, E. J.; Creton, C. Mech. Mater. 1991, 11, 257. (14) Creton, C.; Kramer, E. J.; Hui, C. Y.; Brown, H. R. Macromolecules 1992, 25, 3075. (15) Hui, C. Y.; Ruina, A.; Creton, C.; Kramer, E. J. Macromolecules 1992, 25, 3948. (16) Brown, H. R.; Char, K.; Deline, V. R.; Green, P. F. Macromolecules 1993, 26, 4155. (17) Char, K.; Brown, H. R.; Deline, V. R. Macromolecules 1993, 26, 4155. (18) Washiyama, J.; Kramer, E. J.; Hui, C. Y. Macromolecules 1993, 26, 2928.
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